Subgroups — #11736937

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90239	C₂	1	1	yes	maximal, characteristic
2	C₇×#435	C₂	1	1	yes	maximal, characteristic, complemented
3	#89901	C₂	1	1	yes	maximal, characteristic, complemented
4	#89900	C₂	1	1	yes	maximal, characteristic, complemented
5	#89882	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#89783	C₂	1	1	yes	maximal, characteristic, complemented
8	#89608	C₂	1	1	yes	maximal, characteristic, complemented
9	#89605	C₂	1	1	yes	maximal, characteristic, complemented
10	#89501	C₂	1	1	yes	maximal, characteristic, complemented
11	#89499	C₂	1	1	yes	maximal, characteristic, complemented
12	#89499	C₂	1	1	yes	maximal, characteristic, complemented
13	(C₂²⋊D₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
14	#89478	C₂	1	1	yes	maximal, characteristic, complemented
15	#89477	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#1157	C₂²	1	1	yes	characteristic
17	C₂×#1157	C₂²	1	1	yes	characteristic
18	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	#6503	C₄	1	1	yes	characteristic
22	#6501	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊D₇	C₂²	1	1	yes	characteristic
24	C₄×C₂²⋊D₇	C₂²	1	1	yes	characteristic
25	C₂×#1136	C₂²	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×#1131	C₂²	1	1	yes	characteristic
28	C₂×#1131	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
30	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
31	#6458	C₄	1	1	yes	characteristic
32	#6451	C₄	1	1	yes	characteristic
33	#6448	C₄	1	1	yes	characteristic
34	#6448	C₄	1	1	yes	characteristic
35	#6447	C₄	1	1	yes	characteristic
36	#6447	C₄	1	1	yes	characteristic
37	C₄×D₂₈	C₂²	1	1	yes	characteristic
38	C₄²×D₇	C₂²	1	1	yes	characteristic
39	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₇×#97	C₂²	1	1	yes	characteristic
43	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
44	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
45	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
46	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
47	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
52	(C₂×D₇)⋊C₈	C₂²	1	1	yes	characteristic
53	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₇	C₂²	1	1	yes	characteristic
56	C₈×Dic₇	C₂²	1	1	yes	characteristic
57	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
58	#6377	C₂²	1	1	yes	characteristic
59	#2981	—	7	1	no	maximal, Sylow
60	C₂×C₇×D₄	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₇	C₂×C₄	1	1	yes	characteristic
69	#1157	—	4	1	no	—
70	#1157	—	4	1	no	—
71	#1157	—	4	1	no	—
72	#1157	—	4	1	no	—
73	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
74	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
75	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄×D₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
81	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
82	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
