Subgroups — C₇×#18649 #13195865

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3523	C₂	1	1	yes	maximal, characteristic
2	C₇×#3248	C₂	1	1	yes	maximal, characteristic
3	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1653	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1652	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1651	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1650	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1649	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1648	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1648	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1552	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1494	C₂	1	1	yes	maximal, characteristic
15	C₂×C₇×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₇×#557	—	2	1	no	—
17	C₇×#557	C₂²	1	1	yes	characteristic
18	C₇×#557	C₂²	1	1	yes	characteristic
19	C₇×#556	—	2	1	no	—
20	C₇×#556	C₂²	1	1	yes	characteristic
21	C₇×#552	—	2	2	no	—
22	C₇×#552	—	2	1	no	—
23	C₇×D₄×Q₈	—	2	1	no	—
24	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
25	C₇×#541	—	2	2	no	—
26	C₇×#541	—	2	1	no	—
27	C₇×#533	—	2	1	no	—
28	C₇×#533	C₂²	1	1	yes	characteristic
29	C₇×#533	C₂²	1	1	yes	characteristic
30	C₇×#529	—	2	1	no	—
31	C₇×#529	C₂²	1	1	yes	characteristic
32	C₇×#519	—	2	1	no	—
33	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
34	C₇×#500	C₂²	1	1	yes	characteristic
35	C₇×#500	C₂²	1	1	yes	characteristic
36	C₇×#499	—	2	2	no	—
37	C₇×#498	C₂²	1	1	yes	characteristic
38	C₇×#498	C₂²	1	1	yes	characteristic
39	C₇×#483	—	2	1	no	—
40	C₇×#479	—	2	1	no	—
41	C₇×#475	C₂²	1	1	yes	characteristic
42	C₇×#475	C₂²	1	1	yes	characteristic
43	C₇×#475	C₂²	1	1	yes	characteristic
44	C₇×#474	C₂²	1	1	yes	characteristic
45	C₇×#467	—	2	1	no	—
46	C₇×#465	—	2	1	no	—
47	C₇×#464	—	2	1	no	—
48	C₇×#462	C₂²	1	1	yes	characteristic
49	C₇×#462	C₂²	1	1	yes	characteristic
50	C₇×#462	C₂²	1	1	yes	characteristic
51	C₇×#461	C₂²	1	1	yes	characteristic
52	C₇×#451	—	2	1	no	—
53	C₇×#441	—	2	1	no	—
54	C₇×#441	—	2	1	no	—
55	C₇×#432	C₂²	1	1	yes	characteristic
56	C₇×#427	—	2	2	no	—
57	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
59	C₇×#332	C₂²	1	1	yes	characteristic
60	C₇×#332	C₂²	1	1	yes	characteristic
61	C₇×#332	C₂²	1	1	yes	characteristic
62	C₇×#332	C₂²	1	1	yes	characteristic
63	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	#18649	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂²×C₇×Q₈	—	2	1	no	—
73	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
74	C₂×C₇×Q₁₆	—	2	1	no	—
75	C₂×C₇×Q₁₆	—	2	1	no	—
76	C₂×C₇×Q₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂²×C₇×C₈	—	2	1	no	—
79	C₂²×C₇×C₈	—	2	1	no	—
80	C₇×C₄⋊Q₈	—	2	2	no	—
81	C₇×C₄⋊Q₈	—	2	1	no	—
82	C₇×C₄⋊Q₈	—	2	1	no	—
83	C₇×C₄⋊Q₈	—	2	1	no	—
84	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₇×#126	—	2	2	no	—
89	C₇×#126	—	2	2	no	—
90	C₇×#126	—	2	2	no	—
91	C₇×#126	—	2	2	no	—
92	C₇×#126	—	2	1	no	—
93	C₇×#125	—	2	2	no	—
94	C₇×#125	—	2	2	no	—
