Subgroups — C₇×#60431 #13237647

Order 1792.

2782 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#577	C₂	1	2	yes	maximal, complemented
2	C₇×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3045	C₂	1	6	yes	maximal, complemented
5	C₇×#3043	C₂	1	6	yes	maximal, complemented
6	C₇×#3038	C₂	1	3	yes	maximal, complemented
7	C₇×#3037	C₂	1	3	yes	maximal, complemented
8	C₇×#3035	C₂	1	3	yes	maximal, complemented
9	C₂×C₇×#460	C₂	1	3	yes	maximal, complemented
10	C₇×#2906	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
13	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₇×#577	—	2	16	no	—
15	C₂×C₇×#137	C₂²	1	6	yes	complemented
16	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
17	C₂×C₇×#135	C₂²	1	6	yes	complemented
18	C₂²×C₇×SD₁₆	C₂²	1	6	yes	complemented
19	C₂×C₇×#131	C₂²	1	2	yes	—
20	C₇×#550	—	2	1	no	—
21	C₇×#550	—	2	1	no	—
22	C₇×D₄²	—	2	3	no	—
23	C₇×D₄²	—	2	3	no	—
24	C₇×#535	C₂²	1	6	yes	—
25	C₇×#532	C₂²	1	3	yes	—
26	C₂×C₇×#127	C₂²	1	3	yes	—
27	C₇×#520	—	2	3	no	—
28	C₇×#520	C₂²	1	1	yes	characteristic
29	C₇×#520	C₂²	1	3	yes	complemented
30	C₇×#519	—	2	1	no	—
31	C₇×#519	C₂²	1	3	yes	complemented
32	C₄×C₇×#41	—	2	3	no	—
33	C₄×C₇×#41	C₂²	1	1	yes	characteristic
34	C₄×C₇×#41	C₂²	1	3	yes	complemented
35	C₂×C₄×C₇×Q₈	—	2	1	no	—
36	C₂×C₄×C₇×Q₈	C₂²	1	3	yes	complemented
37	C₇×#464	C₂²	1	6	yes	complemented
38	C₇×#464	C₂²	1	12	yes	complemented
39	C₇×#463	C₂²	1	6	yes	complemented
40	C₇×#462	C₂²	1	3	yes	complemented
41	C₇×#461	C₂²	1	1	yes	characteristic, complemented
42	C₇×#460	C₂²	1	3	yes	complemented
43	C₇×#460	C₂²	1	6	yes	complemented
44	C₇×#460	C₂²	1	12	yes	complemented
45	C₇×#460	C₂²	1	6	yes	complemented
46	C₇×#459	C₂²	1	3	yes	complemented
47	C₇×#459	C₂²	1	6	yes	complemented
48	C₇×#424	C₂²	1	6	yes	complemented
49	C₇×#423	C₂²	1	3	yes	complemented
50	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	complemented
51	C₇×#418	C₂²	1	6	yes	complemented
52	C₇×#417	C₂²	1	6	yes	complemented
53	C₇×#416	C₂²	1	6	yes	complemented
54	C₂×C₇×#102	C₂²	1	6	yes	complemented
55	#60431	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₇×2₋¹⁺⁴	—	2	4	no	—
57	C₇×2₋¹⁺⁴	D₄	1	8	yes	complemented
58	C₇×H(5,𝔽₂)	—	2	8	no	—
59	C₇×H(5,𝔽₂)	—	2	8	no	—
60	C₂×C₇×#41	—	2	2	no	—
61	C₂×C₇×#41	—	2	3	no	—
62	C₂×C₇×#41	C₂³	1	6	yes	—
63	C₂×C₇×#41	C₂³	1	3	yes	—
64	C₂×C₇×#41	C₂³	1	1	yes	characteristic
65	C₂²×C₇×Q₈	—	2	1	no	—
66	C₂²×C₇×Q₈	C₂³	1	3	yes	—
67	C₂²×C₇×D₄	—	2	6	no	—
68	C₂²×C₇×D₄	—	2	3	no	—
69	C₂²×C₇×D₄	—	2	3	no	—
70	C₂²×C₇×D₄	C₂³	1	6	yes	—
