Subgroups — C₇×#61466 #13238682

Order 1792.

2102 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3482	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3392	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×Q₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#3359	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3353	C₂	1	3	yes	maximal, complemented
8	C₇×#3330	C₂	1	3	yes	maximal, complemented
9	C₇×#3324	C₂	1	3	yes	maximal, complemented
10	C₇×#3272	C₂	1	3	yes	maximal, complemented
11	C₇×#3182	C₂	1	3	yes	maximal, complemented
12	C₇×#3115	C₂	1	3	yes	maximal, complemented
13	C₇×#3110	C₂	1	3	yes	maximal, complemented
14	C₇×#2971	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
16	C₂²×C₇×#41	—	2	1	no	—
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	3	yes	—
18	C₇×#557	C₂²	1	1	yes	characteristic, complemented
19	C₇×#556	C₂²	1	3	yes	complemented
20	C₇×#552	—	2	3	no	—
21	C₇×#550	C₂²	1	1	yes	characteristic
22	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
23	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₇×#548	—	2	1	no	—
25	C₇×#548	—	2	1	no	—
26	C₇×#547	C₂²	1	3	yes	—
27	C₇×#547	C₂²	1	3	yes	complemented
28	C₇×#546	—	2	3	no	—
29	C₇×D₄²	C₂²	1	3	yes	—
30	C₇×#533	C₂²	1	3	yes	complemented
31	C₇×#532	C₂²	1	3	yes	—
32	C₇×#529	—	2	3	no	—
33	C₇×#525	—	2	3	no	—
34	C₄×C₇×#41	C₂²	1	1	yes	characteristic
35	C₄×C₇×#41	C₂²	1	3	yes	complemented
36	C₂×C₄×C₇×Q₈	—	2	1	no	—
37	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
38	C₇×#500	C₂²	1	3	yes	complemented
39	C₇×#492	C₂²	1	3	yes	complemented
40	C₇×#487	C₂²	1	3	yes	complemented
41	C₇×#486	C₂²	1	3	yes	complemented
42	C₇×#484	C₂²	1	6	yes	complemented
43	C₇×#482	C₂²	1	6	yes	complemented
44	C₇×#475	C₂²	1	3	yes	complemented
45	C₇×#474	C₂²	1	3	yes	—
46	C₇×#474	C₂²	1	3	yes	complemented
47	C₇×#474	C₂²	1	3	yes	complemented
48	C₇×#468	C₂²	1	6	yes	complemented
49	C₇×#468	C₂²	1	6	yes	complemented
50	C₇×#461	C₂²	1	3	yes	—
51	C₇×#461	C₂²	1	3	yes	complemented
52	C₇×#460	C₂²	1	3	yes	complemented
53	C₇×#459	C₂²	1	3	yes	complemented
54	C₇×#450	C₂²	1	6	yes	—
55	C₇×#449	C₂²	1	6	yes	complemented
56	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
57	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
58	C₇×#442	C₂²	1	1	yes	characteristic
59	C₇×#442	C₂²	1	3	yes	complemented
60	C₇×#440	C₂²	1	3	yes	—
61	C₇×#440	C₂²	1	3	yes	complemented
62	C₇×#440	C₂²	1	6	yes	complemented
63	C₄×C₇×SD₁₆	C₂²	1	3	yes	—
64	C₄×C₇×SD₁₆	C₂²	1	3	yes	complemented
65	C₄×C₇×D₈	C₂²	1	3	yes	—
66	C₄×C₇×D₈	C₂²	1	3	yes	complemented
67	C₇×#436	C₂²	1	3	yes	complemented
68	C₇×#436	C₂²	1	3	yes	complemented
69	C₇×#432	C₂²	1	3	yes	complemented
70	C₇×#428	C₂²	1	3	yes	complemented
71	C₇×#418	C₂²	1	3	yes	—
72	C₇×#417	C₂²	1	3	yes	complemented
73	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
74	#61466	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇×#41	—	2	1	no	—
76	C₂×C₇×#41	—	2	3	no	—
77	C₂×C₇×#41	—	2	3	no	—
78	C₂×C₇×#41	—	4	8	no	—
79	C₂×C₇×#41	D₄	1	1	yes	characteristic
80	C₂×C₇×#41	C₂³	1	3	yes	—
