Subgroups — C₇×#60433 #13237649

Order 1792.

2678 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#577	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#3046	C₂	1	3	yes	maximal, complemented
6	C₇×#3045	C₂	1	3	yes	maximal, complemented
7	C₇×#3044	C₂	1	3	yes	maximal, complemented
8	C₇×#3043	C₂	1	3	yes	maximal, complemented
9	C₇×#3040	C₂	1	3	yes	maximal, complemented
10	C₇×#3037	C₂	1	3	yes	maximal, complemented
11	C₇×#3035	C₂	1	3	yes	maximal, complemented
12	C₂×C₇×#464	C₂	1	3	yes	maximal, complemented
13	C₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#2898	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#2897	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
18	C₂×C₇×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
19	C₇×#578	—	2	8	no	—
20	C₇×#577	—	2	8	no	—
21	C₂×C₇×#137	C₂²	1	3	yes	—
22	C₂×C₇×#137	C₂²	1	3	yes	complemented
23	C₂×C₇×#137	C₂²	1	3	yes	complemented
24	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	3	yes	—
25	C₂×C₇×#135	C₂²	1	3	yes	—
26	C₂×C₇×#135	C₂²	1	3	yes	complemented
27	C₂²×C₇×Q₁₆	C₂²	1	3	yes	complemented
28	C₂²×C₇×SD₁₆	C₂²	1	3	yes	—
29	C₂×C₇×#131	C₂²	1	1	yes	characteristic
30	C₂×C₇×#131	C₂²	1	1	yes	characteristic
31	C₇×#548	—	2	1	no	—
32	C₇×#548	—	2	1	no	—
33	C₇×#546	—	2	3	no	—
34	C₇×#546	—	2	3	no	—
35	C₇×#536	C₂²	1	3	yes	—
36	C₇×#535	C₂²	1	3	yes	—
37	C₇×#532	C₂²	1	3	yes	—
38	C₂×C₇×#124	C₂²	1	3	yes	—
39	C₇×#520	—	2	3	no	—
40	C₇×#520	C₂²	1	1	yes	characteristic
41	C₇×#520	C₂²	1	3	yes	complemented
42	C₇×#519	—	2	1	no	—
43	C₇×#519	C₂²	1	3	yes	complemented
44	C₄×C₇×#41	—	2	3	no	—
45	C₄×C₇×#41	C₂²	1	1	yes	characteristic
46	C₄×C₇×#41	C₂²	1	3	yes	complemented
47	C₂×C₄×C₇×Q₈	—	2	1	no	—
48	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	complemented
49	C₇×#464	C₂²	1	3	yes	complemented
50	C₇×#464	C₂²	1	6	yes	complemented
51	C₇×#464	C₂²	1	6	yes	complemented
52	C₇×#464	C₂²	1	6	yes	complemented
53	C₇×#464	C₂²	1	3	yes	complemented
54	C₇×#464	C₂²	1	6	yes	complemented
55	C₇×#463	C₂²	1	1	yes	characteristic, complemented
56	C₇×#463	C₂²	1	3	yes	complemented
57	C₇×#463	C₂²	1	6	yes	complemented
58	C₇×#462	C₂²	1	3	yes	complemented
59	C₇×#462	C₂²	1	6	yes	complemented
60	C₇×#461	C₂²	1	3	yes	complemented
61	C₇×#460	C₂²	1	3	yes	complemented
62	C₇×#460	C₂²	1	6	yes	complemented
63	C₇×#459	C₂²	1	3	yes	complemented
64	C₇×#424	C₂²	1	3	yes	complemented
65	C₇×#424	C₂²	1	3	yes	complemented
66	C₇×#423	C₂²	1	3	yes	complemented
67	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	complemented
68	C₇×#419	C₂²	1	3	yes	complemented
69	C₇×#418	C₂²	1	3	yes	—
70	C₇×#417	C₂²	1	3	yes	—
71	C₇×#417	C₂²	1	3	yes	complemented
72	C₇×#416	C₂²	1	3	yes	—
73	C₇×#416	C₂²	1	3	yes	complemented
74	C₂×C₇×#103	C₂²	1	3	yes	complemented
75	C₂×C₇×#102	C₂²	1	3	yes	—
76	#60433	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₇×2₋¹⁺⁴	—	2	4	no	—
78	C₇×2₋¹⁺⁴	—	2	4	no	—
79	C₇×2₋¹⁺⁴	—	2	4	no	—
80	C₇×2₋¹⁺⁴	D₄	1	8	yes	complemented
81	C₇×H(5,𝔽₂)	—	2	4	no	—
82	C₇×H(5,𝔽₂)	—	2	4	no	—
83	C₂×C₇×#41	—	2	1	no	—
84	C₂×C₇×#41	—	2	3	no	—
85	C₂×C₇×#41	—	2	3	no	—
86	C₂×C₇×#41	C₂³	1	3	yes	—
87	C₂×C₇×#41	C₂³	1	3	yes	—
88	C₂×C₇×#41	C₂³	1	3	yes	—
89	C₂×C₇×#41	C₂³	1	1	yes	characteristic
90	C₂²×C₇×Q₈	—	2	1	no	—