71	C₇×#137	—	2	24	no	—
72	C₇×AGL(1,ℤ/8)	—	2	24	no	—
73	C₇×#135	—	2	24	no	—
74	C₂×C₇×Q₁₆	C₂³	1	6	yes	—
75	C₂×C₇×SD₁₆	—	2	24	no	—
76	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
77	C₂×C₇×SD₁₆	C₂³	1	12	yes	—
78	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
79	C₂×C₇×D₈	C₂³	1	6	yes	—
80	C₇×#131	—	2	8	no	—
81	C₂×C₇×M₁₆	C₂³	1	6	yes	—
82	C₂²×C₇×C₈	C₂³	1	6	yes	—
83	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₇×#127	—	2	3	no	—
85	C₇×#127	—	2	3	no	—
86	C₇×#127	C₂³	1	3	yes	—
87	C₇×#127	C₂³	1	6	yes	—
88	C₇×#124	C₂³	1	6	yes	—
89	C₇×#121	—	2	3	no	—
90	C₇×#121	—	2	3	no	—
91	C₇×#121	—	2	6	no	—
92	C₇×#121	—	2	6	no	—
93	C₇×#120	—	2	12	no	—
94	C₄×C₇×Q₈	—	2	1	no	—
95	C₄×C₇×Q₈	—	2	3	no	—
96	C₄×C₇×Q₈	—	2	3	no	—
97	C₄×C₇×Q₈	—	2	3	no	—
98	C₄×C₇×Q₈	—	2	3	no	—
99	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
100	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
101	C₄×C₇×Q₈	C₂³	1	3	yes	—
102	C₄×C₇×Q₈	C₂³	1	6	yes	—
103	C₄×C₇×Q₈	C₂³	1	3	yes	—
104	C₄×C₇×D₄	—	2	3	no	—
105	C₄×C₇×D₄	—	2	3	no	—
106	C₄×C₇×D₄	—	2	3	no	—
107	C₄×C₇×D₄	—	2	6	no	—
108	C₄×C₇×D₄	—	2	3	no	—
109	C₄×C₇×D₄	—	2	3	no	—
110	C₄×C₇×D₄	D₄	1	3	yes	—
111	C₄×C₇×D₄	D₄	1	3	yes	—
112	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
113	C₄×C₇×D₄	C₂³	1	3	yes	—
114	C₇×#117	—	2	3	no	—
115	C₇×#117	—	2	6	no	—
116	C₇×#117	—	2	3	no	—
117	C₇×#117	C₂³	1	3	yes	—
118	C₇×#117	C₂³	1	3	yes	—
119	C₂×C₇×C₄⋊C₄	—	2	3	no	—
120	C₂×C₇×C₄⋊C₄	—	2	3	no	—
121	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
122	C₂×C₄²×C₇	—	2	3	no	—
123	C₂×C₄²×C₇	—	2	3	no	—
124	C₂×C₄²×C₇	C₂³	1	3	yes	—
125	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
126	C₇×C₄⋊C₈	C₂³	1	3	yes	—
127	C₇×C₄⋊C₈	C₂³	1	6	yes	—
128	C₇×C₄⋊C₈	C₂³	1	6	yes	—
129	C₇×#103	C₂³	1	2	yes	—
130	C₇×#103	C₂³	1	6	yes	—
131	C₇×#102	C₂³	1	6	yes	—
132	C₇×#102	C₂³	1	12	yes	—
133	C₇×#102	C₂³	1	6	yes	—
134	C₂×#577	C₂×C₇	1	2	yes	complemented
135	#3498	C₂×C₇	1	1	yes	characteristic, complemented
136	C₄×2₋¹⁺⁴	C₂×C₇	1	1	yes	characteristic, complemented
137	#3045	C₂×C₇	1	6	yes	complemented
138	#3043	C₂×C₇	1	6	yes	complemented
139	#3038	C₂×C₇	1	3	yes	complemented
140	#3037	C₂×C₇	1	3	yes	complemented
141	#3035	C₂×C₇	1	3	yes	complemented
142	C₂×#460	C₂×C₇	1	3	yes	complemented
143	#2906	C₂×C₇	1	1	yes	characteristic, complemented
144	#2897	C₂×C₇	1	2	yes	complemented
145	C₂⁴×C₇	—	2	6	no	—
146	C₇×#41	—	2	24	no	—
147	C₇×#41	—	2	6	no	—
148	C₇×#41	—	2	12	no	—
149	C₇×#41	—	2	12	no	—
150	C₇×#41	—	4	8	no	—
151	C₇×#41	C₂×D₄	1	4	yes	—
