Subgroups — C₇×#60373 #13237589

Order 1792.

5058 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#577	C₂	1	2	yes	maximal, complemented
3	C₇×#3489	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3012	C₂	1	6	yes	maximal, complemented
5	C₇×#3009	C₂	1	6	yes	maximal, complemented
6	C₇×#3007	C₂	1	3	yes	maximal, complemented
7	C₇×#3003	C₂	1	3	yes	maximal, complemented
8	C₇×#3002	C₂	1	3	yes	maximal, complemented
9	C₂×C₇×#450	C₂	1	3	yes	maximal, complemented
10	C₇×#2897	C₂	1	2	yes	maximal, complemented
11	C₇×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×2₋¹⁺⁴	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₇×H(5,𝔽₂)	—	2	16	no	—
14	C₂×C₇×H(5,𝔽₂)	—	2	3	no	—
15	C₂×C₇×H(5,𝔽₂)	—	2	1	no	—
16	C₂×C₇×H(5,𝔽₂)	C₂²	1	6	yes	—
17	C₂×C₇×H(5,𝔽₂)	C₂²	1	2	yes	—
18	C₂²×C₇×#41	—	2	1	no	—
19	C₂²×C₇×#41	C₂²	1	3	yes	—
20	C₂²×C₇×#41	C₂²	1	1	yes	characteristic
21	C₂³×C₇×D₄	—	2	3	no	—
22	C₂³×C₇×D₄	C₂²	1	3	yes	—
23	C₇×#577	—	2	16	no	—
24	C₂×C₇×#137	C₂²	1	6	yes	complemented
25	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
26	C₂×C₇×#135	C₂²	1	6	yes	complemented
27	C₂²×C₇×SD₁₆	C₂²	1	6	yes	complemented
28	C₂×C₇×#131	C₂²	1	2	yes	complemented
29	C₇×D₄×Q₈	—	2	1	no	—
30	C₇×#548	—	2	1	no	—
31	C₇×#547	—	2	3	no	—
32	C₇×#546	—	2	3	no	—
33	C₇×#537	C₂²	1	3	yes	complemented
34	C₇×#536	C₂²	1	3	yes	complemented
35	C₇×#525	C₂²	1	3	yes	complemented
36	C₂×C₇×C₂²⋊Q₈	C₂²	1	3	yes	complemented
37	C₇×#520	C₂²	1	1	yes	characteristic, complemented
38	C₇×#452	C₂²	1	6	yes	complemented
39	C₇×#452	C₂²	1	12	yes	complemented
40	C₇×#451	C₂²	1	3	yes	complemented
41	C₇×#450	C₂²	1	3	yes	complemented
42	C₇×#450	C₂²	1	6	yes	complemented
43	C₇×#450	C₂²	1	12	yes	complemented
44	C₇×#450	C₂²	1	6	yes	complemented
45	C₇×#449	C₂²	1	6	yes	complemented
46	C₇×#448	C₂²	1	1	yes	characteristic, complemented
47	C₇×#447	C₂²	1	3	yes	complemented
48	C₇×#447	C₂²	1	6	yes	complemented
49	C₇×#418	C₂²	1	6	yes	complemented
50	C₇×#417	C₂²	1	6	yes	complemented
51	C₇×#416	C₂²	1	6	yes	complemented
52	C₂×C₇×#102	C₂²	1	6	yes	complemented
53	C₇×#408	C₂²	1	6	yes	complemented
54	C₇×#407	C₂²	1	3	yes	complemented
55	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	complemented
56	#60373	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁵×C₇	—	2	3	no	—
58	C₇×2₋¹⁺⁴	—	2	8	no	—
59	C₇×H(5,𝔽₂)	—	2	16	no	—
60	C₇×H(5,𝔽₂)	—	2	48	no	—
61	C₇×H(5,𝔽₂)	—	2	24	no	—
62	C₇×H(5,𝔽₂)	—	2	8	no	—
63	C₇×H(5,𝔽₂)	—	2	24	no	—
64	C₇×H(5,𝔽₂)	D₄	1	16	yes	complemented
65	C₂×C₇×#41	—	2	16	no	—
66	C₂×C₇×#41	—	2	24	no	—
67	C₂×C₇×#41	—	2	8	no	—
68	C₂×C₇×#41	—	2	6	no	—
69	C₂×C₇×#41	—	2	2	no	—
70	C₂×C₇×#41	—	2	3	no	—
71	C₂×C₇×#41	—	2	3	no	—
72	C₂×C₇×#41	—	2	3	no	—
73	C₂×C₇×#41	D₄	1	3	yes	—
74	C₂×C₇×#41	D₄	1	1	yes	characteristic
75	C₂×C₇×#41	C₂³	1	6	yes	—
76	C₂×C₇×#41	C₂³	1	6	yes	—
77	C₂×C₇×#41	C₂³	1	3	yes	—
78	C₂×C₇×#41	C₂³	1	1	yes	characteristic
79	C₂×C₇×#41	C₂³	1	3	yes	complemented
80	C₂²×C₇×Q₈	—	2	1	no	—
81	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
82	C₂²×C₇×Q₈	C₂³	1	1	yes	characteristic
