Subgroups — #63513

Order 288.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#3697	C₂	1	1	yes	maximal, characteristic, complemented
3	#3675	C₂	1	1	yes	maximal, characteristic, complemented
4	#3674	C₂	1	1	yes	maximal, characteristic, complemented
5	#3673	C₂	1	1	yes	maximal, characteristic, complemented
6	#3672	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#950	—	3	1	no	maximal
9	Q₈×C₉	C₂²	1	1	yes	characteristic
10	D₄×C₉	—	2	1	no	—
11	D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₉	—	2	1	no	—
13	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
14	#611	—	2	1	no	—
15	D₃₆	C₂²	1	1	yes	characteristic
16	C₄×D₉	—	2	1	no	—
17	Dic₁₈	C₂²	1	1	yes	characteristic
18	C₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×#41	D₃	1	1	yes	characteristic
21	#235	—	3	1	no	—
22	#216	—	3	1	no	—
23	#215	—	3	1	no	—
24	#214	—	3	1	no	—
25	#213	—	3	1	no	—
26	C₂×C₃⋊C₈	—	3	1	no	—
27	C₂²×C₉	—	2	1	no	—
28	C₂²×C₉	D₄	1	1	yes	characteristic
29	C₂×D₉	—	2	1	no	—
30	C₄×C₉	—	2	1	no	—
31	C₄×C₉	D₄	1	1	yes	characteristic
32	C₄×C₉	C₂³	1	1	yes	characteristic
33	Dic₉	—	2	1	no	—
34	#135	—	9	1	no	Sylow
35	C₃×Q₈	C₂×D₃	1	1	yes	characteristic
36	C₃×D₄	—	2	1	no	—
37	C₃×D₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃×C₄	—	2	1	no	—
39	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
40	#67	—	6	1	no	—
41	D₁₂	—	3	1	no	—
42	C₄×D₃	—	6	1	no	—
43	Dic₆	—	3	1	no	—
44	C₃⋊C₈	—	3	1	no	—
45	C₃⋊C₈	—	3	1	no	—
46	C₂×C₉	—	2	1	no	—
47	C₂×C₉	—	4	1	no	—
48	C₂×C₉	C₂×D₄	1	1	yes	characteristic
49	D₉	—	4	1	no	—
50	#41	—	9	1	no	—
51	#41	D₉	1	1	yes	characteristic, complemented
52	Q₁₆	—	9	1	no	—
53	SD₁₆	—	9	1	no	—
54	SD₁₆	—	9	1	no	—
55	D₈	—	9	1	no	—
56	C₂×C₈	—	9	1	no	—
57	C₂²×C₃	—	2	1	no	—
58	C₂²×C₃	#67	1	1	yes	characteristic
59	C₂×D₃	—	6	1	no	—
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	#67	1	1	yes	characteristic
62	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
63	Dic₃	—	6	1	no	—
64	C₉	#135	1	1	yes	characteristic, complemented, Sylow
65	Q₈	—	9	1	no	—
66	Q₈	C₂×D₉	1	1	yes	characteristic
67	D₄	—	2	1	no	—
68	D₄	—	9	1	no	—
69	D₄	—	18	1	no	—
70	D₄	C₂×D₉	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	18	1	no	—
73	C₂×C₄	C₂×D₉	1	1	yes	characteristic
74	C₈	—	9	1	no	—
75	C₈	—	9	1	no	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	—	4	1	no	—
78	C₂×C₃	C₂×#67	1	1	yes	characteristic
79	D₃	—	12	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	18	1	no	—
82	C₂²	#611	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	18	1	no	—
85	C₄	#611	1	1	yes	characteristic
86	C₄	C₂²×D₉	1	1	yes	characteristic
87	C₃	#950	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	4	1	no	—
90	C₂	—	36	1	no	—
91	C₂	C₂×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#611
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₉×#41	C₂
socle	C₂×C₃	C₂×#67
hypercenter	#41	D₉
hypocenter	C₉	#135
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₉×#41	C₂	semidirect
2. #3697	C₂	semidirect
3. #3675	C₂	semidirect
4. #3674	C₂	semidirect
5. #3673	C₂	semidirect
6. #3672	C₂	semidirect
7. C₂×C₉⋊C₈	C₂	semidirect
51. #41	D₉	semidirect
64. C₉	#135	semidirect

Maximal subgroups

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

subgroup	index
C₉×#41	2
#3697	2
#3675	2
#3674	2
#3673	2
#3672	2
C₂×C₉⋊C₈	2
#950	3