152	C₇×#41	C₂×D₄	1	12	yes	complemented
153	C₇×#41	#41	1	4	yes	—
154	C₂×C₇×Q₈	—	2	8	no	—
155	C₂×C₇×Q₈	—	2	3	no	—
156	C₂×C₇×Q₈	—	2	6	no	—
157	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
158	C₂×C₇×Q₈	C₂×D₄	1	12	yes	complemented
159	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
160	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
161	C₂×C₇×D₄	—	2	3	no	—
162	C₂×C₇×D₄	—	2	6	no	—
163	C₂×C₇×D₄	—	2	12	no	—
164	C₂×C₇×D₄	—	2	3	no	—
165	C₂×C₇×D₄	—	2	3	no	—
166	C₂×C₇×D₄	—	2	6	no	—
167	C₂×C₇×D₄	—	2	6	no	—
168	C₂×C₇×D₄	—	2	6	no	—
169	C₂×C₇×D₄	—	2	12	no	—
170	C₂×C₇×D₄	—	4	24	no	—
171	C₂×C₇×D₄	—	8	6	no	—
172	C₂×C₇×D₄	—	8	6	no	—
173	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
174	C₂×C₇×D₄	C₂⁴	1	6	yes	—
175	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₇	—	2	3	no	—
177	C₂²×C₄×C₇	—	2	3	no	—
178	C₂²×C₄×C₇	—	2	3	no	—
179	C₂²×C₄×C₇	—	2	1	no	—
180	C₂²×C₄×C₇	—	2	6	no	—
181	C₂²×C₄×C₇	—	2	3	no	—
182	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
183	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
184	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
185	C₇×Q₁₆	—	2	12	no	—
186	C₇×SD₁₆	—	2	12	no	—
187	C₇×SD₁₆	—	2	24	no	—
188	C₇×SD₁₆	—	2	4	no	—
189	C₇×D₈	—	2	12	no	—
190	C₇×M₁₆	—	2	12	no	—
191	C₂×C₇×C₈	—	2	12	no	—
192	C₂×C₇×C₈	C₂⁴	1	2	yes	—
193	C₂×C₇×C₈	C₂⁴	1	6	yes	—
194	C₇×C₄⋊C₄	—	2	6	no	—
195	C₇×C₄⋊C₄	—	2	3	no	—
196	C₇×C₄⋊C₄	—	2	3	no	—
197	C₇×C₄⋊C₄	—	2	3	no	—
198	C₇×C₄⋊C₄	—	2	1	no	—
199	C₇×C₄⋊C₄	—	2	1	no	—
200	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
201	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
202	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
203	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
204	C₇×C₂²⋊C₄	—	2	3	no	—
205	C₇×C₂²⋊C₄	—	2	1	no	—
206	C₇×C₂²⋊C₄	—	2	3	no	—
207	C₇×C₂²⋊C₄	—	2	6	no	—
208	C₇×C₂²⋊C₄	—	2	6	no	—
209	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
210	C₄²×C₇	—	2	3	no	—
211	C₄²×C₇	—	2	3	no	—
212	C₄²×C₇	—	2	6	no	—
213	C₄²×C₇	—	2	3	no	—
214	C₄²×C₇	C₂×D₄	1	3	yes	—
215	C₄²×C₇	C₂×D₄	1	3	yes	—
216	C₄²×C₇	C₂⁴	1	1	yes	characteristic
217	C₄²×C₇	C₂⁴	1	3	yes	—
218	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic
219	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
220	#577	—	2	16	no	—
221	C₂×#137	C₂²×C₇	1	6	yes	complemented
222	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	complemented
223	C₂×#135	C₂²×C₇	1	6	yes	complemented
224	C₂²×SD₁₆	C₂²×C₇	1	6	yes	complemented
225	C₂×#131	C₂²×C₇	1	2	yes	—
226	#550	—	2	1	no	—
227	#550	—	2	1	no	—