95	C₇×#125	—	2	2	no	—
96	C₇×#125	—	2	1	no	—
97	C₇×#125	—	2	2	no	—
98	C₇×#125	—	2	2	no	—
99	C₇×#125	—	2	1	no	—
100	C₇×#125	—	2	1	no	—
101	C₇×#125	D₄	1	1	yes	characteristic
102	C₇×#125	D₄	1	2	yes	—
103	C₇×#124	—	2	1	no	—
104	C₇×#124	D₄	1	1	yes	characteristic
105	C₇×#123	—	2	2	no	—
106	C₇×#123	—	2	1	no	—
107	C₇×C₂²⋊Q₈	—	2	2	no	—
108	C₇×C₂²⋊Q₈	—	2	2	no	—
109	C₇×C₂²⋊Q₈	—	2	1	no	—
110	C₇×C₂²⋊Q₈	—	2	1	no	—
111	C₇×C₂²⋊Q₈	—	2	1	no	—
112	C₇×C₂²⋊Q₈	—	2	1	no	—
113	C₇×C₂²⋊Q₈	—	2	2	no	—
114	C₇×C₂²⋊Q₈	—	2	1	no	—
115	C₇×C₂²⋊Q₈	—	2	1	no	—
116	C₇×C₂²⋊Q₈	D₄	1	1	yes	characteristic
117	C₇×#121	D₄	1	1	yes	characteristic
118	C₄×C₇×Q₈	—	2	2	no	—
119	C₄×C₇×Q₈	—	2	2	no	—
120	C₄×C₇×Q₈	—	2	1	no	—
121	C₄×C₇×Q₈	—	2	1	no	—
122	C₄×C₇×Q₈	—	2	1	no	—
123	C₄×C₇×Q₈	—	2	1	no	—
124	C₄×C₇×Q₈	—	2	1	no	—
125	C₄×C₇×Q₈	—	2	1	no	—
126	C₄×C₇×Q₈	—	2	1	no	—
127	C₄×C₇×Q₈	—	2	1	no	—
128	C₄×C₇×Q₈	—	2	1	no	—
129	C₄×C₇×Q₈	—	2	1	no	—
130	C₄×C₇×Q₈	—	2	1	no	—
131	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
132	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
133	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
134	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
135	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
136	C₄×C₇×D₄	—	2	2	no	—
137	C₄×C₇×D₄	—	2	1	no	—
138	C₄×C₇×D₄	—	2	1	no	—
139	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
140	C₇×#117	—	2	2	no	—
141	C₇×#117	—	2	2	no	—
142	C₇×#117	—	2	2	no	—
143	C₇×#117	—	2	1	no	—
144	C₇×#117	—	2	1	no	—
145	C₇×#117	—	2	1	no	—
146	C₇×#117	D₄	1	2	yes	—
147	C₂×C₇×C₄⋊C₄	—	2	1	no	—
148	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
149	C₂×C₄²×C₇	—	2	1	no	—
150	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
151	C₇×#107	—	2	1	no	—
152	C₇×#107	—	2	2	no	—
153	C₇×#107	—	2	2	no	—
154	C₇×#106	—	2	1	no	—
155	C₇×#106	—	2	1	no	—
156	C₇×#106	—	2	1	no	—
157	C₇×#106	—	2	2	no	—
158	C₇×#106	—	2	2	no	—
159	C₇×C₄⋊C₈	—	2	1	no	—
160	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
164	C₇×#103	—	2	1	no	—
165	C₇×#103	—	2	1	no	—
166	C₇×#103	—	2	1	no	—
167	C₇×#103	—	2	1	no	—
168	C₇×#103	—	2	1	no	—
169	C₇×#103	—	2	1	no	—
170	C₇×#103	—	2	1	no	—
171	C₇×#102	—	2	1	no	—
172	C₇×C₂²⋊C₈	—	2	1	no	—
173	C₇×#97	—	2	1	no	—
174	C₄×C₇×C₈	—	2	1	no	—
175	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
176	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
177	#3523	C₂×C₇	1	1	yes	characteristic
178	#3248	C₂×C₇	1	1	yes	characteristic
179	#3160	C₂×C₇	1	1	yes	characteristic, complemented
180	#3113	C₂×C₇	1	1	yes	characteristic, complemented
181	#1653	C₂×C₇	1	1	yes	characteristic, complemented
182	#1653	C₂×C₇	1	1	yes	characteristic, complemented
183	#1652	C₂×C₇	1	1	yes	characteristic, complemented
184	#1651	C₂×C₇	1	1	yes	characteristic, complemented
185	#1650	C₂×C₇	1	1	yes	characteristic, complemented
186	#1649	C₂×C₇	1	1	yes	characteristic, complemented
187	#1648	C₂×C₇	1	1	yes	characteristic, complemented
188	#1648	C₂×C₇	1	1	yes	characteristic, complemented
