Subgroups — #36377910

Order 22592.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015014	C₂	1	1	yes	maximal, characteristic
2	#26014998	C₂	1	2	yes	maximal, complemented
3	#26014928	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014920	C₂	1	2	yes	maximal, complemented
5	#97×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150737	C₂²	1	2	yes	—
7	Dic₁₄₁₂	C₄	1	1	yes	characteristic, complemented
8	#19150726	C₄	1	2	yes	complemented
9	D₂₈₂₄	C₄	1	1	yes	characteristic, complemented
10	#19150721	C₂²	1	2	yes	—
11	#19150720	—	2	2	no	—
12	C₈×D₃₅₃	—	2	2	no	—
13	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
15	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
16	Dic₇₀₆	C₂×C₄	1	2	yes	—
17	D₁₄₁₂	C₂×C₄	1	2	yes	—
18	C₂²⋊D₃₅₃	—	2	2	no	—
19	C₄×D₃₅₃	—	2	2	no	—
20	#15667725	—	2	2	no	—
21	C₂×C₄×C₃₅₃	—	2	1	no	—
22	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
23	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₅₃	D₄	1	2	yes	—
26	C₂²×C₃₅₃	#41	1	1	yes	characteristic
27	C₂×D₃₅₃	—	2	2	no	—
28	C₄×C₃₅₃	—	4	1	no	—
29	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃₅₃	C₂×D₄	1	1	yes	characteristic
31	Dic₃₅₃	—	2	2	no	—
32	C₂×C₃₅₃	—	2	1	no	—
33	C₂×C₃₅₃	C₄×D₄	1	1	yes	characteristic
34	D₃₅₃	—	4	2	no	—
35	C₃₅₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	353	1	no	maximal, Sylow
37	#135	—	353	1	no	—
38	#131	—	353	2	no	—
39	#108	—	353	1	no	—
40	#104	—	353	2	no	—
41	#97	D₃₅₃	1	1	yes	characteristic, complemented
42	#41	—	353	2	no	—
43	Q₁₆	—	353	1	no	—
44	SD₁₆	—	353	2	no	—
45	D₈	—	353	1	no	—
46	M₁₆	—	353	2	no	—
47	M₁₆	—	706	2	no	—
48	C₂×C₈	—	706	2	no	—
49	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
50	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
51	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
52	Q₈	—	353	2	no	—
53	D₄	—	353	2	no	—
54	D₄	—	706	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	706	2	no	—
57	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
58	C₈	—	706	2	no	—
59	C₈	C₄×D₃₅₃	1	1	yes	characteristic
60	C₈	C₄×D₃₅₃	1	1	yes	characteristic
61	C₈	D₁₄₁₂	1	2	yes	—
62	C₂²	—	706	2	no	—
63	C₂²	#19150737	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	706	2	no	—
66	C₄	C₂×C₄×D₃₅₃	1	1	yes	characteristic
67	C₄	C₂×D₁₄₁₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1412	2	no	—
70	C₂	C₄×D₁₄₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₄₁₂
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#97×C₃₅₃	C₂
socle	C₂×C₃₅₃	C₄×D₄
prime radical	#26015014	C₂
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	#97	D₃₅₃
hypocenter	C₃₅₃	#444
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #26014998	C₂	semidirect
3. #26014928	C₂	semidirect
4. #26014920	C₂	semidirect
5. #97×C₃₅₃	C₂	semidirect
7. Dic₁₄₁₂	C₄	semidirect
8. #19150726	C₄	semidirect
9. D₂₈₂₄	C₄	semidirect
35. C₃₅₃	#444	semidirect
41. #97	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
#26015014	2
#26014998	2
#26014928	2
#26014920	2
#97×C₃₅₃	2
#444	353