Subgroups — #34094405

Order 20032.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₀₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#23756471	C₂	1	2	yes	maximal, complemented
3	#23756429	C₂	1	1	yes	maximal, characteristic, complemented
4	#23756419	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₅₂	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₃₁₃	—	2	2	no	—
8	D₂₅₀₄	—	2	2	no	—
9	C₄⋊Dic₃₁₃	C₂²	1	2	yes	—
10	#19097538	—	2	2	no	—
11	C₂²×C₄×C₃₁₃	C₂²	1	1	yes	characteristic
12	M₁₆×C₃₁₃	—	2	2	no	—
13	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₃	—	2	2	no	—
16	D₁₂₅₂	—	2	2	no	—
17	D₁₂₅₂	—	2	2	no	—
18	C₂²⋊D₃₁₃	—	4	4	no	—
19	C₂×Dic₃₁₃	—	2	2	no	—
20	C₂³×C₃₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₃	—	2	1	no	—
22	C₂×C₄×C₃₁₃	—	2	1	no	—
23	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
25	C₈×C₃₁₃	—	2	1	no	—
26	C₈×C₃₁₃	D₄	1	2	yes	complemented
27	C₂²×C₃₁₃	—	2	1	no	—
28	C₂²×C₃₁₃	—	2	2	no	—
29	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₁₃	—	4	2	no	—
31	C₂×D₃₁₃	—	4	2	no	—
32	C₂×D₃₁₃	—	4	2	no	—
33	C₄×C₃₁₃	—	2	1	no	—
34	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₃₁₃	#41	1	1	yes	characteristic
36	Dic₃₁₃	—	4	2	no	—
37	C₂×C₃₁₃	—	4	1	no	—
38	C₂×C₃₁₃	#121	1	1	yes	characteristic
39	C₂×C₃₁₃	AGL(1,ℤ/8)	1	2	yes	—
40	D₃₁₃	—	8	2	no	—
41	C₃₁₃	#469	1	1	yes	characteristic, complemented, Sylow
42	#469	—	313	1	no	maximal, Sylow
43	C₂×D₈	—	313	1	no	—
44	C₂×M₁₆	D₃₁₃	1	1	yes	characteristic, complemented
45	#121	—	313	2	no	—
46	#106	—	313	1	no	—
47	#102	—	313	2	no	—
48	C₂×D₄	—	313	2	no	—
49	C₂×D₄	—	626	2	no	—
50	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
51	D₈	—	626	2	no	—
52	M₁₆	—	2	2	no	—
53	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
54	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
55	C₄⋊C₄	—	313	2	no	—
56	C₂²⋊C₄	—	626	2	no	—
57	C₂³	—	626	2	no	—
58	C₂³	D₁₂₅₂	1	1	yes	characteristic
59	D₄	—	626	2	no	—
60	D₄	—	626	2	no	—
61	D₄	—	1252	4	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	626	2	no	—
65	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	C₂²⋊D₃₁₃	1	2	yes	complemented
69	C₂²	—	2	1	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	1252	2	no	—
72	C₂²	—	1252	2	no	—
73	C₂²	—	1252	2	no	—
74	C₂²	C₂×D₁₂₅₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	1252	2	no	—
77	C₄	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
78	C₄	#19097561	1	1	yes	characteristic
79	C₂	—	4	1	no	—
80	C₂	—	2504	2	no	—
81	C₂	#23756471	1	1	yes	characteristic
82	C₂	#23756525	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₂
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₂×M₁₆×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
hypercenter	C₂×M₁₆	D₃₁₃
hypocenter	C₃₁₃	#469
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₅₀₄	C₂	semidirect
2. #23756471	C₂	semidirect
3. #23756429	C₂	semidirect
4. #23756419	C₂	semidirect
5. C₂×M₁₆×C₃₁₃	C₂	semidirect
26. C₈×C₃₁₃	D₄	semidirect
41. C₃₁₃	#469	semidirect
44. C₂×M₁₆	D₃₁₃	semidirect
68. C₈	C₂²⋊D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₀₄	2
#23756471	2
#23756429	2
#23756419	2
C₂×M₁₆×C₃₁₃	2
#469	313