Subgroups — #10622331

Order 704.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Dic₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₁₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₁	C₄	1	1	yes	characteristic
8	C₂²×Dic₁₁	C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₁	C₂²	1	2	yes	—
10	C₄×Dic₁₁	C₄	1	1	yes	characteristic
11	C₄×Dic₁₁	C₄	1	1	yes	characteristic
12	C₄×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₄×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₁⋊C₈	C₂²	1	2	yes	—
15	C₂³×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₁	—	2	2	no	—
17	C₂×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₁	—	2	2	no	—
20	C₂×Dic₁₁	C₈	1	4	yes	complemented
21	C₂×Dic₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₁	—	2	2	no	—
26	C₁₁⋊C₈	—	2	2	no	—
27	#431	—	11	1	no	maximal, Sylow
28	C₂²×C₁₁	—	2	1	no	—
29	C₂²×C₁₁	—	2	1	no	—
30	C₂²×C₁₁	C₂×C₈	1	2	yes	—
31	C₂²×C₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₁₁	—	2	1	no	—
33	C₄×C₁₁	#41	1	2	yes	—
34	Dic₁₁	—	2	1	no	—
35	Dic₁₁	C₂×C₈	1	2	yes	—
36	Dic₁₁	C₂×C₈	1	2	yes	—
37	Dic₁₁	M₁₆	1	2	yes	—
38	C₂×C₄²	—	11	1	no	—
39	C₄⋊C₈	—	11	2	no	—
40	C₂²⋊C₈	—	11	1	no	—
41	C₂²⋊C₈	D₁₁	1	1	yes	characteristic
42	C₄×C₈	—	11	2	no	—
43	C₂×C₁₁	—	2	2	no	—
44	C₂×C₁₁	#117	1	1	yes	characteristic
45	C₂×C₁₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₁₁	C₂×M₁₆	1	1	yes	characteristic
47	C₂²×C₄	—	11	1	no	—
48	C₂²×C₄	—	11	1	no	—
49	C₂²×C₄	C₂×D₁₁	1	1	yes	characteristic
50	C₂×C₈	—	11	2	no	—
51	C₂×C₈	C₂×D₁₁	1	2	yes	—
52	C₄²	—	11	1	no	—
53	C₄²	—	11	1	no	—
54	C₄²	—	11	1	no	—
55	C₄²	—	11	1	no	—
56	C₁₁	#431	1	1	yes	characteristic, complemented, Sylow
57	C₂³	C₄×D₁₁	1	1	yes	characteristic
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	11	4	no	—
60	C₂×C₄	—	11	1	no	—
61	C₂×C₄	—	11	1	no	—
62	C₂×C₄	—	11	1	no	—
63	C₂×C₄	—	11	1	no	—
64	C₂×C₄	—	22	2	no	—
65	C₂×C₄	C₄×D₁₁	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₁₁	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₈	—	22	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	C₈×D₁₁	1	2	yes	—
72	C₂²	C₂×C₄×D₁₁	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	11	2	no	—
75	C₄	—	11	2	no	—
76	C₄	—	11	2	no	—
77	C₄	—	22	1	no	—
78	C₄	#4341	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	#66930	1	1	yes	characteristic
81	C₂	C₂×C₈×D₁₁	1	1	yes	characteristic
82	C₂	M₁₆×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₁
commutator	C₂×C₁₁	C₂²×C₈
Fitting	C₁₁×C₂²⋊C₈	C₂
socle	C₂²×C₁₁	C₂²×C₄
prime radical	C₂³×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂³
hypercenter	C₂²⋊C₈	D₁₁
hypocenter	C₁₁	#431
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. Dic₁₁⋊C₈	C₂	semidirect
5. C₈×Dic₁₁	C₂	semidirect
20. C₂×Dic₁₁	C₈	semidirect
56. C₁₁	#431	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₁₁	2
C₁₁×C₂²⋊C₈	2
C₂²⋊(C₁₁⋊C₈)	2
Dic₁₁⋊C₈	2
C₈×Dic₁₁	2
#431	11