Subgroups — #11739411

Order 896.

360 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90418	C₂	1	1	yes	maximal, characteristic
2	#90376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	#89863	C₂	1	1	yes	maximal, characteristic, complemented
5	#89838	C₂	1	1	yes	maximal, characteristic
6	#89662	C₂	1	1	yes	maximal, characteristic
7	#89661	C₂	1	1	yes	maximal, characteristic
8	#89647	C₂	1	1	yes	maximal, characteristic, complemented
9	#89646	C₂	1	1	yes	maximal, characteristic, complemented
10	#89643	C₂	1	1	yes	maximal, characteristic
11	#89628	C₂	1	1	yes	maximal, characteristic, complemented
12	#89626	C₂	1	1	yes	maximal, characteristic, complemented
13	#89624	C₂	1	1	yes	maximal, characteristic
14	#89482	C₂	1	1	yes	maximal, characteristic
15	#89473	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₇	—	2	1	no	—
17	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₇×#125	C₂²	1	1	yes	characteristic
19	#6509	—	2	1	no	—
20	C₂×#1132	C₂²	1	1	yes	characteristic
21	#6459	—	2	1	no	—
22	#6459	—	2	1	no	—
23	#6456	C₂²	1	1	yes	characteristic
24	#6456	C₂²	1	1	yes	characteristic
25	#6453	C₂²	1	1	yes	characteristic
26	#6452	—	2	1	no	—
27	#6452	—	2	1	no	—
28	#6451	—	2	1	no	—
29	#6451	C₂²	1	1	yes	characteristic
30	#6451	C₂²	1	1	yes	characteristic
31	#6451	C₂²	1	1	yes	characteristic
32	#6450	—	2	1	no	—
33	#6450	C₂²	1	1	yes	characteristic
34	#6450	C₂²	1	1	yes	characteristic
35	C₄×D₂₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₄	C₂²	1	1	yes	characteristic
38	C₇×#107	C₂²	1	1	yes	characteristic
39	C₇×#107	C₂²	1	1	yes	characteristic
40	C₇×#106	C₂²	1	1	yes	characteristic
41	C₇×#106	C₂²	1	1	yes	characteristic
42	C₇×#97	C₂²	1	1	yes	characteristic
43	#6395	C₂²	1	1	yes	characteristic
44	#6392	C₂²	1	1	yes	characteristic
45	#6390	C₂²	1	1	yes	characteristic
46	#6389	C₂²	1	1	yes	characteristic
47	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
48	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
49	C₈×Dic₇	C₂²	1	1	yes	characteristic
50	#6384	C₂²	1	1	yes	characteristic
51	#6384	C₂²	1	1	yes	characteristic
52	#6383	C₂²	1	1	yes	characteristic
53	#6383	C₂²	1	1	yes	characteristic
54	#6382	C₂²	1	1	yes	characteristic
55	#6381	C₂²	1	1	yes	characteristic
56	#6381	C₂²	1	1	yes	characteristic
57	#6381	C₂²	1	1	yes	characteristic
58	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
59	#3386	—	7	1	no	maximal, Sylow
60	C₂×C₇×Q₈	—	2	1	no	—
61	Q₈×D₇	—	4	4	no	—
62	C₂×D₂₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₇	—	2	1	no	—
64	C₂×C₄×D₇	—	2	1	no	—
65	C₂×C₄×D₇	—	2	1	no	—
66	C₂×Dic₁₄	—	2	1	no	—
67	C₂×Dic₁₄	—	2	1	no	—
68	C₂×Dic₁₄	—	2	1	no	—
69	C₂×Dic₁₄	D₄	1	1	yes	characteristic
70	C₂×Dic₁₄	C₂³	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
73	C₇×C₄⋊C₄	—	2	1	no	—
74	C₇×C₄⋊C₄	—	2	1	no	—
75	C₇×C₄⋊C₄	—	2	1	no	—
76	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
77	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
78	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₄²×C₇	C₂³	1	1	yes	characteristic
81	(C₂×D₇)⋊C₄	—	2	1	no	—
82	(C₂×D₇)⋊C₄	—	2	1	no	—
83	(C₂×D₇)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₇	—	2	1	no	—
