Subgroups — #10622697

Order 704.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄₄	C₂	1	1	yes	maximal, characteristic
3	#66873	C₂	1	2	yes	maximal
4	#66872	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₁₁	—	2	1	no	—
7	C₂×D₄₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₁₁	—	2	1	no	—
9	C₁₁×C₄⋊C₄	—	2	1	no	—
10	C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄²×C₁₁	C₂²	1	1	yes	characteristic
12	(C₂×D₁₁)⋊C₄	—	2	1	no	—
13	C₄⋊Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₁⋊C₈	C₂²	1	2	yes	—
15	C₂²×D₁₁	—	2	1	no	—
16	Q₈×C₁₁	—	4	2	no	—
17	C₂×C₄×C₁₁	—	2	2	no	—
18	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₁	—	2	1	no	—
22	D₄₄	—	2	1	no	—
23	D₄₄	Q₈	1	2	yes	complemented
24	C₄×D₁₁	—	4	2	no	—
25	C₁₁⋊C₈	—	2	2	no	—
26	#476	—	11	1	no	maximal, Sylow
27	C₂²×C₁₁	C₂×D₄	1	1	yes	characteristic
28	C₂×D₁₁	—	2	2	no	—
29	C₂×D₁₁	—	4	1	no	—
30	C₂×D₁₁	—	4	1	no	—
31	C₄×C₁₁	—	2	1	no	—
32	C₄×C₁₁	—	4	2	no	—
33	C₄×C₁₁	SD₁₆	1	2	yes	—
34	C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₁₁	#41	1	1	yes	characteristic
36	Dic₁₁	—	4	1	no	—
37	C₄⋊Q₈	D₁₁	1	1	yes	characteristic, complemented
38	C₄×D₄	—	11	1	no	—
39	#106	—	11	2	no	—
40	C₄⋊C₈	—	11	1	no	—
41	#102	—	11	2	no	—
42	C₂×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
44	C₂×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₁₁	—	4	2	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×D₄	—	11	1	no	—
48	C₂²×C₄	—	22	1	no	—
49	C₂×C₈	—	11	2	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	—	11	1	no	—
52	C₄⋊C₄	C₂×D₁₁	1	2	yes	—
53	C₂²⋊C₄	—	22	1	no	—
54	C₄²	C₂×D₁₁	1	1	yes	characteristic
55	C₁₁	#476	1	1	yes	characteristic, complemented, Sylow
56	C₂³	—	22	1	no	—
57	Q₈	—	4	2	no	—
58	D₄	—	11	2	no	—
59	D₄	—	22	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	22	1	no	—
62	C₂×C₄	—	44	2	no	—
63	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
64	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₁₁	1	1	yes	characteristic
66	C₈	—	22	2	no	—
67	C₂²	—	22	2	no	—
68	C₂²	—	44	1	no	—
69	C₂²	—	44	1	no	—
70	C₂²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	4	2	no	—
73	C₄	—	44	1	no	—
74	C₄	#4325	1	2	yes	—
75	C₄	Q₈×D₁₁	1	1	yes	characteristic
76	C₄	#4343	1	1	yes	characteristic
77	C₂	—	44	2	no	—
78	C₂	#66985	1	1	yes	characteristic
79	C₂	C₂×#4325	1	1	yes	characteristic
80	C₂	#66999	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₁
commutator, cosocle	C₂×C₄×C₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₁
Fitting	C₁₁×C₄⋊Q₈	C₂
socle	C₂²×C₁₁	C₂×D₄
prime radical	C₂×D₄₄	C₂²
hypercenter	C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#476
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₁₁×C₄⋊Q₈	C₂	semidirect
4. #66872	C₂	semidirect
5. C₄⋊(C₁₁⋊C₈)	C₂	semidirect
23. D₄₄	Q₈	semidirect
37. C₄⋊Q₈	D₁₁	semidirect
55. C₁₁	#476	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×C₄⋊Q₈	2
C₄×D₄₄	2
#66873	2
#66872	2
C₄⋊(C₁₁⋊C₈)	2
#476	11