Subgroups — #36377969

Order 22592.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015040	C₂	1	1	yes	maximal, characteristic
2	#26015023	C₂	1	2	yes	maximal, complemented
3	#26014920	C₂	1	2	yes	maximal, complemented
4	#26014911	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150738	—	2	2	no	—
7	#19150737	—	2	2	no	—
8	#19150737	C₂²	1	2	yes	—
9	Q₈×D₃₅₃	—	2	2	no	—
10	#19150727	—	2	2	no	—
11	#19150724	—	2	2	no	—
12	#19150721	C₂²	1	2	yes	—
13	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₃₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₅₃	—	2	2	no	—
16	C₄²×C₃₅₃	C₂²	1	1	yes	characteristic
17	Dic₇₀₆	—	2	2	no	—
18	Dic₇₀₆	D₄	1	2	yes	—
19	D₁₄₁₂	—	2	2	no	—
20	D₁₄₁₂	D₄	1	2	yes	—
21	C₂²⋊D₃₅₃	—	2	2	no	—
22	C₄×D₃₅₃	—	2	4	no	—
23	C₄×D₃₅₃	—	2	2	no	—
24	#15667725	—	2	2	no	—
25	Q₈×C₃₅₃	—	2	2	no	—
26	Q₈×C₃₅₃	—	2	2	no	—
27	C₂×C₄×C₃₅₃	—	2	1	no	—
28	C₂×C₄×C₃₅₃	—	2	1	no	—
29	C₂×C₄×C₃₅₃	D₄	1	2	yes	—
30	C₂×C₄×C₃₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₃₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₅₃	—	2	2	no	—
33	C₄×C₃₅₃	—	2	2	no	—
34	C₄×C₃₅₃	—	2	2	no	—
35	C₄×C₃₅₃	—	4	1	no	—
36	C₄×C₃₅₃	C₂×D₄	1	2	yes	—
37	Dic₃₅₃	—	2	2	no	—
38	C₂×C₃₅₃	—	2	1	no	—
39	C₂×C₃₅₃	#120	1	1	yes	characteristic
40	D₃₅₃	—	4	2	no	—
41	C₃₅₃	#456	1	1	yes	characteristic, complemented, Sylow
42	#456	—	353	1	no	maximal, Sylow
43	2₋¹⁺⁴	—	353	1	no	—
44	#137	—	353	2	no	—
45	C₄⋊Q₈	D₃₅₃	1	1	yes	characteristic, complemented
46	#104	—	353	2	no	—
47	#101	—	353	1	no	—
48	#41	—	353	2	no	—
49	#41	—	706	2	no	—
50	#41	—	706	2	no	—
51	C₂×Q₈	—	706	2	no	—
52	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
53	C₂×Q₈	C₂×D₃₅₃	1	1	yes	characteristic
54	Q₁₆	—	706	2	no	—
55	SD₁₆	—	706	2	no	—
56	M₁₆	—	353	2	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄²	C₂×D₃₅₃	1	1	yes	characteristic
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	353	2	no	—
62	Q₈	—	706	2	no	—
63	D₄	—	353	2	no	—
64	D₄	—	706	2	no	—
65	D₄	—	706	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	706	4	no	—
69	C₂×C₄	—	706	2	no	—
70	C₂×C₄	C₂²⋊D₃₅₃	1	2	yes	—
71	C₂×C₄	C₂²×D₃₅₃	1	1	yes	characteristic
72	C₈	—	706	2	no	—
73	C₂²	—	706	2	no	—
74	C₂²	C₂×C₂²⋊D₃₅₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	706	2	no	—
79	C₄	D₄×D₃₅₃	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	1412	2	no	—
82	C₂	#26014960	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26014960
commutator, cosocle	C₂×C₄×C₃₅₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₃
Fitting	C₄⋊Q₈×C₃₅₃	C₂
socle	C₂×C₃₅₃	#120
prime radical	#26015040	C₂
hypercenter	C₄⋊Q₈	D₃₅₃
hypocenter	C₃₅₃	#456
LCS	G ▷ C₂×C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #26015023	C₂	semidirect
3. #26014920	C₂	semidirect
4. #26014911	C₂	semidirect
5. C₄⋊Q₈×C₃₅₃	C₂	semidirect
41. C₃₅₃	#456	semidirect
45. 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
#26015040	2
#26015023	2
#26014920	2
#26014911	2
C₄⋊Q₈×C₃₅₃	2
#456	353