Subgroups — #11736948

Order 896.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90241	C₂	1	1	yes	maximal, characteristic
2	#90234	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#435	C₂	1	1	yes	maximal, characteristic, complemented
4	#89906	C₂	1	1	yes	maximal, characteristic, complemented
5	#89905	C₂	1	1	yes	maximal, characteristic, complemented
6	#89888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#6395	C₂	1	1	yes	maximal, characteristic, complemented
8	#89618	C₂	1	1	yes	maximal, characteristic, complemented
9	#89617	C₂	1	1	yes	maximal, characteristic, complemented
10	#89508	C₂	1	1	yes	maximal, characteristic, complemented
11	#89505	C₂	1	1	yes	maximal, characteristic, complemented
12	#89503	C₂	1	1	yes	maximal, characteristic, complemented
13	#89494	C₂	1	1	yes	maximal, characteristic, complemented
14	#89486	C₂	1	1	yes	maximal, characteristic, complemented
15	#89482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#1157	C₂²	1	1	yes	characteristic
17	C₂²×D₂₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
21	#6500	—	2	1	no	—
22	D₄×Dic₇	—	2	1	no	—
23	#6493	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
26	#6487	C₂²	1	1	yes	characteristic
27	#6487	C₂²	1	1	yes	characteristic
28	#6487	C₂²	1	1	yes	characteristic, complemented
29	C₂×Dic₂₈	C₂²	1	1	yes	characteristic, complemented
30	C₂×#1132	C₂²	1	1	yes	characteristic
31	C₂×#1132	C₂²	1	1	yes	characteristic
32	#6457	—	2	1	no	—
33	#6453	—	2	1	no	—
34	#6447	—	2	1	no	—
35	D₇×C₂²⋊C₄	—	2	1	no	—
36	#6441	—	2	1	no	—
37	#6441	—	2	1	no	—
38	#6438	C₂²	1	1	yes	characteristic
39	C₄×D₂₈	C₂²	1	1	yes	characteristic
40	#6432	C₂²	1	1	yes	characteristic, complemented
41	C₄×Dic₁₄	C₂²	1	1	yes	characteristic, complemented
42	C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic
44	C₇×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₇×#97	C₂²	1	1	yes	characteristic, complemented
46	#6395	C₂²	1	1	yes	characteristic
47	#6395	C₂²	1	1	yes	characteristic
48	#6395	C₂²	1	1	yes	characteristic
49	#6395	C₂²	1	1	yes	characteristic
50	#6395	C₂²	1	1	yes	characteristic
51	#6395	C₂²	1	1	yes	characteristic
52	#6392	C₂²	1	1	yes	characteristic, complemented
53	#6392	C₂²	1	1	yes	characteristic, complemented
54	#6392	C₂²	1	1	yes	characteristic, complemented
55	#6390	C₂²	1	1	yes	characteristic, complemented
56	#6390	C₂²	1	1	yes	characteristic, complemented
57	#6390	C₂²	1	1	yes	characteristic, complemented
58	#6390	C₂²	1	1	yes	characteristic, complemented
59	#3314	—	7	1	no	maximal, Sylow
60	C₂³×D₇	—	2	1	no	—
61	C₂×C₇×D₄	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇	—	2	1	no	—
65	C₂×C₂²⋊D₇	—	2	1	no	—
66	C₂²×Dic₇	—	2	1	no	—
67	C₂²×Dic₇	—	2	1	no	—
68	#1157	—	2	2	no	—
69	#1157	—	4	1	no	—
70	C₂×D₂₈	—	2	2	no	—
71	C₂×D₂₈	—	2	1	no	—
72	C₂×D₂₈	—	4	1	no	—
73	C₂×D₂₈	C₂³	1	1	yes	characteristic
