Subgroups — #36377570

Order 22592.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014998	C₂	1	1	yes	maximal, characteristic
2	#26014932	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#19150737	C₄	1	1	yes	characteristic
5	#19150721	C₄	1	1	yes	characteristic
6	#19150720	—	2	1	no	—
7	C₈×D₃₅₃	—	2	1	no	—
8	#19150712	—	2	1	no	—
9	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₅₃	—	2	1	no	—
11	Dic₇₀₆	C₈	1	1	yes	characteristic
12	D₁₄₁₂	C₈	1	1	yes	characteristic
13	C₂²⋊D₃₅₃	—	2	1	no	—
14	C₄×D₃₅₃	—	2	1	no	—
15	#15667725	—	2	1	no	—
16	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₅₃	D₄	1	1	yes	characteristic
18	C₈×C₃₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₃₅₃	M₁₆	1	1	yes	characteristic
20	C₂×D₃₅₃	—	2	1	no	—
21	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₅₃	C₂×C₈	1	1	yes	characteristic
23	Dic₃₅₃	—	2	1	no	—
24	C₂×C₃₅₃	—	2	1	no	—
25	C₂×C₃₅₃	C₂²⋊C₈	1	1	yes	characteristic
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	D₃₅₃	1	1	yes	characteristic, complemented
32	#41	—	353	1	no	—
33	M₁₆	—	353	1	no	—
34	M₁₆	—	706	1	no	—
35	C₂×C₈	—	706	1	no	—
36	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	706	1	no	—
39	Q₈	—	353	1	no	—
40	D₄	—	353	1	no	—
41	D₄	—	706	1	no	—
42	C₂×C₄	—	706	1	no	—
43	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
44	C₈	—	706	1	no	—
45	C₈	C₂²⋊D₃₅₃	1	1	yes	characteristic
46	C₈	D₁₄₁₂	1	1	yes	characteristic
47	C₂²	—	706	1	no	—
48	C₂²	#19150720	1	1	yes	characteristic
49	C₄	—	706	1	no	—
50	C₄	#19150714	1	1	yes	characteristic
51	C₄	C₈×D₃₅₃	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1412	1	no	—
54	C₂	#26014904	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₄₁₂
commutator	C₄×C₃₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₃₅₃
Fitting	C₂×C₁₆×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₂²⋊C₈
prime radical	#19150737	C₄
cosocle	C₂×C₈×C₃₅₃	C₂²
hypercenter	C₂×C₁₆	D₃₅₃
hypocenter	C₃₅₃	#350
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26014932	C₂	semidirect
3. C₂×C₁₆×C₃₅₃	C₂	semidirect
27. C₃₅₃	#350	semidirect
31. C₂×C₁₆	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26014998	2
#26014932	2
C₂×C₁₆×C₃₅₃	2
#350	353