Subgroups — #41006345

Order 26973.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582488	C₃	1	6	yes	maximal, complemented
3	#22582465	C₃	1	3	yes	maximal
4	#22582464	C₃	1	2	yes	maximal, complemented
5	C₃₇×#6864	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×C₉×C₃₇⋊C₃	C₉	1	1	yes	characteristic
7	C₃×C₉×C₃₇⋊C₃	C₉	1	2	yes	—
8	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758601	C₃²	1	6	yes	—
10	C₉×#66524	C₉	1	2	yes	—
11	C₉×#66524	C₉	1	1	yes	characteristic
12	C₉×#66524	C₉	1	2	yes	—
13	C₉×#66524	C₉	1	1	yes	characteristic
14	C₉×#66524	C₃²	1	2	yes	—
15	C₃×C₂₇×C₃₇	C₃²	1	3	yes	—
16	C₉²×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	C₃×#66524	C₃×C₉	1	1	yes	characteristic
22	C₃×#66524	C₃×C₉	1	1	yes	characteristic
23	C₃×#66524	C₃×C₉	1	2	yes	—
24	C₃×#66524	C₃×C₉	1	2	yes	—
25	C₃×#66524	C₃×C₉	1	2	yes	—
26	C₉×C₃₇⋊C₃	—	3	3	no	—
27	C₉×C₃₇⋊C₃	—	3	3	no	—
28	C₉×C₃₇⋊C₃	—	3	6	no	—
29	C₂₇×C₃₇	—	3	3	no	—
30	#11758601	—	3	6	no	—
31	#10624711	—	37	1	no	maximal, Sylow
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	3	1	no	—
34	C₃×C₃₇⋊C₃	—	3	2	no	—
35	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	3	yes	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	3	2	no	—
38	C₉×C₃₇	#737	1	3	yes	—
39	#66524	—	3	1	no	—
40	#66524	—	3	1	no	—
41	#66524	—	3	2	no	—
42	#66524	—	3	2	no	—
43	#66524	C₂₇⋊C₃	1	6	yes	—
44	C₃×C₉²	—	37	1	no	—
45	C₉⋊C₂₇	—	37	6	no	—
46	C₃²⋊C₂₇	—	37	3	no	—
47	#6864	—	37	2	no	—
48	#6864	C₃₇⋊C₃	1	1	yes	characteristic, complemented
49	C₃×C₃₇	#6887	1	1	yes	characteristic
50	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
51	C₃×C₃₇	C₃×C₂₇⋊C₃	1	2	yes	—
52	C₃₇⋊C₃	—	3	3	no	—
53	C₃²×C₉	—	37	2	no	—
54	C₃²×C₉	—	37	1	no	—
55	C₃²×C₉	—	37	1	no	—
56	C₃×C₂₇	—	37	6	no	—
57	C₃×C₂₇	C₃×C₃₇⋊C₃	1	3	yes	—
58	C₉²	—	37	1	no	—
59	C₉²	—	37	2	no	—
60	C₉²	—	37	2	no	—
61	C₉²	—	37	1	no	—
62	C₉²	—	37	2	no	—
63	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
64	C₃₇	#10624711	1	1	yes	characteristic, complemented, Sylow
65	C₃³	—	37	1	no	—
66	C₃×C₉	—	37	2	no	—
67	C₃×C₉	—	37	1	no	—
68	C₃×C₉	—	37	1	no	—
69	C₃×C₉	—	37	2	no	—
70	C₃×C₉	—	37	2	no	—
71	C₃×C₉	—	111	6	no	—
72	C₃×C₉	—	111	3	no	—
73	C₃×C₉	—	111	3	no	—
74	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
75	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
76	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
77	C₂₇	—	3	3	no	—
78	C₂₇	—	111	6	no	—
79	C₃²	—	37	3	no	—
80	C₃²	—	111	1	no	—
81	C₃²	—	111	2	no	—
82	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
83	C₉	—	3	2	no	—
84	C₉	—	3	1	no	—
85	C₉	—	37	6	no	—
86	C₉	—	111	1	no	—
87	C₉	—	111	1	no	—
88	C₉	—	111	2	no	—
89	C₉	—	111	2	no	—
90	C₉	#15688531	1	3	yes	—
91	C₃	—	111	3	no	—
92	C₃	#22582536	1	1	yes	characteristic
93	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	2	yes	—
94	C₃	C₂₇⋊C₃×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6864	C₃
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	#6864	C₃₇⋊C₃
hypocenter	C₃₇	#10624711
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ #6864

Complements

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

N	H ≅ G/N	type
2. #22582488	C₃	semidirect
4. #22582464	C₃	semidirect
5. C₃₇×#6864	C₃	semidirect
48. #6864	C₃₇⋊C₃	semidirect
64. C₃₇	#10624711	semidirect

Maximal subgroups

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

subgroup	index
C₉²×C₃₇⋊C₃	3
#22582488	3
#22582465	3
#22582464	3
C₃₇×#6864	3
#10624711	37