81	C₂²×C₇×Q₈	—	2	1	no	—
82	C₂²×C₇×Q₈	—	2	1	no	—
83	C₂²×C₇×D₄	—	2	3	no	—
84	C₂²×C₇×D₄	C₂³	1	3	yes	—
85	C₂³×C₄×C₇	—	2	3	no	—
86	C₇×AGL(1,ℤ/8)	—	2	12	no	—
87	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
88	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
89	C₂×C₇×SD₁₆	C₂³	1	6	yes	—
90	C₂×C₇×D₈	C₂³	1	3	yes	—
91	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
92	C₂×C₇×M₁₆	C₂³	1	3	yes	—
93	C₂×C₇×M₁₆	C₂³	1	3	yes	—
94	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
95	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
96	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
97	C₇×#127	C₂³	1	3	yes	—
98	C₇×#126	—	2	6	no	—
99	C₇×#125	—	2	3	no	—
100	C₇×#125	—	2	3	no	—
101	C₇×#124	—	2	3	no	—
102	C₇×#124	—	2	3	no	—
103	C₇×#123	—	2	6	no	—
104	C₇×#123	—	2	6	no	—
105	C₇×C₂²⋊Q₈	—	2	3	no	—
106	C₇×C₂²⋊Q₈	—	2	3	no	—
107	C₇×C₂²⋊Q₈	—	2	3	no	—
108	C₇×C₂²⋊Q₈	—	2	3	no	—
109	C₇×C₂²⋊Q₈	C₂³	1	6	yes	—
110	C₇×#121	—	2	3	no	—
111	C₇×#121	—	2	3	no	—
112	C₇×#121	—	2	3	no	—
113	C₇×#121	—	2	3	no	—
114	C₇×#121	C₂³	1	6	yes	—
115	C₇×#120	—	2	6	no	—
116	C₄×C₇×Q₈	—	2	1	no	—
117	C₄×C₇×Q₈	—	2	3	no	—
118	C₄×C₇×Q₈	—	2	3	no	—
119	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
120	C₄×C₇×Q₈	C₂³	1	3	yes	—
121	C₄×C₇×Q₈	C₂³	1	3	yes	—
122	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
123	C₄×C₇×D₄	—	2	3	no	—
124	C₄×C₇×D₄	—	2	3	no	—
125	C₄×C₇×D₄	—	2	3	no	—
126	C₄×C₇×D₄	—	2	3	no	—
127	C₄×C₇×D₄	—	2	3	no	—
128	C₄×C₇×D₄	D₄	1	3	yes	—
129	C₄×C₇×D₄	C₂³	1	3	yes	—
130	C₄×C₇×D₄	C₂³	1	3	yes	—
131	C₄×C₇×D₄	C₂³	1	3	yes	—
132	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
133	C₄×C₇×D₄	C₂³	1	3	yes	—
134	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
135	C₄×C₇×D₄	C₂³	1	3	yes	—
136	C₇×#117	—	2	3	no	—
137	C₇×#117	—	2	3	no	—
138	C₇×#117	D₄	1	3	yes	—
139	C₇×#117	C₂³	1	3	yes	—
140	C₂×C₇×C₄⋊C₄	—	2	3	no	—
141	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
142	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
143	C₂×C₄²×C₇	—	2	3	no	—
144	C₂×C₄²×C₇	C₂³	1	3	yes	—
145	C₇×#107	C₂³	1	3	yes	—
146	C₇×#107	C₂³	1	6	yes	—
147	C₇×#107	C₂³	1	3	yes	—
148	C₇×#106	C₂³	1	3	yes	—
149	C₇×#106	C₂³	1	1	yes	characteristic
150	C₇×C₄⋊C₈	C₂³	1	3	yes	—
151	C₇×C₄⋊C₈	C₂³	1	3	yes	—
152	C₇×#103	C₂³	1	3	yes	—
153	C₇×#103	C₂³	1	6	yes	—
154	C₇×#103	C₂³	1	3	yes	—
155	C₇×#102	C₂³	1	3	yes	—
156	C₇×#102	C₂³	1	6	yes	—
157	C₇×#102	C₂³	1	3	yes	—
158	C₇×#102	C₂³	1	3	yes	—
159	C₇×#102	C₂³	1	3	yes	—
160	C₇×#102	C₂³	1	1	yes	characteristic
161	C₇×#102	C₂³	1	1	yes	characteristic
162	C₇×C₂²⋊C₈	C₂³	1	6	yes	—
163	C₇×#97	C₂³	1	1	yes	characteristic
164	C₇×#97	C₂³	1	3	yes	—
165	C₄×C₇×C₈	C₂³	1	3	yes	—
166	C₄×C₇×C₈	C₂³	1	3	yes	—
167	#3482	C₂×C₇	1	1	yes	characteristic, complemented
168	D₄×#41	C₂×C₇	1	1	yes	characteristic, complemented
169	#3392	C₂×C₇	1	1	yes	characteristic, complemented
170	Q₈×D₈	C₂×C₇	1	1	yes	characteristic, complemented
171	#3359	C₂×C₇	1	1	yes	characteristic, complemented
172	#3358	C₂×C₇	1	1	yes	characteristic, complemented
173	#3353	C₂×C₇	1	3	yes	complemented
