Subgroups — #10622706

Order 704.

98 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
2	#66922	C₂	1	1	yes	maximal, characteristic
3	#66874	C₂	1	4	yes	maximal
4	C₄×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₁₁	—	2	1	no	—
6	C₂×Dic₂₂	C₂²	1	2	yes	—
7	C₁₁×C₄⋊C₄	—	2	1	no	—
8	C₁₁×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₁₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₁	—	2	1	no	—
11	C₄⋊Dic₁₁	—	2	1	no	—
12	C₂×C₁₁⋊C₈	C₂²	1	2	yes	—
13	Q₈×C₁₁	—	4	2	no	—
14	C₂×C₄×C₁₁	—	2	2	no	—
15	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₁₁	—	2	2	no	—
19	Dic₂₂	—	2	4	no	—
20	C₁₁⋊C₈	—	2	2	no	—
21	#487	—	11	1	no	maximal, Sylow
22	C₂²×C₁₁	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₁	—	4	2	no	—
24	C₄×C₁₁	SD₁₆	1	2	yes	—
25	C₄×C₁₁	Q₁₆	1	2	yes	—
26	C₄×C₁₁	#41	1	2	yes	—
27	Dic₁₁	—	4	2	no	—
28	C₄⋊Q₈	—	11	1	no	—
29	C₄⋊Q₈	D₁₁	1	1	yes	characteristic
30	#103	—	11	4	no	—
31	C₄×C₈	—	11	1	no	—
32	C₂×C₁₁	#124	1	1	yes	characteristic
33	C₂×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
34	C₂×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×Q₈	—	2	1	no	—
36	C₂×Q₈	—	11	2	no	—
37	C₂×C₈	—	11	2	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	22	1	no	—
40	C₄⋊C₄	—	22	1	no	—
41	C₄⋊C₄	C₂×D₁₁	1	2	yes	—
42	C₄²	C₂×D₁₁	1	1	yes	characteristic
43	C₁₁	#487	1	1	yes	characteristic, complemented, Sylow
44	Q₈	—	4	2	no	—
45	Q₈	—	22	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	22	2	no	—
48	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₁	1	1	yes	characteristic
51	C₈	—	22	2	no	—
52	C₂²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	44	2	no	—
55	C₄	#4324	1	2	yes	—
56	C₄	#4326	1	2	yes	—
57	C₄	#4343	1	2	yes	—
58	C₂	C₂×#4324	1	1	yes	characteristic
59	C₂	C₂×#4326	1	1	yes	characteristic
60	C₂	#67000	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, prime radical	C₂²×C₁₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#487
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
43. C₁₁	#487	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
#66922	2
#66874	2
C₄×C₁₁⋊C₈	2
#487	11