83	C₂²×C₇×Q₈	C₂³	1	3	yes	complemented
84	C₂²×C₇×D₄	—	2	48	no	—
85	C₂²×C₇×D₄	—	2	3	no	—
86	C₂²×C₇×D₄	—	2	12	no	—
87	C₂²×C₇×D₄	—	2	6	no	—
88	C₂²×C₇×D₄	—	2	6	no	—
89	C₂²×C₇×D₄	—	2	24	no	—
90	C₂²×C₇×D₄	—	2	3	no	—
91	C₂²×C₇×D₄	—	2	3	no	—
92	C₂²×C₇×D₄	—	2	3	no	—
93	C₂²×C₇×D₄	D₄	1	3	yes	—
94	C₂²×C₇×D₄	C₂³	1	3	yes	—
95	C₂²×C₇×D₄	C₂³	1	6	yes	—
96	C₂²×C₇×D₄	C₂³	1	6	yes	—
97	C₂³×C₄×C₇	—	2	3	no	—
98	C₂³×C₄×C₇	C₂³	1	3	yes	—
99	C₇×#137	—	2	24	no	—
100	C₇×AGL(1,ℤ/8)	—	2	24	no	—
101	C₇×#135	—	2	24	no	—
102	C₂×C₇×Q₁₆	C₂³	1	6	yes	complemented
103	C₂×C₇×SD₁₆	—	2	24	no	—
104	C₂×C₇×SD₁₆	C₂³	1	6	yes	complemented
105	C₂×C₇×SD₁₆	C₂³	1	12	yes	complemented
106	C₂×C₇×SD₁₆	C₂³	1	2	yes	complemented
107	C₂×C₇×D₈	C₂³	1	6	yes	complemented
108	C₇×#131	—	2	8	no	—
109	C₂×C₇×M₁₆	C₂³	1	6	yes	complemented
110	C₂²×C₇×C₈	C₂³	1	6	yes	complemented
111	C₇×C₄⋊Q₈	—	2	3	no	—
112	C₇×#124	—	2	3	no	—
113	C₇×#123	—	2	6	no	—
114	C₇×#123	—	2	3	no	—
115	C₇×C₂²⋊Q₈	—	2	3	no	—
116	C₇×C₂²⋊Q₈	—	2	3	no	—
117	C₇×C₂²⋊Q₈	—	2	3	no	—
118	C₇×C₂²⋊Q₈	C₂³	1	3	yes	—
119	C₇×C₂²⋊Q₈	C₂³	1	6	yes	complemented
120	C₇×C₂²⋊Q₈	C₂³	1	3	yes	complemented
121	C₇×#121	—	2	6	no	—
122	C₇×#121	—	2	3	no	—
123	C₇×#121	C₂³	1	1	yes	characteristic
124	C₇×#121	C₂³	1	3	yes	complemented
125	C₇×#120	—	2	3	no	—
126	C₄×C₇×Q₈	—	2	1	no	—
127	C₄×C₇×D₄	—	2	3	no	—
128	C₄×C₇×D₄	—	2	3	no	—
129	C₄×C₇×D₄	—	2	3	no	—
130	C₇×#117	C₂³	1	3	yes	complemented
131	C₂×C₇×C₄⋊C₄	—	2	3	no	—
132	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	complemented
133	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
134	C₇×#103	C₂³	1	2	yes	complemented
135	C₇×#103	C₂³	1	6	yes	complemented
136	C₇×#102	C₂³	1	6	yes	complemented
137	C₇×#102	C₂³	1	12	yes	complemented
138	C₇×#102	C₂³	1	6	yes	complemented
139	C₇×C₂²⋊C₈	C₂³	1	1	yes	characteristic
140	C₇×C₂²⋊C₈	C₂³	1	3	yes	—
141	C₇×C₂²⋊C₈	C₂³	1	6	yes	complemented
142	C₇×C₂²⋊C₈	C₂³	1	6	yes	complemented
143	C₂²×H(5,𝔽₂)	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂×#577	C₂×C₇	1	2	yes	complemented
145	#3489	C₂×C₇	1	1	yes	characteristic, complemented
146	#3012	C₂×C₇	1	6	yes	complemented
147	#3009	C₂×C₇	1	6	yes	complemented
148	#3007	C₂×C₇	1	3	yes	complemented
149	#3003	C₂×C₇	1	3	yes	complemented
150	#3002	C₂×C₇	1	3	yes	complemented
151	C₂×#450	C₂×C₇	1	3	yes	complemented
152	#2897	C₂×C₇	1	2	yes	complemented
153	#2880	C₂×C₇	1	1	yes	characteristic, complemented
154	C₂⁴×C₇	—	2	24	no	—
155	C₂⁴×C₇	—	2	3	no	—
156	C₂⁴×C₇	—	2	6	no	—
157	C₂⁴×C₇	—	2	6	no	—
158	C₂⁴×C₇	—	8	6	no	—
159	C₂⁴×C₇	—	8	6	no	—
160	C₂⁴×C₇	C₂×D₄	1	3	yes	—
161	C₇×#41	—	2	24	no	—
162	C₇×#41	—	2	48	no	—
163	C₇×#41	—	2	24	no	—
164	C₇×#41	—	2	16	no	—
165	C₇×#41	—	2	4	no	—
166	C₇×#41	—	2	12	no	—
167	C₇×#41	—	2	12	no	—
168	C₇×#41	—	2	12	no	—
169	C₇×#41	—	2	12	no	—
170	C₇×#41	—	2	12	no	—
171	C₇×#41	—	2	12	no	—
172	C₇×#41	—	4	12	no	—
173	C₇×#41	C₂×D₄	1	24	yes	—
174	C₇×#41	C₂×D₄	1	8	yes	—
175	C₂×C₇×Q₈	—	2	4	no	—
176	C₂×C₇×Q₈	—	2	4	no	—
177	C₂×C₇×Q₈	—	2	1	no	—
178	C₂×C₇×Q₈	—	2	3	no	—
179	C₂×C₇×Q₈	—	2	12	no	—