174	#3330	C₂×C₇	1	3	yes	complemented
175	#3324	C₂×C₇	1	3	yes	complemented
176	#3272	C₂×C₇	1	3	yes	complemented
177	#3182	C₂×C₇	1	3	yes	complemented
178	#3115	C₂×C₇	1	3	yes	complemented
179	#3110	C₂×C₇	1	3	yes	complemented
180	#2971	C₂×C₇	1	1	yes	characteristic, complemented
181	C₄×AGL(1,ℤ/8)	C₂×C₇	1	3	yes	complemented
182	C₂⁴×C₇	—	2	3	no	—
183	C₇×#41	—	2	4	no	—
184	C₇×#41	—	2	6	no	—
185	C₇×#41	—	4	3	no	—
186	C₇×#41	—	4	12	no	—
187	C₇×#41	—	4	12	no	—
188	C₇×#41	—	4	4	no	—
189	C₂×C₇×Q₈	—	2	3	no	—
190	C₂×C₇×Q₈	—	2	3	no	—
191	C₂×C₇×Q₈	—	4	4	no	—
192	C₂×C₇×Q₈	—	4	4	no	—
193	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₇×Q₈	C₂⁴	1	3	yes	—
195	C₂×C₇×D₄	—	2	6	no	—
196	C₂×C₇×D₄	—	2	3	no	—
197	C₂×C₇×D₄	—	2	6	no	—
198	C₂×C₇×D₄	—	2	3	no	—
199	C₂×C₇×D₄	—	2	3	no	—
200	C₂×C₇×D₄	—	2	3	no	—
201	C₂×C₇×D₄	—	4	3	no	—
202	C₂×C₇×D₄	—	4	12	no	—
203	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
204	C₂×C₇×D₄	C₂⁴	1	3	yes	—
205	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
206	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
207	C₂²×C₄×C₇	—	2	3	no	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	2	3	no	—
210	C₂²×C₄×C₇	—	2	3	no	—
211	C₂²×C₄×C₇	—	2	1	no	—
212	C₂²×C₄×C₇	—	2	3	no	—
213	C₂²×C₄×C₇	—	2	3	no	—
214	C₂²×C₄×C₇	—	2	3	no	—
215	C₂²×C₄×C₇	—	4	12	no	—
216	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
217	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
218	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
219	C₇×SD₁₆	—	2	6	no	—
220	C₇×SD₁₆	—	2	6	no	—
221	C₇×SD₁₆	—	4	6	no	—
222	C₇×D₈	—	2	6	no	—
223	C₇×D₈	#41	1	4	yes	complemented
224	C₇×M₁₆	—	2	6	no	—
225	C₇×M₁₆	—	4	3	no	—
226	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
227	C₂×C₇×C₈	C₂⁴	1	3	yes	—
228	C₂×C₇×C₈	C₂⁴	1	3	yes	—
229	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
230	C₇×C₄⋊C₄	—	2	3	no	—
231	C₇×C₄⋊C₄	—	2	6	no	—
232	C₇×C₄⋊C₄	—	2	3	no	—
233	C₇×C₄⋊C₄	—	2	3	no	—
234	C₇×C₄⋊C₄	—	2	3	no	—
235	C₇×C₄⋊C₄	—	2	3	no	—
236	C₇×C₄⋊C₄	—	2	3	no	—
237	C₇×C₄⋊C₄	—	2	3	no	—
238	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
239	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
240	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
241	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
242	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
243	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
244	C₇×C₂²⋊C₄	—	2	6	no	—
245	C₇×C₂²⋊C₄	—	2	3	no	—
246	C₇×C₂²⋊C₄	—	2	1	no	—
247	C₇×C₂²⋊C₄	—	2	3	no	—
248	C₇×C₂²⋊C₄	—	2	3	no	—
249	C₇×C₂²⋊C₄	—	2	3	no	—
250	C₇×C₂²⋊C₄	—	2	1	no	—
251	C₇×C₂²⋊C₄	—	2	3	no	—
252	C₇×C₂²⋊C₄	—	2	6	no	—
253	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
254	C₄²×C₇	—	2	3	no	—
255	C₄²×C₇	—	2	3	no	—
256	C₄²×C₇	—	2	3	no	—
257	C₄²×C₇	C₂×D₄	1	3	yes	—
258	C₄²×C₇	C₂×D₄	1	3	yes	—
259	C₄²×C₇	C₂⁴	1	3	yes	—
260	C₄²×C₇	C₂⁴	1	1	yes	characteristic
261	C₂²×#41	—	2	1	no	—
262	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	3	yes	—
263	#557	C₂²×C₇	1	1	yes	characteristic, complemented
