Subgroups — #36378702

Order 22592.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015065	C₂	1	1	yes	maximal, characteristic, complemented
2	#26015063	C₂	1	1	yes	maximal, characteristic, complemented
3	#26015015	C₂	1	1	yes	maximal, characteristic
4	#19150747	—	2	1	no	—
5	#19150740	—	2	1	no	—
6	#19150738	—	2	1	no	—
7	#19150724	—	2	1	no	—
8	D₂₈₂₄	C₄	1	1	yes	characteristic
9	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₅₃	C₄	1	1	yes	characteristic
11	#15667732	—	4	1	no	—
12	D₁₄₁₂	—	2	1	no	—
13	D₁₄₁₂	—	4	1	no	—
14	C₄×D₃₅₃	—	4	1	no	—
15	C₄×D₃₅₃	D₄	1	1	yes	characteristic
16	#15667725	D₄	1	1	yes	characteristic
17	Q₈×C₃₅₃	—	2	1	no	—
18	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₅₃	—	4	1	no	—
20	C₂×D₃₅₃	SD₁₆	1	1	yes	characteristic
21	C₄×C₃₅₃	—	4	1	no	—
22	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₅₃	D₈	1	1	yes	characteristic
24	C₂×C₃₅₃	#102	1	1	yes	characteristic
25	D₃₅₃	—	2	1	no	—
26	D₃₅₃	—	8	1	no	—
27	C₃₅₃	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	353	1	no	maximal, Sylow
29	#135	—	353	1	no	—
30	C₂×C₁₆	—	353	1	no	—
31	#108	—	353	1	no	—
32	#41	—	706	1	no	—
33	Q₁₆	#13083443	1	1	yes	characteristic
34	SD₁₆	—	706	1	no	—
35	D₈	—	353	1	no	—
36	M₁₆	—	706	1	no	—
37	C₂×C₈	—	353	1	no	—
38	C₁₆	—	706	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	1412	1	no	—
41	D₄	—	706	1	no	—
42	C₂×C₄	—	353	1	no	—
43	C₂×C₄	—	1412	1	no	—
44	C₈	—	353	1	no	—
45	C₈	—	1412	1	no	—
46	C₈	C₂×#13083443	1	1	yes	characteristic
47	C₂²	—	353	1	no	—
48	C₂²	—	1412	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	353	1	no	—
51	C₄	C₂²⋊#13083443	1	1	yes	characteristic
52	C₂	—	2824	1	no	—
53	C₂	—	706	1	no	—
54	C₂	#26015054	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26015054
commutator	C₈×C₃₅₃	C₂×C₄
Frattini	C₈	C₂×#13083443
Fitting	Q₁₆×C₃₅₃	C₄
socle	C₂×C₃₅₃	#102
prime radical	#26015015	C₂
cosocle	C₈×D₃₅₃	C₂²
hypercenter	Q₁₆	#13083443
hypocenter	C₃₅₃	#359
LCS	G ▷ C₈×C₃₅₃ ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #26015065	C₂	semidirect
2. #26015063	C₂	semidirect
27. C₃₅₃	#359	semidirect

Maximal subgroups

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

subgroup	index
#26015065	2
#26015063	2
#26015015	2
#359	353