91	C₂²×C₇×Q₈	—	2	1	no	—
92	C₂²×C₇×Q₈	C₂³	1	3	yes	—
93	C₂²×C₇×Q₈	C₂³	1	3	yes	—
94	C₂²×C₇×D₄	—	2	3	no	—
95	C₂²×C₇×D₄	C₂³	1	3	yes	—
96	C₇×#137	—	2	12	no	—
97	C₇×#137	—	2	12	no	—
98	C₇×#137	—	2	12	no	—
99	C₇×AGL(1,ℤ/8)	—	2	12	no	—
100	C₇×#135	—	2	12	no	—
101	C₇×#135	—	2	12	no	—
102	C₂×C₇×Q₁₆	—	2	12	no	—
103	C₂×C₇×Q₁₆	C₂³	1	3	yes	—
104	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
105	C₂×C₇×Q₁₆	C₂³	1	3	yes	—
106	C₂×C₇×SD₁₆	—	2	12	no	—
107	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
108	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
109	C₂×C₇×SD₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
111	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
112	C₂×C₇×D₈	C₂³	1	3	yes	—
113	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
114	C₇×#131	—	2	4	no	—
115	C₇×#131	—	2	4	no	—
116	C₂×C₇×M₁₆	C₂³	1	3	yes	—
117	C₂×C₇×M₁₆	C₂³	1	3	yes	—
118	C₂²×C₇×C₈	C₂³	1	3	yes	—
119	C₂²×C₇×C₈	C₂³	1	3	yes	—
120	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
121	C₇×#127	C₂³	1	3	yes	—
122	C₇×#124	—	2	3	no	—
123	C₇×#124	—	2	3	no	—
124	C₇×#124	C₂³	1	3	yes	—
125	C₇×#124	C₂³	1	6	yes	—
126	C₇×#124	C₂³	1	1	yes	characteristic
127	C₇×#123	—	2	6	no	—
128	C₇×C₂²⋊Q₈	—	2	3	no	—
129	C₇×C₂²⋊Q₈	—	2	3	no	—
130	C₇×#121	—	2	3	no	—
131	C₇×#121	—	2	3	no	—
132	C₇×#121	—	2	3	no	—
133	C₇×#121	—	2	3	no	—
134	C₇×#120	—	2	6	no	—
135	C₄×C₇×Q₈	—	2	1	no	—
136	C₄×C₇×Q₈	—	2	3	no	—
137	C₄×C₇×Q₈	—	2	3	no	—
138	C₄×C₇×Q₈	—	2	3	no	—
139	C₄×C₇×Q₈	—	2	3	no	—
140	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
141	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
142	C₄×C₇×Q₈	C₂³	1	3	yes	—
143	C₄×C₇×Q₈	C₂³	1	6	yes	—
144	C₄×C₇×Q₈	C₂³	1	3	yes	—
145	C₄×C₇×D₄	—	2	3	no	—
146	C₄×C₇×D₄	—	2	3	no	—
147	C₄×C₇×D₄	—	2	3	no	—
148	C₄×C₇×D₄	—	2	6	no	—
149	C₄×C₇×D₄	—	2	3	no	—
150	C₄×C₇×D₄	—	2	3	no	—
151	C₄×C₇×D₄	D₄	1	3	yes	—
152	C₄×C₇×D₄	D₄	1	3	yes	—
153	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
154	C₄×C₇×D₄	C₂³	1	3	yes	—
155	C₇×#117	—	2	3	no	—
156	C₇×#117	—	2	6	no	—
157	C₇×#117	—	2	3	no	—
158	C₇×#117	C₂³	1	3	yes	—
159	C₇×#117	C₂³	1	3	yes	—
160	C₂×C₇×C₄⋊C₄	—	2	3	no	—
161	C₂×C₇×C₄⋊C₄	—	2	3	no	—
162	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
163	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
164	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
165	C₂×C₄²×C₇	—	2	3	no	—
166	C₂×C₄²×C₇	—	2	3	no	—
167	C₂×C₄²×C₇	C₂³	1	3	yes	—
168	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
169	C₇×C₄⋊C₈	C₂³	1	3	yes	—
170	C₇×C₄⋊C₈	C₂³	1	3	yes	—
171	C₇×C₄⋊C₈	C₂³	1	3	yes	—
172	C₇×C₄⋊C₈	C₂³	1	6	yes	—
173	C₇×#103	C₂³	1	3	yes	—
174	C₇×#103	C₂³	1	6	yes	—
175	C₇×#103	C₂³	1	1	yes	characteristic
176	C₇×#103	C₂³	1	3	yes	—
177	C₇×#103	C₂³	1	3	yes	—
178	C₇×#102	C₂³	1	3	yes	—
179	C₇×#102	C₂³	1	6	yes	—
180	C₇×#102	C₂³	1	3	yes	—
181	C₇×#102	C₂³	1	1	yes	characteristic
182	C₇×#102	C₂³	1	3	yes	—
183	C₂×#578	C₂×C₇	1	1	yes	characteristic, complemented
184	C₂×#577	C₂×C₇	1	1	yes	characteristic, complemented
185	#3500	C₂×C₇	1	1	yes	characteristic, complemented
186	C₄×2₋¹⁺⁴	C₂×C₇	1	1	yes	characteristic, complemented
187	#3046	C₂×C₇	1	3	yes	complemented
188	#3045	C₂×C₇	1	3	yes	complemented