264	#556	C₂²×C₇	1	3	yes	complemented
265	#552	—	2	3	no	—
266	#550	C₂²×C₇	1	1	yes	characteristic
267	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
268	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
269	#548	—	2	1	no	—
270	#548	—	2	1	no	—
271	#547	C₂²×C₇	1	3	yes	—
272	#547	C₂²×C₇	1	3	yes	complemented
273	#546	—	2	3	no	—
274	D₄²	C₂²×C₇	1	3	yes	—
275	#533	C₂²×C₇	1	3	yes	complemented
276	#532	C₂²×C₇	1	3	yes	—
277	#529	—	2	3	no	—
278	#525	—	2	3	no	—
279	C₄×#41	C₂²×C₇	1	1	yes	characteristic
280	C₄×#41	C₂²×C₇	1	3	yes	complemented
281	C₂×C₄×Q₈	—	2	1	no	—
282	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
283	#500	C₂²×C₇	1	3	yes	complemented
284	#492	C₂²×C₇	1	3	yes	complemented
285	#487	C₂²×C₇	1	3	yes	complemented
286	#486	C₂²×C₇	1	3	yes	complemented
287	#484	C₂²×C₇	1	6	yes	complemented
288	#482	C₂²×C₇	1	6	yes	complemented
289	#475	C₂²×C₇	1	3	yes	complemented
290	#474	C₂²×C₇	1	3	yes	—
291	#474	C₂²×C₇	1	3	yes	complemented
292	#474	C₂²×C₇	1	3	yes	complemented
293	#468	C₂²×C₇	1	6	yes	complemented
294	#468	C₂²×C₇	1	6	yes	complemented
295	#461	C₂²×C₇	1	3	yes	—
296	#461	C₂²×C₇	1	3	yes	complemented
297	#460	C₂²×C₇	1	3	yes	complemented
298	#459	C₂²×C₇	1	3	yes	complemented
299	#450	C₂²×C₇	1	6	yes	—
300	#449	C₂²×C₇	1	6	yes	complemented
301	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
302	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
303	#442	C₂²×C₇	1	1	yes	characteristic
304	#442	C₂²×C₇	1	3	yes	complemented
305	#440	C₂²×C₇	1	3	yes	—
306	#440	C₂²×C₇	1	6	yes	complemented
307	#440	C₂²×C₇	1	3	yes	complemented
308	C₄×SD₁₆	C₂²×C₇	1	3	yes	—
309	C₄×SD₁₆	C₂²×C₇	1	3	yes	complemented
310	C₄×D₈	C₂²×C₇	1	3	yes	—
311	C₄×D₈	C₂²×C₇	1	3	yes	complemented
312	#436	C₂²×C₇	1	3	yes	complemented
313	#436	C₂²×C₇	1	3	yes	complemented
314	#432	C₂²×C₇	1	3	yes	complemented
315	#428	C₂²×C₇	1	3	yes	complemented
316	#418	C₂²×C₇	1	3	yes	—
317	#417	C₂²×C₇	1	3	yes	complemented
318	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
319	C₂³×C₇	—	2	1	no	—
320	C₂³×C₇	—	2	1	no	—
321	C₂³×C₇	—	2	3	no	—
322	C₂³×C₇	—	4	6	no	—
323	C₂³×C₇	—	4	6	no	—
324	C₂³×C₇	—	8	3	no	—
325	C₂³×C₇	C₂²×D₄	1	3	yes	—
326	C₇×Q₈	—	2	3	no	—
327	C₇×Q₈	—	2	3	no	—
328	C₇×Q₈	—	4	6	no	—
329	C₇×Q₈	—	4	6	no	—
330	C₇×Q₈	AGL(1,ℤ/8)	1	4	yes	complemented
331	C₇×D₄	—	2	6	no	—
332	C₇×D₄	—	2	3	no	—
333	C₇×D₄	—	2	1	no	—
334	C₇×D₄	—	2	1	no	—
335	C₇×D₄	—	2	3	no	—
336	C₇×D₄	—	4	6	no	—
337	C₇×D₄	—	4	6	no	—
338	C₇×D₄	—	4	12	no	—
339	C₇×D₄	—	4	6	no	—
340	C₇×D₄	—	8	3	no	—
341	C₇×D₄	C₂×#41	1	2	yes	—
342	C₇×D₄	C₂×#41	1	2	yes	complemented
343	C₂×C₄×C₇	—	2	6	no	—
344	C₂×C₄×C₇	—	2	3	no	—
345	C₂×C₄×C₇	—	2	3	no	—
346	C₂×C₄×C₇	—	2	1	no	—
347	C₂×C₄×C₇	—	2	3	no	—
348	C₂×C₄×C₇	—	2	3	no	—
349	C₂×C₄×C₇	—	2	1	no	—
350	C₂×C₄×C₇	—	2	3	no	—
351	C₂×C₄×C₇	—	4	3	no	—
352	C₂×C₄×C₇	—	4	6	no	—
353	C₂×C₄×C₇	—	4	2	no	—
354	C₂×C₄×C₇	—	4	2	no	—
355	C₂×C₄×C₇	—	4	6	no	—
356	C₂×C₄×C₇	—	4	6	no	—
357	C₂×C₄×C₇	—	4	6	no	—
358	C₂×C₄×C₇	—	4	6	no	—
359	C₂×C₄×C₇	—	4	6	no	—
360	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
