Subgroups — #34094560

Order 20032.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756496	C₂	1	1	yes	maximal, characteristic
2	#23756488	C₂	1	1	yes	maximal, characteristic
3	#23756430	C₂	1	1	yes	maximal, characteristic
4	#23756429	C₂	1	1	yes	maximal, characteristic
5	#23756428	C₂	1	1	yes	maximal, characteristic
6	#23756407	C₂	1	1	yes	maximal, characteristic
7	#107×C₃₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₂₆	—	2	1	no	—
9	C₂×#14578809	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₁₃	—	2	1	no	—
11	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
13	#19097540	—	2	1	no	—
14	#19097540	—	2	1	no	—
15	C₄×Dic₃₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₁₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₁₃	C₂²	1	1	yes	characteristic
19	Dic₆₂₆	—	4	2	no	—
20	C₂×Dic₃₁₃	—	2	1	no	—
21	C₂×Dic₃₁₃	—	2	1	no	—
22	C₂×Dic₃₁₃	D₄	1	1	yes	characteristic
23	C₂×Dic₃₁₃	D₄	1	1	yes	characteristic
24	#14578809	Q₈	1	2	yes	complemented
25	C₂×C₄×C₃₁₃	—	2	1	no	—
26	C₂×C₄×C₃₁₃	—	2	1	no	—
27	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
28	C₈×C₃₁₃	Q₈	1	2	yes	complemented
29	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
30	C₄×C₃₁₃	—	4	1	no	—
31	C₄×C₃₁₃	—	4	1	no	—
32	C₄×C₃₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₃₁₃	C₂×Q₈	1	1	yes	characteristic
34	Dic₃₁₃	—	2	1	no	—
35	Dic₃₁₃	—	2	1	no	—
36	Dic₃₁₃	—	4	1	no	—
37	Dic₃₁₃	—	4	1	no	—
38	C₂×C₃₁₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₃₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₃₁₃	#137	1	1	yes	characteristic
41	C₃₁₃	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	313	1	no	maximal, Sylow
43	C₄⋊Q₈	—	313	1	no	—
44	#125	—	313	1	no	—
45	#107	—	313	1	no	—
46	#107	D₃₁₃	1	1	yes	characteristic
47	#106	—	313	1	no	—
48	#106	—	313	1	no	—
49	#97	—	313	1	no	—
50	C₂×Q₈	—	626	1	no	—
51	C₂×C₈	—	313	1	no	—
52	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
53	C₄⋊C₄	—	313	1	no	—
54	C₄⋊C₄	—	313	1	no	—
55	C₄⋊C₄	—	626	1	no	—
56	C₄⋊C₄	—	626	1	no	—
57	C₄⋊C₄	—	626	1	no	—
58	C₄⋊C₄	C₂×D₃₁₃	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₃₁₃	1	1	yes	characteristic
60	C₄²	—	313	1	no	—
61	Q₈	—	1252	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	313	1	no	—
65	C₂×C₄	—	313	1	no	—
66	C₂×C₄	—	626	1	no	—
67	C₂×C₄	—	626	1	no	—
68	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
69	C₈	—	313	2	no	—
70	C₈	Dic₆₂₆	1	2	yes	complemented
71	C₂²	D₄×D₃₁₃	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	626	1	no	—
75	C₄	—	626	1	no	—
76	C₄	—	1252	1	no	—
77	C₄	—	1252	1	no	—
78	C₄	Q₈×D₃₁₃	1	1	yes	characteristic
79	C₄	C₂×Dic₆₂₆	1	1	yes	characteristic
80	C₂	#23756496	1	1	yes	characteristic
81	C₂	#23756521	1	1	yes	characteristic
82	C₂	#23756529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₃
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	#107×C₃₁₃	C₂
socle, prime radical	C₂²×C₃₁₃	C₂×D₄
hypercenter	#107	D₃₁₃
hypocenter	C₃₁₃	#501
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
24. #14578809	Q₈	semidirect
28. C₈×C₃₁₃	Q₈	semidirect
41. C₃₁₃	#501	semidirect
70. C₈	Dic₆₂₆	semidirect

Maximal subgroups

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

subgroup	index
#23756496	2
#23756488	2
#23756430	2
#23756429	2
#23756428	2
#23756407	2
#107×C₃₁₃	2
#501	313