Subgroups — #36378834

Order 22592.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26015079	C₂	1	2	yes	maximal, complemented
2	#26015062	C₂	1	1	yes	maximal, characteristic, complemented
3	#26015056	C₂	1	2	yes	maximal, complemented
4	#26015046	C₂	1	1	yes	maximal, characteristic, complemented
5	#26015011	C₂	1	1	yes	maximal, characteristic
6	#19150747	C₂²	1	2	yes	—
7	#19150746	—	2	2	no	—
8	#19150745	C₂²	1	1	yes	characteristic
9	C₂×#15667732	—	2	2	no	—
10	C₄×#13083443	C₂²	1	1	yes	characteristic
11	#19150736	C₂²	1	2	yes	—
12	Dic₁₄₁₂	C₄	1	1	yes	characteristic, complemented
13	#19150725	C₄	1	2	yes	complemented
14	C₈×D₃₅₃	C₂²	1	1	yes	characteristic
15	D₈×C₃₅₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#13083443	—	2	1	no	—
17	#15667732	—	2	2	no	—
18	#15667732	D₄	1	2	yes	—
19	Dic₇₀₆	C₂×C₄	1	2	yes	—
20	C₂²⋊D₃₅₃	—	2	2	no	—
21	C₂×Dic₃₅₃	—	2	2	no	—
22	C₄×D₃₅₃	C₂³	1	1	yes	characteristic
23	#15667725	C₂×C₄	1	1	yes	characteristic
24	D₄×C₃₅₃	C₂×C₄	1	2	yes	—
25	C₈×C₃₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₅₃	—	2	2	no	—
27	C₂×D₃₅₃	#41	1	1	yes	characteristic
28	#13083443	—	4	1	no	—
29	C₄×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
30	Dic₃₅₃	—	2	2	no	—
31	Dic₃₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₅₃	—	4	2	no	—
33	C₂×C₃₅₃	C₄×D₄	1	1	yes	characteristic
34	D₃₅₃	—	2	1	no	—
35	C₃₅₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	353	1	no	maximal, Sylow
37	#135	—	353	1	no	—
38	#131	—	353	2	no	—
39	#108	—	353	1	no	—
40	#104	—	353	2	no	—
41	#97	—	353	1	no	—
42	#41	—	353	2	no	—
43	Q₁₆	—	353	1	no	—
44	SD₁₆	—	353	2	no	—
45	D₈	#13083443	1	1	yes	characteristic, complemented
46	M₁₆	—	353	2	no	—
47	M₁₆	—	706	2	no	—
48	C₂×C₈	—	353	1	no	—
49	C₂×C₈	—	353	1	no	—
50	C₂×C₈	—	706	2	no	—
51	C₄²	—	353	1	no	—
52	Q₈	—	353	2	no	—
53	D₄	—	706	2	no	—
54	D₄	C₂×#13083443	1	2	yes	—
55	C₂×C₄	—	353	1	no	—
56	C₂×C₄	—	706	2	no	—
57	C₂×C₄	—	706	1	no	—
58	C₈	—	353	2	no	—
59	C₈	—	353	1	no	—
60	C₈	—	706	2	no	—
61	C₈	C₂×#13083443	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	353	1	no	—
64	C₄	—	353	1	no	—
65	C₄	—	1412	1	no	—
66	C₄	—	706	2	no	—
67	C₄	C₂²×#13083443	1	1	yes	characteristic
68	C₂	—	4	2	no	—
69	C₂	—	706	1	no	—
70	C₂	D₄×#13083443	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13083443
commutator	C₄×C₃₅₃	C₂²×C₄
Frattini	C₄	C₂²×#13083443
Fitting	D₈×C₃₅₃	C₄
socle	C₂×C₃₅₃	C₄×D₄
prime radical	#26015011	C₂
cosocle	C₄×D₃₅₃	C₂³
hypercenter	D₈	#13083443
hypocenter	C₃₅₃	#444
LCS	G ▷ C₄×C₃₅₃ ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26015079	C₂	semidirect
2. #26015062	C₂	semidirect
3. #26015056	C₂	semidirect
4. #26015046	C₂	semidirect
12. Dic₁₄₁₂	C₄	semidirect
13. #19150725	C₄	semidirect
15. D₈×C₃₅₃	C₄	semidirect
35. C₃₅₃	#444	semidirect
45. D₈	#13083443	semidirect

Maximal subgroups

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

subgroup	index
#26015079	2
#26015062	2
#26015056	2
#26015046	2
#26015011	2
#444	353