180	C₂×C₇×Q₈	—	4	4	no	—
181	C₂×C₇×Q₈	C₂×D₄	1	3	yes	—
182	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
183	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
184	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
185	C₂×C₇×D₄	—	2	24	no	—
186	C₂×C₇×D₄	—	2	24	no	—
187	C₂×C₇×D₄	—	2	96	no	—
188	C₂×C₇×D₄	—	2	24	no	—
189	C₂×C₇×D₄	—	2	48	no	—
190	C₂×C₇×D₄	—	2	24	no	—
191	C₂×C₇×D₄	—	2	6	no	—
192	C₂×C₇×D₄	—	2	6	no	—
193	C₂×C₇×D₄	—	2	12	no	—
194	C₂×C₇×D₄	—	2	48	no	—
195	C₂×C₇×D₄	—	2	12	no	—
196	C₂×C₇×D₄	—	2	24	no	—
197	C₂×C₇×D₄	—	2	1	no	—
198	C₂×C₇×D₄	—	2	3	no	—
199	C₂×C₇×D₄	—	2	3	no	—
200	C₂×C₇×D₄	—	2	6	no	—
201	C₂×C₇×D₄	—	2	3	no	—
202	C₂×C₇×D₄	—	2	6	no	—
203	C₂×C₇×D₄	—	2	12	no	—
204	C₂×C₇×D₄	C₂×D₄	1	6	yes	—
205	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
206	C₂×C₇×D₄	C₂×D₄	1	24	yes	—
207	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
208	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
209	C₂×C₇×D₄	C₂⁴	1	6	yes	—
210	C₂²×C₄×C₇	—	2	24	no	—
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	6	no	—
213	C₂²×C₄×C₇	—	2	6	no	—
214	C₂²×C₄×C₇	—	2	12	no	—
215	C₂²×C₄×C₇	—	2	3	no	—
216	C₂²×C₄×C₇	—	2	3	no	—
217	C₂²×C₄×C₇	—	2	3	no	—
218	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
219	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
220	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
221	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
222	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
223	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
224	C₇×Q₁₆	—	2	12	no	—
225	C₇×SD₁₆	—	2	4	no	—
226	C₇×SD₁₆	—	2	12	no	—
227	C₇×SD₁₆	—	2	24	no	—
228	C₇×D₈	—	2	12	no	—
229	C₇×M₁₆	—	2	12	no	—
230	C₂×C₇×C₈	—	2	12	no	—
231	C₂×C₇×C₈	C₂⁴	1	2	yes	—
232	C₂×C₇×C₈	C₂⁴	1	6	yes	—
233	C₇×C₄⋊C₄	—	2	3	no	—
234	C₇×C₄⋊C₄	—	2	6	no	—
235	C₇×C₄⋊C₄	—	2	3	no	—
236	C₇×C₄⋊C₄	—	2	3	no	—
237	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
238	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
239	C₇×C₂²⋊C₄	—	2	3	no	—
240	C₇×C₂²⋊C₄	—	2	6	no	—
241	C₇×C₂²⋊C₄	—	2	3	no	—
242	C₇×C₂²⋊C₄	—	2	3	no	—
243	C₇×C₂²⋊C₄	—	2	1	no	—
244	C₇×C₂²⋊C₄	—	2	3	no	—
245	C₄²×C₇	—	2	3	no	—
246	C₂×2₋¹⁺⁴	C₂²×C₇	1	1	yes	characteristic, complemented
247	C₂×H(5,𝔽₂)	—	2	16	no	—
248	C₂×H(5,𝔽₂)	—	2	3	no	—
249	C₂×H(5,𝔽₂)	—	2	1	no	—
250	C₂×H(5,𝔽₂)	C₂²×C₇	1	6	yes	—
251	C₂×H(5,𝔽₂)	C₂²×C₇	1	2	yes	—
252	C₂²×#41	—	2	1	no	—
253	C₂²×#41	C₂²×C₇	1	3	yes	—
254	C₂²×#41	C₂²×C₇	1	1	yes	characteristic
255	C₂³×D₄	—	2	3	no	—
256	C₂³×D₄	C₂²×C₇	1	3	yes	—
257	#577	—	2	16	no	—
258	C₂×#137	C₂²×C₇	1	6	yes	complemented
259	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	6	yes	complemented
260	C₂×#135	C₂²×C₇	1	6	yes	complemented
261	C₂²×SD₁₆	C₂²×C₇	1	6	yes	complemented
262	C₂×#131	C₂²×C₇	1	2	yes	complemented
263	D₄×Q₈	—	2	1	no	—
264	#548	—	2	1	no	—
265	#547	—	2	3	no	—
266	#546	—	2	3	no	—
267	#537	C₂²×C₇	1	3	yes	complemented
268	#536	C₂²×C₇	1	3	yes	complemented
269	#525	C₂²×C₇	1	3	yes	complemented
270	C₂×C₂²⋊Q₈	C₂²×C₇	1	3	yes	complemented
