Subgroups — #10589411

Order 576.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#127	C₂	1	1	yes	maximal, characteristic
2	#63661	C₂	1	2	yes	maximal, complemented
3	#4545	—	3	4	no	maximal
4	C₂×C₃²×D₄	—	2	2	no	—
5	C₂×C₃²×D₄	C₄	1	1	yes	characteristic, complemented
6	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
7	C₃²×C₄²	C₄	1	1	yes	characteristic, complemented
8	#3757	—	2	2	no	—
9	C₃×#127	D₃	1	4	yes	—
10	#834	—	3	8	no	—
11	C₂³×C₃²	—	2	1	no	—
12	C₂³×C₃²	D₄	1	2	yes	—
13	C₃²×D₄	—	4	1	no	—
14	C₃²×D₄	—	4	1	no	—
15	C₃²×D₄	—	4	4	no	—
16	C₂×C₃²×C₄	—	2	1	no	—
17	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃⋊Dic₃	—	4	2	no	—
19	#353	—	9	1	no	Sylow
20	C₂×C₃×D₄	—	2	8	no	—
21	C₂×C₃×D₄	Dic₃	1	4	yes	complemented
22	C₂×C₃×D₄	C₂×D₃	1	4	yes	—
23	C₃×C₄²	Dic₃	1	4	yes	complemented
24	#217	—	6	8	no	—
25	C₂²×C₃²	—	2	2	no	—
26	C₂²×C₃²	—	4	2	no	—
27	C₂²×C₃²	—	4	2	no	—
28	C₂²×C₃²	—	4	1	no	—
29	C₂²×C₃²	C₂²⋊C₄	1	1	yes	characteristic
30	C₃²×C₄	—	2	2	no	—
31	C₃²×C₄	—	2	1	no	—
32	C₃⋊Dic₃	—	8	2	no	—
33	#127	C₃⋊D₃	1	1	yes	characteristic
34	#99	—	9	2	no	—
35	C₂³×C₃	—	2	4	no	—
36	C₂³×C₃	#67	1	8	yes	—
37	C₃×D₄	—	4	4	no	—
38	C₃×D₄	—	4	4	no	—
39	C₃×D₄	—	4	16	no	—
40	C₂×C₃×C₄	—	2	4	no	—
41	C₂×C₃×C₄	C₂×Dic₃	1	4	yes	—
42	C₂×Dic₃	—	12	8	no	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	—	4	2	no	—
45	C₂×C₃²	—	8	1	no	—
46	C₂×C₃²	#99	1	1	yes	characteristic
47	C₂×D₄	—	2	2	no	—
48	C₂×D₄	C₃⋊Dic₃	1	1	yes	characteristic, complemented
49	C₂×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₂²⋊C₄	—	18	2	no	—
51	C₄²	C₃⋊Dic₃	1	1	yes	characteristic, complemented
52	C₂²×C₃	—	2	8	no	—
53	C₂²×C₃	—	4	8	no	—
54	C₂²×C₃	—	4	8	no	—
55	C₂²×C₃	—	4	4	no	—
56	C₂²×C₃	#217	1	4	yes	—
57	C₃×C₄	—	2	8	no	—
58	C₃×C₄	—	2	4	no	—
59	Dic₃	—	24	8	no	—
60	C₃²	#353	1	1	yes	characteristic, complemented, Sylow
61	C₂³	—	2	1	no	—
62	C₂³	#638	1	2	yes	—
63	D₄	—	4	1	no	—
64	D₄	—	4	1	no	—
65	D₄	—	4	4	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	36	2	no	—
68	C₂×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
69	C₂×C₃	—	2	4	no	—
70	C₂×C₃	—	4	8	no	—
71	C₂×C₃	—	8	4	no	—
72	C₂×C₃	#834	1	4	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	—	4	1	no	—
77	C₂²	#3757	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	1	no	—
80	C₄	—	72	2	no	—
81	C₃	#4545	1	4	yes	complemented
82	C₂	—	2	1	no	—
83	C₂	—	4	2	no	—
84	C₂	—	8	1	no	—
85	C₂	#63661	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#63661
commutator	C₂×C₃²×C₄	C₂×C₄
Frattini	C₂×D₄	C₂×C₃⋊D₃
Fitting, prime radical	C₃²×#127	C₂
socle	C₂×C₃²	#99
cosocle	C₂×C₃²×D₄	C₂²
hypercenter	#127	C₃⋊D₃
hypocenter	C₃²	#353
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #127

Complements

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

N	H ≅ G/N	type
2. #63661	C₂	semidirect
5. C₂×C₃²×D₄	C₄	semidirect
7. C₃²×C₄²	C₄	semidirect
21. C₂×C₃×D₄	Dic₃	semidirect
23. C₃×C₄²	Dic₃	semidirect
48. C₂×D₄	C₃⋊Dic₃	semidirect
51. C₄²	C₃⋊Dic₃	semidirect
60. C₃²	#353	semidirect
81. C₃	#4545	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#127	2
#63661	2
#4545	3