361	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
362	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
363	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
364	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
365	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
366	C₇×C₈	—	2	3	no	—
367	C₇×C₈	—	2	1	no	—
368	C₇×C₈	—	2	3	no	—
369	C₇×C₈	C₂×#41	1	2	yes	complemented
370	C₂×#41	—	2	1	no	—
371	C₂×#41	—	2	3	no	—
372	C₂×#41	—	2	3	no	—
373	C₂×#41	—	4	8	no	—
374	C₂×#41	C₇×D₄	1	1	yes	characteristic
375	C₂×#41	C₂³×C₇	1	3	yes	—
376	C₂²×Q₈	—	2	1	no	—
377	C₂²×Q₈	—	2	1	no	—
378	C₂²×D₄	—	2	3	no	—
379	C₂²×D₄	C₂³×C₇	1	3	yes	—
380	C₂³×C₄	—	2	3	no	—
381	AGL(1,ℤ/8)	—	2	12	no	—
382	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
383	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
384	C₂×SD₁₆	C₂³×C₇	1	6	yes	—
385	C₂×D₈	C₂³×C₇	1	3	yes	—
386	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
387	C₂×M₁₆	C₂³×C₇	1	3	yes	—
388	C₂×M₁₆	C₂³×C₇	1	3	yes	—
389	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
390	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
391	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
392	#127	C₂³×C₇	1	3	yes	—
393	#126	—	2	6	no	—
394	#125	—	2	3	no	—
395	#125	—	2	3	no	—
396	#124	—	2	3	no	—
397	#124	—	2	3	no	—
398	#123	—	2	6	no	—
399	#123	—	2	6	no	—
400	C₂²⋊Q₈	—	2	3	no	—
401	C₂²⋊Q₈	—	2	3	no	—
402	C₂²⋊Q₈	—	2	3	no	—
403	C₂²⋊Q₈	—	2	3	no	—
404	C₂²⋊Q₈	C₂³×C₇	1	6	yes	—
405	#121	—	2	3	no	—
406	#121	—	2	3	no	—
407	#121	—	2	3	no	—
408	#121	—	2	3	no	—
409	#121	C₂³×C₇	1	6	yes	—
410	#120	—	2	6	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	—	2	3	no	—
413	C₄×Q₈	—	2	3	no	—
414	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
415	C₄×Q₈	C₂³×C₇	1	3	yes	—
416	C₄×Q₈	C₂³×C₇	1	3	yes	—
417	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
418	C₄×D₄	—	2	3	no	—
419	C₄×D₄	—	2	3	no	—
420	C₄×D₄	—	2	3	no	—
421	C₄×D₄	—	2	3	no	—
422	C₄×D₄	—	2	3	no	—
423	C₄×D₄	C₇×D₄	1	3	yes	—
424	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
425	C₄×D₄	C₂³×C₇	1	3	yes	—
426	C₄×D₄	C₂³×C₇	1	3	yes	—
427	C₄×D₄	C₂³×C₇	1	3	yes	—
428	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
429	C₄×D₄	C₂³×C₇	1	3	yes	—
430	C₄×D₄	C₂³×C₇	1	3	yes	—
431	#117	—	2	3	no	—
432	#117	—	2	3	no	—
433	#117	C₇×D₄	1	3	yes	—
434	#117	C₂³×C₇	1	3	yes	—
435	C₂×C₄⋊C₄	—	2	3	no	—
436	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
437	C₂×C₂²⋊C₄	—	2	3	no	—
438	C₂×C₄²	—	2	3	no	—
439	C₂×C₄²	C₂³×C₇	1	3	yes	—
440	#107	C₂³×C₇	1	3	yes	—
441	#107	C₂³×C₇	1	3	yes	—
442	#107	C₂³×C₇	1	6	yes	—
443	#106	C₂³×C₇	1	1	yes	characteristic
444	#106	C₂³×C₇	1	3	yes	—
445	C₄⋊C₈	C₂³×C₇	1	3	yes	—
446	C₄⋊C₈	C₂³×C₇	1	3	yes	—
447	#103	C₂³×C₇	1	3	yes	—
448	#103	C₂³×C₇	1	3	yes	—
449	#103	C₂³×C₇	1	6	yes	—
450	#102	C₂³×C₇	1	1	yes	characteristic
451	#102	C₂³×C₇	1	3	yes	—
452	#102	C₂³×C₇	1	1	yes	characteristic
453	#102	C₂³×C₇	1	3	yes	—
454	#102	C₂³×C₇	1	3	yes	—
455	#102	C₂³×C₇	1	3	yes	—
456	#102	C₂³×C₇	1	6	yes	—
457	C₂²⋊C₈	C₂³×C₇	1	6	yes	—
