Subgroups — #36378662

Order 22592.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₅₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150745	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₃₅₃	C₂	1	1	yes	maximal, characteristic
4	#19150747	C₄	1	2	yes	—
5	#19150746	—	2	1	no	—
6	#19150745	—	2	1	no	—
7	#19150745	—	2	1	no	—
8	#19150745	C₄	1	2	yes	—
9	C₂×#15667732	—	2	1	no	—
10	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
11	#19150721	C₄	1	1	yes	characteristic
12	#19150720	—	2	1	no	—
13	C₈×D₃₅₃	—	2	1	no	—
14	M₁₆×C₃₅₃	C₄	1	1	yes	characteristic
15	#15667732	—	2	2	no	—
16	#15667732	—	2	2	no	—
17	C₂²×D₃₅₃	Q₈	1	1	yes	characteristic
18	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
19	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×D₃₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×D₃₅₃	D₄	1	1	yes	characteristic
22	C₄×D₃₅₃	D₄	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	C₄⋊C₄	1	1	yes	characteristic
27	C₂×D₃₅₃	—	2	2	no	—
28	C₂×D₃₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₃₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃₅₃	C₄²	1	1	yes	characteristic
31	C₄×C₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₃₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃₅₃	—	2	1	no	—
35	C₂×C₃₅₃	#95	1	1	yes	characteristic
36	D₃₅₃	—	2	1	no	—
37	D₃₅₃	#108	1	2	yes	—
38	C₃₅₃	#341	1	1	yes	characteristic, complemented, Sylow
39	#341	—	353	1	no	maximal, Sylow
40	C₂×M₁₆	—	353	1	no	—
41	C₂×M₁₆	—	353	1	no	—
42	C₂²×C₈	—	353	1	no	—
43	C₂²×C₄	—	353	1	no	—
44	M₁₆	—	353	2	no	—
45	M₁₆	—	353	1	no	—
46	M₁₆	—	706	1	no	—
47	M₁₆	—	706	1	no	—
48	M₁₆	#13083443	1	1	yes	characteristic
49	C₂×C₈	—	353	2	no	—
50	C₂×C₈	—	706	1	no	—
51	C₂×C₈	—	706	1	no	—
52	C₂×C₈	—	706	1	no	—
53	C₂×C₈	—	706	1	no	—
54	C₂³	—	353	1	no	—
55	C₂×C₄	—	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	C₂×#13083443	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	706	2	no	—
63	C₈	—	706	2	no	—
64	C₈	—	706	1	no	—
65	C₂²	—	353	1	no	—
66	C₂²	—	353	1	no	—
67	C₂²	—	706	2	no	—
68	C₂²	C₄⋊#13083443	1	1	yes	characteristic
69	C₄	—	353	1	no	—
70	C₄	—	353	1	no	—
71	C₄	C₄×#13083443	1	1	yes	characteristic
72	C₄	C₂²⋊#13083443	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	353	2	no	—
75	C₂	—	706	1	no	—
76	C₂	(C₂×#13083443)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13083443
commutator	C₄×C₃₅₃	C₄²
Frattini	C₂×C₄	C₂×#13083443
Fitting	M₁₆×C₃₅₃	C₄
socle	C₂×C₃₅₃	#95
prime radical	C₂²×D₃₅₃	Q₈
cosocle	C₂×C₄×D₃₅₃	C₂²
hypercenter	M₁₆	#13083443
hypocenter	C₃₅₃	#341
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
38. C₃₅₃	#341	semidirect

Maximal subgroups

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

subgroup	index
D₃₅₃⋊M₁₆	2
C₂×#19150745	2
M₁₆×D₃₅₃	2
#341	353