Subgroups — #36378888

Order 22592.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015089	C₂	1	1	yes	maximal, characteristic, complemented
2	#26015087	C₂	1	1	yes	maximal, characteristic, complemented
3	#26015080	C₂	1	1	yes	maximal, characteristic
4	#19150749	—	2	1	no	—
5	#19150749	—	2	1	no	—
6	#19150747	—	2	1	no	—
7	#19150747	C₄	1	1	yes	characteristic
8	#19150745	—	2	1	no	—
9	#19150745	C₂²	1	1	yes	characteristic
10	#19150738	C₄	1	1	yes	characteristic
11	#15667732	—	2	1	no	—
12	#15667732	D₄	1	1	yes	characteristic
13	#15667732	D₄	1	1	yes	characteristic
14	D₁₄₁₂	—	2	1	no	—
15	D₁₄₁₂	C₈	1	1	yes	characteristic
16	C₄×D₃₅₃	—	2	1	no	—
17	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
18	Q₈×C₃₅₃	C₈	1	1	yes	characteristic
19	C₂×D₃₅₃	—	2	1	no	—
20	C₂×D₃₅₃	M₁₆	1	1	yes	characteristic
21	C₄×C₃₅₃	—	2	1	no	—
22	C₄×C₃₅₃	C₂×C₈	1	1	yes	characteristic
23	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₃₅₃	C₂²⋊C₈	1	1	yes	characteristic
25	D₃₅₃	—	2	1	no	—
26	D₃₅₃	—	4	1	no	—
27	C₃₅₃	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	353	1	no	maximal, Sylow
29	#131	—	353	1	no	—
30	M₃₂	—	353	1	no	—
31	C₂×C₁₆	—	353	1	no	—
32	#41	—	353	1	no	—
33	M₁₆	—	353	1	no	—
34	M₁₆	—	706	1	no	—
35	C₂×C₈	—	353	1	no	—
36	C₂×C₈	—	706	1	no	—
37	C₁₆	—	706	1	no	—
38	C₁₆	—	706	1	no	—
39	Q₈	#15667734	1	1	yes	characteristic
40	D₄	—	353	1	no	—
41	D₄	—	706	1	no	—
42	C₂×C₄	—	353	1	no	—
43	C₂×C₄	—	706	1	no	—
44	C₈	—	353	1	no	—
45	C₈	—	353	1	no	—
46	C₈	—	706	1	no	—
47	C₂²	—	353	1	no	—
48	C₂²	—	706	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	353	1	no	—
51	C₄	C₂×#15667734	1	1	yes	characteristic
52	C₂	—	1412	1	no	—
53	C₂	—	706	1	no	—
54	C₂	C₂²⋊#15667734	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#15667734
commutator	C₄×C₃₅₃	C₂×C₈
Frattini	C₄	C₂×#15667734
Fitting	Q₈×C₃₅₃	C₈
socle	C₂×C₃₅₃	C₂²⋊C₈
prime radical	#19150738	C₄
cosocle	#19150745	C₂²
hypercenter	Q₈	#15667734
hypocenter	C₃₅₃	#350
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #26015089	C₂	semidirect
2. #26015087	C₂	semidirect
27. C₃₅₃	#350	semidirect

Maximal subgroups

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

subgroup	index
#26015089	2
#26015087	2
#26015080	2
#350	353