Subgroups — #11741202

Order 896.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#90482	C₂	1	1	yes	maximal, characteristic
3	C₂×#6470	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#89901	C₂	1	1	yes	maximal, characteristic, complemented
6	#89900	C₂	1	2	yes	maximal, complemented
7	#89899	C₂	1	2	yes	maximal, complemented
8	#89894	C₂	1	1	yes	maximal, characteristic, complemented
9	#89891	C₂	1	1	yes	maximal, characteristic, complemented
10	#89883	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊D₇	C₂	1	2	yes	maximal, complemented
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂×#1157	C₂²	1	1	yes	characteristic
14	C₂×C₇×M₁₆	—	2	4	no	—
15	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₇×M₁₆	C₂²	1	2	yes	—
17	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₇×C₈	C₂²	1	2	yes	—
19	#6516	C₄	1	2	yes	—
20	#6493	C₄	1	1	yes	characteristic
21	#6493	C₄	1	1	yes	characteristic
22	#6492	C₄	1	2	yes	—
23	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
24	C₄×C₂²⋊D₇	C₂²	1	2	yes	—
25	#6489	C₂²	1	1	yes	characteristic
26	#6487	C₄	1	1	yes	characteristic
27	#6487	C₄	1	1	yes	characteristic
28	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
29	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
30	#6470	—	2	4	no	—
31	C₂×#1131	C₂²	1	2	yes	complemented
32	C₂×C₈×D₇	C₂²	1	2	yes	complemented
33	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
34	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
36	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
37	(C₂×D₇)⋊C₈	C₂²	1	2	yes	complemented
38	(C₂×D₇)⋊C₈	C₂²	1	2	yes	complemented
39	#6389	C₂²	1	2	yes	complemented
40	Dic₇⋊C₈	C₂²	1	2	yes	complemented
41	Dic₇⋊C₈	C₂²	1	2	yes	complemented
42	C₈×Dic₇	C₂²	1	2	yes	complemented
43	#2933	—	7	1	no	maximal, Sylow
44	C₂⁴×C₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	2	no	—
46	C₂²×C₄×C₇	—	2	2	no	—
47	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
56	#1157	—	2	2	no	—
57	#1157	—	2	2	no	—
58	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×D₇	C₂³	1	2	yes	—
60	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
61	C₇×M₁₆	—	2	2	no	—
62	C₇×M₁₆	—	2	4	no	—
63	C₇×M₁₆	D₄	1	4	yes	complemented
64	C₂×C₇×C₈	—	2	4	no	—
65	C₂×C₇×C₈	C₂³	1	2	yes	—
66	C₂×C₇×C₈	C₂³	1	2	yes	—
67	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
69	C₂²⋊Dic₇	C₂×C₄	1	2	yes	—
70	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
71	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
72	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₇	C₂×C₄	1	1	yes	characteristic
74	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
75	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
76	C₄×Dic₇	C₂³	1	2	yes	—
77	C₂×C₇⋊C₈	—	2	2	no	—
78	C₂×C₇⋊C₈	—	2	2	no	—
79	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
83	#1131	—	2	4	no	—
84	C₈×D₇	—	2	4	no	—
85	C₂×#131	—	7	1	no	—
86	C₂²×M₁₆	D₇	1	1	yes	characteristic, complemented
87	#525	—	7	1	no	—