228	D₄²	—	2	3	no	—
229	D₄²	—	2	3	no	—
230	#535	C₂²×C₇	1	6	yes	—
231	#532	C₂²×C₇	1	3	yes	—
232	C₂×#127	C₂²×C₇	1	3	yes	—
233	#520	—	2	3	no	—
234	#520	C₂²×C₇	1	1	yes	characteristic
235	#520	C₂²×C₇	1	3	yes	complemented
236	#519	—	2	1	no	—
237	#519	C₂²×C₇	1	3	yes	complemented
238	C₄×#41	—	2	3	no	—
239	C₄×#41	C₂²×C₇	1	1	yes	characteristic
240	C₄×#41	C₂²×C₇	1	3	yes	complemented
241	C₂×C₄×Q₈	—	2	1	no	—
242	C₂×C₄×Q₈	C₂²×C₇	1	3	yes	complemented
243	#464	C₂²×C₇	1	6	yes	complemented
244	#464	C₂²×C₇	1	12	yes	complemented
245	#463	C₂²×C₇	1	6	yes	complemented
246	#462	C₂²×C₇	1	3	yes	complemented
247	#461	C₂²×C₇	1	1	yes	characteristic, complemented
248	#460	C₂²×C₇	1	3	yes	complemented
249	#460	C₂²×C₇	1	6	yes	complemented
250	#460	C₂²×C₇	1	12	yes	complemented
251	#460	C₂²×C₇	1	6	yes	complemented
252	#459	C₂²×C₇	1	3	yes	complemented
253	#459	C₂²×C₇	1	6	yes	complemented
254	#424	C₂²×C₇	1	6	yes	complemented
255	#423	C₂²×C₇	1	3	yes	complemented
256	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	complemented
257	#418	C₂²×C₇	1	6	yes	complemented
258	#417	C₂²×C₇	1	6	yes	complemented
259	#416	C₂²×C₇	1	6	yes	complemented
260	C₂×#102	C₂²×C₇	1	6	yes	complemented
261	C₂³×C₇	—	2	1	no	—
262	C₂³×C₇	—	2	6	no	—
263	C₂³×C₇	—	4	12	no	—
264	C₂³×C₇	—	8	6	no	—
265	C₂³×C₇	—	8	6	no	—
266	C₂³×C₇	C₂²×D₄	1	3	yes	—
267	C₇×Q₈	—	2	12	no	—
268	C₇×Q₈	—	2	3	no	—
269	C₇×Q₈	—	2	2	no	—
270	C₇×Q₈	—	2	2	no	—
271	C₇×Q₈	#121	1	4	yes	—
272	C₇×Q₈	C₂²×D₄	1	6	yes	—
273	C₇×D₄	—	2	12	no	—
274	C₇×D₄	—	2	6	no	—
275	C₇×D₄	—	2	1	no	—
276	C₇×D₄	—	2	6	no	—
277	C₇×D₄	—	4	6	no	—
278	C₇×D₄	—	4	12	no	—
279	C₇×D₄	—	4	12	no	—
280	C₇×D₄	—	4	6	no	—
281	C₇×D₄	—	8	6	no	—
282	C₇×D₄	—	8	6	no	—
283	C₇×D₄	#121	1	12	yes	—
284	C₇×D₄	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₇	—	2	12	no	—
286	C₂×C₄×C₇	—	2	3	no	—
287	C₂×C₄×C₇	—	2	12	no	—
288	C₂×C₄×C₇	—	2	6	no	—
289	C₂×C₄×C₇	—	2	1	no	—
290	C₂×C₄×C₇	—	2	3	no	—
291	C₂×C₄×C₇	—	2	2	no	—
292	C₂×C₄×C₇	—	4	3	no	—
293	C₂×C₄×C₇	—	4	12	no	—
294	C₂×C₄×C₇	—	4	2	no	—
295	C₂×C₄×C₇	—	8	3	no	—
296	C₂×C₄×C₇	—	8	3	no	—
297	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
298	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
299	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
301	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
302	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₇	C₂×#41	1	6	yes	—
304	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
305	C₇×C₈	—	2	2	no	—
306	C₇×C₈	—	2	6	no	—
307	2₋¹⁺⁴	—	2	4	no	—
