Subgroups — #34095071

Order 20032.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23756572	C₂	1	1	yes	maximal, characteristic
2	#23756567	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₁₃	C₂	1	1	yes	maximal, characteristic
4	#19097571	—	2	1	no	—
5	#19097564	—	2	1	no	—
6	Q₈×D₃₁₃	—	2	1	no	—
7	Dic₁₂₅₂	C₄	1	1	yes	characteristic
8	#19097551	—	2	1	no	—
9	C₈×D₃₁₃	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₁₃	C₄	1	1	yes	characteristic
11	#14578816	—	4	1	no	—
12	Dic₆₂₆	—	2	1	no	—
13	Dic₆₂₆	—	4	1	no	—
14	C₄×D₃₁₃	—	4	1	no	—
15	C₄×D₃₁₃	D₄	1	1	yes	characteristic
16	#14578809	D₄	1	1	yes	characteristic
17	Q₈×C₃₁₃	—	2	1	no	—
18	C₈×C₃₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₁₃	SD₁₆	1	1	yes	characteristic
20	C₄×C₃₁₃	—	4	1	no	—
21	C₄×C₃₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₃₁₃	—	4	1	no	—
23	Dic₃₁₃	D₈	1	1	yes	characteristic
24	C₂×C₃₁₃	#102	1	1	yes	characteristic
25	D₃₁₃	—	2	1	no	—
26	C₃₁₃	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	313	1	no	maximal, Sylow
28	C₂×Q₁₆	—	313	1	no	—
29	M₃₂	—	313	1	no	—
30	#108	—	313	1	no	—
31	C₂×Q₈	—	626	1	no	—
32	Q₁₆	—	313	1	no	—
33	Q₁₆	—	626	1	no	—
34	Q₁₆	#11929411	1	1	yes	characteristic
35	M₁₆	—	626	1	no	—
36	C₂×C₈	—	313	1	no	—
37	C₁₆	—	626	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	626	1	no	—
40	Q₈	—	1252	1	no	—
41	C₂×C₄	—	313	1	no	—
42	C₂×C₄	—	1252	1	no	—
43	C₈	—	313	1	no	—
44	C₈	—	1252	1	no	—
45	C₈	C₂×#11929411	1	1	yes	characteristic
46	C₂²	—	313	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	313	1	no	—
49	C₄	—	1252	1	no	—
50	C₄	C₂²⋊#11929411	1	1	yes	characteristic
51	C₂	—	626	1	no	—
52	C₂	#23756559	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23756559
commutator	C₈×C₃₁₃	C₂×C₄
Frattini	C₈	C₂×#11929411
Fitting	Q₁₆×C₃₁₃	C₄
socle	C₂×C₃₁₃	#102
prime radical	C₂×D₃₁₃	SD₁₆
cosocle	C₈×D₃₁₃	C₂²
hypercenter	Q₁₆	#11929411
hypocenter	C₃₁₃	#362
LCS	G ▷ C₈×C₃₁₃ ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₃₁₃	#362	semidirect

Maximal subgroups

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

subgroup	index
#23756572	2
#23756567	2
Q₁₆×D₃₁₃	2
#362	313