Subgroups — #41013003

Order 27008.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213526	C₂	1	1	yes	maximal, characteristic, complemented
2	#27213492	C₂	1	4	yes	maximal, complemented
3	C₈×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₈₄₄	—	2	2	no	—
6	#20269977	C₂²	1	2	yes	—
7	#20269898	—	2	8	no	—
8	C₄×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	C₂×D₈×C₂₁₁	—	2	2	no	—
10	#127×C₂₁₁	C₂²	1	2	yes	—
11	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₂₂	—	2	2	no	—
13	Dic₈₄₄	—	4	4	no	—
14	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
15	C₄⋊Dic₂₁₁	—	2	4	no	—
16	C₂×D₄×C₂₁₁	—	2	2	no	—
17	C₂×D₄×C₂₁₁	—	2	4	no	—
18	D₈×C₂₁₁	—	4	4	no	—
19	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
20	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
21	Dic₄₂₂	—	4	4	no	—
22	C₂×Dic₂₁₁	—	4	2	no	—
23	C₂₁₁⋊C₈	—	2	4	no	—
24	C₂³×C₂₁₁	—	4	2	no	—
25	D₄×C₂₁₁	—	4	4	no	—
26	D₄×C₂₁₁	—	4	8	no	—
27	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₁₁	D₈	1	4	yes	—
30	C₂²×C₂₁₁	—	8	2	no	—
31	C₂²×C₂₁₁	—	8	4	no	—
32	C₂²×C₂₁₁	#127	1	1	yes	characteristic
33	C₄×C₂₁₁	C₂×D₈	1	2	yes	—
34	C₄×C₂₁₁	#135	1	4	yes	—
35	Dic₂₁₁	—	8	2	no	—
36	C₂×C₂₁₁	—	16	2	no	—
37	C₂×C₂₁₁	#493	1	1	yes	characteristic
38	C₂×C₂₁₁	#495	1	2	yes	—
39	C₂₁₁	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	211	1	no	maximal, Sylow
41	#494	—	211	1	no	—
42	#493	D₂₁₁	1	1	yes	characteristic
43	#486	—	211	4	no	—
44	C₈²	—	211	1	no	—
45	C₂×Q₁₆	—	422	2	no	—
46	C₂×D₈	—	2	2	no	—
47	C₄⋊Q₈	—	211	2	no	—
48	#127	C₂×D₂₁₁	1	2	yes	—
49	#102	—	422	8	no	—
50	C₄×C₈	—	211	2	no	—
51	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
52	C₂×Q₈	—	422	2	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	2	2	no	—
55	Q₁₆	—	844	4	no	—
56	D₈	—	4	4	no	—
57	C₂×C₈	—	211	4	no	—
58	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
59	C₄⋊C₄	—	422	4	no	—
60	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
61	C₂³	—	4	2	no	—
62	Q₈	—	844	4	no	—
63	D₄	—	4	4	no	—
64	D₄	—	4	8	no	—
65	C₂×C₄	—	844	2	no	—
66	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₈	—	422	4	no	—
69	C₈	#16809947	1	4	yes	—
70	C₂²	—	8	2	no	—
71	C₂²	—	8	4	no	—
72	C₂²	#20269973	1	1	yes	characteristic
73	C₄	—	1688	2	no	—
74	C₄	C₂×#16809947	1	2	yes	—
75	C₄	#20269996	1	4	yes	—
76	C₂	—	16	2	no	—
77	C₂	#27213522	1	1	yes	characteristic
78	C₂	#27213529	1	2	yes	—

	subgroup	G/H
center	C₂²	#20269973
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting, prime radical	#493×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#127
hypercenter	#493	D₂₁₁
hypocenter	C₂₁₁	#1709
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

Complements

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

N	H ≅ G/N	type
1. #27213526	C₂	semidirect
2. #27213492	C₂	semidirect
3. C₈×C₂₁₁⋊C₈	C₂	semidirect
39. C₂₁₁	#1709	semidirect

Maximal subgroups

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

subgroup	index
#27213526	2
#27213492	2
C₈×C₂₁₁⋊C₈	2
#493×C₂₁₁	2
#1709	211