Subgroups — #44525013

Order 30861.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11765917	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×C₁₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
4	C₃²⋊#11765917	C₃	1	1	yes	maximal, characteristic, complemented
5	#23779831	C₃	1	6	yes	maximal, complemented
6	C₃²⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₁₂₇⋊C₃	C₃²	1	3	yes	—
8	C₃²⋊(C₁₂₇⋊C₃)	C₉	1	9	yes	complemented
9	C₃×#11765917	—	3	3	no	—
10	C₃×#11765917	C₃²	1	3	yes	—
11	C₃×#11765917	C₃²	1	2	yes	—
12	C₃×#11765917	C₃²	1	1	yes	characteristic
13	C₉×C₁₂₇⋊C₃	—	3	9	no	—
14	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₁₂₇	C₃²	1	3	yes	—
16	C₃²×C₁₂₇	—	3	1	no	—
17	C₃²×C₁₂₇	—	3	2	no	—
18	C₃²×C₁₂₇	C₃×C₉	1	3	yes	complemented
19	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
20	C₃×C₁₂₇⋊C₃	—	3	3	no	—
21	C₃×C₁₂₇⋊C₃	—	3	6	no	—
22	C₃×C₁₂₇⋊C₃	C₃×C₉	1	9	yes	complemented
23	C₉×C₁₂₇	—	3	3	no	—
24	#11765917	—	3	3	no	—
25	#11765917	—	3	2	no	—
26	#11765917	H(3,𝔽₃)	1	3	yes	complemented
27	C₃×C₁₂₇	—	3	3	no	—
28	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
29	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₃×C₁₂₇	#737	1	2	yes	—
31	C₁₂₇⋊C₃	—	3	9	no	—
32	C₉×H(3,𝔽₃)	—	127	1	no	maximal, Sylow
33	C₁₂₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
34	C₃×H(3,𝔽₃)	—	127	1	no	—
35	C₃²×C₉	—	127	1	no	—
36	C₃²×C₉	—	127	3	no	—
37	C₃²⋊C₉	—	127	1	no	—
38	C₃²⋊C₉	—	127	6	no	—
39	C₃²⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
40	C₃³	—	127	3	no	—
41	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
42	H(3,𝔽₃)	—	127	9	no	—
43	C₃×C₉	—	381	9	no	—
44	C₃×C₉	—	381	3	no	—
45	C₃×C₉	—	127	3	no	—
46	C₃×C₉	—	127	2	no	—
47	C₃×C₉	—	127	1	no	—
48	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	—
49	C₃²	—	3	1	no	—
50	C₃²	—	3	2	no	—
51	C₃²	—	381	3	no	—
52	C₃²	—	381	6	no	—
53	C₃²	—	127	9	no	—
54	C₃²	C₉×C₁₂₇⋊C₃	1	3	yes	complemented
55	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	381	3	no	—
58	C₉	—	381	2	no	—
59	C₉	—	127	3	no	—
60	C₃	—	3	3	no	—
61	C₃	—	381	9	no	—
62	C₃	C₃×C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
63	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic
64	C₃	#23779861	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator	C₃×C₁₂₇	C₃²×C₉
Fitting	C₃²⋊C₉×C₁₂₇	C₃
socle, cosocle	C₃²×C₁₂₇	C₃³
prime radical	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃
hypercenter	C₃²⋊C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₉×H(3,𝔽₃)
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×#11765917	C₃	semidirect
3. C₃×C₉×C₁₂₇⋊C₃	C₃	semidirect
4. C₃²⋊#11765917	C₃	semidirect
5. #23779831	C₃	semidirect
6. C₃²⋊C₉×C₁₂₇	C₃	semidirect
8. C₃²⋊(C₁₂₇⋊C₃)	C₉	semidirect
18. C₃²×C₁₂₇	C₃×C₉	semidirect
22. C₃×C₁₂₇⋊C₃	C₃×C₉	semidirect
26. #11765917	H(3,𝔽₃)	semidirect
33. C₁₂₇	C₉×H(3,𝔽₃)	semidirect
39. C₃²⋊C₉	C₁₂₇⋊C₃	semidirect
54. C₃²	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×C₃²⋊(C₁₂₇⋊C₃)	3
C₃²×#11765917	3
C₃×C₉×C₁₂₇⋊C₃	3
C₃²⋊#11765917	3
#23779831	3
C₃²⋊C₉×C₁₂₇	3
C₉×H(3,𝔽₃)	127