Subgroups — #36378759

Order 22592.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150746	C₂	1	2	yes	maximal, complemented
2	#26015049	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	#19150746	—	2	4	no	—
5	C₂×#15667732	C₂²	1	4	yes	complemented
6	C₂³×D₃₅₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₃₅₃	—	2	4	no	—
9	C₂×C₄×D₃₅₃	C₄	1	2	yes	—
10	C₂×C₄×D₃₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₅₃	C₄	1	1	yes	characteristic
12	#15667732	—	2	4	no	—
13	C₂²×D₃₅₃	—	2	2	no	—
14	C₂²×D₃₅₃	—	2	4	no	—
15	C₂²×D₃₅₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₅₃	D₄	1	4	yes	—
18	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
19	C₄×D₃₅₃	—	2	4	no	—
20	C₄×D₃₅₃	—	2	4	no	—
21	C₂³×C₃₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₅₃	—	2	2	no	—
23	C₂×C₄×C₃₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₃₅₃	—	2	2	no	—
25	C₂²×C₃₅₃	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₅₃	—	2	8	no	—
28	C₂×D₃₅₃	—	2	2	no	—
29	C₂×D₃₅₃	—	2	2	no	—
30	C₂×D₃₅₃	M₁₆	1	4	yes	complemented
31	C₄×C₃₅₃	—	2	2	no	—
32	Dic₃₅₃	C₂²⋊C₄	1	2	yes	—
33	Dic₃₅₃	C₂×D₄	1	2	yes	—
34	C₂×C₃₅₃	—	2	2	no	—
35	C₂×C₃₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₃₅₃	C₂×M₁₆	1	2	yes	—
37	D₃₅₃	—	2	4	no	—
38	C₃₅₃	#407	1	1	yes	characteristic, complemented, Sylow
39	#407	—	353	1	no	maximal, Sylow
40	C₂³×C₄	—	353	1	no	—
41	C₂×M₁₆	—	353	2	no	—
42	C₂²⋊C₈	—	353	4	no	—
43	C₂⁴	—	353	1	no	—
44	C₂²×C₄	—	353	1	no	—
45	C₂²×C₄	—	353	2	no	—
46	C₂²×C₄	—	353	2	no	—
47	C₂²×C₄	—	706	4	no	—
48	C₂²×C₄	#13083443	1	1	yes	characteristic
49	M₁₆	—	706	4	no	—
50	C₂×C₈	—	353	4	no	—
51	C₂³	—	353	2	no	—
52	C₂³	—	706	4	no	—
53	C₂³	—	706	2	no	—
54	C₂³	C₂×#13083443	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	353	4	no	—
57	C₂×C₄	—	353	1	no	—
58	C₂×C₄	—	353	1	no	—
59	C₂×C₄	—	706	4	no	—
60	C₂×C₄	—	706	4	no	—
61	C₂×C₄	C₂×#13083443	1	2	yes	—
62	C₈	—	706	4	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	353	4	no	—
65	C₂²	—	706	8	no	—
66	C₂²	—	706	2	no	—
67	C₂²	—	706	2	no	—
68	C₂²	C₂²⋊#13083443	1	2	yes	—
69	C₂²	C₂²×#13083443	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	353	2	no	—
72	C₄	—	353	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	706	4	no	—
75	C₂	C₂×C₂²⋊#13083443	1	1	yes	characteristic
76	C₂	D₃₅₃⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13083443
commutator, socle	C₂²×C₃₅₃	C₂²×C₄
Fitting	C₂²×C₄×C₃₅₃	C₄
prime radical	C₂³×D₃₅₃	C₄
cosocle	C₂×Dic₃₅₃	C₂³
hypercenter	C₂²×C₄	#13083443
hypocenter	C₃₅₃	#407
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19150746	C₂	semidirect
2. #26015049	C₂	semidirect
5. C₂×#15667732	C₂²	semidirect
30. C₂×D₃₅₃	M₁₆	semidirect
38. C₃₅₃	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150746	2
#26015049	2
C₂²×C₄×D₃₅₃	2
#407	353