74	C₂×D₂₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₇	—	2	1	no	—
76	C₂×C₄×D₇	—	2	1	no	—
77	C₂×Dic₁₄	—	2	1	no	—
78	C₂×Dic₁₄	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄	C₂³	1	1	yes	characteristic
80	C₇×M₁₆	—	4	1	no	—
81	C₂×C₇×C₈	—	4	1	no	—
82	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
83	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
86	C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
87	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
88	C₇×C₂²⋊C₄	D₄	1	1	yes	characteristic
89	C₄²×C₇	C₂³	1	1	yes	characteristic
90	C₂²⋊Dic₇	—	2	1	no	—
91	C₂²⋊Dic₇	—	2	1	no	—
92	C₂²⋊Dic₇	—	2	1	no	—
93	(C₂×D₇)⋊C₄	—	2	1	no	—
94	(C₂×D₇)⋊C₄	—	2	1	no	—
95	(C₂×D₇)⋊C₄	—	2	1	no	—
96	(C₂×D₇)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₇	—	2	1	no	—
98	C₄⋊Dic₇	—	2	1	no	—
99	C₄⋊Dic₇	—	2	1	no	—
100	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
102	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
103	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
104	Dic₇⋊C₄	—	2	1	no	—
105	Dic₇⋊C₄	—	2	1	no	—
106	Dic₇⋊C₄	—	2	1	no	—
107	Dic₇⋊C₄	—	2	1	no	—
108	C₄×Dic₇	—	2	1	no	—
109	Dic₂₈	—	4	1	no	—
110	#1132	—	2	2	no	—
111	#1132	—	4	1	no	—
112	#554	—	7	1	no	—
113	#546	—	7	1	no	—
114	#489	—	7	1	no	—
115	#484	—	7	1	no	—
116	#480	—	7	1	no	—
117	#474	—	7	1	no	—
118	#468	—	7	1	no	—
119	#467	—	7	1	no	—
120	#464	—	7	1	no	—
121	#452	—	7	1	no	—
122	#450	—	7	1	no	—
123	#440	—	7	1	no	—
124	#435	D₇	1	1	yes	characteristic, complemented
125	#417	—	7	1	no	—
126	C₂×#102	—	7	1	no	—
127	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×D₇	—	2	1	no	—
130	C₂²×D₇	—	2	1	no	—
131	C₂²×D₇	—	4	2	no	—
132	C₂²×D₇	—	4	2	no	—
133	C₂²×D₇	—	8	1	no	—
134	C₇×D₄	—	2	2	no	—
135	C₂×C₄×C₇	—	2	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	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
144	C₂²⋊D₇	—	4	2	no	—
145	C₂²⋊D₇	—	8	1	no	—
146	C₂×Dic₇	—	2	1	no	—
147	C₂×Dic₇	—	2	1	no	—
148	C₂×Dic₇	—	2	1	no	—
149	C₂×Dic₇	—	2	1	no	—
150	C₂×Dic₇	—	2	1	no	—
151	C₂×Dic₇	—	2	1	no	—
152	C₂×Dic₇	—	8	1	no	—
153	C₂×Dic₇	—	8	1	no	—
154	D₂₈	—	2	1	no	—
155	D₂₈	—	2	1	no	—
156	D₂₈	—	2	1	no	—
157	D₂₈	—	4	2	no	—
158	D₂₈	#41	1	2	yes	—
159	C₄×D₇	—	4	2	no	—
160	C₄×D₇	—	8	1	no	—
161	Dic₁₄	—	2	1	no	—
162	Dic₁₄	—	2	1	no	—
163	Dic₁₄	—	2	1	no	—
164	Dic₁₄	—	2	1	no	—
165	Dic₁₄	—	8	1	no	—
166	C₇×C₈	—	2	1	no	—
167	C₇×C₈	—	2	1	no	—
168	C₇×C₈	—	2	1	no	—
169	C₇×C₈	—	2	1	no	—
170	C₂×#41	—	7	1	no	—
171	C₂²×D₄	—	7	1	no	—
172	C₂×Q₁₆	—	7	1	no	—
173	C₂×SD₁₆	—	7	1	no	—
174	C₂×SD₁₆	—	7	1	no	—
175	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic, complemented
176	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	7	1	no	—
178	#125	—	14	1	no	—
179	#124	—	7	1	no	—
180	#123	—	14	1	no	—
181	C₂²⋊Q₈	—	7	1	no	—
182	C₂²⋊Q₈	—	7	1	no	—
183	C₂²⋊Q₈	—	7	1	no	—
184	C₂²⋊Q₈	—	14	1	no	—
185	C₂²⋊Q₈	—	14	1	no	—
186	#121	—	7	1	no	—