271	#520	C₂²×C₇	1	1	yes	characteristic, complemented
272	#452	C₂²×C₇	1	6	yes	complemented
273	#452	C₂²×C₇	1	12	yes	complemented
274	#451	C₂²×C₇	1	3	yes	complemented
275	#450	C₂²×C₇	1	3	yes	complemented
276	#450	C₂²×C₇	1	6	yes	complemented
277	#450	C₂²×C₇	1	12	yes	complemented
278	#450	C₂²×C₇	1	6	yes	complemented
279	#449	C₂²×C₇	1	6	yes	complemented
280	#448	C₂²×C₇	1	1	yes	characteristic, complemented
281	#447	C₂²×C₇	1	3	yes	complemented
282	#447	C₂²×C₇	1	6	yes	complemented
283	#418	C₂²×C₇	1	6	yes	complemented
284	#417	C₂²×C₇	1	6	yes	complemented
285	#416	C₂²×C₇	1	6	yes	complemented
286	C₂×#102	C₂²×C₇	1	6	yes	complemented
287	#408	C₂²×C₇	1	6	yes	complemented
288	#407	C₂²×C₇	1	3	yes	complemented
289	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	complemented
290	C₂³×C₇	—	2	12	no	—
291	C₂³×C₇	—	2	24	no	—
292	C₂³×C₇	—	2	6	no	—
293	C₂³×C₇	—	2	6	no	—
294	C₂³×C₇	—	2	6	no	—
295	C₂³×C₇	—	2	24	no	—
296	C₂³×C₇	—	2	1	no	—
297	C₂³×C₇	—	4	3	no	—
298	C₂³×C₇	—	4	6	no	—
299	C₂³×C₇	—	4	6	no	—
300	C₂³×C₇	—	8	6	no	—
301	C₂³×C₇	—	8	6	no	—
302	C₂³×C₇	—	8	6	no	—
303	C₂³×C₇	—	8	6	no	—
304	C₂³×C₇	—	8	48	no	—
305	C₂³×C₇	C₂²×D₄	1	3	yes	—
306	C₂³×C₇	C₂²×D₄	1	3	yes	—
307	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
308	C₇×Q₈	—	2	4	no	—
309	C₇×Q₈	—	2	6	no	—
310	C₇×Q₈	—	2	2	no	—
311	C₇×Q₈	—	2	6	no	—
312	C₇×Q₈	—	4	6	no	—
313	C₇×Q₈	#120	1	4	yes	—
314	C₇×Q₈	C₂²×D₄	1	4	yes	—
315	C₇×D₄	—	2	24	no	—
316	C₇×D₄	—	2	24	no	—
317	C₇×D₄	—	2	12	no	—
318	C₇×D₄	—	2	24	no	—
319	C₇×D₄	—	2	12	no	—
320	C₇×D₄	—	2	6	no	—
321	C₇×D₄	—	2	24	no	—
322	C₇×D₄	—	2	6	no	—
323	C₇×D₄	—	2	2	no	—
324	C₇×D₄	—	4	6	no	—
325	C₇×D₄	—	4	2	no	—
326	C₇×D₄	—	8	3	no	—
327	C₇×D₄	#120	1	12	yes	—
328	C₇×D₄	C₂²×D₄	1	12	yes	—
329	C₂×C₄×C₇	—	2	4	no	—
330	C₂×C₄×C₇	—	2	24	no	—
331	C₂×C₄×C₇	—	2	6	no	—
332	C₂×C₄×C₇	—	2	2	no	—
333	C₂×C₄×C₇	—	2	6	no	—
334	C₂×C₄×C₇	—	2	24	no	—
335	C₂×C₄×C₇	—	2	6	no	—
336	C₂×C₄×C₇	—	2	2	no	—
337	C₂×C₄×C₇	—	2	6	no	—
338	C₂×C₄×C₇	—	2	1	no	—
339	C₂×C₄×C₇	—	2	3	no	—
340	C₂×C₄×C₇	—	2	3	no	—
341	C₂×C₄×C₇	—	4	6	no	—
342	C₂×C₄×C₇	—	4	6	no	—
343	C₂×C₄×C₇	—	8	3	no	—
344	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
345	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
347	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
348	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
349	C₇×C₈	—	2	2	no	—
350	C₇×C₈	—	2	6	no	—
351	C₂⁵	—	2	3	no	—
352	2₋¹⁺⁴	—	2	8	no	—
353	H(5,𝔽₂)	—	2	16	no	—
354	H(5,𝔽₂)	—	2	48	no	—
355	H(5,𝔽₂)	—	2	24	no	—
356	H(5,𝔽₂)	—	2	8	no	—
357	H(5,𝔽₂)	—	2	24	no	—
358	H(5,𝔽₂)	C₇×D₄	1	16	yes	complemented
359	C₂×#41	—	2	16	no	—
360	C₂×#41	—	2	24	no	—
361	C₂×#41	—	2	8	no	—
362	C₂×#41	—	2	6	no	—
363	C₂×#41	—	2	2	no	—
364	C₂×#41	—	2	3	no	—
365	C₂×#41	—	2	3	no	—
366	C₂×#41	—	2	3	no	—
367	C₂×#41	C₇×D₄	1	3	yes	—
368	C₂×#41	C₇×D₄	1	1	yes	characteristic
369	C₂×#41	C₂³×C₇	1	6	yes	—
370	C₂×#41	C₂³×C₇	1	6	yes	—
371	C₂×#41	C₂³×C₇	1	3	yes	—
372	C₂×#41	C₂³×C₇	1	1	yes	characteristic
373	C₂×#41	C₂³×C₇	1	3	yes	complemented