88	#436	—	7	2	no	—
89	#435	—	7	2	no	—
90	#435	—	7	2	no	—
91	C₈×D₄	—	7	2	no	—
92	#424	—	7	1	no	—
93	#408	—	7	1	no	—
94	C₂×C₂²⋊C₈	—	7	1	no	—
95	#405	—	7	1	no	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	—	2	2	no	—
98	C₂³×C₇	—	4	1	no	—
99	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
100	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×D₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	1	no	—
110	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₇	#41	1	2	yes	—
115	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₇	—	2	4	no	—
117	C₂²⋊D₇	—	2	4	no	—
118	C₂²⋊D₇	—	4	2	no	—
119	C₂×Dic₇	C₂²×C₄	1	2	yes	—
120	C₂×Dic₇	C₂²×C₄	1	2	yes	—
121	C₂×Dic₇	C₂²×C₄	1	2	yes	—
122	D₂₈	—	2	2	no	—
123	C₄×D₇	—	2	2	no	—
124	C₄×D₇	—	2	2	no	—
125	Dic₁₄	—	2	2	no	—
126	C₇×C₈	—	2	2	no	—
127	C₇×C₈	C₂×D₄	1	4	yes	—
128	C₇⋊C₈	—	2	1	no	—
129	C₇⋊C₈	—	2	1	no	—
130	C₇⋊C₈	—	2	1	no	—
131	C₇⋊C₈	—	2	1	no	—
132	C₂×#41	—	7	1	no	—
133	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
134	#131	—	14	4	no	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	7	2	no	—
137	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
138	C₂×M₁₆	C₂×D₇	1	2	yes	—
139	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
140	C₂²×C₈	—	7	1	no	—
141	C₂²×C₈	—	7	1	no	—
142	C₂²×C₈	—	7	2	no	—
143	C₂²×C₈	C₂×D₇	1	2	yes	—
144	#123	—	7	2	no	—
145	C₂²⋊Q₈	—	7	1	no	—
146	C₂²⋊Q₈	—	7	1	no	—
147	#121	—	7	1	no	—
148	#121	—	7	1	no	—
149	#120	—	7	2	no	—
150	C₄×D₄	—	7	2	no	—
151	C₄×D₄	—	7	2	no	—
152	#117	—	7	1	no	—
153	C₄⋊C₈	—	7	2	no	—
154	C₄⋊C₈	—	7	2	no	—
155	C₂²⋊C₈	—	7	1	no	—
156	C₂²⋊C₈	—	7	1	no	—
157	C₂²⋊C₈	—	7	1	no	—
158	C₂²⋊C₈	—	7	1	no	—
159	C₂²⋊C₈	—	7	2	no	—
160	C₂²⋊C₈	—	7	2	no	—
161	#97	—	7	2	no	—
162	C₄×C₈	—	7	2	no	—
163	C₂²×C₇	—	2	1	no	—
164	C₂²×C₇	—	2	1	no	—
165	C₂²×C₇	—	2	1	no	—
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	1	no	—
168	C₂²×C₇	—	2	1	no	—
169	C₂²×C₇	—	2	1	no	—
170	C₂²×C₇	—	4	4	no	—
171	C₂²×C₇	C₄×D₄	1	2	yes	—
172	C₂²×C₇	#131	1	2	yes	complemented
173	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
174	C₂×D₇	—	2	2	no	—
175	C₂×D₇	—	2	2	no	—
176	C₂×D₇	—	2	2	no	—
177	C₄×C₇	—	2	1	no	—
178	C₄×C₇	—	2	1	no	—
179	C₄×C₇	C₄×D₄	1	2	yes	—
180	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₇	C₂×#41	1	1	yes	characteristic
182	Dic₇	—	2	2	no	—
183	Dic₇	—	2	2	no	—
184	Dic₇	—	2	2	no	—
185	C₂⁴	C₄×D₇	1	1	yes	characteristic
186	#41	—	14	2	no	—
187	#41	—	14	2	no	—
188	C₂×Q₈	—	7	1	no	—
189	C₂×D₄	—	7	2	no	—
190	C₂×D₄	—	7	2	no	—
191	C₂×D₄	—	7	2	no	—
192	C₂×D₄	—	7	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	7	2	no	—
196	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
197	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
198	C₂²×C₄	C₄×D₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	14	4	no	—
205	M₁₆	C₂²⋊D₇	1	4	yes	complemented
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	7	1	no	—
208	C₂×C₈	—	7	1	no	—
209	C₂×C₈	—	7	1	no	—