458	#97	C₂³×C₇	1	1	yes	characteristic
459	#97	C₂³×C₇	1	3	yes	—
460	C₄×C₈	C₂³×C₇	1	3	yes	—
461	C₄×C₈	C₂³×C₇	1	3	yes	—
462	C₂²×C₇	—	2	3	no	—
463	C₂²×C₇	—	2	3	no	—
464	C₂²×C₇	—	2	2	no	—
465	C₂²×C₇	—	2	2	no	—
466	C₂²×C₇	—	2	3	no	—
467	C₂²×C₇	—	4	1	no	—
468	C₂²×C₇	—	4	1	no	—
469	C₂²×C₇	—	4	1	no	—
470	C₂²×C₇	—	4	1	no	—
471	C₂²×C₇	—	4	3	no	—
472	C₂²×C₇	—	4	3	no	—
473	C₂²×C₇	—	4	3	no	—
474	C₂²×C₇	—	8	12	no	—
475	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
476	C₄×C₇	—	2	1	no	—
477	C₄×C₇	—	2	3	no	—
478	C₄×C₇	—	2	3	no	—
479	C₄×C₇	—	2	1	no	—
480	C₄×C₇	—	4	1	no	—
481	C₄×C₇	—	4	3	no	—
482	C₄×C₇	—	4	1	no	—
483	C₄×C₇	—	4	3	no	—
484	C₄×C₇	—	4	3	no	—
485	C₄×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
486	C₄×C₇	C₂²×#41	1	1	yes	characteristic
487	C₄×C₇	#585	1	1	yes	characteristic
488	C₂⁴	—	2	3	no	—
489	#41	—	2	4	no	—
490	#41	—	2	6	no	—
491	#41	—	4	3	no	—
492	#41	—	4	12	no	—
493	#41	—	4	12	no	—
494	#41	—	4	4	no	—
495	C₂×Q₈	—	2	3	no	—
496	C₂×Q₈	—	2	3	no	—
497	C₂×Q₈	—	4	4	no	—
498	C₂×Q₈	—	4	4	no	—
499	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
500	C₂×Q₈	C₂⁴×C₇	1	3	yes	—
501	C₂×D₄	—	2	6	no	—
502	C₂×D₄	—	2	3	no	—
503	C₂×D₄	—	2	6	no	—
504	C₂×D₄	—	2	3	no	—
505	C₂×D₄	—	2	3	no	—
506	C₂×D₄	—	2	3	no	—
507	C₂×D₄	—	4	3	no	—
508	C₂×D₄	—	4	12	no	—
509	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
510	C₂×D₄	C₂⁴×C₇	1	3	yes	—
511	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
512	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
513	C₂²×C₄	—	2	3	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	3	no	—
516	C₂²×C₄	—	2	3	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	3	no	—
519	C₂²×C₄	—	2	3	no	—
520	C₂²×C₄	—	2	3	no	—
521	C₂²×C₄	—	4	12	no	—
522	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
523	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
524	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
525	SD₁₆	—	2	6	no	—
526	SD₁₆	—	2	6	no	—
527	SD₁₆	—	4	6	no	—
528	D₈	—	2	6	no	—
529	D₈	C₇×#41	1	4	yes	complemented
530	M₁₆	—	2	6	no	—
531	M₁₆	—	4	3	no	—
532	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
533	C₂×C₈	C₂⁴×C₇	1	3	yes	—
534	C₂×C₈	C₂⁴×C₇	1	3	yes	—
535	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
536	C₄⋊C₄	—	2	3	no	—
537	C₄⋊C₄	—	2	6	no	—
538	C₄⋊C₄	—	2	3	no	—
539	C₄⋊C₄	—	2	3	no	—
540	C₄⋊C₄	—	2	3	no	—
541	C₄⋊C₄	—	2	3	no	—
542	C₄⋊C₄	—	2	3	no	—
543	C₄⋊C₄	—	2	3	no	—
544	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
545	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
546	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
547	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
548	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
549	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
550	C₂²⋊C₄	—	2	6	no	—
551	C₂²⋊C₄	—	2	3	no	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	3	no	—
554	C₂²⋊C₄	—	2	3	no	—
555	C₂²⋊C₄	—	2	3	no	—
556	C₂²⋊C₄	—	2	1	no	—
557	C₂²⋊C₄	—	2	3	no	—
558	C₂²⋊C₄	—	2	6	no	—
559	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
560	C₄²	—	2	3	no	—