189	#3044	C₂×C₇	1	3	yes	complemented
190	#3043	C₂×C₇	1	3	yes	complemented
191	#3040	C₂×C₇	1	3	yes	complemented
192	#3037	C₂×C₇	1	3	yes	complemented
193	#3035	C₂×C₇	1	3	yes	complemented
194	C₂×#464	C₂×C₇	1	3	yes	complemented
195	#2906	C₂×C₇	1	1	yes	characteristic, complemented
196	#2898	C₂×C₇	1	1	yes	characteristic, complemented
197	#2897	C₂×C₇	1	1	yes	characteristic, complemented
198	C₂⁴×C₇	—	2	3	no	—
199	C₇×#41	—	2	24	no	—
200	C₇×#41	—	2	6	no	—
201	C₇×#41	—	2	6	no	—
202	C₇×#41	—	2	6	no	—
203	C₇×#41	—	2	6	no	—
204	C₇×#41	—	2	6	no	—
205	C₇×#41	—	4	4	no	—
206	C₇×#41	—	4	12	no	—
207	C₇×#41	C₂×D₄	1	12	yes	—
208	C₇×#41	C₂×D₄	1	4	yes	—
209	C₇×#41	#41	1	4	yes	—
210	C₂×C₇×Q₈	—	2	8	no	—
211	C₂×C₇×Q₈	—	2	3	no	—
212	C₂×C₇×Q₈	—	2	3	no	—
213	C₂×C₇×Q₈	—	2	6	no	—
214	C₂×C₇×Q₈	—	2	6	no	—
215	C₂×C₇×Q₈	—	2	6	no	—
216	C₂×C₇×Q₈	—	4	4	no	—
217	C₂×C₇×Q₈	C₂×D₄	1	12	yes	—
218	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
219	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
220	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
221	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
222	C₂×C₇×D₄	—	2	3	no	—
223	C₂×C₇×D₄	—	2	3	no	—
224	C₂×C₇×D₄	—	2	3	no	—
225	C₂×C₇×D₄	—	2	6	no	—
226	C₂×C₇×D₄	—	2	3	no	—
227	C₂×C₇×D₄	—	2	3	no	—
228	C₂×C₇×D₄	—	2	3	no	—
229	C₂×C₇×D₄	—	2	6	no	—
230	C₂×C₇×D₄	—	4	12	no	—
231	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
232	C₂×C₇×D₄	C₂⁴	1	3	yes	—
233	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
234	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
235	C₂²×C₄×C₇	—	2	3	no	—
236	C₂²×C₄×C₇	—	2	3	no	—
237	C₂²×C₄×C₇	—	2	3	no	—
238	C₂²×C₄×C₇	—	2	3	no	—
239	C₂²×C₄×C₇	—	2	1	no	—
240	C₂²×C₄×C₇	—	2	3	no	—
241	C₂²×C₄×C₇	—	2	3	no	—
242	C₂²×C₄×C₇	—	2	3	no	—
243	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
244	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
245	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
246	C₇×Q₁₆	—	2	6	no	—
247	C₇×Q₁₆	—	2	12	no	—
248	C₇×Q₁₆	—	2	6	no	—
249	C₇×SD₁₆	—	2	6	no	—
250	C₇×SD₁₆	—	2	12	no	—
251	C₇×SD₁₆	—	2	6	no	—
252	C₇×SD₁₆	—	2	2	no	—
253	C₇×SD₁₆	—	2	6	no	—
254	C₇×D₈	—	2	2	no	—
255	C₇×D₈	—	2	6	no	—
256	C₇×M₁₆	—	2	6	no	—
257	C₇×M₁₆	—	2	6	no	—
258	C₂×C₇×C₈	—	2	6	no	—
259	C₂×C₇×C₈	—	2	6	no	—
260	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
261	C₂×C₇×C₈	C₂⁴	1	3	yes	—
262	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
263	C₂×C₇×C₈	C₂⁴	1	3	yes	—
264	C₇×C₄⋊C₄	—	2	6	no	—
265	C₇×C₄⋊C₄	—	2	3	no	—
266	C₇×C₄⋊C₄	—	2	3	no	—
267	C₇×C₄⋊C₄	—	2	3	no	—
268	C₇×C₄⋊C₄	—	2	3	no	—
269	C₇×C₄⋊C₄	—	2	3	no	—
270	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
271	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
272	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
273	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
274	C₇×C₂²⋊C₄	—	2	3	no	—
275	C₇×C₂²⋊C₄	—	2	3	no	—
276	C₇×C₂²⋊C₄	—	2	3	no	—
277	C₇×C₂²⋊C₄	—	2	6	no	—
278	C₇×C₂²⋊C₄	—	2	6	no	—
279	C₇×C₂²⋊C₄	—	2	1	no	—
280	C₇×C₂²⋊C₄	—	2	3	no	—
281	C₇×C₂²⋊C₄	—	2	1	no	—
282	C₇×C₂²⋊C₄	—	2	1	no	—
283	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
284	C₄²×C₇	—	2	3	no	—
285	C₄²×C₇	—	2	3	no	—
286	C₄²×C₇	—	2	6	no	—
287	C₄²×C₇	—	2	3	no	—
288	C₄²×C₇	C₂×D₄	1	3	yes	—
