Subgroups — #11727092

Order 896.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₅₆	C₂	1	1	yes	maximal, characteristic
2	#89645	C₂	1	2	yes	maximal
3	C₇×#367	C₂	1	1	yes	maximal, characteristic
4	#89297	C₂	1	1	yes	maximal, characteristic
5	#89287	C₂	1	2	yes	maximal
6	C₂×Dic₂₈	C₂²	1	2	yes	—
7	#6450	—	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	#6384	—	2	2	no	—
13	Dic₅₆	C₄	1	4	yes	complemented
14	#2176	—	7	1	no	maximal, Sylow
15	C₂×Dic₁₄	—	2	2	no	—
16	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
17	C₇×C₄⋊C₄	—	2	2	no	—
18	Dic₇⋊C₄	—	4	2	no	—
19	C₄×Dic₇	—	4	2	no	—
20	C₄×Dic₇	D₄	1	1	yes	characteristic
21	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
22	Dic₂₈	—	2	2	no	—
23	Dic₂₈	C₂×C₄	1	4	yes	—
24	C₇×C₁₆	C₂×C₄	1	2	yes	—
25	C₇⋊C₁₆	—	2	1	no	—
26	C₂×Q₃₂	—	7	1	no	—
27	C₄×Q₁₆	—	7	2	no	—
28	#367	D₇	1	1	yes	characteristic
29	#358	—	7	2	no	—
30	#346	—	7	1	no	—
31	C₂×C₄×C₇	—	4	2	no	—
32	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇	—	4	2	no	—
34	C₂×Dic₇	D₈	1	2	yes	—
35	Dic₁₄	—	2	4	no	—
36	Dic₁₄	—	4	2	no	—
37	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₇×C₈	#41	1	1	yes	characteristic
39	C₇⋊C₈	—	2	1	no	—
40	C₂×Q₁₆	—	7	2	no	—
41	C₄×Q₈	—	14	2	no	—
42	Q₃₂	—	7	4	no	—
43	C₂×C₁₆	—	7	1	no	—
44	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
45	#107	C₂×D₇	1	2	yes	—
46	#103	—	14	2	no	—
47	C₄×C₈	—	7	1	no	—
48	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
49	C₄×C₇	—	8	2	no	—
50	C₄×C₇	C₄×D₄	1	1	yes	characteristic
51	C₄×C₇	#135	1	1	yes	characteristic
52	Dic₇	—	2	2	no	—
53	Dic₇	—	4	4	no	—
54	C₂×Q₈	—	14	2	no	—
55	Q₁₆	—	7	4	no	—
56	Q₁₆	—	14	2	no	—
57	C₂×C₈	—	7	1	no	—
58	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	—	28	2	no	—
61	C₄²	—	7	1	no	—
62	C₄²	—	28	2	no	—
63	C₁₆	—	14	1	no	—
64	C₁₆	C₄×D₇	1	2	yes	—
65	C₂×C₇	C₄×D₈	1	1	yes	characteristic
66	C₂×C₇	#510	1	2	yes	—
67	Q₈	—	14	4	no	—
68	Q₈	—	28	2	no	—
69	C₂×C₄	—	4	2	no	—
70	C₂×C₄	—	7	2	no	—
71	C₂×C₄	—	28	2	no	—
72	C₂×C₄	D₄×D₇	1	1	yes	characteristic
73	C₈	—	14	1	no	—
74	C₈	C₂×C₄×D₇	1	1	yes	characteristic
75	C₈	#1161	1	1	yes	characteristic
76	C₇	#2176	1	1	yes	characteristic, complemented, Sylow
77	C₂²	D₇×D₈	1	1	yes	characteristic
78	C₄	—	8	2	no	—
79	C₄	—	14	2	no	—
80	C₄	—	28	4	no	—
81	C₄	#6456	1	1	yes	characteristic
82	C₄	#6482	1	1	yes	characteristic
83	C₂	#89644	1	1	yes	characteristic
84	C₂	#89687	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, prime radical	C₂²×C₇	C₂×D₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	#367	D₇
hypocenter	C₇	#2176
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
13. Dic₅₆	C₄	semidirect
76. C₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₅₆	2
#89645	2
C₇×#367	2
#89297	2
#89287	2
#2176	7