Subgroups — #27213913

Order 13504.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269984	C₂	1	8	yes	maximal, complemented
2	C₂×#16809946	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#16809946	C₂²	1	4	yes	—
5	#16809946	C₂²	1	12	yes	complemented
6	C₂×C₂₁₁⋊C₈	C₂²	1	12	yes	complemented
7	#41×C₂₁₁	C₄	1	4	yes	—
8	#41×C₂₁₁	C₂²	1	4	yes	—
9	C₂×Q₈×C₂₁₁	C₄	1	1	yes	characteristic
10	C₂×D₄×C₂₁₁	C₄	1	3	yes	—
11	C₂²×C₄×C₂₁₁	C₂²	1	3	yes	—
12	C₂₁₁⋊C₈	C₂³	1	8	yes	—
13	C₂³×C₂₁₁	C₂×C₄	1	3	yes	—
14	Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
15	D₄×C₂₁₁	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₂₁₁	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₂₁₁	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₁₁	C₂³	1	6	yes	—
20	C₂²×C₂₁₁	—	4	3	no	—
21	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₂₁₁	C₂²×C₄	1	6	yes	—
23	C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₁₁	C₂²×C₄	1	6	yes	—
25	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
26	C₂×C₂₁₁	—	2	1	no	—
27	C₂×C₂₁₁	—	2	6	no	—
28	C₂×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
29	C₂₁₁	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	211	1	no	maximal, Sylow
31	C₂×#41	D₂₁₁	1	1	yes	characteristic
32	#131	—	211	8	no	—
33	C₂×M₁₆	—	211	6	no	—
34	#41	Dic₂₁₁	1	4	yes	—
35	#41	C₂×D₂₁₁	1	4	yes	—
36	C₂×Q₈	Dic₂₁₁	1	1	yes	characteristic
37	C₂×D₄	Dic₂₁₁	1	3	yes	—
38	C₂²×C₄	C₂×D₂₁₁	1	3	yes	—
39	M₁₆	—	211	4	no	—
40	M₁₆	—	211	12	no	—
41	C₂×C₈	—	211	12	no	—
42	C₂³	C₂×Dic₂₁₁	1	3	yes	—
43	Q₈	C₂×Dic₂₁₁	1	4	yes	—
44	D₄	C₂×Dic₂₁₁	1	12	yes	—
45	C₂×C₄	C₂×Dic₂₁₁	1	3	yes	—
46	C₂×C₄	C₂×Dic₂₁₁	1	6	yes	—
47	C₂×C₄	C₂²×D₂₁₁	1	6	yes	—
48	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
49	C₈	—	211	8	no	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
52	C₂²	C₂²×Dic₂₁₁	1	6	yes	—
53	C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
54	C₄	C₂²×Dic₂₁₁	1	6	yes	—
55	C₄	C₂³×D₂₁₁	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₂₁₁
commutator, socle	C₂×C₂₁₁	C₂³×C₄
Fitting, prime radical	C₂×#41×C₂₁₁	C₂
cosocle	C₄×C₂₁₁	C₂⁴
hypercenter	C₂×#41	D₂₁₁
hypocenter	C₂₁₁	#568
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20269984	C₂	semidirect
2. C₂×#16809946	C₂	semidirect
5. #16809946	C₂²	semidirect
6. C₂×C₂₁₁⋊C₈	C₂²	semidirect
29. C₂₁₁	#568	semidirect

Maximal subgroups

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

subgroup	index
#20269984	2
C₂×#16809946	2
C₂×#41×C₂₁₁	2
#568	211