85	C₄⋊Dic₇	D₄	1	1	yes	characteristic
86	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₇	C₂³	1	1	yes	characteristic
88	Dic₇⋊C₄	—	2	1	no	—
89	Dic₇⋊C₄	—	2	1	no	—
90	Dic₇⋊C₄	—	2	1	no	—
91	Dic₇⋊C₄	—	2	1	no	—
92	Dic₇⋊C₄	—	2	1	no	—
93	Dic₇⋊C₄	—	2	1	no	—
94	Dic₇⋊C₄	—	2	1	no	—
95	Dic₇⋊C₄	D₄	1	1	yes	characteristic
96	Dic₇⋊C₄	D₄	1	1	yes	characteristic
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	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
104	#1132	Q₈	1	4	yes	complemented
105	#557	—	7	1	no	—
106	D₄×Q₈	—	7	1	no	—
107	#501	—	7	1	no	—
108	#501	D₇	1	1	yes	characteristic, complemented
109	#500	—	7	1	no	—
110	#478	—	7	1	no	—
111	#478	—	7	1	no	—
112	#477	—	7	1	no	—
113	#477	—	7	1	no	—
114	#475	—	7	1	no	—
115	#474	—	7	1	no	—
116	#446	—	7	1	no	—
117	#440	—	7	1	no	—
118	#440	—	7	1	no	—
119	C₄×SD₁₆	—	7	1	no	—
120	C₂²×D₇	—	2	1	no	—
121	C₇×Q₈	—	4	2	no	—
122	C₂×C₄×C₇	—	2	1	no	—
123	C₂×C₄×C₇	—	2	1	no	—
124	C₂×C₄×C₇	—	2	1	no	—
125	C₂×C₄×C₇	—	2	1	no	—
126	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
129	C₂×Dic₇	—	2	1	no	—
130	C₂×Dic₇	—	2	1	no	—
131	C₂×Dic₇	—	2	1	no	—
132	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₇	C₂×D₄	1	1	yes	characteristic
136	D₂₈	—	2	1	no	—
137	D₂₈	C₂×Q₈	1	2	yes	—
138	C₄×D₇	—	4	2	no	—
139	C₄×D₇	—	4	2	no	—
140	C₄×D₇	—	4	2	no	—
141	Dic₁₄	—	2	1	no	—
142	Dic₁₄	—	2	2	no	—
143	Dic₁₄	—	4	2	no	—
144	Dic₁₄	—	4	2	no	—
145	Dic₁₄	—	4	2	no	—
146	Dic₁₄	C₂×Q₈	1	2	yes	complemented
147	C₇×C₈	—	2	1	no	—
148	C₇×C₈	C₂×Q₈	1	2	yes	complemented
149	C₇⋊C₈	—	2	1	no	—
150	C₇⋊C₈	—	2	1	no	—
151	C₂²×Q₈	—	14	1	no	—
152	C₂×SD₁₆	—	7	1	no	—
153	C₄⋊Q₈	—	7	1	no	—
154	C₄⋊Q₈	—	7	1	no	—
155	C₄⋊Q₈	—	7	1	no	—
156	C₄⋊Q₈	—	14	1	no	—
157	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
158	#125	—	7	1	no	—
159	#125	—	14	1	no	—
160	#125	—	14	1	no	—
161	#125	C₂×D₇	1	1	yes	characteristic
162	C₂²⋊Q₈	—	14	1	no	—
163	C₂²⋊Q₈	—	14	1	no	—
164	C₂²⋊Q₈	—	14	1	no	—
165	C₄×Q₈	—	7	1	no	—
166	C₄×Q₈	—	7	1	no	—
167	C₄×Q₈	—	7	1	no	—
168	C₄×Q₈	—	7	1	no	—
169	C₄×Q₈	—	14	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	1	no	—
172	C₄×D₄	—	7	1	no	—
173	#107	—	7	1	no	—
174	#107	—	7	1	no	—
175	#107	—	7	1	no	—
176	#107	—	7	1	no	—
177	#107	C₂×D₇	1	1	yes	characteristic
178	#107	C₂×D₇	1	1	yes	characteristic
179	#106	—	7	1	no	—
180	#106	C₂×D₇	1	1	yes	characteristic
181	#106	C₂×D₇	1	1	yes	characteristic
182	C₄⋊C₈	—	7	1	no	—
183	C₄⋊C₈	—	7	1	no	—
184	C₄⋊C₈	—	7	1	no	—
185	#103	—	7	1	no	—
186	#103	—	7	1	no	—
187	#103	—	7	1	no	—
188	#102	—	7	1	no	—
189	#102	—	7	1	no	—
190	#102	—	7	1	no	—
191	#97	—	7	1	no	—
192	#97	C₂×D₇	1	1	yes	characteristic
193	C₄×C₈	—	7	1	no	—
194	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×D₇	—	2	2	no	—
196	C₂×D₇	—	4	1	no	—
197	C₂×D₇	—	4	1	no	—
198	C₄×C₇	—	2	1	no	—
199	C₄×C₇	—	2	1	no	—
200	C₄×C₇	—	4	1	no	—
201	C₄×C₇	—	4	1	no	—
202	C₄×C₇	—	4	1	no	—
203	C₄×C₇	—	4	1	no	—
204	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
205	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
206	Dic₇	—	2	2	no	—
207	Dic₇	—	2	1	no	—
208	Dic₇	—	2	1	no	—
209	Dic₇	—	2	1	no	—
210	Dic₇	—	4	1	no	—
211	Dic₇	—	4	1	no	—