289	C₄²×C₇	C₂×D₄	1	3	yes	—
290	C₄²×C₇	C₂⁴	1	1	yes	characteristic
291	C₄²×C₇	C₂⁴	1	3	yes	—
292	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
293	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
294	C₂×H(5,𝔽₂)	C₂²×C₇	1	1	yes	characteristic
295	#578	—	2	8	no	—
296	#577	—	2	8	no	—
297	C₂×#137	C₂²×C₇	1	3	yes	—
298	C₂×#137	C₂²×C₇	1	3	yes	complemented
299	C₂×#137	C₂²×C₇	1	3	yes	complemented
300	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	3	yes	—
301	C₂×#135	C₂²×C₇	1	3	yes	—
302	C₂×#135	C₂²×C₇	1	3	yes	complemented
303	C₂²×Q₁₆	C₂²×C₇	1	3	yes	complemented
304	C₂²×SD₁₆	C₂²×C₇	1	3	yes	—
305	C₂×#131	C₂²×C₇	1	1	yes	characteristic
306	C₂×#131	C₂²×C₇	1	1	yes	characteristic
307	#548	—	2	1	no	—
308	#548	—	2	1	no	—
309	#546	—	2	3	no	—
310	#546	—	2	3	no	—
311	#536	C₂²×C₇	1	3	yes	—
312	#535	C₂²×C₇	1	3	yes	—
313	#532	C₂²×C₇	1	3	yes	—
314	C₂×#124	C₂²×C₇	1	3	yes	—
315	#520	—	2	3	no	—
316	#520	C₂²×C₇	1	1	yes	characteristic
317	#520	C₂²×C₇	1	3	yes	complemented
318	#519	—	2	1	no	—
319	#519	C₂²×C₇	1	3	yes	complemented
320	C₄×#41	—	2	3	no	—
321	C₄×#41	C₂²×C₇	1	1	yes	characteristic
322	C₄×#41	C₂²×C₇	1	3	yes	complemented
323	C₂×C₄×Q₈	—	2	1	no	—
324	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	complemented
325	#464	C₂²×C₇	1	3	yes	complemented
326	#464	C₂²×C₇	1	6	yes	complemented
327	#464	C₂²×C₇	1	6	yes	complemented
328	#464	C₂²×C₇	1	6	yes	complemented
329	#464	C₂²×C₇	1	3	yes	complemented
330	#464	C₂²×C₇	1	6	yes	complemented
331	#463	C₂²×C₇	1	1	yes	characteristic, complemented
332	#463	C₂²×C₇	1	3	yes	complemented
333	#463	C₂²×C₇	1	6	yes	complemented
334	#462	C₂²×C₇	1	3	yes	complemented
335	#462	C₂²×C₇	1	6	yes	complemented
336	#461	C₂²×C₇	1	3	yes	complemented
337	#460	C₂²×C₇	1	3	yes	complemented
338	#460	C₂²×C₇	1	6	yes	complemented
339	#459	C₂²×C₇	1	3	yes	complemented
340	#424	C₂²×C₇	1	3	yes	complemented
341	#424	C₂²×C₇	1	3	yes	complemented
342	#423	C₂²×C₇	1	3	yes	complemented
343	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	complemented
344	#419	C₂²×C₇	1	3	yes	complemented
345	#418	C₂²×C₇	1	3	yes	—
346	#417	C₂²×C₇	1	3	yes	—
347	#417	C₂²×C₇	1	3	yes	complemented
348	#416	C₂²×C₇	1	3	yes	—
349	#416	C₂²×C₇	1	3	yes	complemented
350	C₂×#103	C₂²×C₇	1	3	yes	complemented
351	C₂×#102	C₂²×C₇	1	3	yes	—
352	C₂³×C₇	—	2	1	no	—
353	C₂³×C₇	—	2	3	no	—
354	C₂³×C₇	—	2	1	no	—
355	C₂³×C₇	—	4	6	no	—
356	C₂³×C₇	—	8	3	no	—
357	C₂³×C₇	—	8	3	no	—
358	C₂³×C₇	C₂²×D₄	1	3	yes	—
359	C₇×Q₈	—	2	12	no	—
360	C₇×Q₈	—	2	3	no	—
361	C₇×Q₈	—	2	3	no	—
362	C₇×Q₈	—	2	1	no	—
363	C₇×Q₈	—	2	3	no	—
364	C₇×Q₈	—	2	1	no	—
365	C₇×Q₈	—	4	6	no	—
366	C₇×Q₈	#121	1	4	yes	—
367	C₇×Q₈	C₂²×D₄	1	6	yes	—
368	C₇×D₄	—	2	12	no	—
369	C₇×D₄	—	2	3	no	—
370	C₇×D₄	—	2	1	no	—
371	C₇×D₄	—	2	1	no	—
372	C₇×D₄	—	2	3	no	—
373	C₇×D₄	—	2	1	no	—
374	C₇×D₄	—	4	6	no	—
375	C₇×D₄	—	4	6	no	—
376	C₇×D₄	—	4	6	no	—
377	C₇×D₄	—	8	3	no	—
378	C₇×D₄	—	8	3	no	—
379	C₇×D₄	#121	1	12	yes	—
380	C₇×D₄	C₂²×D₄	1	2	yes	—
381	C₂×C₄×C₇	—	2	12	no	—
382	C₂×C₄×C₇	—	2	3	no	—
383	C₂×C₄×C₇	—	2	12	no	—
384	C₂×C₄×C₇	—	2	1	no	—
385	C₂×C₄×C₇	—	2	3	no	—
386	C₂×C₄×C₇	—	2	3	no	—
387	C₂×C₄×C₇	—	2	1	no	—
388	C₂×C₄×C₇	—	4	3	no	—
389	C₂×C₄×C₇	—	4	3	no	—
390	C₂×C₄×C₇	—	4	2	no	—
