Subgroups — #36377824

Order 22592.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₅₃	C₂	1	1	yes	maximal, characteristic
2	#26014950	C₂	1	2	yes	maximal
3	#26014926	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014923	C₂	1	2	yes	maximal, complemented
5	#107×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
7	C₂×#15667725	C₂²	1	1	yes	characteristic
8	C₈×D₃₅₃	C₄	1	4	yes	complemented
9	C₄⋊Dic₃₅₃	C₂²	1	2	yes	—
10	#19150714	—	2	2	no	—
11	C₄×Dic₃₅₃	—	2	2	no	—
12	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₅₃	C₂²	1	2	yes	—
14	C₂²×D₃₅₃	D₄	1	1	yes	characteristic
15	C₂×Dic₃₅₃	—	2	2	no	—
16	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
17	C₄×D₃₅₃	C₂×C₄	1	2	yes	—
18	C₄×D₃₅₃	Q₈	1	2	yes	—
19	#15667725	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₃₅₃	—	2	2	no	—
21	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
22	C₈×C₃₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₅₃	—	2	1	no	—
25	C₂×D₃₅₃	—	2	1	no	—
26	C₂×D₃₅₃	C₄⋊C₄	1	2	yes	—
27	C₄×C₃₅₃	—	4	2	no	—
28	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₅₃	C₂×Q₈	1	1	yes	characteristic
30	Dic₃₅₃	—	4	2	no	—
31	Dic₃₅₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₃₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃₅₃	#135	1	1	yes	characteristic
34	C₂×C₃₅₃	#135	1	1	yes	characteristic
35	D₃₅₃	—	2	2	no	—
36	C₃₅₃	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	353	1	no	maximal, Sylow
38	C₂²×C₈	—	353	1	no	—
39	#117	—	353	2	no	—
40	#107	—	353	1	no	—
41	#107	D₃₅₃	1	1	yes	characteristic, complemented
42	#106	—	353	2	no	—
43	C₂²×C₄	—	353	1	no	—
44	C₂×C₈	—	353	1	no	—
45	C₂×C₈	—	353	4	no	—
46	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
47	C₄⋊C₄	—	353	2	no	—
48	C₄⋊C₄	C₂×D₃₅₃	1	2	yes	—
49	C₂²⋊C₄	—	706	2	no	—
50	C₄²	—	706	2	no	—
51	C₂³	—	353	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	353	1	no	—
54	C₂×C₄	—	353	2	no	—
55	C₂×C₄	—	353	2	no	—
56	C₂×C₄	—	706	2	no	—
57	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
58	C₈	—	353	2	no	—
59	C₈	C₄×D₃₅₃	1	2	yes	—
60	C₂²	—	353	2	no	—
61	C₂²	—	706	1	no	—
62	C₂²	—	706	1	no	—
63	C₂²	D₄×D₃₅₃	1	1	yes	characteristic
64	C₄	—	4	2	no	—
65	C₄	—	353	2	no	—
66	C₄	—	1412	2	no	—
67	C₄	C₂×C₄×D₃₅₃	1	1	yes	characteristic
68	C₄	Q₈×D₃₅₃	1	1	yes	characteristic
69	C₂	—	706	2	no	—
70	C₂	C₄⋊C₄×D₃₅₃	1	1	yes	characteristic
71	C₂	#26015011	1	1	yes	characteristic
72	C₂	#26015015	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₃
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#107×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	C₂²×D₃₅₃	D₄
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	#107	D₃₅₃
hypocenter	C₃₅₃	#427
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
3. #26014926	C₂	semidirect
4. #26014923	C₂	semidirect
5. #107×C₃₅₃	C₂	semidirect
8. C₈×D₃₅₃	C₄	semidirect
36. C₃₅₃	#427	semidirect
41. #107	D₃₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₅₃	2
#26014950	2
#26014926	2
#26014923	2
#107×C₃₅₃	2
#427	353