374	C₂²×Q₈	—	2	1	no	—
375	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
376	C₂²×Q₈	C₂³×C₇	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₇	1	3	yes	complemented
378	C₂²×D₄	—	2	48	no	—
379	C₂²×D₄	—	2	3	no	—
380	C₂²×D₄	—	2	12	no	—
381	C₂²×D₄	—	2	6	no	—
382	C₂²×D₄	—	2	6	no	—
383	C₂²×D₄	—	2	24	no	—
384	C₂²×D₄	—	2	3	no	—
385	C₂²×D₄	—	2	3	no	—
386	C₂²×D₄	—	2	3	no	—
387	C₂²×D₄	C₇×D₄	1	3	yes	—
388	C₂²×D₄	C₂³×C₇	1	3	yes	—
389	C₂²×D₄	C₂³×C₇	1	6	yes	—
390	C₂²×D₄	C₂³×C₇	1	6	yes	—
391	C₂³×C₄	—	2	3	no	—
392	C₂³×C₄	C₂³×C₇	1	3	yes	—
393	#137	—	2	24	no	—
394	AGL(1,ℤ/8)	—	2	24	no	—
395	#135	—	2	24	no	—
396	C₂×Q₁₆	C₂³×C₇	1	6	yes	complemented
397	C₂×SD₁₆	—	2	24	no	—
398	C₂×SD₁₆	C₂³×C₇	1	6	yes	complemented
399	C₂×SD₁₆	C₂³×C₇	1	12	yes	complemented
400	C₂×SD₁₆	C₂³×C₇	1	2	yes	complemented
401	C₂×D₈	C₂³×C₇	1	6	yes	complemented
402	#131	—	2	8	no	—
403	C₂×M₁₆	C₂³×C₇	1	6	yes	complemented
404	C₂²×C₈	C₂³×C₇	1	6	yes	complemented
405	C₄⋊Q₈	—	2	3	no	—
406	#124	—	2	3	no	—
407	#123	—	2	6	no	—
408	#123	—	2	3	no	—
409	C₂²⋊Q₈	—	2	3	no	—
410	C₂²⋊Q₈	—	2	3	no	—
411	C₂²⋊Q₈	—	2	3	no	—
412	C₂²⋊Q₈	C₂³×C₇	1	3	yes	—
413	C₂²⋊Q₈	C₂³×C₇	1	6	yes	complemented
414	C₂²⋊Q₈	C₂³×C₇	1	3	yes	complemented
415	#121	—	2	6	no	—
416	#121	—	2	3	no	—
417	#121	C₂³×C₇	1	1	yes	characteristic
418	#121	C₂³×C₇	1	3	yes	complemented
419	#120	—	2	3	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×D₄	—	2	3	no	—
422	C₄×D₄	—	2	3	no	—
423	C₄×D₄	—	2	3	no	—
424	#117	C₂³×C₇	1	3	yes	complemented
425	C₂×C₄⋊C₄	—	2	3	no	—
426	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	complemented
427	C₂×C₂²⋊C₄	—	2	3	no	—
428	#103	C₂³×C₇	1	2	yes	complemented
429	#103	C₂³×C₇	1	6	yes	complemented
430	#102	C₂³×C₇	1	6	yes	complemented
431	#102	C₂³×C₇	1	12	yes	complemented
432	#102	C₂³×C₇	1	6	yes	complemented
433	C₂²⋊C₈	C₂³×C₇	1	1	yes	characteristic
434	C₂²⋊C₈	C₂³×C₇	1	3	yes	—
435	C₂²⋊C₈	C₂³×C₇	1	6	yes	complemented
436	C₂²⋊C₈	C₂³×C₇	1	6	yes	complemented
437	C₂²×C₇	—	2	3	no	—
438	C₂²×C₇	—	2	1	no	—
439	C₂²×C₇	—	2	1	no	—
440	C₂²×C₇	—	2	3	no	—
441	C₂²×C₇	—	2	3	no	—
442	C₂²×C₇	—	2	3	no	—
443	C₂²×C₇	—	2	12	no	—
444	C₂²×C₇	—	2	3	no	—
445	C₂²×C₇	—	4	12	no	—
446	C₂²×C₇	—	4	24	no	—
447	C₂²×C₇	—	4	6	no	—
448	C₂²×C₇	—	4	6	no	—
449	C₂²×C₇	—	4	1	no	—
450	C₂²×C₇	—	4	1	no	—
451	C₂²×C₇	—	4	1	no	—
452	C₂²×C₇	—	8	24	no	—
453	C₂²×C₇	—	8	24	no	—
454	C₂²×C₇	C₂×#120	1	6	yes	—
455	C₂²×C₇	#577	1	2	yes	complemented
456	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
457	C₄×C₇	—	2	4	no	—
458	C₄×C₇	—	2	3	no	—
459	C₄×C₇	—	2	1	no	—
460	C₄×C₇	—	4	3	no	—
461	C₄×C₇	—	4	3	no	—
462	C₄×C₇	—	4	1	no	—
463	C₄×C₇	C₂×#120	1	6	yes	—
464	C₄×C₇	C₂³×D₄	1	2	yes	—
465	C₂⁴	—	2	24	no	—
466	C₂⁴	—	2	3	no	—
467	C₂⁴	—	2	6	no	—
468	C₂⁴	—	2	6	no	—
469	C₂⁴	—	8	6	no	—
470	C₂⁴	—	8	6	no	—
471	C₂⁴	C₂×C₇×D₄	1	3	yes	—
472	#41	—	2	24	no	—
473	#41	—	2	48	no	—
474	#41	—	2	24	no	—
475	#41	—	2	16	no	—
476	#41	—	2	4	no	—
477	#41	—	2	12	no	—
478	#41	—	2	12	no	—
479	#41	—	2	12	no	—
480	#41	—	2	12	no	—
481	#41	—	2	12	no	—
482	#41	—	2	12	no	—