189	#1552	C₂×C₇	1	1	yes	characteristic, complemented
190	#1494	C₂×C₇	1	1	yes	characteristic
191	C₂×Q₈⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
192	C₂×C₇×Q₈	—	2	2	no	—
193	C₂×C₇×Q₈	—	2	1	no	—
194	C₂×C₇×Q₈	—	2	1	no	—
195	C₂×C₇×Q₈	—	2	1	no	—
196	C₂×C₇×Q₈	—	2	2	no	—
197	C₂×C₇×Q₈	—	2	1	no	—
198	C₂×C₇×Q₈	—	2	1	no	—
199	C₂×C₇×Q₈	—	4	1	no	—
200	C₂×C₇×Q₈	—	4	2	no	—
201	C₂×C₇×Q₈	—	4	2	no	—
202	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
204	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₇×D₄	—	2	1	no	—
206	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₇	—	2	2	no	—
208	C₂²×C₄×C₇	—	2	1	no	—
209	C₂²×C₄×C₇	—	2	1	no	—
210	C₂²×C₄×C₇	—	2	1	no	—
211	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
214	C₇×Q₁₆	—	4	2	no	—
215	C₇×Q₁₆	—	4	2	no	—
216	C₇×Q₁₆	—	8	1	no	—
217	C₇×SD₁₆	—	8	1	no	—
218	C₂×C₇×C₈	—	2	4	no	—
219	C₂×C₇×C₈	—	2	1	no	—
220	C₂×C₇×C₈	—	2	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	—	2	1	no	—
224	C₂×C₇×C₈	—	2	1	no	—
225	C₂×C₇×C₈	—	8	1	no	—
226	C₇×C₄⋊C₄	—	2	2	no	—
227	C₇×C₄⋊C₄	—	2	2	no	—
228	C₇×C₄⋊C₄	—	2	2	no	—
229	C₇×C₄⋊C₄	—	2	2	no	—
230	C₇×C₄⋊C₄	—	2	2	no	—
231	C₇×C₄⋊C₄	—	2	2	no	—
232	C₇×C₄⋊C₄	—	2	2	no	—
233	C₇×C₄⋊C₄	—	2	2	no	—
234	C₇×C₄⋊C₄	—	2	1	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	1	no	—
237	C₇×C₄⋊C₄	—	2	2	no	—
238	C₇×C₄⋊C₄	—	2	1	no	—
239	C₇×C₄⋊C₄	—	2	1	no	—
240	C₇×C₄⋊C₄	—	2	1	no	—
241	C₇×C₄⋊C₄	—	2	2	no	—
242	C₇×C₄⋊C₄	—	2	1	no	—
243	C₇×C₄⋊C₄	—	2	1	no	—
244	C₇×C₄⋊C₄	—	2	1	no	—
245	C₇×C₄⋊C₄	—	2	1	no	—
246	C₇×C₄⋊C₄	—	2	1	no	—
247	C₇×C₄⋊C₄	—	2	1	no	—
248	C₇×C₄⋊C₄	—	2	1	no	—
249	C₇×C₄⋊C₄	—	2	1	no	—
250	C₇×C₄⋊C₄	—	2	1	no	—
251	C₇×C₄⋊C₄	—	4	1	no	—
252	C₇×C₄⋊C₄	—	4	1	no	—
253	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
254	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
255	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
256	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
257	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
258	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
259	C₇×C₂²⋊C₄	—	2	2	no	—
260	C₇×C₂²⋊C₄	—	2	2	no	—
261	C₇×C₂²⋊C₄	—	2	2	no	—
262	C₇×C₂²⋊C₄	—	2	1	no	—
263	C₇×C₂²⋊C₄	—	2	2	no	—
264	C₇×C₂²⋊C₄	—	2	1	no	—
265	C₇×C₂²⋊C₄	—	2	1	no	—
266	C₇×C₂²⋊C₄	—	2	1	no	—
267	C₇×C₂²⋊C₄	—	2	1	no	—
268	C₇×C₂²⋊C₄	—	4	1	no	—
269	C₄²×C₇	—	2	2	no	—
270	C₄²×C₇	—	2	2	no	—
271	C₄²×C₇	—	2	2	no	—
272	C₄²×C₇	—	2	1	no	—
273	C₄²×C₇	—	2	1	no	—
274	C₄²×C₇	—	2	2	no	—
275	C₄²×C₇	—	2	1	no	—
276	C₄²×C₇	—	2	1	no	—
277	C₄²×C₇	—	2	1	no	—
278	C₄²×C₇	—	4	1	no	—
279	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₇	C₂⁴	1	1	yes	characteristic
283	#557	—	2	1	no	—
284	#557	C₂²×C₇	1	1	yes	characteristic
285	#557	C₂²×C₇	1	1	yes	characteristic
286	#556	—	2	1	no	—
287	#556	C₂²×C₇	1	1	yes	characteristic
288	#552	—	2	2	no	—
289	#552	—	2	1	no	—
290	D₄×Q₈	—	2	1	no	—
291	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
292	#541	—	2	2	no	—
293	#541	—	2	1	no	—
294	#533	—	2	1	no	—
295	#533	C₂²×C₇	1	1	yes	characteristic
