Subgroups — #34094541

Order 20032.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097550	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₅₀₄	C₂	1	2	yes	maximal, complemented
3	#23756494	C₂	1	1	yes	maximal, characteristic, complemented
4	#23756485	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₂₆	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₅₂	—	2	2	no	—
8	C₂×D₁₂₅₂	C₂²	1	1	yes	characteristic
9	C₂×D₁₂₅₂	C₂²	1	2	yes	—
10	#19097550	—	2	4	no	—
11	D₂₅₀₄	—	2	4	no	—
12	#19097538	—	2	2	no	—
13	C₂×C₈×C₃₁₃	C₂²	1	2	yes	—
14	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₃	—	2	1	no	—
16	C₂²×D₃₁₃	—	2	2	no	—
17	Dic₆₂₆	—	2	2	no	—
18	D₁₂₅₂	—	2	2	no	—
19	D₁₂₅₂	—	2	2	no	—
20	D₁₂₅₂	—	2	2	no	—
21	D₁₂₅₂	—	4	4	no	—
22	C₂×Dic₃₁₃	—	2	1	no	—
23	C₂×C₄×C₃₁₃	D₄	1	2	yes	—
24	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
25	C₈×C₃₁₃	D₄	1	4	yes	complemented
26	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₃₁₃	—	4	1	no	—
28	C₂×D₃₁₃	—	4	2	no	—
29	C₂×D₃₁₃	—	4	4	no	—
30	C₂×D₃₁₃	—	4	2	no	—
31	C₄×C₃₁₃	—	2	2	no	—
32	C₄×C₃₁₃	C₂×D₄	1	2	yes	—
33	Dic₃₁₃	—	4	1	no	—
34	C₂×C₃₁₃	#127	1	1	yes	characteristic
35	C₂×C₃₁₃	AGL(1,ℤ/8)	1	2	yes	—
36	D₃₁₃	—	8	2	no	—
37	D₃₁₃	—	8	1	no	—
38	C₃₁₃	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	313	1	no	maximal, Sylow
40	C₂×SD₁₆	—	313	2	no	—
41	C₂×D₈	—	313	2	no	—
42	#127	—	313	1	no	—
43	#124	—	313	1	no	—
44	#97	D₃₁₃	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	313	1	no	—
46	C₂×D₄	—	313	1	no	—
47	C₂×D₄	—	313	2	no	—
48	C₂×D₄	—	626	2	no	—
49	SD₁₆	—	626	4	no	—
50	D₈	—	626	4	no	—
51	C₂×C₈	C₂×D₃₁₃	1	2	yes	—
52	C₂²⋊C₄	—	626	2	no	—
53	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
54	C₂³	—	626	1	no	—
55	C₂³	—	626	2	no	—
56	Q₈	—	626	2	no	—
57	D₄	—	626	2	no	—
58	D₄	—	626	2	no	—
59	D₄	—	626	2	no	—
60	D₄	—	1252	4	no	—
61	C₂×C₄	—	626	1	no	—
62	C₂×C₄	D₁₂₅₂	1	2	yes	—
63	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
64	C₈	D₁₂₅₂	1	4	yes	complemented
65	C₂²	—	1252	1	no	—
66	C₂²	—	1252	2	no	—
67	C₂²	—	1252	4	no	—
68	C₂²	—	1252	2	no	—
69	C₂²	C₂×D₁₂₅₂	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	1252	1	no	—
72	C₄	C₂×D₁₂₅₂	1	2	yes	—
73	C₂	—	2504	2	no	—
74	C₂	—	2504	1	no	—
75	C₂	#23756494	1	1	yes	characteristic
76	C₂	#23756525	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₂
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	#97×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
hypercenter	#97	D₃₁₃
hypocenter	C₃₁₃	#496
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#19097550	C₂	semidirect
2. C₂×D₂₅₀₄	C₂	semidirect
3. #23756494	C₂	semidirect
4. #23756485	C₂	semidirect
5. #97×C₃₁₃	C₂	semidirect
25. C₈×C₃₁₃	D₄	semidirect
38. C₃₁₃	#496	semidirect
44. #97	D₃₁₃	semidirect
64. C₈	D₁₂₅₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097550	2
C₂×D₂₅₀₄	2
#23756494	2
#23756485	2
#97×C₃₁₃	2
#496	313