391	C₂×C₄×C₇	—	4	6	no	—
392	C₂×C₄×C₇	—	4	6	no	—
393	C₂×C₄×C₇	—	4	6	no	—
394	C₂×C₄×C₇	—	8	3	no	—
395	C₂×C₄×C₇	—	8	3	no	—
396	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
397	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
398	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
399	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
400	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
401	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
402	C₂×C₄×C₇	C₂×#41	1	6	yes	—
403	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
404	C₇×C₈	—	2	1	no	—
405	C₇×C₈	—	2	1	no	—
406	C₇×C₈	—	2	3	no	—
407	C₇×C₈	—	2	3	no	—
408	2₋¹⁺⁴	—	2	4	no	—
409	2₋¹⁺⁴	—	2	4	no	—
410	2₋¹⁺⁴	—	2	4	no	—
411	2₋¹⁺⁴	C₇×D₄	1	8	yes	complemented
412	H(5,𝔽₂)	—	2	4	no	—
413	H(5,𝔽₂)	—	2	4	no	—
414	C₂×#41	—	2	1	no	—
415	C₂×#41	—	2	3	no	—
416	C₂×#41	—	2	3	no	—
417	C₂×#41	C₂³×C₇	1	3	yes	—
418	C₂×#41	C₂³×C₇	1	3	yes	—
419	C₂×#41	C₂³×C₇	1	3	yes	—
420	C₂×#41	C₂³×C₇	1	1	yes	characteristic
421	C₂²×Q₈	—	2	1	no	—
422	C₂²×Q₈	—	2	1	no	—
423	C₂²×Q₈	C₂³×C₇	1	3	yes	—
424	C₂²×Q₈	C₂³×C₇	1	3	yes	—
425	C₂²×D₄	—	2	3	no	—
426	C₂²×D₄	C₂³×C₇	1	3	yes	—
427	#137	—	2	12	no	—
428	#137	—	2	12	no	—
429	#137	—	2	12	no	—
430	AGL(1,ℤ/8)	—	2	12	no	—
431	#135	—	2	12	no	—
432	#135	—	2	12	no	—
433	C₂×Q₁₆	—	2	12	no	—
434	C₂×Q₁₆	C₂³×C₇	1	3	yes	—
435	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
436	C₂×Q₁₆	C₂³×C₇	1	3	yes	—
437	C₂×SD₁₆	—	2	12	no	—
438	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
439	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
440	C₂×SD₁₆	C₂³×C₇	1	1	yes	characteristic
441	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
442	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
443	C₂×D₈	C₂³×C₇	1	3	yes	—
444	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
445	#131	—	2	4	no	—
446	#131	—	2	4	no	—
447	C₂×M₁₆	C₂³×C₇	1	3	yes	—
448	C₂×M₁₆	C₂³×C₇	1	3	yes	—
449	C₂²×C₈	C₂³×C₇	1	3	yes	—
450	C₂²×C₈	C₂³×C₇	1	3	yes	—
451	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
452	#127	C₂³×C₇	1	3	yes	—
453	#124	—	2	3	no	—
454	#124	—	2	3	no	—
455	#124	C₂³×C₇	1	3	yes	—
456	#124	C₂³×C₇	1	6	yes	—
457	#124	C₂³×C₇	1	1	yes	characteristic
458	#123	—	2	6	no	—
459	C₂²⋊Q₈	—	2	3	no	—
460	C₂²⋊Q₈	—	2	3	no	—
461	#121	—	2	3	no	—
462	#121	—	2	3	no	—
463	#121	—	2	3	no	—
464	#121	—	2	3	no	—
465	#120	—	2	6	no	—
466	C₄×Q₈	—	2	1	no	—
467	C₄×Q₈	—	2	3	no	—
468	C₄×Q₈	—	2	3	no	—
469	C₄×Q₈	—	2	3	no	—
470	C₄×Q₈	—	2	3	no	—
471	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
472	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
473	C₄×Q₈	C₂³×C₇	1	3	yes	—
474	C₄×Q₈	C₂³×C₇	1	6	yes	—
475	C₄×Q₈	C₂³×C₇	1	3	yes	—
476	C₄×D₄	—	2	3	no	—
477	C₄×D₄	—	2	3	no	—
478	C₄×D₄	—	2	3	no	—
479	C₄×D₄	—	2	6	no	—
480	C₄×D₄	—	2	3	no	—
481	C₄×D₄	—	2	3	no	—
482	C₄×D₄	C₇×D₄	1	3	yes	—
483	C₄×D₄	C₇×D₄	1	3	yes	—
484	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
485	C₄×D₄	C₂³×C₇	1	3	yes	—
486	#117	—	2	3	no	—
487	#117	—	2	6	no	—
488	#117	—	2	3	no	—
489	#117	C₂³×C₇	1	3	yes	—
490	#117	C₂³×C₇	1	3	yes	—
491	C₂×C₄⋊C₄	—	2	3	no	—
492	C₂×C₄⋊C₄	—	2	3	no	—
493	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
494	C₂×C₂²⋊C₄	—	2	3	no	—
495	C₂×C₂²⋊C₄	—	2	3	no	—
