Subgroups — #37526742

Order 22976.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040006	C₂	1	1	yes	maximal, characteristic, complemented
2	#26039977	C₂	1	1	yes	maximal, characteristic
3	#26039948	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#26039937	C₂	1	1	yes	maximal, characteristic
6	#26039935	C₂	1	1	yes	maximal, characteristic
7	#106×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₅₉	—	2	1	no	—
10	C₂×C₃₅₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₅₉	—	2	1	no	—
12	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
13	Dic₃₅₉⋊C₄	—	2	1	no	—
14	(C₂×D₃₅₉)⋊C₄	—	2	1	no	—
15	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₃₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₅₉	C₂²	1	1	yes	characteristic
19	C₂²×D₃₅₉	—	2	1	no	—
20	D₁₄₃₆	—	2	1	no	—
21	D₁₄₃₆	Q₈	1	2	yes	—
22	C₂×Dic₃₅₉	—	2	1	no	—
23	C₂×Dic₃₅₉	D₄	1	1	yes	characteristic
24	C₂×Dic₃₅₉	D₄	1	1	yes	characteristic
25	C₄×D₃₅₉	—	4	2	no	—
26	C₃₅₉⋊C₈	—	2	1	no	—
27	C₂×C₄×C₃₅₉	—	2	1	no	—
28	C₂×C₄×C₃₅₉	—	2	1	no	—
29	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
30	C₈×C₃₅₉	—	2	1	no	—
31	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₅₉	—	2	2	no	—
33	C₂×D₃₅₉	—	4	1	no	—
34	C₂×D₃₅₉	—	4	1	no	—
35	C₄×C₃₅₉	—	4	1	no	—
36	C₄×C₃₅₉	—	4	1	no	—
37	C₄×C₃₅₉	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₃₅₉	#41	1	1	yes	characteristic
39	Dic₃₅₉	—	2	1	no	—
40	Dic₃₅₉	—	2	1	no	—
41	Dic₃₅₉	—	4	1	no	—
42	C₂×C₃₅₉	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₃₅₉	#135	1	1	yes	characteristic
44	C₂×C₃₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₃₅₉	—	4	2	no	—
46	C₃₅₉	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	359	1	no	maximal, Sylow
48	#125	—	359	1	no	—
49	C₄×D₄	—	359	1	no	—
50	#107	—	359	1	no	—
51	#106	D₃₅₉	1	1	yes	characteristic, complemented
52	C₄⋊C₈	—	359	1	no	—
53	#102	—	359	1	no	—
54	#102	—	359	1	no	—
55	C₂×D₄	—	359	1	no	—
56	C₂²×C₄	—	718	1	no	—
57	C₂×C₈	—	359	1	no	—
58	C₂×C₈	C₂×D₃₅₉	1	1	yes	characteristic
59	C₄⋊C₄	—	359	1	no	—
60	C₄⋊C₄	—	718	1	no	—
61	C₄⋊C₄	—	718	1	no	—
62	C₄⋊C₄	C₂×D₃₅₉	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₃₅₉	1	1	yes	characteristic
64	C₂²⋊C₄	—	718	1	no	—
65	C₄²	—	359	1	no	—
66	C₂³	—	718	1	no	—
67	D₄	—	359	2	no	—
68	D₄	—	718	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	359	1	no	—
72	C₂×C₄	—	359	1	no	—
73	C₂×C₄	—	1436	2	no	—
74	C₂×C₄	—	718	1	no	—
75	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	718	1	no	—
78	C₂²	—	1436	1	no	—
79	C₂²	—	1436	1	no	—
80	C₂²	—	718	2	no	—
81	C₂²	D₄×D₃₅₉	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	1436	1	no	—
85	C₄	—	718	1	no	—
86	C₄	—	718	1	no	—
87	C₄	Q₈×D₃₅₉	1	1	yes	characteristic
88	C₄	#19153341	1	1	yes	characteristic
89	C₂	—	1436	2	no	—
90	C₂	#26039993	1	1	yes	characteristic
91	C₂	#26040036	1	1	yes	characteristic
92	C₂	#26040042	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₉
commutator, cosocle	C₂×C₄×C₃₅₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	#106×C₃₅₉	C₂
socle	C₂²×C₃₅₉	C₂×D₄
prime radical	C₂×D₁₄₃₆	C₂²
hypercenter	#106	D₃₅₉
hypocenter	C₃₅₉	#478
LCS	G ▷ C₂×C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. #26040006	C₂	semidirect
3. #26039948	C₂	semidirect
4. Dic₃₅₉⋊C₈	C₂	semidirect
7. #106×C₃₅₉	C₂	semidirect
46. C₃₅₉	#478	semidirect
51. #106	D₃₅₉	semidirect

Maximal subgroups

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

subgroup	index
#26040006	2
#26039977	2
#26039948	2
Dic₃₅₉⋊C₈	2
#26039937	2
#26039935	2
#106×C₃₅₉	2
#478	359