Subgroups — #27212944

Order 13504.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269975	C₂	1	1	yes	maximal, characteristic
2	#20269896	C₂	1	1	yes	maximal, characteristic
3	#101×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₂	C₄	1	1	yes	characteristic
5	#16809946	—	2	1	no	—
6	Dic₂₁₁⋊C₄	—	2	1	no	—
7	Dic₂₁₁⋊C₄	—	2	1	no	—
8	C₄×Dic₂₁₁	C₄	1	1	yes	characteristic
9	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	M₁₆×C₂₁₁	—	2	1	no	—
11	Dic₄₂₂	—	4	1	no	—
12	C₂×Dic₂₁₁	—	2	1	no	—
13	C₂×Dic₂₁₁	—	2	1	no	—
14	C₂₁₁⋊C₈	—	4	1	no	—
15	Q₈×C₂₁₁	—	4	1	no	—
16	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
19	C₈×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₁₁	—	2	1	no	—
22	C₄×C₂₁₁	—	2	1	no	—
23	C₄×C₂₁₁	—	2	1	no	—
24	Dic₂₁₁	—	2	2	no	—
25	Dic₂₁₁	—	4	1	no	—
26	C₂×C₂₁₁	—	2	1	no	—
27	C₂×C₂₁₁	#99	1	1	yes	characteristic
28	C₂₁₁	#356	1	1	yes	characteristic, complemented, Sylow
29	#356	—	211	1	no	maximal, Sylow
30	C₄⋊Q₈	—	211	1	no	—
31	#101	—	211	1	no	—
32	#101	D₂₁₁	1	1	yes	characteristic
33	C₂×Q₈	—	211	1	no	—
34	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
35	M₁₆	—	2	1	no	—
36	M₁₆	—	422	1	no	—
37	C₄⋊C₄	—	422	1	no	—
38	C₄⋊C₄	—	422	1	no	—
39	C₄²	—	211	1	no	—
40	Q₈	—	4	1	no	—
41	Q₈	—	844	1	no	—
42	C₂×C₄	—	422	1	no	—
43	C₂×C₄	—	422	1	no	—
44	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
45	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
46	C₂×C₄	D₈₄₄	1	1	yes	characteristic
47	C₈	—	4	1	no	—
48	C₈	—	844	1	no	—
49	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	844	1	no	—
54	C₄	—	422	2	no	—
55	C₂	—	2	1	no	—
56	C₂	#20269891	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269891
commutator	C₂×C₄×C₂₁₁	C₂×C₄
Frattini	C₂×Q₈	C₂×D₂₁₁
Fitting	#101×C₂₁₁	C₂
socle	C₂×C₂₁₁	#99
prime radical	C₂²×C₂₁₁	C₂²⋊C₄
cosocle	C₂×Q₈×C₂₁₁	C₂²
hypercenter	#101	D₂₁₁
hypocenter	C₂₁₁	#356
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ #101

Complements

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

N	H ≅ G/N	type
28. C₂₁₁	#356	semidirect

Maximal subgroups

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

subgroup	index
#20269975	2
#20269896	2
#101×C₂₁₁	2
#356	211