296	#533	C₂²×C₇	1	1	yes	characteristic
297	#529	—	2	1	no	—
298	#529	C₂²×C₇	1	1	yes	characteristic
299	#519	—	2	1	no	—
300	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
301	#500	C₂²×C₇	1	1	yes	characteristic
302	#500	C₂²×C₇	1	1	yes	characteristic
303	#499	—	2	2	no	—
304	#498	C₂²×C₇	1	1	yes	characteristic
305	#498	C₂²×C₇	1	1	yes	characteristic
306	#483	—	2	1	no	—
307	#479	—	2	1	no	—
308	#475	C₂²×C₇	1	1	yes	characteristic
309	#475	C₂²×C₇	1	1	yes	characteristic
310	#475	C₂²×C₇	1	1	yes	characteristic
311	#474	C₂²×C₇	1	1	yes	characteristic
312	#467	—	2	1	no	—
313	#465	—	2	1	no	—
314	#464	—	2	1	no	—
315	#462	C₂²×C₇	1	1	yes	characteristic
316	#462	C₂²×C₇	1	1	yes	characteristic
317	#462	C₂²×C₇	1	1	yes	characteristic
318	#461	C₂²×C₇	1	1	yes	characteristic
319	#451	—	2	1	no	—
320	#441	—	2	1	no	—
321	#441	—	2	1	no	—
322	#432	C₂²×C₇	1	1	yes	characteristic
323	#427	—	2	2	no	—
324	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
326	#332	C₂²×C₇	1	1	yes	characteristic
327	#332	C₂²×C₇	1	1	yes	characteristic
328	#332	C₂²×C₇	1	1	yes	characteristic
329	#332	C₂²×C₇	1	1	yes	characteristic
330	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
331	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
332	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
333	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
334	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
335	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic
336	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
337	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
338	C₂³×C₇	—	2	1	no	—
339	C₂³×C₇	#120	1	1	yes	characteristic
340	C₇×Q₈	—	2	1	no	—
341	C₇×Q₈	—	2	1	no	—
342	C₇×Q₈	—	2	1	no	—
343	C₇×Q₈	—	2	1	no	—
344	C₇×Q₈	—	2	1	no	—
345	C₇×Q₈	—	4	2	no	—
346	C₇×Q₈	—	4	4	no	—
347	C₇×Q₈	—	4	1	no	—
348	C₇×Q₈	—	4	1	no	—
349	C₇×Q₈	—	4	1	no	—
350	C₇×Q₈	—	4	1	no	—
351	C₇×Q₈	—	4	1	no	—
352	C₇×Q₈	—	4	1	no	—
353	C₇×Q₈	—	4	1	no	—
354	C₇×Q₈	—	4	1	no	—
355	C₇×Q₈	#135	1	2	yes	—
356	C₇×D₄	—	2	1	no	—
357	C₇×D₄	—	2	1	no	—
358	C₇×D₄	—	4	2	no	—
359	C₂×C₄×C₇	—	2	1	no	—
360	C₂×C₄×C₇	—	2	2	no	—
361	C₂×C₄×C₇	—	2	1	no	—
362	C₂×C₄×C₇	—	2	1	no	—
363	C₂×C₄×C₇	—	2	1	no	—
364	C₂×C₄×C₇	—	2	1	no	—
365	C₂×C₄×C₇	—	2	1	no	—
366	C₂×C₄×C₇	—	2	1	no	—
367	C₂×C₄×C₇	—	2	2	no	—
368	C₂×C₄×C₇	—	2	1	no	—
369	C₂×C₄×C₇	—	2	1	no	—
370	C₂×C₄×C₇	—	2	1	no	—
371	C₂×C₄×C₇	—	2	1	no	—
372	C₂×C₄×C₇	—	2	2	no	—
373	C₂×C₄×C₇	—	4	2	no	—
374	C₂×C₄×C₇	—	4	4	no	—
375	C₂×C₄×C₇	—	4	2	no	—
376	C₂×C₄×C₇	—	4	1	no	—
377	C₂×C₄×C₇	—	4	1	no	—
378	C₂×C₄×C₇	—	4	1	no	—
379	C₂×C₄×C₇	—	4	1	no	—
380	C₂×C₄×C₇	#120	1	1	yes	characteristic
381	C₂×C₄×C₇	#120	1	1	yes	characteristic
382	C₂×C₄×C₇	#120	1	1	yes	characteristic
383	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
384	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
385	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
386	C₇×C₈	—	2	2	no	—
387	C₇×C₈	—	2	2	no	—
388	C₇×C₈	—	4	1	no	—
389	C₇×C₈	—	4	1	no	—