308	2₋¹⁺⁴	C₇×D₄	1	8	yes	complemented
309	H(5,𝔽₂)	—	2	8	no	—
310	H(5,𝔽₂)	—	2	8	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	3	no	—
313	C₂×#41	C₂³×C₇	1	6	yes	—
314	C₂×#41	C₂³×C₇	1	3	yes	—
315	C₂×#41	C₂³×C₇	1	1	yes	characteristic
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×Q₈	C₂³×C₇	1	3	yes	—
318	C₂²×D₄	—	2	6	no	—
319	C₂²×D₄	—	2	3	no	—
320	C₂²×D₄	—	2	3	no	—
321	C₂²×D₄	C₂³×C₇	1	6	yes	—
322	#137	—	2	24	no	—
323	AGL(1,ℤ/8)	—	2	24	no	—
324	#135	—	2	24	no	—
325	C₂×Q₁₆	C₂³×C₇	1	6	yes	—
326	C₂×SD₁₆	—	2	24	no	—
327	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
328	C₂×SD₁₆	C₂³×C₇	1	12	yes	—
329	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
330	C₂×D₈	C₂³×C₇	1	6	yes	—
331	#131	—	2	8	no	—
332	C₂×M₁₆	C₂³×C₇	1	6	yes	—
333	C₂²×C₈	C₂³×C₇	1	6	yes	—
334	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
335	#127	—	2	3	no	—
336	#127	—	2	3	no	—
337	#127	C₂³×C₇	1	3	yes	—
338	#127	C₂³×C₇	1	6	yes	—
339	#124	C₂³×C₇	1	6	yes	—
340	#121	—	2	3	no	—
341	#121	—	2	3	no	—
342	#121	—	2	6	no	—
343	#121	—	2	6	no	—
344	#120	—	2	12	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×Q₈	—	2	3	no	—
347	C₄×Q₈	—	2	3	no	—
348	C₄×Q₈	—	2	3	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
351	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
352	C₄×Q₈	C₂³×C₇	1	3	yes	—
353	C₄×Q₈	C₂³×C₇	1	6	yes	—
354	C₄×Q₈	C₂³×C₇	1	3	yes	—
355	C₄×D₄	—	2	3	no	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	6	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	C₇×D₄	1	3	yes	—
362	C₄×D₄	C₇×D₄	1	3	yes	—
363	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
364	C₄×D₄	C₂³×C₇	1	3	yes	—
365	#117	—	2	3	no	—
366	#117	—	2	6	no	—
367	#117	—	2	3	no	—
368	#117	C₂³×C₇	1	3	yes	—
369	#117	C₂³×C₇	1	3	yes	—
370	C₂×C₄⋊C₄	—	2	3	no	—
371	C₂×C₄⋊C₄	—	2	3	no	—
372	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
373	C₂×C₄²	—	2	3	no	—
374	C₂×C₄²	—	2	3	no	—
375	C₂×C₄²	C₂³×C₇	1	3	yes	—
376	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
377	C₄⋊C₈	C₂³×C₇	1	3	yes	—
378	C₄⋊C₈	C₂³×C₇	1	6	yes	—
379	C₄⋊C₈	C₂³×C₇	1	6	yes	—
380	#103	C₂³×C₇	1	2	yes	—
381	#103	C₂³×C₇	1	6	yes	—
382	#102	C₂³×C₇	1	6	yes	—
383	#102	C₂³×C₇	1	12	yes	—
384	#102	C₂³×C₇	1	6	yes	—
385	C₂²×C₇	—	2	3	no	—
386	C₂²×C₇	—	2	3	no	—
387	C₂²×C₇	—	2	2	no	—
388	C₂²×C₇	—	4	1	no	—
389	C₂²×C₇	—	4	1	no	—
390	C₂²×C₇	—	4	6	no	—
391	C₂²×C₇	—	4	6	no	—
392	C₂²×C₇	—	4	6	no	—
393	C₂²×C₇	—	16	12	no	—
394	C₂²×C₇	C₂×#121	1	6	yes	—
395	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
396	C₄×C₇	—	2	1	no	—