83	C₇×M₁₆	—	4	1	no	—
84	C₂×C₇×C₈	—	4	1	no	—
85	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
89	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
91	C₇×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
92	C₄²×C₇	C₂³	1	1	yes	characteristic
93	C₂²⋊Dic₇	C₂×C₄	1	1	yes	characteristic
94	C₂²⋊Dic₇	C₂×C₄	1	1	yes	characteristic
95	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
96	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
97	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
98	(C₂×D₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
99	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
100	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
101	Dic₇⋊C₄	C₂×C₄	1	1	yes	characteristic
102	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
103	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
104	C₄×Dic₇	C₂³	1	1	yes	characteristic
105	#1136	—	4	1	no	—
106	C₂×C₇⋊C₈	—	4	1	no	—
107	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
110	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
111	#1131	—	4	1	no	—
112	#1131	—	4	1	no	—
113	C₈×D₇	—	4	1	no	—
114	C₈×D₇	—	4	1	no	—
115	#532	—	7	1	no	—
116	#436	—	7	1	no	—
117	#436	—	7	1	no	—
118	#435	—	7	1	no	—
119	#435	—	7	1	no	—
120	#435	—	7	1	no	—
121	#435	D₇	1	1	yes	characteristic, complemented
122	C₈×D₄	—	7	1	no	—
123	C₈×D₄	—	7	1	no	—
124	#432	—	7	1	no	—
125	#431	—	7	1	no	—
126	#408	—	7	1	no	—
127	#408	—	7	1	no	—
128	#408	—	7	1	no	—
129	#408	—	7	1	no	—
130	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
131	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×D₇	C₂²×C₄	1	1	yes	characteristic
135	C₇×D₄	—	4	1	no	—
136	C₂×C₄×C₇	—	2	1	no	—
137	C₂×C₄×C₇	—	2	1	no	—
138	C₂×C₄×C₇	—	2	1	no	—
139	C₂×C₄×C₇	—	2	1	no	—
140	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
145	C₂²⋊D₇	—	2	2	no	—
146	C₂²⋊D₇	—	2	2	no	—
147	C₂²⋊D₇	—	4	1	no	—
148	C₂²⋊D₇	—	4	1	no	—
149	C₂²⋊D₇	—	4	1	no	—
150	C₂²⋊D₇	—	4	1	no	—
151	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
153	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
154	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
155	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
156	D₂₈	—	4	1	no	—
157	D₂₈	—	4	1	no	—
158	D₂₈	—	4	1	no	—
159	C₄×D₇	—	2	1	no	—
160	C₄×D₇	—	2	1	no	—
161	C₄×D₇	—	2	1	no	—
162	C₄×D₇	—	2	1	no	—
163	C₄×D₇	—	2	1	no	—
164	C₄×D₇	—	2	1	no	—
165	C₄×D₇	—	2	2	no	—
166	C₄×D₇	—	4	1	no	—
167	Dic₁₄	—	4	1	no	—
168	Dic₁₄	—	4	1	no	—
169	C₇×C₈	—	2	1	no	—
170	C₇×C₈	—	2	1	no	—
171	C₇×C₈	—	2	1	no	—
172	C₇×C₈	—	2	1	no	—
173	C₇⋊C₈	—	2	1	no	—
174	C₇⋊C₈	—	2	1	no	—
175	C₇⋊C₈	—	2	1	no	—
176	C₇⋊C₈	—	2	1	no	—
177	C₂×#41	—	7	1	no	—
178	C₂×#41	—	7	1	no	—
179	C₂×M₁₆	—	7	1	no	—
180	C₂×M₁₆	—	7	1	no	—
181	C₂×M₁₆	—	7	1	no	—
182	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
183	C₂²×C₈	—	7	1	no	—
184	C₂²×C₈	—	7	1	no	—
185	C₂²×C₈	—	7	1	no	—