496	C₂×C₄²	—	2	3	no	—
497	C₂×C₄²	—	2	3	no	—
498	C₂×C₄²	C₂³×C₇	1	3	yes	—
499	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
500	C₄⋊C₈	C₂³×C₇	1	3	yes	—
501	C₄⋊C₈	C₂³×C₇	1	3	yes	—
502	C₄⋊C₈	C₂³×C₇	1	3	yes	—
503	C₄⋊C₈	C₂³×C₇	1	6	yes	—
504	#103	C₂³×C₇	1	3	yes	—
505	#103	C₂³×C₇	1	6	yes	—
506	#103	C₂³×C₇	1	1	yes	characteristic
507	#103	C₂³×C₇	1	3	yes	—
508	#103	C₂³×C₇	1	3	yes	—
509	#102	C₂³×C₇	1	3	yes	—
510	#102	C₂³×C₇	1	6	yes	—
511	#102	C₂³×C₇	1	3	yes	—
512	#102	C₂³×C₇	1	1	yes	characteristic
513	#102	C₂³×C₇	1	3	yes	—
514	C₂²×C₇	—	2	3	no	—
515	C₂²×C₇	—	2	3	no	—
516	C₂²×C₇	—	2	2	no	—
517	C₂²×C₇	—	4	1	no	—
518	C₂²×C₇	—	4	1	no	—
519	C₂²×C₇	—	4	1	no	—
520	C₂²×C₇	—	4	3	no	—
521	C₂²×C₇	—	4	1	no	—
522	C₂²×C₇	—	4	3	no	—
523	C₂²×C₇	—	4	1	no	—
524	C₂²×C₇	—	4	3	no	—
525	C₂²×C₇	—	16	6	no	—
526	C₂²×C₇	C₂×#121	1	6	yes	—
527	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
528	C₄×C₇	—	2	1	no	—
529	C₄×C₇	—	2	3	no	—
530	C₄×C₇	—	2	1	no	—
531	C₄×C₇	—	2	3	no	—
532	C₄×C₇	—	2	6	no	—
533	C₄×C₇	—	4	3	no	—
534	C₄×C₇	—	4	1	no	—
535	C₄×C₇	—	4	1	no	—
536	C₄×C₇	—	4	3	no	—
537	C₄×C₇	C₂×#121	1	6	yes	—
538	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
539	C₄×C₇	C₂²×#41	1	1	yes	characteristic
540	C₂⁴	—	2	3	no	—
541	#41	—	2	24	no	—
542	#41	—	2	6	no	—
543	#41	—	2	6	no	—
544	#41	—	2	6	no	—
545	#41	—	2	6	no	—
546	#41	—	2	6	no	—
547	#41	—	4	4	no	—
548	#41	—	4	12	no	—
549	#41	C₂×C₇×D₄	1	12	yes	—
550	#41	C₂×C₇×D₄	1	4	yes	—
551	#41	C₇×#41	1	4	yes	—
552	C₂×Q₈	—	2	8	no	—
553	C₂×Q₈	—	2	3	no	—
554	C₂×Q₈	—	2	3	no	—
555	C₂×Q₈	—	2	6	no	—
556	C₂×Q₈	—	2	6	no	—
557	C₂×Q₈	—	2	6	no	—
558	C₂×Q₈	—	4	4	no	—
559	C₂×Q₈	C₂×C₇×D₄	1	12	yes	—
560	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
561	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
562	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
563	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
564	C₂×D₄	—	2	3	no	—
565	C₂×D₄	—	2	3	no	—
566	C₂×D₄	—	2	3	no	—
567	C₂×D₄	—	2	6	no	—
568	C₂×D₄	—	2	3	no	—
569	C₂×D₄	—	2	3	no	—
570	C₂×D₄	—	2	3	no	—
571	C₂×D₄	—	2	6	no	—
572	C₂×D₄	—	4	12	no	—
573	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
574	C₂×D₄	C₂⁴×C₇	1	3	yes	—
575	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
576	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
577	C₂²×C₄	—	2	3	no	—
578	C₂²×C₄	—	2	3	no	—
579	C₂²×C₄	—	2	3	no	—
580	C₂²×C₄	—	2	3	no	—
581	C₂²×C₄	—	2	1	no	—
582	C₂²×C₄	—	2	3	no	—
583	C₂²×C₄	—	2	3	no	—
584	C₂²×C₄	—	2	3	no	—
585	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
586	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
587	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
588	Q₁₆	—	2	6	no	—
589	Q₁₆	—	2	12	no	—
590	Q₁₆	—	2	6	no	—
591	SD₁₆	—	2	6	no	—
592	SD₁₆	—	2	12	no	—
593	SD₁₆	—	2	6	no	—
594	SD₁₆	—	2	2	no	—
595	SD₁₆	—	2	6	no	—
596	D₈	—	2	2	no	—
597	D₈	—	2	6	no	—
598	M₁₆	—	2	6	no	—
599	M₁₆	—	2	6	no	—
600	C₂×C₈	—	2	6	no	—
601	C₂×C₈	—	2	6	no	—
602	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
603	C₂×C₈	C₂⁴×C₇	1	3	yes	—
604	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
605	C₂×C₈	C₂⁴×C₇	1	3	yes	—
606	C₄⋊C₄	—	2	6	no	—
607	C₄⋊C₄	—	2	3	no	—
608	C₄⋊C₄	—	2	3	no	—
609	C₄⋊C₄	—	2	3	no	—
