Subgroups — #36378670

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150747	C₂	1	1	yes	maximal, characteristic
2	#26015071	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	#19150747	—	2	1	no	—
5	#19150747	—	2	1	no	—
6	C₂×#15667732	C₄	1	2	yes	complemented
7	C₄⋊#13083443	—	2	1	no	—
8	C₂²⋊#13083443	C₄	1	2	yes	—
9	C₄×#13083443	—	2	1	no	—
10	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
11	#19150721	C₄	1	1	yes	characteristic, complemented
12	#19150720	—	2	1	no	—
13	C₈×D₃₅₃	—	2	1	no	—
14	M₁₆×C₃₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#13083443	—	2	2	no	—
16	#15667732	—	2	2	no	—
17	C₂²×D₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₅₃	D₄	1	1	yes	characteristic
20	C₄×D₃₅₃	D₄	1	1	yes	characteristic
21	C₄×D₃₅₃	D₄	1	1	yes	characteristic
22	C₄×D₃₅₃	Q₈	1	1	yes	characteristic
23	#15667725	—	2	1	no	—
24	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₅₃	—	2	1	no	—
26	C₂²×C₃₅₃	C₄²	1	1	yes	characteristic
27	C₂×D₃₅₃	—	4	1	no	—
28	C₂×D₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₃₅₃	C₄⋊C₄	1	1	yes	characteristic
30	#13083443	—	4	2	no	—
31	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₃₅₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₃₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₃₅₃	—	2	1	no	—
36	C₂×C₃₅₃	#95	1	1	yes	characteristic
37	D₃₅₃	—	2	1	no	—
38	D₃₅₃	—	2	1	no	—
39	C₃₅₃	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	353	1	no	maximal, Sylow
41	C₂×M₁₆	—	353	1	no	—
42	C₂×M₁₆	—	353	1	no	—
43	#117	—	353	1	no	—
44	C₂²×C₄	—	353	1	no	—
45	M₁₆	—	353	1	no	—
46	M₁₆	—	706	1	no	—
47	M₁₆	—	706	1	no	—
48	M₁₆	—	706	1	no	—
49	M₁₆	#13083443	1	1	yes	characteristic, complemented
50	C₂×C₈	—	353	2	no	—
51	C₂×C₈	—	706	1	no	—
52	C₄⋊C₄	—	706	1	no	—
53	C₂²⋊C₄	—	353	2	no	—
54	C₄²	—	706	1	no	—
55	C₂³	—	353	1	no	—
56	C₂×C₄	—	353	1	no	—
57	C₂×C₄	—	353	1	no	—
58	C₂×C₄	—	353	1	no	—
59	C₂×C₄	—	353	1	no	—
60	C₂×C₄	—	353	1	no	—
61	C₂×C₄	—	706	2	no	—
62	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	706	2	no	—
65	C₈	—	706	1	no	—
66	C₂²	—	353	1	no	—
67	C₂²	—	353	1	no	—
68	C₂²	—	1412	1	no	—
69	C₂²	C₄×#13083443	1	1	yes	characteristic
70	C₄	—	353	1	no	—
71	C₄	—	353	1	no	—
72	C₄	—	1412	2	no	—
73	C₄	C₂²⋊#13083443	1	1	yes	characteristic
74	C₄	C₄⋊#13083443	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	706	1	no	—
77	C₂	—	706	1	no	—
78	C₂	(C₂×#13083443)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#13083443)⋊C₄
commutator	C₂²×C₃₅₃	C₄²
Frattini	C₂×C₄	C₂×#13083443
Fitting	M₁₆×C₃₅₃	C₄
socle	C₂×C₃₅₃	#95
prime radical	C₂²×D₃₅₃	C₂×C₄
cosocle	C₂×C₄×D₃₅₃	C₂²
hypercenter	M₁₆	#13083443
hypocenter	C₃₅₃	#344
LCS	G ▷ C₂²×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. C₂×#15667732	C₄	semidirect
11. #19150721	C₄	semidirect
14. M₁₆×C₃₅₃	C₄	semidirect
39. C₃₅₃	#344	semidirect
49. M₁₆	#13083443	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150747	2
#26015071	2
M₁₆×D₃₅₃	2
#344	353