Subgroups — #36377879

Order 22592.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150727	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
3	#26014957	C₂	1	1	yes	maximal, characteristic
4	#26014925	C₂	1	1	yes	maximal, characteristic
5	#26014917	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
9	#19150727	C₄	1	4	yes	complemented
10	C₂×#15667725	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₅₃	—	2	1	no	—
12	C₄⋊Dic₃₅₃	C₂²	1	1	yes	characteristic
13	#19150716	—	2	1	no	—
14	C₄×Dic₃₅₃	—	2	1	no	—
15	C₄×Dic₃₅₃	—	2	1	no	—
16	C₄×Dic₃₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₅₃	C₂²	1	1	yes	characteristic
20	Dic₇₀₆	—	2	1	no	—
21	Dic₇₀₆	C₂×C₄	1	2	yes	—
22	C₂×Dic₃₅₃	—	2	1	no	—
23	C₂×Dic₃₅₃	—	2	1	no	—
24	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
25	C₂×Dic₃₅₃	D₄	1	1	yes	characteristic
26	#15667725	C₂×C₄	1	2	yes	—
27	Q₈×C₃₅₃	—	2	1	no	—
28	Q₈×C₃₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₅₃	—	2	1	no	—
30	C₂×C₄×C₃₅₃	—	2	1	no	—
31	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
32	C₈×C₃₅₃	—	2	1	no	—
33	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₅₃	—	2	2	no	—
35	C₄×C₃₅₃	—	4	1	no	—
36	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₃₅₃	#41	1	1	yes	characteristic
38	Dic₃₅₃	—	2	2	no	—
39	Dic₃₅₃	—	2	1	no	—
40	Dic₃₅₃	—	4	1	no	—
41	Dic₃₅₃	Q₁₆	1	2	yes	complemented
42	C₂×C₃₅₃	C₄×D₄	1	1	yes	characteristic
43	C₂×C₃₅₃	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₃₅₃	#135	1	1	yes	characteristic
45	C₃₅₃	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	353	1	no	maximal, Sylow
47	C₂×Q₁₆	—	353	1	no	—
48	C₄×Q₈	—	353	1	no	—
49	C₄×Q₈	—	353	1	no	—
50	#107	—	353	1	no	—
51	#103	—	353	1	no	—
52	#103	D₃₅₃	1	1	yes	characteristic
53	C₄×C₈	—	353	1	no	—
54	C₂×Q₈	—	353	1	no	—
55	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
56	Q₁₆	—	353	4	no	—
57	C₂×C₈	—	353	1	no	—
58	C₂×C₈	C₂×D₃₅₃	1	1	yes	characteristic
59	C₄⋊C₄	—	353	1	no	—
60	C₄⋊C₄	—	706	1	no	—
61	C₄⋊C₄	—	706	1	no	—
62	C₄⋊C₄	C₂×D₃₅₃	1	1	yes	characteristic
63	C₄²	—	353	1	no	—
64	C₄²	—	706	1	no	—
65	C₄²	—	706	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	353	2	no	—
68	Q₈	—	706	1	no	—
69	Q₈	C₄×D₃₅₃	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	353	1	no	—
73	C₂×C₄	—	353	1	no	—
74	C₂×C₄	—	706	1	no	—
75	C₂×C₄	—	706	1	no	—
76	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	353	2	no	—
79	C₂²	D₄×D₃₅₃	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	353	2	no	—
83	C₄	—	1412	1	no	—
84	C₄	—	706	1	no	—
85	C₄	—	706	2	no	—
86	C₄	C₂×C₄×D₃₅₃	1	1	yes	characteristic
87	C₄	#19150736	1	1	yes	characteristic
88	C₂	#26014952	1	1	yes	characteristic
89	C₂	Q₁₆×D₃₅₃	1	1	yes	characteristic
90	C₂	#26015013	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₃
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	#103×C₃₅₃	C₂
socle, prime radical	C₂²×C₃₅₃	C₂×D₄
cosocle	C₂×C₄×C₃₅₃	C₂³
hypercenter	#103	D₃₅₃
hypocenter	C₃₅₃	C₄×Q₁₆
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #19150727	C₄	semidirect
41. Dic₃₅₃	Q₁₆	semidirect
45. C₃₅₃	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150727	2
Q₈×Dic₃₅₃	2
#26014957	2
#26014925	2
#26014917	2
C₈×Dic₃₅₃	2
#103×C₃₅₃	2
C₄×Q₁₆	353