561	C₄²	—	2	3	no	—
562	C₄²	—	2	3	no	—
563	C₄²	C₂×C₇×D₄	1	3	yes	—
564	C₄²	C₂×C₇×D₄	1	3	yes	—
565	C₄²	C₂⁴×C₇	1	3	yes	—
566	C₄²	C₂⁴×C₇	1	1	yes	characteristic
567	C₂×C₇	—	4	3	no	—
568	C₂×C₇	—	4	2	no	—
569	C₂×C₇	—	4	2	no	—
570	C₂×C₇	—	8	3	no	—
571	C₂×C₇	D₄×#41	1	1	yes	characteristic
572	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
573	C₂×C₇	#3586	1	1	yes	characteristic
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	3	no	—
577	C₂³	—	4	6	no	—
578	C₂³	—	4	6	no	—
579	C₂³	—	8	3	no	—
580	C₂³	C₂²×C₇×D₄	1	3	yes	—
581	Q₈	—	2	3	no	—
582	Q₈	—	2	3	no	—
583	Q₈	—	4	6	no	—
584	Q₈	—	4	6	no	—
585	Q₈	C₇×AGL(1,ℤ/8)	1	4	yes	complemented
586	D₄	—	2	3	no	—
587	D₄	—	2	1	no	—
588	D₄	—	2	1	no	—
589	D₄	—	2	3	no	—
590	D₄	—	2	6	no	—
591	D₄	—	4	6	no	—
592	D₄	—	4	6	no	—
593	D₄	—	4	12	no	—
594	D₄	—	4	6	no	—
595	D₄	—	8	3	no	—
596	D₄	C₂×C₇×#41	1	2	yes	—
597	D₄	C₂×C₇×#41	1	2	yes	complemented
598	C₂×C₄	—	2	3	no	—
599	C₂×C₄	—	2	3	no	—
600	C₂×C₄	—	2	6	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	3	no	—
603	C₂×C₄	—	2	3	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	3	no	—
606	C₂×C₄	—	4	3	no	—
607	C₂×C₄	—	4	6	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	6	no	—
611	C₂×C₄	—	4	6	no	—
612	C₂×C₄	—	4	6	no	—
613	C₂×C₄	—	4	6	no	—
614	C₂×C₄	—	4	6	no	—
615	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
616	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
617	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
618	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
619	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
620	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
621	C₈	—	2	3	no	—
622	C₈	—	2	1	no	—
623	C₈	—	2	3	no	—
624	C₈	C₂×C₇×#41	1	2	yes	complemented
625	C₇	#61466	1	1	yes	characteristic, direct product factor, Sylow
626	C₂²	—	2	3	no	—
627	C₂²	—	2	3	no	—
628	C₂²	—	2	3	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	3	no	—
636	C₂²	—	4	3	no	—
637	C₂²	—	4	3	no	—
638	C₂²	—	8	12	no	—
639	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
640	C₄	—	2	1	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	3	no	—
643	C₄	—	2	3	no	—
644	C₄	—	4	1	no	—
645	C₄	—	4	3	no	—
646	C₄	—	4	3	no	—
647	C₄	—	4	1	no	—
648	C₄	—	4	3	no	—
649	C₄	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
650	C₄	C₂²×C₇×#41	1	1	yes	characteristic
651	C₄	C₇×#585	1	1	yes	characteristic
652	C₂	—	4	3	no	—
653	C₂	—	4	2	no	—
654	C₂	—	4	2	no	—
655	C₂	—	8	3	no	—
656	C₂	C₇×D₄×#41	1	1	yes	characteristic
657	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
658	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. C₇×#3482	C₂	semidirect
2. C₇×D₄×#41	C₂	semidirect
3. C₇×#3392	C₂	semidirect
4. C₇×Q₈×D₈	C₂	semidirect
5. C₇×#3359	C₂	semidirect
6. C₇×#3358	C₂	semidirect
7. C₇×#3353	C₂	semidirect
8. C₇×#3330	C₂	semidirect
9. C₇×#3324	C₂	semidirect
10. C₇×#3272	C₂	semidirect
11. C₇×#3182	C₂	semidirect
12. C₇×#3115	C₂	semidirect
13. C₇×#3110	C₂	semidirect
14. C₇×#2971	C₂	semidirect
15. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