390	C₇×C₈	—	4	1	no	—
391	C₇×C₈	—	4	1	no	—
392	C₂²×Q₈	—	2	1	no	—
393	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
394	C₂×Q₁₆	—	2	1	no	—
395	C₂×Q₁₆	—	2	1	no	—
396	C₂×Q₁₆	—	2	1	no	—
397	C₂×SD₁₆	—	2	1	no	—
398	C₂²×C₈	—	2	1	no	—
399	C₂²×C₈	—	2	1	no	—
400	C₄⋊Q₈	—	2	2	no	—
401	C₄⋊Q₈	—	2	1	no	—
402	C₄⋊Q₈	—	2	1	no	—
403	C₄⋊Q₈	—	2	1	no	—
404	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
405	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
406	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
407	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
408	#126	—	2	2	no	—
409	#126	—	2	2	no	—
410	#126	—	2	2	no	—
411	#126	—	2	2	no	—
412	#126	—	2	1	no	—
413	#125	—	2	2	no	—
414	#125	—	2	2	no	—
415	#125	—	2	2	no	—
416	#125	—	2	1	no	—
417	#125	—	2	2	no	—
418	#125	—	2	2	no	—
419	#125	—	2	1	no	—
420	#125	—	2	1	no	—
421	#125	C₇×D₄	1	1	yes	characteristic
422	#125	C₇×D₄	1	2	yes	—
423	#124	—	2	1	no	—
424	#124	C₇×D₄	1	1	yes	characteristic
425	#123	—	2	2	no	—
426	#123	—	2	1	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	1	no	—
430	C₂²⋊Q₈	—	2	1	no	—
431	C₂²⋊Q₈	—	2	1	no	—
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	—	2	1	no	—
435	C₂²⋊Q₈	—	2	1	no	—
436	C₂²⋊Q₈	C₇×D₄	1	1	yes	characteristic
437	#121	C₇×D₄	1	1	yes	characteristic
438	C₄×Q₈	—	2	2	no	—
439	C₄×Q₈	—	2	2	no	—
440	C₄×Q₈	—	2	1	no	—
441	C₄×Q₈	—	2	1	no	—
442	C₄×Q₈	—	2	1	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	—	2	1	no	—
446	C₄×Q₈	—	2	1	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
452	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
453	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
454	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
455	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
460	#117	—	2	2	no	—
461	#117	—	2	2	no	—
462	#117	—	2	2	no	—
463	#117	—	2	1	no	—
464	#117	—	2	1	no	—
465	#117	—	2	1	no	—
466	#117	C₇×D₄	1	2	yes	—
467	C₂×C₄⋊C₄	—	2	1	no	—
468	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
471	#107	—	2	2	no	—
472	#107	—	2	2	no	—
473	#107	—	2	1	no	—
474	#106	—	2	2	no	—
475	#106	—	2	2	no	—
476	#106	—	2	1	no	—
477	#106	—	2	1	no	—
478	#106	—	2	1	no	—
479	C₄⋊C₈	—	2	1	no	—
480	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
481	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
482	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
483	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
484	#103	—	2	1	no	—
485	#103	—	2	1	no	—
486	#103	—	2	1	no	—
487	#103	—	2	1	no	—
488	#103	—	2	1	no	—
489	#103	—	2	1	no	—
490	#103	—	2	1	no	—
491	#102	—	2	1	no	—
492	C₂²⋊C₈	—	2	1	no	—
493	#97	—	2	1	no	—
494	C₄×C₈	—	2	1	no	—
495	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
496	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
497	C₂²×C₇	—	2	1	no	—
498	C₂²×C₇	—	2	1	no	—
499	C₂²×C₇	—	4	1	no	—
500	C₂²×C₇	—	4	1	no	—
501	C₂²×C₇	—	4	1	no	—
502	C₂²×C₇	#456	1	2	yes	—
503	C₂²×C₇	C₂×#120	1	1	yes	characteristic
504	C₄×C₇	—	2	1	no	—