483	#41	—	4	12	no	—
484	#41	C₂×C₇×D₄	1	24	yes	—
485	#41	C₂×C₇×D₄	1	8	yes	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	3	no	—
490	C₂×Q₈	—	2	12	no	—
491	C₂×Q₈	—	4	4	no	—
492	C₂×Q₈	C₂×C₇×D₄	1	3	yes	—
493	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
494	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
495	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
496	C₂×D₄	—	2	24	no	—
497	C₂×D₄	—	2	24	no	—
498	C₂×D₄	—	2	96	no	—
499	C₂×D₄	—	2	24	no	—
500	C₂×D₄	—	2	48	no	—
501	C₂×D₄	—	2	24	no	—
502	C₂×D₄	—	2	6	no	—
503	C₂×D₄	—	2	6	no	—
504	C₂×D₄	—	2	12	no	—
505	C₂×D₄	—	2	48	no	—
506	C₂×D₄	—	2	12	no	—
507	C₂×D₄	—	2	24	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	3	no	—
510	C₂×D₄	—	2	3	no	—
511	C₂×D₄	—	2	6	no	—
512	C₂×D₄	—	2	3	no	—
513	C₂×D₄	—	2	6	no	—
514	C₂×D₄	—	2	12	no	—
515	C₂×D₄	C₂×C₇×D₄	1	6	yes	—
516	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
517	C₂×D₄	C₂×C₇×D₄	1	24	yes	—
518	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
519	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
520	C₂×D₄	C₂⁴×C₇	1	6	yes	—
521	C₂²×C₄	—	2	24	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	6	no	—
524	C₂²×C₄	—	2	6	no	—
525	C₂²×C₄	—	2	12	no	—
526	C₂²×C₄	—	2	3	no	—
527	C₂²×C₄	—	2	3	no	—
528	C₂²×C₄	—	2	3	no	—
529	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
530	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
531	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
532	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
534	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
535	Q₁₆	—	2	12	no	—
536	SD₁₆	—	2	4	no	—
537	SD₁₆	—	2	12	no	—
538	SD₁₆	—	2	24	no	—
539	D₈	—	2	12	no	—
540	M₁₆	—	2	12	no	—
541	C₂×C₈	—	2	12	no	—
542	C₂×C₈	C₂⁴×C₇	1	2	yes	—
543	C₂×C₈	C₂⁴×C₇	1	6	yes	—
544	C₄⋊C₄	—	2	3	no	—
545	C₄⋊C₄	—	2	6	no	—
546	C₄⋊C₄	—	2	3	no	—
547	C₄⋊C₄	—	2	3	no	—
548	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
549	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
550	C₂²⋊C₄	—	2	3	no	—
551	C₂²⋊C₄	—	2	6	no	—
552	C₂²⋊C₄	—	2	3	no	—
553	C₂²⋊C₄	—	2	3	no	—
554	C₂²⋊C₄	—	2	1	no	—
555	C₂²⋊C₄	—	2	3	no	—
556	C₄²	—	2	3	no	—
557	C₂×C₇	—	2	2	no	—
558	C₂×C₇	—	2	6	no	—
559	C₂×C₇	—	4	3	no	—
560	C₂×C₇	—	4	12	no	—
561	C₂×C₇	—	8	1	no	—
562	C₂×C₇	C₂²×#120	1	1	yes	characteristic
563	C₂×C₇	C₂×#577	1	1	yes	characteristic
564	C₂×C₇	#3585	1	1	yes	characteristic
565	C₂³	—	2	12	no	—
566	C₂³	—	2	24	no	—
567	C₂³	—	2	6	no	—
568	C₂³	—	2	6	no	—
569	C₂³	—	2	6	no	—
570	C₂³	—	2	24	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	4	3	no	—
573	C₂³	—	4	6	no	—
574	C₂³	—	4	6	no	—
575	C₂³	—	8	6	no	—
576	C₂³	—	8	6	no	—
577	C₂³	—	8	6	no	—
578	C₂³	—	8	6	no	—
579	C₂³	—	8	48	no	—
580	C₂³	C₂²×C₇×D₄	1	3	yes	—
581	C₂³	C₂²×C₇×D₄	1	3	yes	—
582	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
583	Q₈	—	2	4	no	—
584	Q₈	—	2	2	no	—
585	Q₈	—	2	6	no	—
586	Q₈	—	2	6	no	—
587	Q₈	—	4	6	no	—
588	Q₈	C₇×#120	1	4	yes	—
589	Q₈	C₂²×C₇×D₄	1	4	yes	—
590	D₄	—	2	24	no	—
591	D₄	—	2	24	no	—
592	D₄	—	2	12	no	—
