Subgroups — #36378179

Order 22592.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150727	C₂	1	2	yes	maximal, complemented
2	C₂×#19150724	C₂	1	2	yes	maximal
3	#26014980	C₂	1	1	yes	maximal, characteristic
4	#26014903	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₀₆	C₂²	1	1	yes	characteristic
7	C₂×D₁₄₁₂	C₂²	1	1	yes	characteristic
8	#19150727	—	2	4	no	—
9	#19150724	—	2	4	no	—
10	C₂×#15667725	C₂²	1	2	yes	—
11	#19150714	—	2	2	no	—
12	C₂×Q₈×C₃₅₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₃₅₃	—	2	2	no	—
14	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
15	C₂²×D₃₅₃	—	2	1	no	—
16	Dic₇₀₆	—	2	2	no	—
17	D₁₄₁₂	—	2	2	no	—
18	C₂×Dic₃₅₃	—	2	1	no	—
19	#15667725	D₄	1	4	yes	—
20	Q₈×C₃₅₃	—	2	2	no	—
21	Q₈×C₃₅₃	—	2	2	no	—
22	C₂×C₄×C₃₅₃	—	2	2	no	—
23	C₂×C₄×C₃₅₃	D₄	1	2	yes	—
24	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₃₅₃	—	4	1	no	—
27	C₂×D₃₅₃	—	4	2	no	—
28	C₄×C₃₅₃	—	2	2	no	—
29	C₄×C₃₅₃	—	4	2	no	—
30	C₄×C₃₅₃	C₂×D₄	1	2	yes	—
31	Dic₃₅₃	—	4	1	no	—
32	C₂×C₃₅₃	#127	1	1	yes	characteristic
33	C₂×C₃₅₃	#137	1	2	yes	—
34	D₃₅₃	—	8	1	no	—
35	C₃₅₃	#497	1	1	yes	characteristic, complemented, Sylow
36	#497	—	353	1	no	maximal, Sylow
37	C₂×Q₁₆	—	353	2	no	—
38	C₂×SD₁₆	—	353	2	no	—
39	C₄⋊Q₈	D₃₅₃	1	1	yes	characteristic, complemented
40	#124	—	353	1	no	—
41	#97	—	353	1	no	—
42	C₂×Q₈	—	353	1	no	—
43	C₂×Q₈	C₂×D₃₅₃	1	2	yes	—
44	C₂×D₄	—	353	1	no	—
45	Q₁₆	—	706	4	no	—
46	SD₁₆	—	706	4	no	—
47	C₂×C₈	—	353	2	no	—
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	—	706	2	no	—
50	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
51	C₂³	—	706	1	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	Q₈	—	706	2	no	—
55	D₄	—	706	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	706	1	no	—
58	C₂×C₄	C₂²⋊D₃₅₃	1	2	yes	—
59	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
60	C₈	—	353	4	no	—
61	C₂²	—	1412	1	no	—
62	C₂²	—	1412	2	no	—
63	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	2	no	—
66	C₄	—	1412	1	no	—
67	C₄	D₄×D₃₅₃	1	2	yes	—
68	C₂	—	2824	1	no	—
69	C₂	#26014988	1	1	yes	characteristic
70	C₂	#26015023	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₃
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₄⋊Q₈×C₃₅₃	C₂
socle	C₂²×C₃₅₃	C₂×D₄
prime radical	C₂×D₁₄₁₂	C₂²
hypercenter	C₄⋊Q₈	D₃₅₃
hypocenter	C₃₅₃	#497
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19150727	C₂	semidirect
4. #26014903	C₂	semidirect
5. C₄⋊Q₈×C₃₅₃	C₂	semidirect
35. C₃₅₃	#497	semidirect
39. C₄⋊Q₈	D₃₅₃	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
C₂×#19150724	2
#26014980	2
#26014903	2
C₄⋊Q₈×C₃₅₃	2
#497	353