397	C₄×C₇	—	2	3	no	—
398	C₄×C₇	—	2	6	no	—
399	C₄×C₇	—	2	3	no	—
400	C₄×C₇	—	4	3	no	—
401	C₄×C₇	—	4	2	no	—
402	C₄×C₇	—	4	1	no	—
403	C₄×C₇	C₂×#121	1	6	yes	—
404	C₄×C₇	#577	1	2	yes	complemented
405	C₄×C₇	C₂³×D₄	1	1	yes	characteristic
406	C₄×C₇	C₂²×#41	1	1	yes	characteristic
407	C₂⁴	—	2	6	no	—
408	#41	—	2	24	no	—
409	#41	—	2	6	no	—
410	#41	—	2	12	no	—
411	#41	—	2	12	no	—
412	#41	—	4	8	no	—
413	#41	C₂×C₇×D₄	1	4	yes	—
414	#41	C₂×C₇×D₄	1	12	yes	complemented
415	#41	C₇×#41	1	4	yes	—
416	C₂×Q₈	—	2	8	no	—
417	C₂×Q₈	—	2	3	no	—
418	C₂×Q₈	—	2	6	no	—
419	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
420	C₂×Q₈	C₂×C₇×D₄	1	12	yes	complemented
421	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
422	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
423	C₂×D₄	—	2	3	no	—
424	C₂×D₄	—	2	6	no	—
425	C₂×D₄	—	2	12	no	—
426	C₂×D₄	—	2	3	no	—
427	C₂×D₄	—	2	3	no	—
428	C₂×D₄	—	2	6	no	—
429	C₂×D₄	—	2	6	no	—
430	C₂×D₄	—	2	6	no	—
431	C₂×D₄	—	2	12	no	—
432	C₂×D₄	—	4	24	no	—
433	C₂×D₄	—	8	6	no	—
434	C₂×D₄	—	8	6	no	—
435	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
436	C₂×D₄	C₂⁴×C₇	1	6	yes	—
437	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
438	C₂²×C₄	—	2	3	no	—
439	C₂²×C₄	—	2	3	no	—
440	C₂²×C₄	—	2	3	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	6	no	—
443	C₂²×C₄	—	2	3	no	—
444	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
445	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
446	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
447	Q₁₆	—	2	12	no	—
448	SD₁₆	—	2	12	no	—
449	SD₁₆	—	2	24	no	—
450	SD₁₆	—	2	4	no	—
451	D₈	—	2	12	no	—
452	M₁₆	—	2	12	no	—
453	C₂×C₈	—	2	12	no	—
454	C₂×C₈	C₂⁴×C₇	1	2	yes	—
455	C₂×C₈	C₂⁴×C₇	1	6	yes	—
456	C₄⋊C₄	—	2	6	no	—
457	C₄⋊C₄	—	2	3	no	—
458	C₄⋊C₄	—	2	3	no	—
459	C₄⋊C₄	—	2	3	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
463	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
464	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
465	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
466	C₂²⋊C₄	—	2	3	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	3	no	—
469	C₂²⋊C₄	—	2	6	no	—
470	C₂²⋊C₄	—	2	6	no	—
471	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
472	C₄²	—	2	3	no	—
473	C₄²	—	2	3	no	—
474	C₄²	—	2	6	no	—
475	C₄²	—	2	3	no	—
476	C₄²	C₂×C₇×D₄	1	3	yes	—
477	C₄²	C₂×C₇×D₄	1	3	yes	—
478	C₄²	C₂⁴×C₇	1	1	yes	characteristic
479	C₄²	C₂⁴×C₇	1	3	yes	—
480	C₂×C₇	—	2	6	no	—
481	C₂×C₇	—	4	2	no	—
482	C₂×C₇	—	8	6	no	—
483	C₂×C₇	C₂²×#121	1	1	yes	characteristic