187	#121	—	14	1	no	—
188	#120	—	14	1	no	—
189	C₄×Q₈	—	7	1	no	—
190	C₄×D₄	—	7	1	no	—
191	C₄×D₄	—	14	1	no	—
192	C₄×D₄	C₂×D₇	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	7	1	no	—
194	C₂×C₄⋊C₄	—	7	1	no	—
195	C₂×C₂²⋊C₄	—	14	1	no	—
196	#107	—	7	1	no	—
197	#107	—	7	1	no	—
198	#107	—	7	1	no	—
199	C₄⋊C₈	C₂×D₇	1	1	yes	characteristic, complemented
200	#103	—	7	1	no	—
201	#103	—	7	1	no	—
202	#103	—	7	1	no	—
203	#103	—	7	1	no	—
204	#102	—	7	1	no	—
205	#102	—	7	1	no	—
206	#102	—	7	1	no	—
207	#102	—	7	1	no	—
208	#102	—	7	1	no	—
209	#102	—	7	1	no	—
210	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic
211	C₂²⋊C₈	C₂×D₇	1	1	yes	characteristic, complemented
212	#97	C₂×D₇	1	1	yes	characteristic, complemented
213	C₂²×C₇	—	2	1	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	#137	1	2	yes	—
219	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×D₇	—	2	2	no	—
221	C₂×D₇	—	4	1	no	—
222	C₂×D₇	—	4	1	no	—
223	C₂×D₇	—	4	1	no	—
224	C₂×D₇	—	4	1	no	—
225	C₂×D₇	—	4	1	no	—
226	C₂×D₇	—	8	4	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	C₂×#41	1	1	yes	characteristic
232	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
233	Dic₇	—	4	1	no	—
234	Dic₇	—	4	1	no	—
235	Dic₇	—	4	1	no	—
236	Dic₇	—	4	1	no	—
237	Dic₇	—	4	1	no	—
238	Dic₇	—	4	1	no	—
239	C₂⁴	—	14	1	no	—
240	#41	—	14	2	no	—
241	#41	—	28	1	no	—
242	C₂×Q₈	—	7	1	no	—
243	C₂×Q₈	—	7	1	no	—
244	C₂×Q₈	—	14	1	no	—
245	C₂×D₄	—	7	1	no	—
246	C₂×D₄	—	7	1	no	—
247	C₂×D₄	—	14	2	no	—
248	C₂×D₄	—	14	1	no	—
249	C₂×D₄	—	14	1	no	—
250	C₂×D₄	—	14	1	no	—
251	C₂×D₄	—	28	1	no	—
252	C₂×D₄	D₂₈	1	1	yes	characteristic
253	C₂²×C₄	—	14	1	no	—
254	C₂²×C₄	—	14	1	no	—
255	C₂²×C₄	—	14	1	no	—
256	C₂²×C₄	—	14	1	no	—
257	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
259	Q₁₆	—	28	1	no	—
260	SD₁₆	—	14	2	no	—
261	SD₁₆	—	28	1	no	—
262	M₁₆	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
268	C₄⋊C₄	—	7	1	no	—
269	C₄⋊C₄	—	7	1	no	—
270	C₄⋊C₄	—	7	1	no	—
271	C₄⋊C₄	—	7	1	no	—
272	C₄⋊C₄	—	14	1	no	—
273	C₄⋊C₄	—	14	1	no	—
274	C₄⋊C₄	—	14	1	no	—
275	C₄⋊C₄	—	14	1	no	—
276	C₄⋊C₄	—	14	1	no	—
277	C₄⋊C₄	—	14	1	no	—
278	C₄⋊C₄	—	14	1	no	—
279	C₄⋊C₄	D₂₈	1	1	yes	characteristic
280	C₂²⋊C₄	—	14	1	no	—
281	C₂²⋊C₄	—	14	1	no	—
282	C₂²⋊C₄	—	14	1	no	—
283	C₂²⋊C₄	—	14	1	no	—
284	C₂²⋊C₄	—	14	1	no	—
285	C₂²⋊C₄	—	14	1	no	—
286	C₂²⋊C₄	—	14	1	no	—
287	C₂²⋊C₄	D₂₈	1	1	yes	characteristic
288	C₂²⋊C₄	D₂₈	1	1	yes	characteristic
289	C₄²	—	14	1	no	—
290	C₄²	C₂²×D₇	1	1	yes	characteristic
291	C₂×C₇	—	2	2	no	—
292	C₂×C₇	—	4	1	no	—
293	C₂×C₇	#546	1	1	yes	characteristic
294	C₂×C₇	C₂×#137	1	1	yes	characteristic
295	C₂×C₇	#576	1	1	yes	characteristic
296	D₇	—	4	2	no	—
297	D₇	—	8	1	no	—
298	C₂³	—	14	1	no	—
299	C₂³	—	14	1	no	—
300	C₂³	—	28	2	no	—
301	C₂³	—	28	2	no	—
302	C₂³	—	56	1	no	—
303	C₂³	C₂×D₂₈	1	1	yes	characteristic
304	C₂³	C₂×D₂₈	1	1	yes	characteristic
305	Q₈	—	14	1	no	—
306	Q₈	—	14	1	no	—
