Subgroups — #27212941

Order 13504.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269892	C₂	1	2	yes	maximal
2	#124×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₂₁₁	—	2	2	no	—
4	C₂×Q₈×C₂₁₁	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
6	C₂²⋊C₄×C₂₁₁	—	2	2	no	—
7	C₄²×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂×Dic₂₁₁	—	4	2	no	—
9	C₂³×C₂₁₁	D₄	1	2	yes	—
10	Q₈×C₂₁₁	—	4	1	no	—
11	D₄×C₂₁₁	—	4	1	no	—
12	C₂×C₄×C₂₁₁	—	2	1	no	—
13	C₂×C₄×C₂₁₁	—	2	1	no	—
14	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₂₁₁	—	2	2	no	—
16	C₂²×C₂₁₁	—	4	2	no	—
17	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₂₁₁	—	2	1	no	—
19	C₄×C₂₁₁	—	4	1	no	—
20	C₄×C₂₁₁	—	4	1	no	—
21	Dic₂₁₁	—	8	2	no	—
22	C₂×C₂₁₁	—	2	1	no	—
23	C₂×C₂₁₁	—	4	2	no	—
24	C₂×C₂₁₁	#99	1	1	yes	characteristic
25	C₂₁₁	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	211	1	no	maximal, Sylow
27	#124	D₂₁₁	1	1	yes	characteristic
28	#99	—	211	2	no	—
29	C₂×Q₈	Dic₂₁₁	1	1	yes	characteristic, complemented
30	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
31	C₂²⋊C₄	—	2	2	no	—
32	C₂²⋊C₄	—	422	2	no	—
33	C₄²	Dic₂₁₁	1	1	yes	characteristic, complemented
34	C₂³	C₂²⋊D₂₁₁	1	2	yes	—
35	Q₈	—	4	1	no	—
36	D₄	—	4	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	844	2	no	—
40	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
41	C₂²	—	2	2	no	—
42	C₂²	—	4	2	no	—
43	C₂²	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	4	1	no	—
47	C₄	—	1688	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	4	2	no	—
50	C₂	#20269892	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269892
commutator	C₂×C₄×C₂₁₁	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₁₁
Fitting	#124×C₂₁₁	C₂
socle	C₂×C₂₁₁	#99
prime radical, cosocle	C₂×D₄×C₂₁₁	C₂²
hypercenter	#124	D₂₁₁
hypocenter	C₂₁₁	#354
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #124

Complements

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

N	H ≅ G/N	type
4. C₂×Q₈×C₂₁₁	C₄	semidirect
7. C₄²×C₂₁₁	C₄	semidirect
25. C₂₁₁	#354	semidirect
29. C₂×Q₈	Dic₂₁₁	semidirect
33. C₄²	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269892	2
#124×C₂₁₁	2
#354	211