Subgroups — #37526833

Order 22976.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153328	C₂	1	2	yes	maximal, complemented
2	C₂×#19153326	C₂	1	2	yes	maximal, complemented
3	#26040012	C₂	1	1	yes	maximal, characteristic, complemented
4	#26039926	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₃₆	—	2	2	no	—
7	C₂×D₁₄₃₆	C₂²	1	2	yes	—
8	#19153328	—	2	4	no	—
9	#19153326	—	2	4	no	—
10	C₂×C₃₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₃₅₉	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₃₅₉	—	2	2	no	—
14	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
15	C₂²×D₃₅₉	—	2	2	no	—
16	D₁₄₃₆	—	2	2	no	—
17	D₁₄₃₆	—	2	2	no	—
18	D₁₄₃₆	—	4	4	no	—
19	C₃₅₉⋊C₈	D₄	1	4	yes	complemented
20	C₂³×C₃₅₉	—	2	1	no	—
21	Q₈×C₃₅₉	—	2	2	no	—
22	D₄×C₃₅₉	—	2	2	no	—
23	C₂×C₄×C₃₅₉	—	2	1	no	—
24	C₂×C₄×C₃₅₉	D₄	1	2	yes	—
25	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
26	C₂²×C₃₅₉	—	4	1	no	—
27	C₂²×C₃₅₉	—	4	2	no	—
28	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
29	C₂×D₃₅₉	—	4	2	no	—
30	C₂×D₃₅₉	—	4	4	no	—
31	C₄×C₃₅₉	—	2	2	no	—
32	C₄×C₃₅₉	—	4	1	no	—
33	C₄×C₃₅₉	C₂×D₄	1	2	yes	—
34	C₂×C₃₅₉	—	8	1	no	—
35	C₂×C₃₅₉	#127	1	1	yes	characteristic
36	C₂×C₃₅₉	AGL(1,ℤ/8)	1	2	yes	—
37	D₃₅₉	—	8	2	no	—
38	C₃₅₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	359	1	no	maximal, Sylow
40	C₂×SD₁₆	—	359	2	no	—
41	C₂×D₈	—	359	2	no	—
42	#127	—	359	1	no	—
43	#124	D₃₅₉	1	1	yes	characteristic, complemented
44	#97	—	359	1	no	—
45	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
46	C₂×D₄	—	359	2	no	—
47	C₂×D₄	—	718	2	no	—
48	C₂×D₄	C₂×D₃₅₉	1	1	yes	characteristic
49	SD₁₆	—	718	4	no	—
50	D₈	—	718	4	no	—
51	C₂×C₈	—	359	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	718	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	1436	4	no	—
59	D₄	—	718	2	no	—
60	D₄	—	718	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₃₅₉	1	2	yes	—
63	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
64	C₈	—	359	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	1436	2	no	—
68	C₂²	—	1436	4	no	—
69	C₂²	C₂×C₂²⋊D₃₅₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₃₅₉	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	2872	2	no	—
75	C₂	#26040011	1	1	yes	characteristic
76	C₂	#26040040	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₅₉
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	#124×C₃₅₉	C₂
socle	C₂²×C₃₅₉	C₂×D₄
hypercenter	#124	D₃₅₉
hypocenter	C₃₅₉	#496
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#19153328	C₂	semidirect
2. C₂×#19153326	C₂	semidirect
3. #26040012	C₂	semidirect
4. #26039926	C₂	semidirect
5. #124×C₃₅₉	C₂	semidirect
19. C₃₅₉⋊C₈	D₄	semidirect
38. C₃₅₉	#496	semidirect
43. #124	D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153328	2
C₂×#19153326	2
#26040012	2
#26039926	2
#124×C₃₅₉	2
#496	359