Subgroups — #37526794

Order 22976.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040015	C₂	1	1	yes	maximal, characteristic
2	#26040015	C₂	1	1	yes	maximal, characteristic
3	#26039940	C₂	1	4	yes	maximal
4	C₄×C₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₁₈	—	2	1	no	—
6	C₂×Dic₇₁₈	C₂²	1	2	yes	—
7	C₄⋊Dic₃₅₉	—	2	1	no	—
8	C₄⋊Dic₃₅₉	—	2	1	no	—
9	C₄⋊Dic₃₅₉	—	2	1	no	—
10	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
11	C₂×C₈×C₃₅₉	C₂²	1	2	yes	—
12	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
13	Dic₇₁₈	—	2	4	no	—
14	Dic₇₁₈	—	4	2	no	—
15	C₂×Dic₃₅₉	—	2	2	no	—
16	C₂×Dic₃₅₉	—	2	2	no	—
17	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₅₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
20	C₈×C₃₅₉	—	2	2	no	—
21	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₃₅₉	SD₁₆	1	2	yes	—
23	C₄×C₃₅₉	Q₁₆	1	2	yes	—
24	C₄×C₃₅₉	#41	1	2	yes	—
25	Dic₃₅₉	—	4	2	no	—
26	Dic₃₅₉	—	4	2	no	—
27	C₂×C₃₅₉	#124	1	1	yes	characteristic
28	C₂×C₃₅₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₃₅₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₃₅₉	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	359	1	no	maximal, Sylow
32	C₄⋊Q₈	—	359	1	no	—
33	C₄⋊Q₈	—	359	1	no	—
34	#103	—	359	4	no	—
35	C₄×C₈	D₃₅₉	1	1	yes	characteristic
36	C₂×Q₈	—	359	2	no	—
37	C₂×Q₈	—	718	1	no	—
38	C₂×C₈	C₂×D₃₅₉	1	2	yes	—
39	C₄⋊C₄	—	359	2	no	—
40	C₄⋊C₄	—	718	1	no	—
41	C₄⋊C₄	—	718	1	no	—
42	C₄⋊C₄	—	718	1	no	—
43	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
44	Q₈	—	1436	2	no	—
45	Q₈	—	718	4	no	—
46	C₂×C₄	—	718	2	no	—
47	C₂×C₄	—	718	2	no	—
48	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
49	C₂×C₄	D₁₄₃₆	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₂²	C₂×D₁₄₃₆	1	1	yes	characteristic
53	C₄	—	1436	2	no	—
54	C₄	—	1436	2	no	—
55	C₄	#19153330	1	2	yes	—
56	C₄	Dic₁₄₃₆	1	2	yes	—
57	C₄	#19153341	1	2	yes	—
58	C₂	#26040003	1	1	yes	characteristic
59	C₂	C₂×#19153330	1	1	yes	characteristic
60	C₂	C₂×Dic₁₄₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₃₆
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	C₄×C₈×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₂×D₄
hypercenter	C₄×C₈	D₃₅₉
hypocenter	C₃₅₉	#487
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
30. C₃₅₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#26040015	2
#26039940	2
C₄×C₈×C₃₅₉	2
#487	359