Subgroups — #34094465

Order 20032.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756471	C₂	1	1	yes	maximal, characteristic
2	#23756430	C₂	1	2	yes	maximal, complemented
3	#23756417	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14578809	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₅₂	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₃₁₃	—	2	1	no	—
8	C₂×#14578809	C₂²	1	2	yes	—
9	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
10	#19097538	—	2	1	no	—
11	C₂²×C₄×C₃₁₃	—	2	1	no	—
12	C₂²×C₄×C₃₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₁₃	—	2	1	no	—
14	C₄⋊C₄×C₃₁₃	C₂²	1	2	yes	—
15	C₂²×D₃₁₃	—	2	1	no	—
16	D₁₂₅₂	—	2	2	no	—
17	C₂²⋊D₃₁₃	—	4	2	no	—
18	C₂×Dic₃₁₃	—	2	1	no	—
19	#14578809	—	2	2	no	—
20	C₂³×C₃₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₃	—	2	2	no	—
22	C₂×C₄×C₃₁₃	—	2	2	no	—
23	C₂×C₄×C₃₁₃	—	4	2	no	—
24	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₃₁₃	—	2	1	no	—
27	C₂²×C₃₁₃	—	2	1	no	—
28	C₂²×C₃₁₃	D₈	1	2	yes	—
29	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₁₃	—	4	1	no	—
31	C₂×D₃₁₃	—	4	1	no	—
32	C₂×D₃₁₃	—	4	1	no	—
33	C₄×C₃₁₃	—	2	1	no	—
34	C₄×C₃₁₃	—	4	2	no	—
35	C₄×C₃₁₃	#41	1	2	yes	—
36	Dic₃₁₃	—	4	1	no	—
37	C₂×C₃₁₃	—	2	2	no	—
38	C₂×C₃₁₃	#123	1	1	yes	characteristic
39	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
40	C₂×C₃₁₃	#137	1	1	yes	characteristic
41	D₃₁₃	—	8	1	no	—
42	C₃₁₃	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	313	1	no	maximal, Sylow
44	#121	—	313	1	no	—
45	C₂×C₄⋊C₄	D₃₁₃	1	1	yes	characteristic, complemented
46	#107	—	313	2	no	—
47	#102	—	313	2	no	—
48	C₂²⋊C₈	—	313	1	no	—
49	C₂×D₄	—	313	1	no	—
50	C₂×D₄	—	626	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
53	C₂×C₈	—	313	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	313	1	no	—
56	C₄⋊C₄	C₂×D₃₁₃	1	2	yes	—
57	C₂²⋊C₄	—	626	1	no	—
58	C₂³	—	626	1	no	—
59	C₂³	C₂²⋊D₃₁₃	1	1	yes	characteristic
60	D₄	—	626	2	no	—
61	D₄	—	1252	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	626	1	no	—
66	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
68	C₈	—	626	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1252	1	no	—
72	C₂²	—	1252	1	no	—
73	C₂²	—	1252	1	no	—
74	C₂²	#19097546	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	1252	1	no	—
79	C₄	#19097561	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	2504	1	no	—
82	C₂	#23756480	1	1	yes	characteristic
83	C₂	C₂×#19097546	1	1	yes	characteristic
84	C₂	#23756528	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁₃
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₂×C₄⋊C₄×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
prime radical	#23756471	C₂
hypercenter	C₂×C₄⋊C₄	D₃₁₃
hypocenter	C₃₁₃	#480
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #23756430	C₂	semidirect
3. #23756417	C₂	semidirect
4. C₂²⋊#14578809	C₂	semidirect
5. C₂×C₄⋊C₄×C₃₁₃	C₂	semidirect
42. C₃₁₃	#480	semidirect
45. C₂×C₄⋊C₄	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756471	2
#23756430	2
#23756417	2
C₂²⋊#14578809	2
C₂×C₄⋊C₄×C₃₁₃	2
#480	313