307	Q₈	—	14	1	no	—
308	Q₈	—	14	1	no	—
309	Q₈	—	56	1	no	—
310	D₄	—	2	2	no	—
311	D₄	—	7	2	no	—
312	D₄	—	14	1	no	—
313	D₄	—	14	1	no	—
314	D₄	—	14	1	no	—
315	D₄	—	28	2	no	—
316	D₄	—	28	2	no	—
317	D₄	—	56	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	14	1	no	—
323	C₂×C₄	—	14	1	no	—
324	C₂×C₄	—	14	1	no	—
325	C₂×C₄	—	14	1	no	—
326	C₂×C₄	—	14	1	no	—
327	C₂×C₄	—	14	1	no	—
328	C₂×C₄	—	28	2	no	—
329	C₂×C₄	—	56	1	no	—
330	C₂×C₄	—	56	1	no	—
331	C₂×C₄	—	56	1	no	—
332	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
333	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
334	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
335	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
336	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₇	#3314	1	1	yes	characteristic, complemented, Sylow
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	14	2	no	—
348	C₂²	—	28	1	no	—
349	C₂²	—	28	1	no	—
350	C₂²	—	28	1	no	—
351	C₂²	—	28	1	no	—
352	C₂²	—	28	1	no	—
353	C₂²	—	56	4	no	—
354	C₂²	#6472	1	2	yes	—
355	C₂²	C₂²×D₂₈	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	28	1	no	—
361	C₄	—	28	1	no	—
362	C₄	—	28	1	no	—
363	C₄	—	28	1	no	—
364	C₄	—	28	1	no	—
365	C₄	—	28	1	no	—
366	C₄	#6548	1	1	yes	characteristic
367	C₄	D₇×#41	1	1	yes	characteristic
368	C₂	—	2	2	no	—
369	C₂	—	4	1	no	—
370	C₂	—	28	2	no	—
371	C₂	—	56	1	no	—
372	C₂	#90245	1	1	yes	characteristic
373	C₂	C₂×#6472	1	1	yes	characteristic
374	C₂	#90443	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#435	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	#90241	C₂
hypercenter	#435	D₇
hypocenter	C₇	#3314
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #90234	C₂	semidirect
3. C₇×#435	C₂	semidirect
4. #89906	C₂	semidirect
5. #89905	C₂	semidirect
6. #89888	C₂	semidirect
7. C₂×#6395	C₂	semidirect
8. #89618	C₂	semidirect
9. #89617	C₂	semidirect
10. #89508	C₂	semidirect
11. #89505	C₂	semidirect
12. #89503	C₂	semidirect
13. #89494	C₂	semidirect
14. #89486	C₂	semidirect
15. #89482	C₂	semidirect
18. C₂×C₇×M₁₆	C₂²	semidirect
25. C₂×C₄⋊Dic₇	C₂²	semidirect
28. #6487	C₂²	semidirect
29. C₂×Dic₂₈	C₂²	semidirect
40. #6432	C₂²	semidirect
41. C₄×Dic₁₄	C₂²	semidirect
42. C₇×C₄⋊C₈	C₂²	semidirect
44. C₇×C₂²⋊C₈	C₂²	semidirect
45. C₇×#97	C₂²	semidirect
52. #6392	C₂²	semidirect
53. #6392	C₂²	semidirect
54. #6392	C₂²	semidirect
55. #6390	C₂²	semidirect
56. #6390	C₂²	semidirect
57. #6390	C₂²	semidirect
58. #6390	C₂²	semidirect
124. #435	D₇	semidirect
175. C₂×M₁₆	C₂×D₇	semidirect
199. C₄⋊C₈	C₂×D₇	semidirect
211. C₂²⋊C₈	C₂×D₇	semidirect
212. #97	C₂×D₇	semidirect
341. C₇	#3314	semidirect

Maximal subgroups

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

subgroup	index
#90241	2
#90234	2
C₇×#435	2
#89906	2
#89905	2
#89888	2
C₂×#6395	2
#89618	2
#89617	2
#89508	2
#89505	2
#89503	2
#89494	2
#89486	2
#89482	2
#3314	7