Subgroups — #36378205

Order 22592.

84 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	#26014931	C₂	1	2	yes	maximal, complemented
3	C₂×#19150712	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×D₃₅₃	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150737	C₄	1	1	yes	characteristic
7	#19150721	C₄	1	1	yes	characteristic
8	#19150720	C₄	1	2	yes	—
9	C₈×D₃₅₃	C₂²	1	2	yes	—
10	#19150712	C₂²	1	1	yes	characteristic
11	#19150712	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₁₆×C₃₅₃	C₂²	1	2	yes	—
14	Dic₇₀₆	C₈	1	1	yes	characteristic
15	D₁₄₁₂	C₈	1	1	yes	characteristic
16	C₂²⋊D₃₅₃	C₈	1	2	yes	—
17	C₄×D₃₅₃	C₂×C₄	1	2	yes	—
18	#15667725	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
21	C₈×C₃₅₃	C₂³	1	1	yes	characteristic
22	C₂²×C₃₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×D₃₅₃	C₂×C₈	1	2	yes	—
24	C₄×C₃₅₃	C₂×C₈	1	1	yes	characteristic
25	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
26	Dic₃₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₃₅₃	—	2	1	no	—
28	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
29	D₃₅₃	—	2	2	no	—
30	C₃₅₃	#504	1	1	yes	characteristic, complemented, Sylow
31	#504	—	353	1	no	maximal, Sylow
32	#131	—	353	1	no	—
33	M₃₂	—	353	2	no	—
34	M₃₂	D₃₅₃	1	1	yes	characteristic, complemented
35	C₂×C₁₆	—	353	1	no	—
36	C₂×C₁₆	—	353	2	no	—
37	#41	—	353	1	no	—
38	M₁₆	—	353	1	no	—
39	M₁₆	—	353	2	no	—
40	C₂×C₈	—	353	2	no	—
41	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
42	C₁₆	—	353	1	no	—
43	C₁₆	—	353	1	no	—
44	C₁₆	C₂×D₃₅₃	1	2	yes	—
45	Q₈	—	353	1	no	—
46	D₄	—	353	2	no	—
47	D₄	—	353	1	no	—
48	C₂×C₄	—	353	2	no	—
49	C₂×C₄	C₄×D₃₅₃	1	1	yes	characteristic
50	C₈	—	353	2	no	—
51	C₈	C₄×D₃₅₃	1	1	yes	characteristic
52	C₈	C₂²×D₃₅₃	1	1	yes	characteristic
53	C₂²	—	353	2	no	—
54	C₂²	C₈×D₃₅₃	1	1	yes	characteristic
55	C₄	—	353	2	no	—
56	C₄	C₈×D₃₅₃	1	1	yes	characteristic
57	C₄	C₂×C₄×D₃₅₃	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	706	2	no	—
60	C₂	C₂×C₈×D₃₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×D₃₅₃
commutator, socle	C₂×C₃₅₃	C₂²×C₈
Fitting	M₃₂×C₃₅₃	C₂
prime radical	#19150737	C₄
cosocle	C₈×C₃₅₃	C₂³
hypercenter	M₃₂	D₃₅₃
hypocenter	C₃₅₃	#504
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #26014931	C₂	semidirect
3. C₂×#19150712	C₂	semidirect
4. C₁₆×D₃₅₃	C₂	semidirect
5. M₃₂×C₃₅₃	C₂	semidirect
30. C₃₅₃	#504	semidirect
34. M₃₂	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
#26014931	2
C₂×#19150712	2
C₁₆×D₃₅₃	2
M₃₂×C₃₅₃	2
#504	353