Subgroups — #34094443

Order 20032.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₂₅₂	C₂	1	1	yes	maximal, characteristic
3	#23756429	C₂	1	2	yes	maximal, complemented
4	#23756419	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₂₆	—	2	1	no	—
7	C₂×D₁₂₅₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₁₃	—	2	1	no	—
9	C₄⋊Dic₃₁₃	—	2	1	no	—
10	C₄⋊Dic₃₁₃	C₂²	1	2	yes	—
11	C₄⋊Dic₃₁₃	C₂²	1	1	yes	characteristic
12	#19097538	—	2	1	no	—
13	C₂×C₈×C₃₁₃	C₂²	1	2	yes	—
14	C₄²×C₃₁₃	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₃	—	2	1	no	—
16	Dic₆₂₆	—	4	2	no	—
17	D₁₂₅₂	—	2	1	no	—
18	D₁₂₅₂	Q₈	1	2	yes	complemented
19	C₂×Dic₃₁₃	—	2	1	no	—
20	C₂×Dic₃₁₃	—	2	2	no	—
21	C₄×D₃₁₃	—	4	2	no	—
22	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
25	C₈×C₃₁₃	—	2	2	no	—
26	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₃₁₃	—	2	2	no	—
28	C₂×D₃₁₃	—	4	1	no	—
29	C₂×D₃₁₃	—	4	1	no	—
30	C₄×C₃₁₃	—	2	1	no	—
31	C₄×C₃₁₃	SD₁₆	1	2	yes	—
32	C₄×C₃₁₃	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₃₁₃	#41	1	1	yes	characteristic
34	Dic₃₁₃	—	4	1	no	—
35	Dic₃₁₃	—	4	2	no	—
36	C₂×C₃₁₃	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₃₁₃	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₃₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₃₁₃	—	4	2	no	—
40	C₃₁₃	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	313	1	no	maximal, Sylow
42	C₄⋊Q₈	—	313	1	no	—
43	C₄×D₄	—	313	1	no	—
44	#106	—	313	2	no	—
45	C₄⋊C₈	D₃₁₃	1	1	yes	characteristic, complemented
46	#102	—	313	2	no	—
47	C₂×Q₈	—	626	1	no	—
48	C₂×D₄	—	313	1	no	—
49	C₂²×C₄	—	626	1	no	—
50	C₂×C₈	C₂×D₃₁₃	1	2	yes	—
51	C₄⋊C₄	—	313	2	no	—
52	C₄⋊C₄	—	313	1	no	—
53	C₄⋊C₄	—	626	1	no	—
54	C₂²⋊C₄	—	626	1	no	—
55	C₄²	C₂×D₃₁₃	1	1	yes	characteristic
56	C₂³	—	626	1	no	—
57	Q₈	—	1252	2	no	—
58	D₄	—	313	2	no	—
59	D₄	—	626	1	no	—
60	C₂×C₄	—	626	1	no	—
61	C₂×C₄	—	626	2	no	—
62	C₂×C₄	—	1252	2	no	—
63	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
64	C₂×C₄	D₁₂₅₂	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	626	2	no	—
68	C₂²	—	1252	1	no	—
69	C₂²	—	1252	1	no	—
70	C₂²	C₂×D₁₂₅₂	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	1252	1	no	—
73	C₄	—	1252	2	no	—
74	C₄	#19097550	1	2	yes	—
75	C₄	Q₈×D₃₁₃	1	1	yes	characteristic
76	C₄	#19097560	1	1	yes	characteristic
77	C₂	—	1252	2	no	—
78	C₂	#23756477	1	1	yes	characteristic
79	C₂	C₂×#19097550	1	1	yes	characteristic
80	C₂	#23756525	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₅₂
commutator, cosocle	C₂×C₄×C₃₁₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₄⋊C₈×C₃₁₃	C₂
socle	C₂²×C₃₁₃	C₂×D₄
prime radical	C₂×D₁₂₅₂	C₂²
hypercenter	C₄⋊C₈	D₃₁₃
hypocenter	C₃₁₃	#476
LCS	G ▷ C₂×C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #23756497	C₂	semidirect
3. #23756429	C₂	semidirect
5. C₄⋊C₈×C₃₁₃	C₂	semidirect
18. D₁₂₅₂	Q₈	semidirect
40. C₃₁₃	#476	semidirect
45. C₄⋊C₈	D₃₁₃	semidirect

Maximal subgroups

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

subgroup	index
#23756497	2
C₄×D₁₂₅₂	2
#23756429	2
#23756419	2
C₄⋊C₈×C₃₁₃	2
#476	313