Subgroups — #34094388

Order 20032.

134 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	#23756431	C₂	1	1	yes	maximal, characteristic, complemented
4	#23756419	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×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	C₂×C₈×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	1	no	—
29	C₂²×C₃₁₃	D₈	1	2	yes	complemented
30	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
31	C₂×D₃₁₃	—	4	2	no	—
32	C₂×D₃₁₃	—	4	2	no	—
33	C₂×D₃₁₃	—	4	2	no	—
34	C₄×C₃₁₃	—	2	1	no	—
35	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₁₃	#41	1	1	yes	characteristic
37	Dic₃₁₃	—	4	2	no	—
38	C₂×C₃₁₃	—	2	2	no	—
39	C₂×C₃₁₃	#121	1	1	yes	characteristic
40	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
41	C₂×C₃₁₃	#135	1	1	yes	characteristic
42	D₃₁₃	—	8	2	no	—
43	C₃₁₃	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	313	1	no	maximal, Sylow
45	C₂×D₈	—	313	1	no	—
46	C₂²×C₈	D₃₁₃	1	1	yes	characteristic, complemented
47	#121	—	313	2	no	—
48	#107	—	313	1	no	—
49	#102	—	313	2	no	—
50	C₂×D₄	—	313	2	no	—
51	C₂×D₄	—	626	2	no	—
52	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
53	D₈	—	626	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₃₁₃	1	1	yes	characteristic
57	C₄⋊C₄	—	313	2	no	—
58	C₂²⋊C₄	—	626	2	no	—
59	C₂³	—	626	2	no	—
60	C₂³	D₁₂₅₂	1	1	yes	characteristic
61	D₄	—	626	2	no	—
62	D₄	—	626	2	no	—
63	D₄	—	1252	4	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	626	2	no	—
67	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊D₃₁₃	1	2	yes	complemented
71	C₂²	—	2	1	no	—
72	C₂²	—	2	1	no	—
73	C₂²	—	1252	2	no	—
74	C₂²	—	1252	2	no	—
75	C₂²	—	1252	2	no	—
76	C₂²	D₂₅₀₄	1	2	yes	complemented
77	C₂²	C₂×D₁₂₅₂	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	1252	2	no	—
80	C₄	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
81	C₄	#19097561	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	—	2504	2	no	—
84	C₂	#23756471	1	1	yes	characteristic
85	C₂	C₂×D₂₅₀₄	1	1	yes	characteristic
86	C₂	#23756519	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₂
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₂²×C₈×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
hypercenter	C₂²×C₈	D₃₁₃
hypocenter	C₃₁₃	#466
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

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. #23756431	C₂	semidirect
4. #23756419	C₂	semidirect
5. C₂²×C₈×C₃₁₃	C₂	semidirect
26. C₈×C₃₁₃	D₄	semidirect
29. C₂²×C₃₁₃	D₈	semidirect
43. C₃₁₃	#466	semidirect
46. C₂²×C₈	D₃₁₃	semidirect
70. C₈	C₂²⋊D₃₁₃	semidirect
76. C₂²	D₂₅₀₄	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
#23756431	2
#23756419	2
C₂²×C₈×C₃₁₃	2
#466	313