Subgroups — C₂×#23756485 #34094669

Order 20032.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₂₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₂₅₂	C₂	1	1	yes	maximal, characteristic
3	#23756485	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19097538	C₂	1	4	yes	maximal, complemented
5	C₂×C₄²×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₁₃	C₂²	1	2	yes	—
7	C₂×Dic₆₂₆	—	2	4	no	—
8	C₂×Dic₆₂₆	C₂²	1	4	yes	complemented
9	C₂×D₁₂₅₂	—	2	4	no	—
10	C₂×D₁₂₅₂	C₂²	1	4	yes	—
11	C₂²×Dic₃₁₃	C₂²	1	2	yes	—
12	#19097538	C₂²	1	16	yes	complemented
13	C₂²×C₄×C₃₁₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₁₃	C₂²	1	1	yes	characteristic
15	C₄²×C₃₁₃	C₂²	1	4	yes	complemented
16	C₂²×D₃₁₃	—	2	4	no	—
17	C₂²×D₃₁₃	—	2	8	no	—
18	C₂²×D₃₁₃	C₂³	1	4	yes	—
19	Dic₆₂₆	—	2	8	no	—
20	D₁₂₅₂	—	2	8	no	—
21	C₂×Dic₃₁₃	—	2	4	no	—
22	C₂×Dic₃₁₃	C₂³	1	4	yes	—
23	C₂³×C₃₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₁₃	—	2	4	no	—
25	C₂×C₄×C₃₁₃	D₄	1	4	yes	—
26	C₂×C₄×C₃₁₃	C₂³	1	4	yes	—
27	C₂×C₄×C₃₁₃	C₂³	1	2	yes	—
28	C₂²×C₃₁₃	C₂×D₄	1	2	yes	—
29	C₂²×C₃₁₃	#41	1	4	yes	—
30	C₂²×C₃₁₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₁₃	—	2	4	no	—
32	C₂×D₃₁₃	—	2	8	no	—
33	C₂×D₃₁₃	—	4	8	no	—
34	C₄×C₃₁₃	—	2	4	no	—
35	C₄×C₃₁₃	C₂×D₄	1	4	yes	—
36	Dic₃₁₃	—	2	4	no	—
37	C₂×C₃₁₃	#124	1	4	yes	complemented
38	C₂×C₃₁₃	C₂²×D₄	1	1	yes	characteristic
39	C₂×C₃₁₃	C₂×#41	1	2	yes	—
40	D₃₁₃	—	4	4	no	—
41	C₃₁₃	C₂×#124	1	1	yes	characteristic, complemented, Sylow
42	C₂×#124	—	313	1	no	maximal, Sylow
43	C₂²×Q₈	—	313	1	no	—
44	C₂²×D₄	—	313	1	no	—
45	#124	—	313	8	no	—
46	C₂×C₂²⋊C₄	—	313	4	no	—
47	C₂×C₄²	D₃₁₃	1	1	yes	characteristic, complemented
48	C₂⁴	—	313	2	no	—
49	C₂×Q₈	—	313	4	no	—
50	C₂×Q₈	—	626	4	no	—
51	C₂×D₄	—	313	4	no	—
52	C₂×D₄	—	626	4	no	—
53	C₂²×C₄	—	313	2	no	—
54	C₂²×C₄	C₂×D₃₁₃	1	2	yes	—
55	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
56	C₂²⋊C₄	—	313	16	no	—
57	C₄²	C₂×D₃₁₃	1	4	yes	complemented
58	C₂³	—	313	4	no	—
59	C₂³	—	626	4	no	—
60	C₂³	—	626	8	no	—
61	C₂³	C₂²×D₃₁₃	1	1	yes	characteristic
62	Q₈	—	626	8	no	—
63	D₄	—	626	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	313	4	no	—
66	C₂×C₄	—	626	4	no	—
67	C₂×C₄	D₁₂₅₂	1	4	yes	—
68	C₂×C₄	C₂²×D₃₁₃	1	4	yes	—
69	C₂×C₄	C₂²×D₃₁₃	1	2	yes	—
70	C₂²	—	626	4	no	—
71	C₂²	—	626	8	no	—
72	C₂²	—	1252	8	no	—
73	C₂²	C₂×D₁₂₅₂	1	2	yes	—
74	C₂²	#19097561	1	4	yes	—
75	C₂²	C₂³×D₃₁₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	626	4	no	—
78	C₄	C₂×D₁₂₅₂	1	4	yes	—
79	C₂	—	1252	4	no	—
80	C₂	#23756485	1	4	yes	direct product factor
81	C₂	C₂²×D₁₂₅₂	1	1	yes	characteristic
82	C₂	C₂×#19097561	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₃
commutator, cosocle	C₂²×C₃₁₃	C₂⁴
Frattini	C₂²	C₂³×D₃₁₃
Fitting	C₂×C₄²×C₃₁₃	C₂
socle	C₂³×C₃₁₃	C₂³
prime radical	C₂²×D₁₂₅₂	C₂
hypercenter	C₂×C₄²	D₃₁₃
hypocenter	C₃₁₃	C₂×#124
LCS	G ▷ C₂²×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₆₂₆	C₂	semidirect
3. #23756485	C₂	direct (N × H)
4. C₂×#19097538	C₂	semidirect
5. C₂×C₄²×C₃₁₃	C₂	semidirect
8. C₂×Dic₆₂₆	C₂²	semidirect
12. #19097538	C₂²	semidirect
15. C₄²×C₃₁₃	C₂²	semidirect
37. C₂×C₃₁₃	#124	semidirect
41. C₃₁₃	C₂×#124	semidirect
47. C₂×C₄²	D₃₁₃	semidirect
57. C₄²	C₂×D₃₁₃	semidirect
80. C₂	#23756485	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₂₆	2
C₂²×D₁₂₅₂	2
#23756485	2
C₂×#19097538	2
C₂×C₄²×C₃₁₃	2
C₂×#124	313

Subgroups — C2×#23756485 #34094669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23756485 #34094669