Subgroups — #36378125

Order 22592.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014951	C₂	1	1	yes	maximal, characteristic
2	#26014925	C₂	1	1	yes	maximal, characteristic, complemented
3	#26014923	C₂	1	1	yes	maximal, characteristic, complemented
4	#26014916	C₂	1	1	yes	maximal, characteristic, complemented
5	#26014916	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊#15667725	C₂	1	1	yes	maximal, characteristic
7	C₂²⋊Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667725	C₂²	1	1	yes	characteristic
9	C₂×#15667725	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₅₃	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₃₅₃	—	2	1	no	—
13	C₄×Dic₃₅₃	—	2	1	no	—
14	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₃₅₃	—	2	1	no	—
17	C₄⋊C₄×C₃₅₃	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₃₅₃	—	2	1	no	—
19	C₂×Dic₃₅₃	—	2	1	no	—
20	C₂×Dic₃₅₃	—	2	1	no	—
21	#15667725	—	2	1	no	—
22	#15667725	—	2	1	no	—
23	C₂³×C₃₅₃	D₄	1	1	yes	characteristic
24	Q₈×C₃₅₃	—	2	1	no	—
25	Q₈×C₃₅₃	—	2	1	no	—
26	C₂×C₄×C₃₅₃	—	2	1	no	—
27	C₂×C₄×C₃₅₃	—	2	1	no	—
28	C₂×C₄×C₃₅₃	—	2	1	no	—
29	C₂×C₄×C₃₅₃	—	2	1	no	—
30	C₂×C₄×C₃₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₃₅₃	—	2	1	no	—
33	C₂²×C₃₅₃	—	2	1	no	—
34	C₂²×C₃₅₃	—	2	1	no	—
35	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₅₃	—	2	1	no	—
37	C₄×C₃₅₃	—	4	1	no	—
38	C₄×C₃₅₃	—	4	1	no	—
39	C₄×C₃₅₃	#41	1	1	yes	characteristic
40	C₄×C₃₅₃	#41	1	1	yes	characteristic
41	Dic₃₅₃	—	4	1	no	—
42	Dic₃₅₃	—	4	1	no	—
43	C₂×C₃₅₃	—	4	1	no	—
44	C₂×C₃₅₃	#123	1	1	yes	characteristic
45	C₂×C₃₅₃	#135	1	1	yes	characteristic
46	C₂×C₃₅₃	#137	1	1	yes	characteristic
47	C₃₅₃	#485	1	1	yes	characteristic, complemented, Sylow
48	#485	—	353	1	no	maximal, Sylow
49	C₂²⋊Q₈	D₃₅₃	1	1	yes	characteristic
50	#117	—	353	1	no	—
51	#107	—	353	1	no	—
52	#106	—	353	1	no	—
53	#103	—	353	1	no	—
54	#103	—	353	1	no	—
55	C₂²⋊C₈	—	353	1	no	—
56	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
58	C₂×C₈	—	353	1	no	—
59	C₂×C₈	—	353	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	353	1	no	—
62	C₄⋊C₄	—	353	1	no	—
63	C₄⋊C₄	C₂×D₃₅₃	1	1	yes	characteristic
64	C₂²⋊C₄	—	2	1	no	—
65	C₂²⋊C₄	—	706	1	no	—
66	C₄²	—	706	1	no	—
67	C₂³	C₂²⋊D₃₅₃	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	706	1	no	—
75	C₂×C₄	—	706	1	no	—
76	C₂×C₄	C₂²⋊D₃₅₃	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
78	C₈	—	706	1	no	—
79	C₈	—	706	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	1412	1	no	—
88	C₄	—	1412	1	no	—
89	C₄	#19150736	1	1	yes	characteristic
90	C₄	#19150736	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#26014973	1	1	yes	characteristic
93	C₂	#26015012	1	1	yes	characteristic
94	C₂	#26015023	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₃
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₂²⋊Q₈×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	C₂³×C₃₅₃	D₄
hypercenter	C₂²⋊Q₈	D₃₅₃
hypocenter	C₃₅₃	#485
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #26014925	C₂	semidirect
3. #26014923	C₂	semidirect
4. #26014916	C₂	semidirect
5. #26014916	C₂	semidirect
47. C₃₅₃	#485	semidirect

Maximal subgroups

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

subgroup	index
#26014951	2
#26014925	2
#26014923	2
#26014916	2
C₂²⋊#15667725	2
C₂²⋊Q₈×C₃₅₃	2
#485	353