Subgroups — #44525053

Order 30861.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812184	C₃	1	2	yes	maximal
2	#23779835	C₃	1	1	yes	maximal, characteristic, complemented
3	#23779834	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779833	C₃	1	2	yes	maximal, complemented
5	#23779833	C₃	1	2	yes	maximal, complemented
6	#23779833	C₃	1	2	yes	maximal, complemented
7	C₃²⋊#11765917	C₃	1	1	yes	maximal, characteristic
8	C₉×#11765917	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃²⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic
10	#16812184	—	3	6	no	—
11	C₃×#11765917	C₃²	1	2	yes	—
12	C₃×#11765917	C₃²	1	2	yes	—
13	C₃×#11765917	C₃²	1	1	yes	characteristic
14	C₃×#11765917	C₃²	1	2	yes	—
15	C₃×#11765917	C₃²	1	2	yes	—
16	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
19	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
20	C₃²×C₁₂₇	—	3	2	no	—
21	C₃²×C₁₂₇	—	3	1	no	—
22	C₃²×C₁₂₇	—	3	1	no	—
23	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
24	C₉×C₁₂₇	—	3	1	no	—
25	C₉×C₁₂₇	—	3	1	no	—
26	C₉×C₁₂₇	—	3	1	no	—
27	#11765917	—	3	2	no	—
28	#11765917	—	3	2	no	—
29	#11765917	—	3	1	no	—
30	#11765917	—	3	2	no	—
31	#11765917	3₋¹⁺²	1	6	yes	complemented
32	C₃×C₁₂₇	—	9	1	no	—
33	C₃×C₁₂₇	C₃×3₋¹⁺²	1	2	yes	—
34	C₃×C₁₂₇	#737	1	1	yes	characteristic
35	C₃×C₁₂₇	#737	1	1	yes	characteristic
36	#6896	—	127	1	no	maximal, Sylow
37	C₁₂₇	#6896	1	1	yes	characteristic, complemented, Sylow
38	C₃×3₋¹⁺²	—	127	2	no	—
39	C₉⋊C₉	—	127	2	no	—
40	C₉⋊C₉	—	127	2	no	—
41	C₉⋊C₉	—	127	2	no	—
42	C₉⋊C₉	—	127	1	no	—
43	C₉⋊C₉	—	127	1	no	—
44	C₃²⋊C₉	—	127	1	no	—
45	C₃²⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic
46	C₉²	—	127	1	no	—
47	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
48	3₋¹⁺²	—	381	6	no	—
49	C₃×C₉	—	127	1	no	—
50	C₃×C₉	—	127	2	no	—
51	C₃×C₉	—	127	2	no	—
52	C₃×C₉	—	127	2	no	—
53	C₃×C₉	—	127	2	no	—
54	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
55	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
56	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
57	C₃²	—	3	2	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	1	no	—
60	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
61	C₉	—	3	1	no	—
62	C₉	—	3	1	no	—
63	C₉	—	3	1	no	—
64	C₉	—	381	2	no	—
65	C₉	—	381	2	no	—
66	C₉	—	381	1	no	—
67	C₉	—	381	2	no	—
68	C₉	—	127	6	no	—
69	C₃	—	9	1	no	—
70	C₃	3₋¹⁺²×C₁₂₇⋊C₃	1	2	yes	—
71	C₃	#23779859	1	1	yes	characteristic
72	C₃	#23779860	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×C₁₂₇⋊C₃
commutator, socle, cosocle	C₃²×C₁₂₇	C₃³
Fitting	C₃²⋊C₉×C₁₂₇	C₃
prime radical	C₃³×C₁₂₇	C₃²
hypercenter	C₃²⋊C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6896
LCS	G ▷ C₃²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊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
5. #23779833	C₃	semidirect
6. #23779833	C₃	semidirect
8. C₉×#11765917	C₃	semidirect
31. #11765917	3₋¹⁺²	semidirect
37. C₁₂₇	#6896	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812184	3
#23779835	3
#23779834	3
#23779833	3
C₃²⋊#11765917	3
C₉×#11765917	3
C₃²⋊C₉×C₁₂₇	3
#6896	127