505	C₄×C₇	—	2	2	no	—
506	C₄×C₇	—	2	1	no	—
507	C₄×C₇	—	2	1	no	—
508	C₄×C₇	—	2	1	no	—
509	C₄×C₇	—	4	1	no	—
510	C₄×C₇	—	4	2	no	—
511	C₄×C₇	—	4	1	no	—
512	C₄×C₇	—	4	1	no	—
513	C₄×C₇	—	4	2	no	—
514	C₄×C₇	—	4	1	no	—
515	C₄×C₇	—	4	1	no	—
516	C₄×C₇	—	4	1	no	—
517	C₄×C₇	—	8	1	no	—
518	C₄×C₇	—	8	1	no	—
519	C₄×C₇	C₂×#135	1	1	yes	characteristic
520	C₄×C₇	#575	1	1	yes	characteristic
521	C₄×C₇	#577	1	1	yes	characteristic
522	C₄×C₇	#578	1	1	yes	characteristic
523	C₂×Q₈	—	2	2	no	—
524	C₂×Q₈	—	2	1	no	—
525	C₂×Q₈	—	2	1	no	—
526	C₂×Q₈	—	2	1	no	—
527	C₂×Q₈	—	2	2	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	4	1	no	—
531	C₂×Q₈	—	4	2	no	—
532	C₂×Q₈	—	4	2	no	—
533	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
534	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
535	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
536	C₂×D₄	—	2	1	no	—
537	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
538	C₂²×C₄	—	2	2	no	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
543	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
544	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
545	Q₁₆	—	4	2	no	—
546	Q₁₆	—	4	2	no	—
547	Q₁₆	—	8	1	no	—
548	SD₁₆	—	8	1	no	—
549	C₂×C₈	—	2	4	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	8	1	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	2	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	2	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	2	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	1	no	—
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	2	1	no	—
577	C₄⋊C₄	—	2	1	no	—
578	C₄⋊C₄	—	2	1	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	1	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	4	1	no	—
583	C₄⋊C₄	—	4	1	no	—
584	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
585	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
586	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
587	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
588	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
589	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	2	2	no	—
592	C₂²⋊C₄	—	2	2	no	—
593	C₂²⋊C₄	—	2	1	no	—
594	C₂²⋊C₄	—	2	2	no	—
595	C₂²⋊C₄	—	2	1	no	—
596	C₂²⋊C₄	—	2	1	no	—
597	C₂²⋊C₄	—	2	1	no	—
598	C₂²⋊C₄	—	2	1	no	—
599	C₂²⋊C₄	—	4	1	no	—
600	C₄²	—	2	2	no	—
601	C₄²	—	2	2	no	—
602	C₄²	—	2	2	no	—
603	C₄²	—	2	1	no	—
604	C₄²	—	2	1	no	—
605	C₄²	—	2	2	no	—
606	C₄²	—	2	1	no	—
607	C₄²	—	2	1	no	—
608	C₄²	—	2	1	no	—
609	C₄²	—	4	1	no	—
610	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
611	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
612	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
613	C₄²	C₂⁴×C₇	1	1	yes	characteristic
614	C₂×C₇	—	2	2	no	—
615	C₂×C₇	—	8	1	no	—
616	C₂×C₇	#3002	1	1	yes	characteristic
617	C₂×C₇	#3013	1	1	yes	characteristic
618	C₂×C₇	C₂×#456	1	1	yes	characteristic
619	C₂³	—	2	1	no	—
620	C₂³	C₇×#120	1	1	yes	characteristic
621	Q₈	—	2	1	no	—
622	Q₈	—	2	1	no	—
623	Q₈	—	2	1	no	—