610	C₄⋊C₄	—	2	3	no	—
611	C₄⋊C₄	—	2	3	no	—
612	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
613	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
614	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
615	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
616	C₂²⋊C₄	—	2	3	no	—
617	C₂²⋊C₄	—	2	3	no	—
618	C₂²⋊C₄	—	2	3	no	—
619	C₂²⋊C₄	—	2	6	no	—
620	C₂²⋊C₄	—	2	6	no	—
621	C₂²⋊C₄	—	2	1	no	—
622	C₂²⋊C₄	—	2	3	no	—
623	C₂²⋊C₄	—	2	1	no	—
624	C₂²⋊C₄	—	2	1	no	—
625	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
626	C₄²	—	2	3	no	—
627	C₄²	—	2	3	no	—
628	C₄²	—	2	6	no	—
629	C₄²	—	2	3	no	—
630	C₄²	C₂×C₇×D₄	1	3	yes	—
631	C₄²	C₂×C₇×D₄	1	3	yes	—
632	C₄²	C₂⁴×C₇	1	1	yes	characteristic
633	C₄²	C₂⁴×C₇	1	3	yes	—
634	C₂×C₇	—	2	6	no	—
635	C₂×C₇	—	4	2	no	—
636	C₂×C₇	—	8	1	no	—
637	C₂×C₇	—	8	3	no	—
638	C₂×C₇	C₂²×#121	1	1	yes	characteristic
639	C₂×C₇	#3584	1	1	yes	characteristic
640	C₂×C₇	#3586	1	1	yes	characteristic
641	C₂³	—	2	1	no	—
642	C₂³	—	2	3	no	—
643	C₂³	—	2	1	no	—
644	C₂³	—	4	6	no	—
645	C₂³	—	8	3	no	—
646	C₂³	—	8	3	no	—
647	C₂³	C₂²×C₇×D₄	1	3	yes	—
648	Q₈	—	2	3	no	—
649	Q₈	—	2	12	no	—
650	Q₈	—	2	3	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	3	no	—
653	Q₈	—	2	1	no	—
654	Q₈	—	4	6	no	—
655	Q₈	C₇×#121	1	4	yes	—
656	Q₈	C₂²×C₇×D₄	1	6	yes	—
657	D₄	—	2	12	no	—
658	D₄	—	2	3	no	—
659	D₄	—	2	1	no	—
660	D₄	—	2	1	no	—
661	D₄	—	2	3	no	—
662	D₄	—	2	1	no	—
663	D₄	—	4	6	no	—
664	D₄	—	4	6	no	—
665	D₄	—	4	6	no	—
666	D₄	—	8	3	no	—
667	D₄	—	8	3	no	—
668	D₄	C₇×#121	1	12	yes	—
669	D₄	C₂²×C₇×D₄	1	2	yes	—
670	C₂×C₄	—	2	3	no	—
671	C₂×C₄	—	2	12	no	—
672	C₂×C₄	—	2	12	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	3	no	—
675	C₂×C₄	—	2	3	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	4	3	no	—
678	C₂×C₄	—	4	3	no	—
679	C₂×C₄	—	4	2	no	—
680	C₂×C₄	—	4	6	no	—
681	C₂×C₄	—	4	6	no	—
682	C₂×C₄	—	4	6	no	—
683	C₂×C₄	—	8	3	no	—
684	C₂×C₄	—	8	3	no	—
685	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
686	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
687	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
688	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
689	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
690	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
691	C₂×C₄	C₂×C₇×#41	1	6	yes	—
692	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
693	C₈	—	2	1	no	—
694	C₈	—	2	1	no	—
695	C₈	—	2	3	no	—
696	C₈	—	2	3	no	—
697	C₇	#60433	1	1	yes	characteristic, direct product factor, Sylow
698	C₂²	—	2	3	no	—
699	C₂²	—	2	3	no	—
700	C₂²	—	2	2	no	—
701	C₂²	—	4	1	no	—
702	C₂²	—	4	1	no	—
703	C₂²	—	4	1	no	—
704	C₂²	—	4	3	no	—
705	C₂²	—	4	1	no	—
706	C₂²	—	4	3	no	—
707	C₂²	—	4	1	no	—
708	C₂²	—	4	3	no	—
709	C₂²	—	16	6	no	—
710	C₂²	C₂×C₇×#121	1	6	yes	—
711	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
712	C₄	—	2	1	no	—
713	C₄	—	2	3	no	—
714	C₄	—	2	1	no	—
715	C₄	—	2	3	no	—
716	C₄	—	2	6	no	—
717	C₄	—	4	3	no	—
718	C₄	—	4	1	no	—
719	C₄	—	4	1	no	—
720	C₄	—	4	3	no	—
721	C₄	C₂×C₇×#121	1	6	yes	—
722	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
723	C₄	C₂²×C₇×#41	1	1	yes	characteristic
724	C₂	—	2	6	no	—
725	C₂	—	4	2	no	—
726	C₂	—	8	3	no	—
727	C₂	—	8	1	no	—
728	C₂	C₂²×C₇×#121	1	1	yes	characteristic
