Subgroups — #10588749

Order 576.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#63437	C₂	1	1	yes	maximal, characteristic
3	#63369	C₂	1	2	yes	maximal
4	#63368	C₂	1	2	yes	maximal, complemented
5	C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#5068	—	3	1	no	maximal
7	C₂×D₃₆	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₈	C₂²	1	1	yes	characteristic
9	C₉×C₄⋊C₄	—	2	2	no	—
10	C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₄²×C₉	C₂²	1	1	yes	characteristic
12	(C₂×D₉)⋊C₄	—	2	2	no	—
13	C₂×C₉⋊C₈	C₂²	1	2	yes	—
14	C₃×#125	D₃	1	1	yes	characteristic
15	#875	—	3	1	no	—
16	#810	—	3	2	no	—
17	#809	—	3	2	no	—
18	C₄×C₃⋊C₈	—	3	1	no	—
19	C₂²×D₉	—	2	1	no	—
20	C₂×C₄×C₉	—	2	2	no	—
21	C₂×C₄×C₉	D₄	1	2	yes	—
22	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂×Dic₉	—	2	1	no	—
24	D₃₆	—	2	2	no	—
25	Dic₁₈	—	2	2	no	—
26	C₉⋊C₈	—	2	2	no	—
27	#488	—	9	1	no	Sylow
28	C₂×D₁₂	—	3	1	no	—
29	C₂×Dic₆	—	3	1	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
32	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
33	(C₂×D₃)⋊C₄	—	6	2	no	—
34	C₂×C₃⋊C₈	—	3	2	no	—
35	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
36	C₂×D₉	—	4	1	no	—
37	C₂×D₉	—	4	2	no	—
38	C₄×C₉	—	2	2	no	—
39	C₄×C₉	—	4	2	no	—
40	C₄×C₉	#41	1	2	yes	—
41	Dic₉	—	4	1	no	—
42	#125	D₉	1	1	yes	characteristic, complemented
43	#124	—	9	1	no	—
44	#103	—	9	2	no	—
45	#102	—	9	2	no	—
46	C₄×C₈	—	9	1	no	—
47	C₂²×D₃	—	6	1	no	—
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	#67	1	2	yes	—
50	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
51	C₂×Dic₃	—	6	1	no	—
52	D₁₂	—	6	2	no	—
53	Dic₆	—	6	2	no	—
54	C₃⋊C₈	—	6	2	no	—
55	C₂×C₉	#124	1	1	yes	characteristic
56	C₂×C₉	#135	1	2	yes	—
57	D₉	—	8	1	no	—
58	C₂×Q₈	—	9	1	no	—
59	C₂×D₄	—	9	1	no	—
60	C₂×C₈	—	9	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	C₂×D₉	1	2	yes	—
63	C₂²⋊C₄	—	18	2	no	—
64	C₄²	C₂×D₉	1	1	yes	characteristic
65	C₂²×C₃	C₂×#67	1	1	yes	characteristic
66	C₂×D₃	—	12	1	no	—
67	C₂×D₃	—	12	2	no	—
68	C₃×C₄	—	2	2	no	—
69	C₃×C₄	—	4	2	no	—
70	C₃×C₄	#239	1	2	yes	—
71	Dic₃	—	12	1	no	—
72	C₉	#488	1	1	yes	characteristic, complemented, Sylow
73	C₂³	—	18	1	no	—
74	Q₈	—	18	2	no	—
75	D₄	—	18	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	18	1	no	—
78	C₂×C₄	#611	1	2	yes	—
79	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
80	C₈	—	18	2	no	—
81	C₂×C₃	#947	1	1	yes	characteristic
82	C₂×C₃	#950	1	2	yes	—
83	D₃	—	24	1	no	—
84	C₂²	—	36	1	no	—
85	C₂²	—	36	2	no	—
86	C₂²	C₂×#611	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	4	2	no	—
89	C₄	—	36	1	no	—
90	C₄	#3701	1	2	yes	—
91	C₃	#5068	1	1	yes	characteristic
92	C₂	—	72	1	no	—
93	C₂	#63509	1	1	yes	characteristic
94	C₂	#63513	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×D₃
Fitting	C₉×#125	C₂
socle	C₂²×C₃	C₂×#67
prime radical	C₂×D₃₆	C₂²
hypercenter	#125	D₉
hypocenter	C₉	#488
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₉×#125	C₂	semidirect
4. #63368	C₂	semidirect
5. C₄×C₉⋊C₈	C₂	semidirect
42. #125	D₉	semidirect
72. C₉	#488	semidirect

Maximal subgroups

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

subgroup	index
C₉×#125	2
#63437	2
#63369	2
#63368	2
C₄×C₉⋊C₈	2
#5068	3