Subgroups — #34095153

Order 20032.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097569	C₂	1	2	yes	maximal
2	#23756564	C₂	1	2	yes	maximal, complemented
3	C₄⋊#14578816	C₂	1	2	yes	maximal, complemented
4	C₄²×D₃₁₃	C₂	1	1	yes	maximal, characteristic
5	#19097569	—	2	4	no	—
6	C₂×#14578816	C₂²	1	4	yes	—
7	C₂×C₄×D₃₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₁₃	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₁₃	C₄	1	1	yes	characteristic
11	C₄×Dic₃₁₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₁₃	C₂²	1	1	yes	characteristic
13	C₄²×C₃₁₃	C₄	1	1	yes	characteristic
14	#14578816	—	2	4	no	—
15	C₂²×D₃₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₁₃	C₂³	1	1	yes	characteristic
19	C₄×D₃₁₃	—	2	2	no	—
20	C₄×D₃₁₃	C₈	1	4	yes	complemented
21	C₄×D₃₁₃	D₄	1	2	yes	—
22	C₄×D₃₁₃	Q₈	1	2	yes	—
23	C₂×C₄×C₃₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₁₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₁₃	C₄⋊C₄	1	2	yes	—
28	C₂×D₃₁₃	C₂×C₈	1	2	yes	—
29	C₂×D₃₁₃	M₁₆	1	2	yes	—
30	C₄×C₃₁₃	—	2	1	no	—
31	C₄×C₃₁₃	C₄⋊C₄	1	2	yes	—
32	C₄×C₃₁₃	C₂×C₈	1	2	yes	complemented
33	Dic₃₁₃	—	2	1	no	—
34	Dic₃₁₃	C₂×C₈	1	2	yes	complemented
35	Dic₃₁₃	C₂×D₄	1	1	yes	characteristic
36	Dic₃₁₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₃₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₁₃	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₃₁₃	C₂×M₁₆	1	1	yes	characteristic
40	D₃₁₃	C₄⋊C₈	1	4	yes	complemented
41	C₃₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄⋊C₈	—	313	1	no	maximal, Sylow
43	C₂²×C₈	—	313	2	no	—
44	C₂×C₄²	—	313	1	no	—
45	C₄⋊C₈	—	313	2	no	—
46	C₄⋊C₈	—	313	2	no	—
47	C₂²×C₄	—	313	1	no	—
48	C₂²×C₄	—	313	1	no	—
49	C₂²×C₄	—	313	1	no	—
50	C₂×C₈	—	313	4	no	—
51	C₂×C₈	—	626	4	no	—
52	C₄²	—	313	1	no	—
53	C₄²	—	313	1	no	—
54	C₄²	—	313	1	no	—
55	C₄²	#11929411	1	1	yes	characteristic
56	C₂³	—	313	1	no	—
57	C₂×C₄	—	313	1	no	—
58	C₂×C₄	—	313	1	no	—
59	C₂×C₄	—	313	2	no	—
60	C₂×C₄	—	313	2	no	—
61	C₂×C₄	—	313	4	no	—
62	C₂×C₄	—	313	1	no	—
63	C₂×C₄	—	626	2	no	—
64	C₂×C₄	C₂×#11929411	1	1	yes	characteristic
65	C₂×C₄	C₂×#11929411	1	1	yes	characteristic
66	C₂×C₄	C₂×#11929411	1	1	yes	characteristic
67	C₈	—	626	4	no	—
68	C₂²	—	313	2	no	—
69	C₂²	—	313	2	no	—
70	C₂²	—	313	2	no	—
71	C₂²	C₂²×#11929411	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	313	1	no	—
74	C₄	—	313	1	no	—
75	C₄	—	313	2	no	—
76	C₄	—	626	1	no	—
77	C₄	C₄⋊#11929411	1	2	yes	—
78	C₄	#19097569	1	2	yes	complemented
79	C₂	—	313	4	no	—
80	C₂	C₂×C₄⋊#11929411	1	1	yes	characteristic
81	C₂	C₂×#19097569	1	1	yes	characteristic
82	C₂	D₃₁₃⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929411
commutator	C₂×C₃₁₃	C₂²×C₈
Frattini	C₂²	C₂²×#11929411
Fitting	C₄²×C₃₁₃	C₄
socle	C₂²×C₃₁₃	C₂²×C₄
prime radical	C₂²×D₃₁₃	C₂×C₄
cosocle	C₂×Dic₃₁₃	C₂³
hypercenter	C₄²	#11929411
hypocenter	C₃₁₃	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #23756564	C₂	semidirect
3. C₄⋊#14578816	C₂	semidirect
20. C₄×D₃₁₃	C₈	semidirect
32. C₄×C₃₁₃	C₂×C₈	semidirect
34. Dic₃₁₃	C₂×C₈	semidirect
40. D₃₁₃	C₄⋊C₈	semidirect
41. C₃₁₃	C₂×C₄⋊C₈	semidirect
78. C₄	#19097569	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097569	2
#23756564	2
C₄⋊#14578816	2
C₄²×D₃₁₃	2
C₂×C₄⋊C₈	313