Subgroups — #37526535

Order 22976.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153331	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic
3	#26039980	C₂	1	1	yes	maximal, characteristic
4	#26039948	C₂	1	1	yes	maximal, characteristic
5	#26039940	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₈	C₂²	1	1	yes	characteristic
9	#19153331	C₄	1	4	yes	complemented
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	C₄×Dic₃₅₉	—	2	1	no	—
15	C₄×Dic₃₅₉	—	2	1	no	—
16	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₅₉	C₂²	1	1	yes	characteristic
20	Dic₇₁₈	—	2	1	no	—
21	Dic₇₁₈	C₂×C₄	1	2	yes	—
22	C₂×Dic₃₅₉	—	2	1	no	—
23	C₂×Dic₃₅₉	—	2	1	no	—
24	C₂×Dic₃₅₉	D₄	1	1	yes	characteristic
25	C₂×Dic₃₅₉	D₄	1	1	yes	characteristic
26	C₃₅₉⋊C₈	C₂×C₄	1	2	yes	—
27	Q₈×C₃₅₉	—	2	1	no	—
28	Q₈×C₃₅₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₅₉	—	2	1	no	—
30	C₂×C₄×C₃₅₉	—	2	1	no	—
31	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
32	C₈×C₃₅₉	—	2	1	no	—
33	C₂²×C₃₅₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₅₉	—	2	2	no	—
35	C₄×C₃₅₉	—	4	1	no	—
36	C₄×C₃₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₃₅₉	#41	1	1	yes	characteristic
38	Dic₃₅₉	—	2	2	no	—
39	Dic₃₅₉	—	2	1	no	—
40	Dic₃₅₉	—	4	1	no	—
41	Dic₃₅₉	Q₁₆	1	2	yes	complemented
42	C₂×C₃₅₉	C₄×D₄	1	1	yes	characteristic
43	C₂×C₃₅₉	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₃₅₉	#135	1	1	yes	characteristic
45	C₃₅₉	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	359	1	no	maximal, Sylow
47	C₂×Q₁₆	—	359	1	no	—
48	C₄×Q₈	—	359	1	no	—
49	C₄×Q₈	—	359	1	no	—
50	#107	—	359	1	no	—
51	#103	—	359	1	no	—
52	#103	D₃₅₉	1	1	yes	characteristic
53	C₄×C₈	—	359	1	no	—
54	C₂×Q₈	—	359	1	no	—
55	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
56	Q₁₆	—	359	4	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	C₄²	—	359	1	no	—
64	C₄²	—	718	1	no	—
65	C₄²	—	718	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	359	2	no	—
68	Q₈	—	718	1	no	—
69	Q₈	C₄×D₃₅₉	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	359	1	no	—
73	C₂×C₄	—	359	1	no	—
74	C₂×C₄	—	718	1	no	—
75	C₂×C₄	—	718	1	no	—
76	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	359	2	no	—
79	C₂²	D₄×D₃₅₉	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	359	2	no	—
83	C₄	—	1436	1	no	—
84	C₄	—	718	2	no	—
85	C₄	—	718	1	no	—
86	C₄	C₂×C₄×D₃₅₉	1	1	yes	characteristic
87	C₄	#19153340	1	1	yes	characteristic
88	C₂	(C₂²⋊D₃₅₉)⋊C₄	1	1	yes	characteristic
89	C₂	Q₁₆×D₃₅₉	1	1	yes	characteristic
90	C₂	#26040036	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₅₉
commutator	C₄×C₃₅₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₅₉
Fitting	#103×C₃₅₉	C₂
socle, prime radical	C₂²×C₃₅₉	C₂×D₄
cosocle	C₂×C₄×C₃₅₉	C₂³
hypercenter	#103	D₃₅₉
hypocenter	C₃₅₉	C₄×Q₁₆
LCS	G ▷ C₄×C₃₅₉ ▷ C₂×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #19153331	C₄	semidirect
41. Dic₃₅₉	Q₁₆	semidirect
45. C₃₅₉	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153331	2
Q₈×Dic₃₅₉	2
#26039980	2
#26039948	2
#26039940	2
C₈×Dic₃₅₉	2
#103×C₃₅₉	2
C₄×Q₁₆	359