18. C₇×#557	C₂²	semidirect
19. C₇×#556	C₂²	semidirect
23. C₇×D₄×Q₈	C₂²	semidirect
27. C₇×#547	C₂²	semidirect
30. C₇×#533	C₂²	semidirect
35. C₄×C₇×#41	C₂²	semidirect
38. C₇×#500	C₂²	semidirect
39. C₇×#492	C₂²	semidirect
40. C₇×#487	C₂²	semidirect
41. C₇×#486	C₂²	semidirect
42. C₇×#484	C₂²	semidirect
43. C₇×#482	C₂²	semidirect
44. C₇×#475	C₂²	semidirect
46. C₇×#474	C₂²	semidirect
47. C₇×#474	C₂²	semidirect
48. C₇×#468	C₂²	semidirect
49. C₇×#468	C₂²	semidirect
51. C₇×#461	C₂²	semidirect
52. C₇×#460	C₂²	semidirect
53. C₇×#459	C₂²	semidirect
55. C₇×#449	C₂²	semidirect
56. C₇×C₈×Q₈	C₂²	semidirect
57. C₇×C₈×Q₈	C₂²	semidirect
59. C₇×#442	C₂²	semidirect
61. C₇×#440	C₂²	semidirect
62. C₇×#440	C₂²	semidirect
64. C₄×C₇×SD₁₆	C₂²	semidirect
66. C₄×C₇×D₈	C₂²	semidirect
67. C₇×#436	C₂²	semidirect
68. C₇×#436	C₂²	semidirect
69. C₇×#432	C₂²	semidirect
70. C₇×#428	C₂²	semidirect
72. C₇×#417	C₂²	semidirect
73. C₄×C₇×M₁₆	C₂²	semidirect
74. #61466	C₇	direct (N × H)
167. #3482	C₂×C₇	semidirect
168. D₄×#41	C₂×C₇	semidirect
169. #3392	C₂×C₇	semidirect
170. Q₈×D₈	C₂×C₇	semidirect
171. #3359	C₂×C₇	semidirect
172. #3358	C₂×C₇	semidirect
173. #3353	C₂×C₇	semidirect
174. #3330	C₂×C₇	semidirect
175. #3324	C₂×C₇	semidirect
176. #3272	C₂×C₇	semidirect
177. #3182	C₂×C₇	semidirect
178. #3115	C₂×C₇	semidirect
179. #3110	C₂×C₇	semidirect
180. #2971	C₂×C₇	semidirect
181. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
223. C₇×D₈	#41	semidirect
263. #557	C₂²×C₇	semidirect
264. #556	C₂²×C₇	semidirect
268. D₄×Q₈	C₂²×C₇	semidirect
272. #547	C₂²×C₇	semidirect
275. #533	C₂²×C₇	semidirect
280. C₄×#41	C₂²×C₇	semidirect
283. #500	C₂²×C₇	semidirect
284. #492	C₂²×C₇	semidirect
285. #487	C₂²×C₇	semidirect
286. #486	C₂²×C₇	semidirect
287. #484	C₂²×C₇	semidirect
288. #482	C₂²×C₇	semidirect
289. #475	C₂²×C₇	semidirect
291. #474	C₂²×C₇	semidirect
292. #474	C₂²×C₇	semidirect
293. #468	C₂²×C₇	semidirect
294. #468	C₂²×C₇	semidirect
296. #461	C₂²×C₇	semidirect
297. #460	C₂²×C₇	semidirect
298. #459	C₂²×C₇	semidirect
300. #449	C₂²×C₇	semidirect
301. C₈×Q₈	C₂²×C₇	semidirect
302. C₈×Q₈	C₂²×C₇	semidirect
304. #442	C₂²×C₇	semidirect
306. #440	C₂²×C₇	semidirect
307. #440	C₂²×C₇	semidirect
309. C₄×SD₁₆	C₂²×C₇	semidirect
311. C₄×D₈	C₂²×C₇	semidirect
312. #436	C₂²×C₇	semidirect
313. #436	C₂²×C₇	semidirect
314. #432	C₂²×C₇	semidirect
315. #428	C₂²×C₇	semidirect
317. #417	C₂²×C₇	semidirect
318. C₄×M₁₆	C₂²×C₇	semidirect
330. C₇×Q₈	AGL(1,ℤ/8)	semidirect
342. C₇×D₄	C₂×#41	semidirect
369. C₇×C₈	C₂×#41	semidirect
529. D₈	C₇×#41	semidirect
585. Q₈	C₇×AGL(1,ℤ/8)	semidirect
597. D₄	C₂×C₇×#41	semidirect
624. C₈	C₂×C₇×#41	semidirect
625. C₇	#61466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#3482	2
C₇×D₄×#41	2
C₇×#3392	2
C₇×Q₈×D₈	2
C₇×#3359	2
C₇×#3358	2
C₇×#3353	2
C₇×#3330	2
C₇×#3324	2
C₇×#3272	2
C₇×#3182	2
C₇×#3115	2
C₇×#3110	2
C₇×#2971	2
C₄×C₇×AGL(1,ℤ/8)	2
#61466	7

Subgroups — C7×#61466 #13238682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61466 #13238682