729	C₂	C₇×#3584	1	1	yes	characteristic
730	C₂	C₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₇×#578	C₂	semidirect
2. C₂×C₇×#577	C₂	semidirect
3. C₇×#3500	C₂	semidirect
4. C₄×C₇×2₋¹⁺⁴	C₂	semidirect
5. C₇×#3046	C₂	semidirect
6. C₇×#3045	C₂	semidirect
7. C₇×#3044	C₂	semidirect
8. C₇×#3043	C₂	semidirect
9. C₇×#3040	C₂	semidirect
10. C₇×#3037	C₂	semidirect
11. C₇×#3035	C₂	semidirect
12. C₂×C₇×#464	C₂	semidirect
13. C₇×#2906	C₂	semidirect
14. C₇×#2898	C₂	semidirect
15. C₇×#2897	C₂	semidirect
22. C₂×C₇×#137	C₂²	semidirect
23. C₂×C₇×#137	C₂²	semidirect
26. C₂×C₇×#135	C₂²	semidirect
27. C₂²×C₇×Q₁₆	C₂²	semidirect
41. C₇×#520	C₂²	semidirect
43. C₇×#519	C₂²	semidirect
46. C₄×C₇×#41	C₂²	semidirect
48. C₂×C₄×C₇×Q₈	C₂²	semidirect
49. C₇×#464	C₂²	semidirect
50. C₇×#464	C₂²	semidirect
51. C₇×#464	C₂²	semidirect
52. C₇×#464	C₂²	semidirect
53. C₇×#464	C₂²	semidirect
54. C₇×#464	C₂²	semidirect
55. C₇×#463	C₂²	semidirect
56. C₇×#463	C₂²	semidirect
57. C₇×#463	C₂²	semidirect
58. C₇×#462	C₂²	semidirect
59. C₇×#462	C₂²	semidirect
60. C₇×#461	C₂²	semidirect
61. C₇×#460	C₂²	semidirect
62. C₇×#460	C₂²	semidirect
63. C₇×#459	C₂²	semidirect
64. C₇×#424	C₂²	semidirect
65. C₇×#424	C₂²	semidirect
66. C₇×#423	C₂²	semidirect
67. C₂×C₇×C₄⋊C₈	C₂²	semidirect
68. C₇×#419	C₂²	semidirect
71. C₇×#417	C₂²	semidirect
73. C₇×#416	C₂²	semidirect
74. C₂×C₇×#103	C₂²	semidirect
76. #60433	C₇	direct (N × H)
80. C₇×2₋¹⁺⁴	D₄	semidirect
183. C₂×#578	C₂×C₇	semidirect
184. C₂×#577	C₂×C₇	semidirect
185. #3500	C₂×C₇	semidirect
186. C₄×2₋¹⁺⁴	C₂×C₇	semidirect
187. #3046	C₂×C₇	semidirect
188. #3045	C₂×C₇	semidirect
189. #3044	C₂×C₇	semidirect
190. #3043	C₂×C₇	semidirect
191. #3040	C₂×C₇	semidirect
192. #3037	C₂×C₇	semidirect
193. #3035	C₂×C₇	semidirect
194. C₂×#464	C₂×C₇	semidirect
195. #2906	C₂×C₇	semidirect
196. #2898	C₂×C₇	semidirect
197. #2897	C₂×C₇	semidirect
298. C₂×#137	C₂²×C₇	semidirect
299. C₂×#137	C₂²×C₇	semidirect
302. C₂×#135	C₂²×C₇	semidirect
303. C₂²×Q₁₆	C₂²×C₇	semidirect
317. #520	C₂²×C₇	semidirect
319. #519	C₂²×C₇	semidirect
322. C₄×#41	C₂²×C₇	semidirect
324. C₂×C₄×Q₈	C₂²×C₇	semidirect
325. #464	C₂²×C₇	semidirect
326. #464	C₂²×C₇	semidirect
327. #464	C₂²×C₇	semidirect
328. #464	C₂²×C₇	semidirect
329. #464	C₂²×C₇	semidirect
330. #464	C₂²×C₇	semidirect
331. #463	C₂²×C₇	semidirect
332. #463	C₂²×C₇	semidirect
333. #463	C₂²×C₇	semidirect
334. #462	C₂²×C₇	semidirect
335. #462	C₂²×C₇	semidirect
336. #461	C₂²×C₇	semidirect
337. #460	C₂²×C₇	semidirect
338. #460	C₂²×C₇	semidirect
339. #459	C₂²×C₇	semidirect
340. #424	C₂²×C₇	semidirect
341. #424	C₂²×C₇	semidirect
342. #423	C₂²×C₇	semidirect
343. C₂×C₄⋊C₈	C₂²×C₇	semidirect
344. #419	C₂²×C₇	semidirect
347. #417	C₂²×C₇	semidirect
349. #416	C₂²×C₇	semidirect
350. C₂×#103	C₂²×C₇	semidirect
411. 2₋¹⁺⁴	C₇×D₄	semidirect
697. C₇	#60433	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#578	2
C₂×C₇×#577	2
C₇×#3500	2
C₄×C₇×2₋¹⁺⁴	2
C₇×#3046	2
C₇×#3045	2
C₇×#3044	2
C₇×#3043	2
C₇×#3040	2
C₇×#3037	2
C₇×#3035	2
C₂×C₇×#464	2
C₇×#2906	2
C₇×#2898	2
C₇×#2897	2
#60433	7

Subgroups — C7×#60433 #13237649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60433 #13237649