593	D₄	—	2	24	no	—
594	D₄	—	2	12	no	—
595	D₄	—	2	6	no	—
596	D₄	—	2	2	no	—
597	D₄	—	2	6	no	—
598	D₄	—	2	24	no	—
599	D₄	—	4	6	no	—
600	D₄	—	4	2	no	—
601	D₄	—	8	3	no	—
602	D₄	C₇×#120	1	12	yes	—
603	D₄	C₂²×C₇×D₄	1	12	yes	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	24	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	6	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	6	no	—
610	C₂×C₄	—	2	6	no	—
611	C₂×C₄	—	2	24	no	—
612	C₂×C₄	—	2	6	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	3	no	—
615	C₂×C₄	—	2	3	no	—
616	C₂×C₄	—	4	6	no	—
617	C₂×C₄	—	4	6	no	—
618	C₂×C₄	—	8	3	no	—
619	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
620	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
621	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
622	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
623	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
624	C₈	—	2	2	no	—
625	C₈	—	2	6	no	—
626	C₇	#60373	1	1	yes	characteristic, direct product factor, Sylow
627	C₂²	—	2	3	no	—
628	C₂²	—	2	3	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	3	no	—
632	C₂²	—	2	3	no	—
633	C₂²	—	2	3	no	—
634	C₂²	—	2	12	no	—
635	C₂²	—	4	12	no	—
636	C₂²	—	4	24	no	—
637	C₂²	—	4	6	no	—
638	C₂²	—	4	6	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	8	24	no	—
643	C₂²	—	8	24	no	—
644	C₂²	C₂×C₇×#120	1	6	yes	—
645	C₂²	C₇×#577	1	2	yes	complemented
646	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
647	C₄	—	2	4	no	—
648	C₄	—	2	3	no	—
649	C₄	—	2	1	no	—
650	C₄	—	4	1	no	—
651	C₄	—	4	3	no	—
652	C₄	—	4	3	no	—
653	C₄	C₂×C₇×#120	1	6	yes	—
654	C₄	C₂³×C₇×D₄	1	2	yes	—
655	C₂	—	2	2	no	—
656	C₂	—	2	6	no	—
657	C₂	—	4	3	no	—
658	C₂	—	4	12	no	—
659	C₂	—	8	1	no	—
660	C₂	C₂²×C₇×#120	1	1	yes	characteristic
661	C₂	C₂×C₇×#577	1	1	yes	characteristic
662	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 ◁ C₂²×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×H(5,𝔽₂)	C₂	semidirect
2. C₂×C₇×#577	C₂	semidirect
3. C₇×#3489	C₂	semidirect
4. C₇×#3012	C₂	semidirect
5. C₇×#3009	C₂	semidirect
6. C₇×#3007	C₂	semidirect
7. C₇×#3003	C₂	semidirect
8. C₇×#3002	C₂	semidirect
9. C₂×C₇×#450	C₂	semidirect
10. C₇×#2897	C₂	semidirect
11. C₇×#2880	C₂	semidirect
12. C₂×C₇×2₋¹⁺⁴	C₂²	semidirect
24. C₂×C₇×#137	C₂²	semidirect
25. C₂×C₇×AGL(1,ℤ/8)	C₂²	semidirect
26. C₂×C₇×#135	C₂²	semidirect
27. C₂²×C₇×SD₁₆	C₂²	semidirect
28. C₂×C₇×#131	C₂²	semidirect
33. C₇×#537	C₂²	semidirect
34. C₇×#536	C₂²	semidirect
35. C₇×#525	C₂²	semidirect
36. C₂×C₇×C₂²⋊Q₈	C₂²	semidirect
37. C₇×#520	C₂²	semidirect
38. C₇×#452	C₂²	semidirect
39. C₇×#452	C₂²	semidirect
40. C₇×#451	C₂²	semidirect
41. C₇×#450	C₂²	semidirect
42. C₇×#450	C₂²	semidirect
43. C₇×#450	C₂²	semidirect
44. C₇×#450	C₂²	semidirect
45. C₇×#449	C₂²	semidirect
46. C₇×#448	C₂²	semidirect
47. C₇×#447	C₂²	semidirect
48. C₇×#447	C₂²	semidirect
49. C₇×#418	C₂²	semidirect
50. C₇×#417	C₂²	semidirect
51. C₇×#416	C₂²	semidirect
52. C₂×C₇×#102	C₂²	semidirect
53. C₇×#408	C₂²	semidirect
54. C₇×#407	C₂²	semidirect
55. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
56. #60373	C₇	direct (N × H)
64. C₇×H(5,𝔽₂)	D₄	semidirect
79. C₂×C₇×#41	C₂³	semidirect
83. C₂²×C₇×Q₈	C₂³	semidirect
102. C₂×C₇×Q₁₆	C₂³	semidirect
104. C₂×C₇×SD₁₆	C₂³	semidirect
105. C₂×C₇×SD₁₆	C₂³	semidirect
106. C₂×C₇×SD₁₆	C₂³	semidirect
107. C₂×C₇×D₈	C₂³	semidirect
109. C₂×C₇×M₁₆	C₂³	semidirect
110. C₂²×C₇×C₈	C₂³	semidirect
119. C₇×C₂²⋊Q₈	C₂³	semidirect
120. C₇×C₂²⋊Q₈	C₂³	semidirect
124. C₇×#121	C₂³	semidirect
130. C₇×#117	C₂³	semidirect
132. C₂×C₇×C₄⋊C₄	C₂³	semidirect
134. C₇×#103	C₂³	semidirect
135. C₇×#103	C₂³	semidirect
136. C₇×#102	C₂³	semidirect
137. C₇×#102	C₂³	semidirect
138. C₇×#102	C₂³	semidirect
141. C₇×C₂²⋊C₈	C₂³	semidirect
142. C₇×C₂²⋊C₈	C₂³	semidirect
143. C₂²×H(5,𝔽₂)	C₂×C₇	semidirect
144. C₂×#577	C₂×C₇	semidirect
145. #3489	C₂×C₇	semidirect
146. #3012	C₂×C₇	semidirect
147. #3009	C₂×C₇	semidirect
148. #3007	C₂×C₇	semidirect
149. #3003	C₂×C₇	semidirect
150. #3002	C₂×C₇	semidirect
151. C₂×#450	C₂×C₇	semidirect
152. #2897	C₂×C₇	semidirect
153. #2880	C₂×C₇	semidirect
246. C₂×2₋¹⁺⁴	C₂²×C₇	semidirect
258. C₂×#137	C₂²×C₇	semidirect
259. C₂×AGL(1,ℤ/8)	C₂²×C₇	semidirect
260. C₂×#135	C₂²×C₇	semidirect
261. C₂²×SD₁₆	C₂²×C₇	semidirect
262. C₂×#131	C₂²×C₇	semidirect
267. #537	C₂²×C₇	semidirect
268. #536	C₂²×C₇	semidirect
269. #525	C₂²×C₇	semidirect
270. C₂×C₂²⋊Q₈	C₂²×C₇	semidirect
271. #520	C₂²×C₇	semidirect
272. #452	C₂²×C₇	semidirect
273. #452	C₂²×C₇	semidirect
274. #451	C₂²×C₇	semidirect
275. #450	C₂²×C₇	semidirect
276. #450	C₂²×C₇	semidirect
277. #450	C₂²×C₇	semidirect
278. #450	C₂²×C₇	semidirect
279. #449	C₂²×C₇	semidirect
280. #448	C₂²×C₇	semidirect
281. #447	C₂²×C₇	semidirect
282. #447	C₂²×C₇	semidirect
283. #418	C₂²×C₇	semidirect
284. #417	C₂²×C₇	semidirect
285. #416	C₂²×C₇	semidirect
286. C₂×#102	C₂²×C₇	semidirect
287. #408	C₂²×C₇	semidirect
288. #407	C₂²×C₇	semidirect
289. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
358. H(5,𝔽₂)	C₇×D₄	semidirect
373. C₂×#41	C₂³×C₇	semidirect
377. C₂²×Q₈	C₂³×C₇	semidirect
396. C₂×Q₁₆	C₂³×C₇	semidirect
398. C₂×SD₁₆	C₂³×C₇	semidirect
399. C₂×SD₁₆	C₂³×C₇	semidirect
400. C₂×SD₁₆	C₂³×C₇	semidirect
401. C₂×D₈	C₂³×C₇	semidirect
403. C₂×M₁₆	C₂³×C₇	semidirect
404. C₂²×C₈	C₂³×C₇	semidirect
413. C₂²⋊Q₈	C₂³×C₇	semidirect
414. C₂²⋊Q₈	C₂³×C₇	semidirect
418. #121	C₂³×C₇	semidirect
424. #117	C₂³×C₇	semidirect
426. C₂×C₄⋊C₄	C₂³×C₇	semidirect
428. #103	C₂³×C₇	semidirect
429. #103	C₂³×C₇	semidirect
430. #102	C₂³×C₇	semidirect
431. #102	C₂³×C₇	semidirect
432. #102	C₂³×C₇	semidirect
435. C₂²⋊C₈	C₂³×C₇	semidirect
436. C₂²⋊C₈	C₂³×C₇	semidirect
455. C₂²×C₇	#577	semidirect
626. C₇	#60373	direct (N × H)
645. C₂²	C₇×#577	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×H(5,𝔽₂)	2
C₂×C₇×#577	2
C₇×#3489	2
C₇×#3012	2
C₇×#3009	2
C₇×#3007	2
C₇×#3003	2
C₇×#3002	2
C₂×C₇×#450	2
C₇×#2897	2
C₇×#2880	2
#60373	7

Subgroups — C7×#60373 #13237589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60373 #13237589