Subgroups — #34094709

Order 20032.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756497	C₂	1	2	yes	maximal, complemented
2	#23756489	C₂	1	2	yes	maximal, complemented
3	#23756476	C₂	1	4	yes	maximal
4	C₄×Dic₆₂₆	C₂	1	4	yes	maximal, complemented
5	#23756456	C₂	1	2	yes	maximal
6	C₂×C₄²×C₃₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₆₂₆	C₂²	1	4	yes	—
8	C₄⋊Dic₃₁₃	C₂²	1	4	yes	—
9	C₄⋊Dic₃₁₃	C₂²	1	4	yes	—
10	#19097540	C₂²	1	8	yes	—
11	C₂²⋊Dic₃₁₃	C₂²	1	4	yes	—
12	C₄×Dic₃₁₃	C₂²	1	4	yes	—
13	C₂²×C₄×C₃₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₁₃	C₂²	1	1	yes	characteristic
15	C₄²×C₃₁₃	C₂²	1	2	yes	—
16	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
17	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
18	Dic₆₂₆	—	2	8	no	—
19	C₂×Dic₃₁₃	C₂³	1	8	yes	—
20	C₂³×C₃₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₁₃	—	2	4	no	—
22	C₂×C₄×C₃₁₃	Q₈	1	4	yes	complemented
23	C₂×C₄×C₃₁₃	C₂³	1	2	yes	—
24	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₃₁₃	C₂³	1	2	yes	—
26	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₃₁₃	—	2	2	no	—
28	C₂²×C₃₁₃	C₂×Q₈	1	2	yes	—
29	C₂²×C₃₁₃	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₁₃	—	2	2	no	—
31	C₄×C₃₁₃	C₂×Q₈	1	2	yes	—
32	C₄×C₃₁₃	C₂×Q₈	1	2	yes	—
33	C₄×C₃₁₃	#41	1	4	yes	—
34	Dic₃₁₃	—	2	8	no	—
35	C₂×C₃₁₃	—	2	2	no	—
36	C₂×C₃₁₃	C₂²×Q₈	1	1	yes	characteristic
37	C₂×C₃₁₃	C₂×#41	1	2	yes	—
38	C₃₁₃	#533	1	1	yes	characteristic, complemented, Sylow
39	#533	—	313	1	no	maximal, Sylow
40	C₄⋊Q₈	—	313	2	no	—
41	#125	—	313	2	no	—
42	C₂²⋊Q₈	—	313	4	no	—
43	C₄×Q₈	—	313	4	no	—
44	#117	—	313	2	no	—
45	C₂×C₄²	D₃₁₃	1	1	yes	characteristic
46	C₂×Q₈	—	313	4	no	—
47	C₂²×C₄	C₂×D₃₁₃	1	2	yes	—
48	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
49	C₄⋊C₄	—	313	4	no	—
50	C₄⋊C₄	—	313	8	no	—
51	C₄⋊C₄	—	313	4	no	—
52	C₂²⋊C₄	—	313	4	no	—
53	C₄²	—	313	4	no	—
54	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
55	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
56	C₄²	C₂×D₃₁₃	1	2	yes	—
57	C₂³	C₂²×D₃₁₃	1	1	yes	characteristic
58	Q₈	—	626	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	313	8	no	—
61	C₂×C₄	Dic₆₂₆	1	4	yes	complemented
62	C₂×C₄	C₂²×D₃₁₃	1	2	yes	—
63	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃₁₃	1	2	yes	—
65	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₆₂₆	1	2	yes	—
68	C₂²	C₂³×D₃₁₃	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	626	8	no	—
71	C₄	C₂×Dic₆₂₆	1	2	yes	—
72	C₄	C₂×Dic₆₂₆	1	2	yes	—
73	C₄	#19097561	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₆₂₆	1	1	yes	characteristic
76	C₂	C₂×#19097561	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₃
commutator, socle, cosocle	C₂²×C₃₁₃	C₂⁴
Frattini	C₂²	C₂³×D₃₁₃
Fitting	C₂×C₄²×C₃₁₃	C₂
prime radical	C₂³×C₃₁₃	C₂³
hypercenter	C₂×C₄²	D₃₁₃
hypocenter	C₃₁₃	#533
LCS	G ▷ C₂²×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #23756497	C₂	semidirect
2. #23756489	C₂	semidirect
4. C₄×Dic₆₂₆	C₂	semidirect
22. C₂×C₄×C₃₁₃	Q₈	semidirect
38. C₃₁₃	#533	semidirect
61. C₂×C₄	Dic₆₂₆	semidirect

Maximal subgroups

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

subgroup	index
#23756497	2
#23756489	2
#23756476	2
C₄×Dic₆₂₆	2
#23756456	2
C₂×C₄²×C₃₁₃	2
#533	313