212	Dic₇	AGL(1,ℤ/8)	1	2	yes	—
213	C₂×Q₈	—	2	1	no	—
214	C₂×Q₈	—	7	1	no	—
215	C₂×Q₈	—	7	1	no	—
216	C₂×Q₈	—	14	1	no	—
217	C₂×Q₈	—	14	1	no	—
218	C₂×Q₈	—	14	1	no	—
219	C₂×Q₈	—	28	4	no	—
220	C₂×D₄	—	7	1	no	—
221	C₂²×C₄	—	14	1	no	—
222	C₂²×C₄	—	14	1	no	—
223	C₂²×C₄	—	14	1	no	—
224	SD₁₆	—	7	4	no	—
225	C₂×C₈	—	7	1	no	—
226	C₂×C₈	—	7	1	no	—
227	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	7	1	no	—
233	C₄⋊C₄	—	7	1	no	—
234	C₄⋊C₄	—	7	1	no	—
235	C₄⋊C₄	—	7	1	no	—
236	C₄⋊C₄	—	7	1	no	—
237	C₄⋊C₄	—	14	1	no	—
238	C₄⋊C₄	—	14	1	no	—
239	C₄⋊C₄	—	14	1	no	—
240	C₄⋊C₄	—	14	1	no	—
241	C₄⋊C₄	—	14	1	no	—
242	C₄⋊C₄	—	14	1	no	—
243	C₄⋊C₄	—	14	1	no	—
244	C₄⋊C₄	—	14	1	no	—
245	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
246	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
249	C₂²⋊C₄	—	14	1	no	—
250	C₂²⋊C₄	—	14	1	no	—
251	C₂²⋊C₄	—	14	1	no	—
252	C₄²	—	7	1	no	—
253	C₄²	—	7	1	no	—
254	C₄²	—	14	1	no	—
255	C₄²	—	14	1	no	—
256	C₄²	—	14	1	no	—
257	C₄²	C₂²×D₇	1	1	yes	characteristic
258	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
259	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂×C₇	#578	1	1	yes	characteristic
261	D₇	—	4	2	no	—
262	C₂³	—	14	1	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	7	2	no	—
265	Q₈	—	14	2	no	—
266	Q₈	—	14	1	no	—
267	Q₈	—	28	2	no	—
268	Q₈	—	28	2	no	—
269	Q₈	—	28	2	no	—
270	D₄	—	7	2	no	—
271	D₄	—	14	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	7	1	no	—
277	C₂×C₄	—	7	1	no	—
278	C₂×C₄	—	7	1	no	—
279	C₂×C₄	—	7	1	no	—
280	C₂×C₄	—	14	1	no	—
281	C₂×C₄	—	14	1	no	—
282	C₂×C₄	—	14	1	no	—
283	C₂×C₄	—	28	2	no	—
284	C₂×C₄	—	28	2	no	—
285	C₂×C₄	—	28	2	no	—
286	C₂×C₄	D₄×D₇	1	1	yes	characteristic
287	C₂×C₄	D₄×D₇	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	14	1	no	—
291	C₈	—	14	1	no	—
292	C₈	Q₈×D₇	1	2	yes	complemented
293	C₇	#3386	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	14	2	no	—
295	C₂²	—	28	1	no	—
296	C₂²	—	28	1	no	—
297	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	7	2	no	—
305	C₄	—	14	2	no	—
306	C₄	—	14	1	no	—
307	C₄	—	14	1	no	—
308	C₄	—	14	1	no	—
309	C₄	—	28	1	no	—
310	C₄	—	28	1	no	—
311	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
312	C₄	#6551	1	1	yes	characteristic
313	C₂	—	28	2	no	—
314	C₂	#90418	1	1	yes	characteristic
315	C₂	D₇×AGL(1,ℤ/8)	1	1	yes	characteristic
316	C₂	#90470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#501	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	C₂×D₂₈	C₂³
hypercenter	#501	D₇
hypocenter	C₇	#3386
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
2. #90376	C₂	semidirect
3. C₇×#501	C₂	semidirect
4. #89863	C₂	semidirect
8. #89647	C₂	semidirect
9. #89646	C₂	semidirect
11. #89628	C₂	semidirect
12. #89626	C₂	semidirect
15. #89473	C₂	semidirect
104. #1132	Q₈	semidirect
108. #501	D₇	semidirect
146. Dic₁₄	C₂×Q₈	semidirect
148. C₇×C₈	C₂×Q₈	semidirect
292. C₈	Q₈×D₇	semidirect
293. C₇	#3386	semidirect

Maximal subgroups

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

subgroup	index
#90418	2
#90376	2
C₇×#501	2
#89863	2
#89838	2
#89662	2
#89661	2
#89647	2
#89646	2
#89643	2
#89628	2
#89626	2
#89624	2
#89482	2
#89473	2
#3386	7