186	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
187	C₄⋊Q₈	—	7	1	no	—
188	#127	—	7	1	no	—
189	#124	—	7	1	no	—
190	#124	—	7	1	no	—
191	#121	—	7	1	no	—
192	#121	—	7	1	no	—
193	#121	—	7	1	no	—
194	#121	—	7	1	no	—
195	C₄×D₄	—	7	1	no	—
196	C₄×D₄	—	7	1	no	—
197	C₄×D₄	—	7	1	no	—
198	C₄×D₄	C₂×D₇	1	1	yes	characteristic
199	C₂×C₄²	—	7	1	no	—
200	C₄⋊C₈	—	7	1	no	—
201	C₄⋊C₈	—	7	1	no	—
202	C₄⋊C₈	—	7	1	no	—
203	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic
204	C₂²⋊C₈	—	7	1	no	—
205	C₂²⋊C₈	—	7	1	no	—
206	C₂²⋊C₈	—	7	1	no	—
207	C₂²⋊C₈	—	7	1	no	—
208	C₂²⋊C₈	—	7	1	no	—
209	C₂²⋊C₈	—	7	1	no	—
210	C₂²⋊C₈	—	7	1	no	—
211	C₂²⋊C₈	—	7	1	no	—
212	C₂²⋊C₈	—	7	1	no	—
213	C₂²⋊C₈	—	7	1	no	—
214	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
215	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
216	#97	—	7	1	no	—
217	#97	C₂×D₇	1	1	yes	characteristic
218	C₄×C₈	—	7	1	no	—
219	C₄×C₈	—	7	1	no	—
220	C₂²×C₇	—	2	1	no	—
221	C₂²×C₇	—	2	1	no	—
222	C₂²×C₇	—	2	1	no	—
223	C₂²×C₇	—	2	1	no	—
224	C₂²×C₇	—	2	1	no	—
225	C₂²×C₇	—	2	1	no	—
226	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
227	C₂×D₇	—	2	1	no	—
228	C₂×D₇	—	2	1	no	—
229	C₂×D₇	—	2	1	no	—
230	C₂×D₇	—	2	1	no	—
231	C₂×D₇	—	2	1	no	—
232	C₂×D₇	—	2	1	no	—
233	C₂×D₇	—	2	1	no	—
234	C₂×D₇	—	2	1	no	—
235	C₂×D₇	—	2	1	no	—
236	C₄×C₇	—	2	1	no	—
237	C₄×C₇	—	2	1	no	—
238	C₄×C₇	—	2	1	no	—
239	C₄×C₇	—	2	1	no	—
240	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
241	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
242	Dic₇	—	2	1	no	—
243	Dic₇	—	2	1	no	—
244	Dic₇	—	2	1	no	—
245	Dic₇	—	2	1	no	—
246	Dic₇	#131	1	2	yes	—
247	#41	—	28	1	no	—
248	#41	—	28	1	no	—
249	#41	—	28	1	no	—
250	#41	—	28	1	no	—
251	C₂×Q₈	—	7	1	no	—
252	C₂×Q₈	—	7	1	no	—
253	C₂×D₄	—	7	1	no	—
254	C₂×D₄	—	7	1	no	—
255	C₂×D₄	—	7	1	no	—
256	C₂×D₄	—	7	1	no	—
257	C₂×D₄	—	7	1	no	—
258	C₂×D₄	—	7	1	no	—
259	C₂×D₄	—	7	1	no	—
260	C₂×D₄	—	7	1	no	—
261	C₂×D₄	—	7	1	no	—
262	C₂×D₄	C₄×D₇	1	1	yes	characteristic
263	C₂²×C₄	—	7	1	no	—
264	C₂²×C₄	—	7	1	no	—
265	C₂²×C₄	—	7	1	no	—
266	C₂²×C₄	—	7	1	no	—
267	C₂²×C₄	—	7	1	no	—
268	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
269	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
270	M₁₆	—	4	1	no	—
271	M₁₆	—	28	1	no	—
272	M₁₆	—	28	1	no	—
273	M₁₆	—	28	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	7	1	no	—
276	C₂×C₈	—	7	1	no	—
277	C₂×C₈	—	7	1	no	—
278	C₂×C₈	—	7	1	no	—
279	C₂×C₈	—	28	1	no	—
280	C₂×C₈	—	28	1	no	—
281	C₂×C₈	—	28	1	no	—
282	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
283	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
284	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
285	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
286	C₄⋊C₄	—	7	1	no	—
287	C₄⋊C₄	—	7	1	no	—
288	C₄⋊C₄	—	7	1	no	—
289	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
290	C₂²⋊C₄	—	7	1	no	—
291	C₂²⋊C₄	—	7	1	no	—
292	C₂²⋊C₄	—	7	1	no	—
293	C₂²⋊C₄	—	7	1	no	—
