Subgroups — #44525011

Order 30861.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#23779835	C₃	1	1	yes	maximal, characteristic, complemented
3	#23779834	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779833	C₃	1	3	yes	maximal, complemented
5	#23779831	C₃	1	3	yes	maximal
6	C₉×#11765917	C₃	1	3	yes	maximal, complemented
7	C₉⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11765917	C₃²	1	3	yes	—
10	C₃×#11765917	C₃²	1	3	yes	—
11	C₃×#11765917	C₃²	1	1	yes	characteristic
12	C₃×#11765917	C₃²	1	1	yes	characteristic
13	C₉×C₁₂₇⋊C₃	C₉	1	9	yes	complemented
14	C₃×C₉×C₁₂₇	C₃²	1	3	yes	—
15	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
16	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
17	C₃×C₁₂₇⋊C₃	—	3	3	no	—
18	C₃×C₁₂₇⋊C₃	C₃×C₉	1	3	yes	—
19	C₉×C₁₂₇	—	3	3	no	—
20	C₉×C₁₂₇	C₃×C₉	1	3	yes	—
21	#11765917	—	3	3	no	—
22	#11765917	—	3	3	no	—
23	#11765917	C₃×C₉	1	3	yes	—
24	#11765917	C₃×C₉	1	3	yes	—
25	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
26	C₃×C₁₂₇	#737	1	3	yes	—
27	C₁₂₇⋊C₃	—	3	3	no	—
28	#6887	—	127	1	no	maximal, Sylow
29	C₁₂₇	#6887	1	1	yes	characteristic, complemented, Sylow
30	C₃²×C₉	—	127	1	no	—
31	C₉⋊C₉	—	127	3	no	—
32	C₉⋊C₉	—	127	1	no	—
33	C₉⋊C₉	—	127	1	no	—
34	C₉⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
35	C₃²⋊C₉	—	127	3	no	—
36	C₉²	—	127	3	no	—
37	C₃³	—	127	1	no	—
38	C₃×C₉	—	127	3	no	—
39	C₃×C₉	—	127	3	no	—
40	C₃×C₉	—	127	1	no	—
41	C₃×C₉	—	127	1	no	—
42	C₃×C₉	—	127	9	no	—
43	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	—
44	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
45	C₃²	—	381	3	no	—
46	C₃²	—	127	3	no	—
47	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
48	C₉	—	3	3	no	—
49	C₉	—	381	3	no	—
50	C₉	—	381	3	no	—
51	C₉	—	127	3	no	—
52	C₉	—	127	3	no	—
53	C₉	C₉×C₁₂₇⋊C₃	1	3	yes	—
54	C₃	—	381	3	no	—
55	C₃	C₃×C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
56	C₃	#23779861	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator	C₃×C₁₂₇	C₃²×C₉
Fitting	C₉⋊C₉×C₁₂₇	C₃
socle, cosocle	C₃²×C₁₂₇	C₃³
prime radical	C₃²×C₁₂₇⋊C₃	C₃²
hypercenter	C₉⋊C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6887
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. #23779835	C₃	semidirect
3. #23779834	C₃	semidirect
4. #23779833	C₃	semidirect
6. C₉×#11765917	C₃	semidirect
7. C₉⋊C₉×C₁₂₇	C₃	semidirect
13. C₉×C₁₂₇⋊C₃	C₉	semidirect
29. C₁₂₇	#6887	semidirect
34. C₉⋊C₉	C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×C₁₂₇⋊C₃	3
#23779835	3
#23779834	3
#23779833	3
#23779831	3
C₉×#11765917	3
C₉⋊C₉×C₁₂₇	3
#6887	127