Subgroups — #44525206

Order 30861.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779872	C₃	1	3	yes	maximal, complemented
2	#23779871	C₃	1	3	yes	maximal, complemented
3	#23779870	C₃	1	2	yes	maximal, complemented
4	(C₁₂₇⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal, complemented
5	C₃×#16812186	C₃	1	1	yes	maximal, characteristic, complemented
6	#23779861	C₃	1	1	yes	maximal, characteristic
7	#16812186	C₃²	1	1	yes	characteristic
8	#16812186	C₃²	1	1	yes	characteristic
9	#16812186	C₃²	1	1	yes	characteristic
10	#16812186	C₃²	1	6	yes	—
11	#16812184	C₉	1	1	yes	characteristic
12	#16812183	C₉	1	3	yes	—
13	#16812182	C₉	1	3	yes	—
14	C₃²⋊(C₁₂₇⋊C₃)	C₉	1	1	yes	characteristic
15	C₃×#11765917	C₃²	1	1	yes	characteristic
16	C₉×C₁₂₇⋊C₃	C₃²	1	3	yes	—
17	3₋¹⁺²×C₁₂₇	C₉	1	1	yes	characteristic
18	C₃²×C₁₂₇	C₃×C₉	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	C₃×C₉	1	3	yes	—
20	C₉×C₁₂₇	C₃×C₉	1	3	yes	—
21	#11765917	C₃×C₉	1	3	yes	—
22	#11765917	C₃×C₉	1	1	yes	characteristic
23	#11765917	C₃×C₉	1	1	yes	characteristic
24	#11765917	C₃³	1	1	yes	characteristic
25	C₃×C₁₂₇	—	3	1	no	—
26	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
27	C₁₂₇⋊C₃	—	3	3	no	—
28	#6903	—	127	1	no	maximal, Sylow
29	C₁₂₇	#6903	1	1	yes	characteristic, complemented, Sylow
30	#737	—	127	1	no	—
31	C₂₇⋊C₃	—	127	2	no	—
32	C₂₇⋊C₃	—	127	3	no	—
33	C₂₇⋊C₃	—	127	3	no	—
34	C₃×C₂₇	—	127	1	no	—
35	C₃×C₂₇	—	127	3	no	—
36	3₋¹⁺²	—	127	1	no	—
37	3₋¹⁺²	—	127	3	no	—
38	3₋¹⁺²	—	127	3	no	—
39	3₋¹⁺²	#11765919	1	1	yes	characteristic
40	H(3,𝔽₃)	—	127	1	no	—
41	C₃×C₉	—	127	1	no	—
42	C₃×C₉	—	127	3	no	—
43	C₂₇	—	127	1	no	—
44	C₂₇	—	127	1	no	—
45	C₂₇	—	127	1	no	—
46	C₂₇	—	127	6	no	—
47	C₃²	—	127	3	no	—
48	C₃²	C₃×#11765919	1	1	yes	characteristic
49	C₉	—	127	1	no	—
50	C₉	—	127	1	no	—
51	C₉	—	127	1	no	—
52	C₉	—	127	3	no	—
53	C₉	C₃×#11765919	1	3	yes	—
54	C₃	—	3	1	no	—
55	C₃	—	381	3	no	—
56	C₃	C₃²×#11765919	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃	C₃²×#11765919
commutator, socle	C₃×C₁₂₇	C₃²×C₉
Fitting	3₋¹⁺²×C₁₂₇	C₉
prime radical	C₃²⋊(C₁₂₇⋊C₃)	C₉
cosocle	#11765917	C₃³
hypercenter	3₋¹⁺²	#11765919
hypocenter	C₁₂₇	#6903
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ 3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #23779872	C₃	semidirect
2. #23779871	C₃	semidirect
3. #23779870	C₃	semidirect
4. (C₁₂₇⋊C₃)⋊C₂₇	C₃	semidirect
5. C₃×#16812186	C₃	semidirect
29. C₁₂₇	#6903	semidirect

Maximal subgroups

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

subgroup	index
#23779872	3
#23779871	3
#23779870	3
(C₁₂₇⋊C₃)⋊C₂₇	3
C₃×#16812186	3
#23779861	3
#6903	127