294	C₂²⋊C₄	—	7	1	no	—
295	C₂²⋊C₄	—	7	1	no	—
296	C₂²⋊C₄	C₄×D₇	1	1	yes	characteristic
297	C₂²⋊C₄	C₄×D₇	1	1	yes	characteristic
298	C₄²	—	7	1	no	—
299	C₄²	—	7	1	no	—
300	C₄²	—	7	1	no	—
301	C₄²	C₂²×D₇	1	1	yes	characteristic
302	C₂×C₇	—	4	1	no	—
303	C₂×C₇	—	4	1	no	—
304	C₂×C₇	#518	1	1	yes	characteristic
305	C₂×C₇	C₂×#131	1	1	yes	characteristic
306	C₂×C₇	#568	1	1	yes	characteristic
307	D₇	—	4	1	no	—
308	D₇	—	4	1	no	—
309	D₇	—	4	1	no	—
310	C₂³	—	7	1	no	—
311	C₂³	—	7	1	no	—
312	C₂³	—	7	1	no	—
313	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
314	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
315	Q₈	—	28	1	no	—
316	Q₈	—	28	1	no	—
317	D₄	—	4	1	no	—
318	D₄	—	14	2	no	—
319	D₄	—	14	2	no	—
320	D₄	—	28	1	no	—
321	D₄	—	28	1	no	—
322	D₄	—	28	1	no	—
323	D₄	—	28	1	no	—
324	D₄	—	28	1	no	—
325	D₄	—	28	1	no	—
326	D₄	—	28	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	7	1	no	—
332	C₂×C₄	—	7	1	no	—
333	C₂×C₄	—	7	1	no	—
334	C₂×C₄	—	7	1	no	—
335	C₂×C₄	—	7	1	no	—
336	C₂×C₄	—	14	2	no	—
337	C₂×C₄	—	14	1	no	—
338	C₂×C₄	—	14	1	no	—
339	C₂×C₄	—	14	1	no	—
340	C₂×C₄	—	14	1	no	—
341	C₂×C₄	—	14	1	no	—
342	C₂×C₄	—	14	1	no	—
343	C₂×C₄	—	28	1	no	—
344	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
348	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	14	1	no	—
354	C₈	—	14	1	no	—
355	C₈	—	14	1	no	—
356	C₈	—	14	1	no	—
357	C₇	#2981	1	1	yes	characteristic, complemented, Sylow
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	14	1	no	—
365	C₂²	—	14	1	no	—
366	C₂²	—	14	1	no	—
367	C₂²	—	14	1	no	—
368	C₂²	—	14	1	no	—
369	C₂²	—	14	1	no	—
370	C₂²	—	14	1	no	—
371	C₂²	—	14	1	no	—
372	C₂²	—	14	1	no	—
373	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	7	2	no	—
379	C₄	—	14	1	no	—
380	C₄	—	14	1	no	—
381	C₄	—	14	1	no	—
382	C₄	—	14	1	no	—
383	C₄	#6548	1	1	yes	characteristic
384	C₄	#6553	1	1	yes	characteristic
385	C₂	—	4	1	no	—
386	C₂	—	4	1	no	—
387	C₂	—	28	1	no	—
388	C₂	—	28	1	no	—
389	C₂	—	28	1	no	—
390	C₂	#90236	1	1	yes	characteristic
391	C₂	#90436	1	1	yes	characteristic
392	C₂	D₇×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#435	C₂
prime radical	#90239	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#435	D₇
hypocenter	C₇	#2981
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. C₇×#435	C₂	semidirect
3. #89901	C₂	semidirect
4. #89900	C₂	semidirect
5. #89882	C₂	semidirect
6. C₈×C₂²⋊D₇	C₂	semidirect
7. #89783	C₂	semidirect
8. #89608	C₂	semidirect
9. #89605	C₂	semidirect
10. #89501	C₂	semidirect
11. #89499	C₂	semidirect
12. #89499	C₂	semidirect
13. (C₂²⋊D₇)⋊C₈	C₂	semidirect
14. #89478	C₂	semidirect
15. #89477	C₂	semidirect
121. #435	D₇	semidirect
357. C₇	#2981	semidirect

Maximal subgroups

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

subgroup	index
#90239	2
C₇×#435	2
#89901	2
#89900	2
#89882	2
C₈×C₂²⋊D₇	2
#89783	2
#89608	2
#89605	2
#89501	2
#89499	2
(C₂²⋊D₇)⋊C₈	2
#89478	2
#89477	2
#2981	7