Subgroups — #37527210

Order 22976.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040022	C₂	1	8	yes	maximal, complemented
2	C₂×#19153325	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₃₅₉	C₂	1	1	yes	maximal, characteristic
4	#19153325	C₂²	1	4	yes	—
5	#19153325	C₂²	1	12	yes	complemented
6	C₂×C₃₅₉⋊C₈	C₂²	1	12	yes	complemented
7	#41×C₃₅₉	C₄	1	4	yes	—
8	#41×C₃₅₉	C₂²	1	4	yes	—
9	C₂×Q₈×C₃₅₉	C₄	1	1	yes	characteristic
10	C₂×D₄×C₃₅₉	C₄	1	3	yes	—
11	C₂²×C₄×C₃₅₉	C₂²	1	3	yes	—
12	C₃₅₉⋊C₈	C₂³	1	8	yes	—
13	C₂³×C₃₅₉	C₂×C₄	1	3	yes	—
14	Q₈×C₃₅₉	C₂×C₄	1	4	yes	—
15	D₄×C₃₅₉	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₃₅₉	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₃₅₉	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₃₅₉	C₂³	1	6	yes	—
20	C₂²×C₃₅₉	—	4	3	no	—
21	C₂²×C₃₅₉	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₃₅₉	C₂²×C₄	1	6	yes	—
23	C₄×C₃₅₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₅₉	C₂²×C₄	1	6	yes	—
25	C₄×C₃₅₉	C₂⁴	1	1	yes	characteristic
26	C₂×C₃₅₉	—	2	1	no	—
27	C₂×C₃₅₉	—	2	6	no	—
28	C₂×C₃₅₉	C₂³×C₄	1	1	yes	characteristic
29	C₃₅₉	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	359	1	no	maximal, Sylow
31	C₂×#41	D₃₅₉	1	1	yes	characteristic
32	#131	—	359	8	no	—
33	C₂×M₁₆	—	359	6	no	—
34	#41	Dic₃₅₉	1	4	yes	—
35	#41	C₂×D₃₅₉	1	4	yes	—
36	C₂×Q₈	Dic₃₅₉	1	1	yes	characteristic
37	C₂×D₄	Dic₃₅₉	1	3	yes	—
38	C₂²×C₄	C₂×D₃₅₉	1	3	yes	—
39	M₁₆	—	359	4	no	—
40	M₁₆	—	359	12	no	—
41	C₂×C₈	—	359	12	no	—
42	C₂³	C₂×Dic₃₅₉	1	3	yes	—
43	Q₈	C₂×Dic₃₅₉	1	4	yes	—
44	D₄	C₂×Dic₃₅₉	1	12	yes	—
45	C₂×C₄	C₂×Dic₃₅₉	1	3	yes	—
46	C₂×C₄	C₂×Dic₃₅₉	1	6	yes	—
47	C₂×C₄	C₂²×D₃₅₉	1	6	yes	—
48	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
49	C₈	—	359	8	no	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₃₅₉	1	1	yes	characteristic
52	C₂²	C₂²×Dic₃₅₉	1	6	yes	—
53	C₄	C₂²×Dic₃₅₉	1	1	yes	characteristic
54	C₄	C₂²×Dic₃₅₉	1	6	yes	—
55	C₄	C₂³×D₃₅₉	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₃₅₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₃₅₉
commutator, socle	C₂×C₃₅₉	C₂³×C₄
Fitting, prime radical	C₂×#41×C₃₅₉	C₂
cosocle	C₄×C₃₅₉	C₂⁴
hypercenter	C₂×#41	D₃₅₉
hypocenter	C₃₅₉	#568
LCS	G ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #26040022	C₂	semidirect
2. C₂×#19153325	C₂	semidirect
5. #19153325	C₂²	semidirect
6. C₂×C₃₅₉⋊C₈	C₂²	semidirect
29. C₃₅₉	#568	semidirect

Maximal subgroups

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

subgroup	index
#26040022	2
C₂×#19153325	2
C₂×#41×C₃₅₉	2
#568	359