484	C₂×C₇	C₂×#577	1	1	yes	characteristic
485	C₂×C₇	#3585	1	1	yes	characteristic
486	C₂³	—	2	1	no	—
487	C₂³	—	2	6	no	—
488	C₂³	—	4	12	no	—
489	C₂³	—	8	6	no	—
490	C₂³	—	8	6	no	—
491	C₂³	C₂²×C₇×D₄	1	3	yes	—
492	Q₈	—	2	3	no	—
493	Q₈	—	2	12	no	—
494	Q₈	—	2	2	no	—
495	Q₈	—	2	2	no	—
496	Q₈	C₇×#121	1	4	yes	—
497	Q₈	C₂²×C₇×D₄	1	6	yes	—
498	D₄	—	2	12	no	—
499	D₄	—	2	6	no	—
500	D₄	—	2	1	no	—
501	D₄	—	2	6	no	—
502	D₄	—	4	6	no	—
503	D₄	—	4	12	no	—
504	D₄	—	4	12	no	—
505	D₄	—	4	6	no	—
506	D₄	—	8	6	no	—
507	D₄	—	8	6	no	—
508	D₄	C₇×#121	1	12	yes	—
509	D₄	C₂²×C₇×D₄	1	2	yes	—
510	C₂×C₄	—	2	3	no	—
511	C₂×C₄	—	2	12	no	—
512	C₂×C₄	—	2	12	no	—
513	C₂×C₄	—	2	6	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	3	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	4	3	no	—
518	C₂×C₄	—	4	12	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	8	3	no	—
521	C₂×C₄	—	8	3	no	—
522	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
523	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
524	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
525	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
526	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
527	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
528	C₂×C₄	C₂×C₇×#41	1	6	yes	—
529	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
530	C₈	—	2	2	no	—
531	C₈	—	2	6	no	—
532	C₇	#60431	1	1	yes	characteristic, direct product factor, Sylow
533	C₂²	—	2	3	no	—
534	C₂²	—	2	3	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	4	1	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	4	6	no	—
539	C₂²	—	4	6	no	—
540	C₂²	—	4	6	no	—
541	C₂²	—	16	12	no	—
542	C₂²	C₂×C₇×#121	1	6	yes	—
543	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
544	C₄	—	2	3	no	—
545	C₄	—	2	1	no	—
546	C₄	—	2	3	no	—
547	C₄	—	2	6	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	3	no	—
550	C₄	—	4	2	no	—
551	C₄	C₂×C₇×#121	1	6	yes	—
552	C₄	C₇×#577	1	2	yes	complemented
553	C₄	C₂³×C₇×D₄	1	1	yes	characteristic
554	C₄	C₂²×C₇×#41	1	1	yes	characteristic
555	C₂	—	2	6	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	6	no	—
558	C₂	C₂²×C₇×#121	1	1	yes	characteristic
559	C₂	C₂×C₇×#577	1	1	yes	characteristic
560	C₂	C₇×#3585	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×#577	C₂	semidirect
2. C₇×#3498	C₂	semidirect
3. C₄×C₇×2₋¹⁺⁴	C₂	semidirect
4. C₇×#3045	C₂	semidirect
5. C₇×#3043	C₂	semidirect
6. C₇×#3038	C₂	semidirect
7. C₇×#3037	C₂	semidirect
8. C₇×#3035	C₂	semidirect
9. C₂×C₇×#460	C₂	semidirect
