Subgroups — #11727093

Order 896.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₁₂	C₂	1	1	yes	maximal, characteristic
2	#89644	C₂	1	2	yes	maximal, complemented
3	C₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	#89297	C₂	1	1	yes	maximal, characteristic, complemented
5	#89286	C₂	1	2	yes	maximal, complemented
6	C₂×D₅₆	C₂²	1	2	yes	—
7	#6456	—	2	2	no	—
8	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×#107	C₂²	1	2	yes	—
10	C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#6383	—	2	2	no	—
13	D₁₁₂	C₄	1	4	yes	complemented
14	#2177	—	7	1	no	maximal, Sylow
15	C₂×D₂₈	—	2	2	no	—
16	C₂×C₄×D₇	—	4	2	no	—
17	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₄	—	2	2	no	—
19	(C₂×D₇)⋊C₄	—	4	2	no	—
20	C₄×Dic₇	D₄	1	1	yes	characteristic
21	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
22	D₅₆	—	2	2	no	—
23	D₅₆	C₂×C₄	1	4	yes	complemented
24	C₇×C₁₆	C₂×C₄	1	2	yes	complemented
25	C₇⋊C₁₆	—	2	1	no	—
26	C₂×D₁₆	—	7	1	no	—
27	C₄×D₈	—	7	2	no	—
28	#367	D₇	1	1	yes	characteristic, complemented
29	#357	—	7	2	no	—
30	#346	—	7	1	no	—
31	C₂²×D₇	—	4	2	no	—
32	C₂×C₄×C₇	—	4	2	no	—
33	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₇	D₈	1	2	yes	—
35	D₂₈	—	2	4	no	—
36	D₂₈	—	4	2	no	—
37	C₄×D₇	—	8	4	no	—
38	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₇×C₈	#41	1	1	yes	characteristic
40	C₇⋊C₈	—	2	1	no	—
41	C₂×D₈	—	7	2	no	—
42	C₄×D₄	—	14	2	no	—
43	D₁₆	—	7	4	no	—
44	C₂×C₁₆	—	7	1	no	—
45	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
46	#107	C₂×D₇	1	2	yes	—
47	#102	—	14	2	no	—
48	C₄×C₈	—	7	1	no	—
49	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
50	C₂×D₇	—	4	4	no	—
51	C₂×D₇	—	8	2	no	—
52	C₂×D₇	—	8	2	no	—
53	C₄×C₇	—	8	2	no	—
54	C₄×C₇	C₄×D₄	1	1	yes	characteristic
55	C₄×C₇	#135	1	1	yes	characteristic
56	Dic₇	—	2	2	no	—
57	C₂×D₄	—	14	2	no	—
58	C₂²×C₄	—	28	2	no	—
59	D₈	—	7	4	no	—
60	D₈	—	14	2	no	—
61	C₂×C₈	—	7	1	no	—
62	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
63	C₄⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	28	2	no	—
65	C₄²	—	7	1	no	—
66	C₁₆	—	14	1	no	—
67	C₁₆	C₄×D₇	1	2	yes	complemented
68	C₂×C₇	C₄×D₈	1	1	yes	characteristic
69	C₂×C₇	#509	1	2	yes	—
70	D₇	—	8	4	no	—
71	C₂³	—	28	2	no	—
72	D₄	—	14	4	no	—
73	D₄	—	28	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	7	2	no	—
76	C₂×C₄	—	56	4	no	—
77	C₂×C₄	D₄×D₇	1	1	yes	characteristic
78	C₈	—	14	1	no	—
79	C₈	C₂×C₄×D₇	1	1	yes	characteristic
80	C₈	#1161	1	1	yes	characteristic
81	C₇	#2177	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	28	4	no	—
83	C₂²	—	56	2	no	—
84	C₂²	—	56	2	no	—
85	C₂²	D₇×D₈	1	1	yes	characteristic
86	C₄	—	8	2	no	—
87	C₄	—	14	2	no	—
88	C₄	#6456	1	1	yes	characteristic
89	C₄	#6482	1	1	yes	characteristic
90	C₂	—	56	4	no	—
91	C₂	#89644	1	1	yes	characteristic
92	C₂	#89686	1	2	yes	—

	subgroup	G/H
center	C₂²	D₇×D₈
commutator	C₇×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₇×#367	C₂
socle	C₂²×C₇	C₂×D₈
prime radical	C₂×D₁₁₂	C₂
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#367	D₇
hypocenter	C₇	#2177
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
2. #89644	C₂	semidirect
3. C₇×#367	C₂	semidirect
4. #89297	C₂	semidirect
5. #89286	C₂	semidirect
13. D₁₁₂	C₄	semidirect
23. D₅₆	C₂×C₄	semidirect
24. C₇×C₁₆	C₂×C₄	semidirect
28. #367	D₇	semidirect
67. C₁₆	C₄×D₇	semidirect
81. C₇	#2177	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₁₂	2
#89644	2
C₇×#367	2
#89297	2
#89286	2
#2177	7