210	C₂×C₈	—	7	1	no	—
211	C₂×C₈	—	14	2	no	—
212	C₂×C₈	—	14	2	no	—
213	C₂×C₈	—	14	4	no	—
214	C₂×C₈	C₂²×D₇	1	2	yes	—
215	C₂×C₈	C₂²×D₇	1	2	yes	—
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	—	7	2	no	—
218	C₄⋊C₄	—	7	1	no	—
219	C₄⋊C₄	—	7	1	no	—
220	C₂²⋊C₄	—	7	2	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₄²	—	7	2	no	—
226	C₂×C₇	—	2	2	no	—
227	C₂×C₇	—	2	2	no	—
228	C₂×C₇	—	4	1	no	—
229	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂×C₇	C₂×#131	1	1	yes	characteristic
231	C₂×C₇	#568	1	1	yes	characteristic
232	D₇	—	4	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	4	1	no	—
236	C₂³	—	7	2	no	—
237	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
238	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
239	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
240	Q₈	—	14	2	no	—
241	D₄	—	14	4	no	—
242	D₄	—	14	4	no	—
243	D₄	—	14	2	no	—
244	D₄	—	28	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	7	2	no	—
253	C₂×C₄	—	7	2	no	—
254	C₂×C₄	—	7	2	no	—
255	C₂×C₄	—	14	2	no	—
256	C₂×C₄	—	14	2	no	—
257	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
260	C₂×C₄	#1157	1	2	yes	—
261	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
262	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	14	1	no	—
265	C₈	—	14	1	no	—
266	C₈	—	14	1	no	—
267	C₈	—	14	1	no	—
268	C₈	C₂×C₂²⋊D₇	1	4	yes	—
269	C₇	#2933	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	14	2	no	—
279	C₂²	—	14	2	no	—
280	C₂²	—	14	2	no	—
281	C₂²	#6470	1	2	yes	complemented
282	C₂²	C₄×C₂²⋊D₇	1	2	yes	—
283	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	14	2	no	—
287	C₄	—	14	2	no	—
288	C₄	—	14	2	no	—
289	C₄	C₄×C₂²⋊D₇	1	2	yes	—
290	C₄	C₂×#1157	1	1	yes	characteristic
291	C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	28	2	no	—
296	C₂	C₂×#6470	1	1	yes	characteristic
297	C₂	#90436	1	1	yes	characteristic
298	C₂	C₂×C₄×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₂²×C₇×M₁₆	C₂
prime radical	#90482	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	C₂²×M₁₆	D₇
hypocenter	C₇	#2933
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×M₁₆	C₂	semidirect
3. C₂×#6470	C₂	semidirect
4. C₂×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. #89901	C₂	semidirect
6. #89900	C₂	semidirect
7. #89899	C₂	semidirect
8. #89894	C₂	semidirect
9. #89891	C₂	semidirect
10. #89883	C₂	semidirect
11. C₈×C₂²⋊D₇	C₂	semidirect
31. C₂×#1131	C₂²	semidirect
32. C₂×C₈×D₇	C₂²	semidirect
37. (C₂×D₇)⋊C₈	C₂²	semidirect
38. (C₂×D₇)⋊C₈	C₂²	semidirect
39. #6389	C₂²	semidirect
40. Dic₇⋊C₈	C₂²	semidirect
41. Dic₇⋊C₈	C₂²	semidirect
42. C₈×Dic₇	C₂²	semidirect
63. C₇×M₁₆	D₄	semidirect
86. C₂²×M₁₆	D₇	semidirect
172. C₂²×C₇	#131	semidirect
205. M₁₆	C₂²⋊D₇	semidirect
269. C₇	#2933	semidirect
281. C₂²	#6470	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₁₆	2
#90482	2
C₂×#6470	2
C₂×C₂²⋊(C₇⋊C₈)	2
#89901	2
#89900	2
#89899	2
#89894	2
#89891	2
#89883	2
C₈×C₂²⋊D₇	2
#2933	7