10. C₇×#2906	C₂	semidirect
11. C₇×#2897	C₂	semidirect
15. C₂×C₇×#137	C₂²	semidirect
16. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₇×#135	C₂²	semidirect
18. C₂²×C₇×SD₁₆	C₂²	semidirect
29. C₇×#520	C₂²	semidirect
31. C₇×#519	C₂²	semidirect
34. C₄×C₇×#41	C₂²	semidirect
36. C₂×C₄×C₇×Q₈	C₂²	semidirect
37. C₇×#464	C₂²	semidirect
38. C₇×#464	C₂²	semidirect
39. C₇×#463	C₂²	semidirect
40. C₇×#462	C₂²	semidirect
41. C₇×#461	C₂²	semidirect
42. C₇×#460	C₂²	semidirect
43. C₇×#460	C₂²	semidirect
44. C₇×#460	C₂²	semidirect
45. C₇×#460	C₂²	semidirect
46. C₇×#459	C₂²	semidirect
47. C₇×#459	C₂²	semidirect
48. C₇×#424	C₂²	semidirect
49. C₇×#423	C₂²	semidirect
50. C₂×C₇×C₄⋊C₈	C₂²	semidirect
51. C₇×#418	C₂²	semidirect
52. C₇×#417	C₂²	semidirect
53. C₇×#416	C₂²	semidirect
54. C₂×C₇×#102	C₂²	semidirect
55. #60431	C₇	direct (N × H)
57. C₇×2₋¹⁺⁴	D₄	semidirect
134. C₂×#577	C₂×C₇	semidirect
135. #3498	C₂×C₇	semidirect
136. C₄×2₋¹⁺⁴	C₂×C₇	semidirect
137. #3045	C₂×C₇	semidirect
138. #3043	C₂×C₇	semidirect
139. #3038	C₂×C₇	semidirect
140. #3037	C₂×C₇	semidirect
141. #3035	C₂×C₇	semidirect
142. C₂×#460	C₂×C₇	semidirect
143. #2906	C₂×C₇	semidirect
144. #2897	C₂×C₇	semidirect
152. C₇×#41	C₂×D₄	semidirect
158. C₂×C₇×Q₈	C₂×D₄	semidirect
221. C₂×#137	C₂²×C₇	semidirect
222. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
223. C₂×#135	C₂²×C₇	semidirect
224. C₂²×SD₁₆	C₂²×C₇	semidirect
235. #520	C₂²×C₇	semidirect
237. #519	C₂²×C₇	semidirect
240. C₄×#41	C₂²×C₇	semidirect
242. C₂×C₄×Q₈	C₂²×C₇	semidirect
243. #464	C₂²×C₇	semidirect
244. #464	C₂²×C₇	semidirect
245. #463	C₂²×C₇	semidirect
246. #462	C₂²×C₇	semidirect
247. #461	C₂²×C₇	semidirect
248. #460	C₂²×C₇	semidirect
249. #460	C₂²×C₇	semidirect
250. #460	C₂²×C₇	semidirect
251. #460	C₂²×C₇	semidirect
252. #459	C₂²×C₇	semidirect
253. #459	C₂²×C₇	semidirect
254. #424	C₂²×C₇	semidirect
255. #423	C₂²×C₇	semidirect
256. C₂×C₄⋊C₈	C₂²×C₇	semidirect
257. #418	C₂²×C₇	semidirect
258. #417	C₂²×C₇	semidirect
259. #416	C₂²×C₇	semidirect
260. C₂×#102	C₂²×C₇	semidirect
308. 2₋¹⁺⁴	C₇×D₄	semidirect
404. C₄×C₇	#577	semidirect
414. #41	C₂×C₇×D₄	semidirect
420. C₂×Q₈	C₂×C₇×D₄	semidirect
532. C₇	#60431	direct (N × H)
552. C₄	C₇×#577	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#577	2
C₇×#3498	2
C₄×C₇×2₋¹⁺⁴	2
C₇×#3045	2
C₇×#3043	2
C₇×#3038	2
C₇×#3037	2
C₇×#3035	2
C₂×C₇×#460	2
C₇×#2906	2
C₇×#2897	2
#60431	7

Subgroups — C7×#60431 #13237647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60431 #13237647