624	Q₈	—	2	1	no	—
625	Q₈	—	2	1	no	—
626	Q₈	—	4	2	no	—
627	Q₈	—	4	4	no	—
628	Q₈	—	4	1	no	—
629	Q₈	—	4	1	no	—
630	Q₈	—	4	1	no	—
631	Q₈	—	4	1	no	—
632	Q₈	—	4	1	no	—
633	Q₈	—	4	1	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	C₇×#135	1	2	yes	—
637	D₄	—	2	1	no	—
638	D₄	—	2	1	no	—
639	D₄	—	4	2	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	1	no	—
643	C₂×C₄	—	2	1	no	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	1	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	1	no	—
652	C₂×C₄	—	2	1	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	4	2	no	—
655	C₂×C₄	—	4	4	no	—
656	C₂×C₄	—	4	2	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	C₇×#120	1	1	yes	characteristic
662	C₂×C₄	C₇×#120	1	1	yes	characteristic
663	C₂×C₄	C₇×#120	1	1	yes	characteristic
664	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
665	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
666	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
667	C₈	—	2	2	no	—
668	C₈	—	2	2	no	—
669	C₈	—	4	1	no	—
670	C₈	—	4	1	no	—
671	C₈	—	4	1	no	—
672	C₈	—	4	1	no	—
673	C₇	#18649	1	1	yes	characteristic, direct product factor, Sylow
674	C₂²	—	2	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	4	1	no	—
677	C₂²	—	4	1	no	—
678	C₂²	—	4	1	no	—
679	C₂²	C₇×#456	1	2	yes	—
680	C₂²	C₂×C₇×#120	1	1	yes	characteristic
681	C₄	—	2	1	no	—
682	C₄	—	2	2	no	—
683	C₄	—	2	1	no	—
684	C₄	—	2	1	no	—
685	C₄	—	2	1	no	—
686	C₄	—	4	1	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	1	no	—
689	C₄	—	4	1	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	1	no	—
694	C₄	—	8	1	no	—
695	C₄	—	8	1	no	—
696	C₄	C₂×C₇×#135	1	1	yes	characteristic
697	C₄	C₇×#575	1	1	yes	characteristic
698	C₄	C₇×#577	1	1	yes	characteristic
699	C₄	C₇×#578	1	1	yes	characteristic
700	C₂	—	2	2	no	—
701	C₂	—	8	1	no	—
702	C₂	C₇×#3002	1	1	yes	characteristic
703	C₂	C₇×#3013	1	1	yes	characteristic
704	C₂	C₂×C₇×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#3160	C₂	semidirect
4. C₇×#3113	C₂	semidirect
5. C₇×#1653	C₂	semidirect
6. C₇×#1653	C₂	semidirect
7. C₇×#1652	C₂	semidirect
8. C₇×#1651	C₂	semidirect
9. C₇×#1650	C₂	semidirect
10. C₇×#1649	C₂	semidirect
11. C₇×#1648	C₂	semidirect
12. C₇×#1648	C₂	semidirect
13. C₇×#1552	C₂	semidirect
15. C₂×C₇×Q₈⋊C₈	C₂	semidirect
71. #18649	C₇	direct (N × H)
179. #3160	C₂×C₇	semidirect
180. #3113	C₂×C₇	semidirect
181. #1653	C₂×C₇	semidirect
182. #1653	C₂×C₇	semidirect
183. #1652	C₂×C₇	semidirect
184. #1651	C₂×C₇	semidirect
185. #1650	C₂×C₇	semidirect
186. #1649	C₂×C₇	semidirect
187. #1648	C₂×C₇	semidirect
188. #1648	C₂×C₇	semidirect
189. #1552	C₂×C₇	semidirect
191. C₂×Q₈⋊C₈	C₂×C₇	semidirect
673. C₇	#18649	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×#3523	2
C₇×#3248	2
C₇×#3160	2
C₇×#3113	2
C₇×#1653	2
C₇×#1652	2
C₇×#1651	2
C₇×#1650	2
C₇×#1649	2
C₇×#1648	2
C₇×#1552	2
C₇×#1494	2
C₂×C₇×Q₈⋊C₈	2
#18649	7

Subgroups — C7×#18649 #13195865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#18649 #13195865