Subgroups — #14641653

Order 2816.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1423	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₈₈	—	2	1	no	—
5	C₂×C₁₁×C₃₂	C₂²	1	1	yes	characteristic
6	C₁₁×#366	—	2	1	no	—
7	Dic₁₇₆	—	2	1	no	—
8	Dic₁₇₆	C₄	1	2	yes	complemented
9	C₁₁⋊C₆₄	—	2	1	no	—
10	C₂×Dic₄₄	—	4	1	no	—
11	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
12	C₁₁×#107	—	4	1	no	—
13	Dic₈₈	—	2	2	no	—
14	Dic₈₈	—	4	1	no	—
15	C₁₁×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₁₁×C₃₂	D₄	1	1	yes	characteristic
17	#7538	—	11	1	no	maximal, Sylow
18	C₂×Dic₂₂	—	8	1	no	—
19	C₂×C₈×C₁₁	D₈	1	1	yes	characteristic
20	C₁₁×C₄⋊C₄	—	8	1	no	—
21	Dic₄₄	—	4	2	no	—
22	Dic₄₄	—	8	1	no	—
23	C₁₁×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
24	C₁₁×C₁₆	SD₁₆	1	1	yes	characteristic
25	C₂×Q₆₄	—	11	1	no	—
26	C₂×C₆₄	—	11	1	no	—
27	#1423	D₁₁	1	1	yes	characteristic
28	C₂×C₄×C₁₁	—	16	1	no	—
29	C₂×C₄×C₁₁	D₁₆	1	1	yes	characteristic
30	C₂×Dic₁₁	—	16	1	no	—
31	Dic₂₂	—	8	2	no	—
32	Dic₂₂	—	16	1	no	—
33	C₈×C₁₁	#102	1	1	yes	characteristic
34	C₈×C₁₁	SD₃₂	1	1	yes	characteristic
35	C₂×Q₃₂	—	22	1	no	—
36	Q₆₄	—	11	2	no	—
37	Q₆₄	—	22	1	no	—
38	C₂×C₃₂	C₂×D₁₁	1	1	yes	characteristic
39	#366	—	2	1	no	—
40	C₆₄	—	22	1	no	—
41	C₂²×C₁₁	D₃₂	1	1	yes	characteristic
42	C₄×C₁₁	—	32	1	no	—
43	C₄×C₁₁	#357	1	1	yes	characteristic
44	C₄×C₁₁	SD₆₄	1	1	yes	characteristic
45	Dic₁₁	—	16	2	no	—
46	C₂×Q₁₆	—	44	1	no	—
47	Q₃₂	—	22	2	no	—
48	Q₃₂	—	44	1	no	—
49	C₂×C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
50	#107	—	4	1	no	—
51	C₃₂	C₄×D₁₁	1	1	yes	characteristic
52	C₃₂	D₄₄	1	1	yes	characteristic
53	C₂×C₁₁	#1415	1	1	yes	characteristic
54	C₂×C₁₁	SD₁₂₈	1	1	yes	characteristic
55	C₂×C₁₁	Q₁₂₈	1	1	yes	characteristic
56	C₂×Q₈	—	88	1	no	—
57	Q₁₆	—	44	2	no	—
58	Q₁₆	—	88	1	no	—
59	C₂×C₈	#4323	1	1	yes	characteristic
60	C₄⋊C₄	—	8	1	no	—
61	C₁₆	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
62	C₁₆	#4325	1	1	yes	characteristic
63	C₁₁	#7538	1	1	yes	characteristic, complemented, Sylow
64	Q₈	—	88	2	no	—
65	Q₈	—	176	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	176	1	no	—
68	C₂×C₄	#66890	1	1	yes	characteristic
69	C₈	#66873	1	1	yes	characteristic
70	C₈	#66892	1	1	yes	characteristic
71	C₂²	#10622157	1	1	yes	characteristic
72	C₄	—	32	1	no	—
73	C₄	—	176	2	no	—
74	C₄	#10622129	1	1	yes	characteristic
75	C₄	#10622159	1	1	yes	characteristic
76	C₂	#13066793	1	1	yes	characteristic
77	C₂	#13066831	1	1	yes	characteristic
78	C₂	#13066833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622157
commutator	C₁₁×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₁₁
Fitting	C₁₁×#1423	C₂
socle, prime radical	C₂²×C₁₁	D₃₂
cosocle	C₂×C₁₁×C₃₂	C₂²
hypercenter	#1423	D₁₁
hypocenter	C₁₁	#7538
LCS	G ▷ C₁₁×C₃₂ ▷ C₁₁×C₁₆ ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
8. Dic₁₇₆	C₄	semidirect
63. C₁₁	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₇₆	2
C₂×C₁₁⋊C₆₄	2
C₁₁×#1423	2
#7538	11