Subgroups — #27212942

Order 13504.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269966	C₂	1	1	yes	maximal, characteristic
2	#20269896	C₂	1	1	yes	maximal, characteristic, complemented
3	#101×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₄	C₄	1	1	yes	characteristic
5	#16809946	—	2	1	no	—
6	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
7	(C₂×D₂₁₁)⋊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	C₂²×D₂₁₁	—	2	1	no	—
12	D₈₄₄	—	4	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	C₂×D₂₁₁	—	4	2	no	—
22	C₂×D₂₁₁	—	4	1	no	—
23	C₄×C₂₁₁	—	2	1	no	—
24	C₄×C₂₁₁	—	2	1	no	—
25	C₄×C₂₁₁	—	2	1	no	—
26	Dic₂₁₁	—	4	1	no	—
27	C₂×C₂₁₁	—	2	1	no	—
28	C₂×C₂₁₁	#99	1	1	yes	characteristic
29	D₂₁₁	—	8	1	no	—
30	C₂₁₁	#355	1	1	yes	characteristic, complemented, Sylow
31	#355	—	211	1	no	maximal, Sylow
32	#124	—	211	1	no	—
33	#101	—	211	1	no	—
34	#101	D₂₁₁	1	1	yes	characteristic, complemented
35	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
36	C₂×D₄	—	211	1	no	—
37	M₁₆	—	2	1	no	—
38	M₁₆	—	422	1	no	—
39	C₂²⋊C₄	—	422	1	no	—
40	C₂²⋊C₄	—	422	1	no	—
41	C₄²	—	211	1	no	—
42	C₂³	—	422	1	no	—
43	Q₈	—	4	1	no	—
44	D₄	—	844	1	no	—
45	C₂×C₄	—	422	1	no	—
46	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
47	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
48	C₂×C₄	D₈₄₄	1	1	yes	characteristic
49	C₈	—	4	1	no	—
50	C₈	—	844	1	no	—
51	C₂²	—	844	2	no	—
52	C₂²	—	844	1	no	—
53	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
54	C₄	—	2	1	no	—
55	C₄	—	2	1	no	—
56	C₄	—	2	1	no	—
57	C₄	—	844	1	no	—
58	C₂	—	2	1	no	—
59	C₂	—	1688	1	no	—
60	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	C₂×D₈₄₄	C₄
cosocle	C₂×Q₈×C₂₁₁	C₂²
hypercenter	#101	D₂₁₁
hypocenter	C₂₁₁	#355
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
2. #20269896	C₂	semidirect
3. #101×C₂₁₁	C₂	semidirect
30. C₂₁₁	#355	semidirect
34. #101	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269966	2
#20269896	2
#101×C₂₁₁	2
#355	211