Subgroups — #27212937

Order 13504.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269958	C₂	1	1	yes	maximal, characteristic
2	#20269892	C₂	1	1	yes	maximal, characteristic
3	#100×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₁	C₄	1	1	yes	characteristic, complemented
5	Dic₂₁₁⋊C₄	—	2	1	no	—
6	C₂²⋊Dic₂₁₁	—	2	1	no	—
7	C₂²⋊Dic₂₁₁	—	2	1	no	—
8	C₂²⋊Dic₂₁₁	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	M₁₆×C₂₁₁	—	2	1	no	—
11	C₂×Dic₂₁₁	—	2	1	no	—
12	C₂×Dic₂₁₁	—	2	1	no	—
13	C₂×Dic₂₁₁	—	4	1	no	—
14	C₂×Dic₂₁₁	—	4	1	no	—
15	C₂³×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
17	D₄×C₂₁₁	—	4	1	no	—
18	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
19	C₈×C₂₁₁	—	4	1	no	—
20	C₂²×C₂₁₁	—	2	1	no	—
21	C₂²×C₂₁₁	—	2	1	no	—
22	C₂²×C₂₁₁	—	4	1	no	—
23	C₂²×C₂₁₁	—	4	1	no	—
24	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₂₁₁	—	2	1	no	—
26	Dic₂₁₁	—	4	1	no	—
27	Dic₂₁₁	—	4	1	no	—
28	Dic₂₁₁	—	8	1	no	—
29	C₂×C₂₁₁	—	2	1	no	—
30	C₂×C₂₁₁	—	4	1	no	—
31	C₂×C₂₁₁	—	4	1	no	—
32	C₂×C₂₁₁	#99	1	1	yes	characteristic
33	C₂₁₁	#352	1	1	yes	characteristic, complemented, Sylow
34	#352	—	211	1	no	maximal, Sylow
35	#123	—	211	1	no	—
36	#100	D₂₁₁	1	1	yes	characteristic
37	#99	—	211	1	no	—
38	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
39	C₂²×C₄	—	211	1	no	—
40	M₁₆	—	2	1	no	—
41	C₄⋊C₄	—	422	1	no	—
42	C₂²⋊C₄	—	422	1	no	—
43	C₂²⋊C₄	—	422	1	no	—
44	C₂²⋊C₄	—	211	1	no	—
45	C₂³	C₄×D₂₁₁	1	1	yes	characteristic
46	C₂³	D₈₄₄	1	1	yes	characteristic
47	D₄	—	4	1	no	—
48	C₂×C₄	—	844	1	no	—
49	C₂×C₄	—	844	1	no	—
50	C₂×C₄	—	422	1	no	—
51	C₂×C₄	—	422	1	no	—
52	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
53	C₈	—	4	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	844	1	no	—
61	C₄	—	844	1	no	—
62	C₄	—	1688	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	#20269891	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269891
commutator	C₂³×C₂₁₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₁₁
Fitting	#100×C₂₁₁	C₂
socle	C₂×C₂₁₁	#99
prime radical, cosocle	C₂×D₄×C₂₁₁	C₂²
hypercenter	#100	D₂₁₁
hypocenter	C₂₁₁	#352
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #100

Complements

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

N	H ≅ G/N	type
4. C₂²×Dic₂₁₁	C₄	semidirect
8. C₂²⋊Dic₂₁₁	C₄	semidirect
33. C₂₁₁	#352	semidirect

Maximal subgroups

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

subgroup	index
#20269958	2
#20269892	2
#100×C₂₁₁	2
#352	211