Subgroups — #44525016

Order 30861.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812182	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
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	#23779831	C₃	1	2	yes	maximal
7	C₉×#11765917	C₃	1	2	yes	maximal, complemented
8	C₉×#11765917	C₃	1	1	yes	maximal, characteristic, complemented
9	C₉⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
11	#16812182	C₉	1	9	yes	complemented
12	C₃×#11765917	C₃²	1	1	yes	characteristic
13	C₃×#11765917	C₃²	1	2	yes	—
14	C₃×#11765917	C₃²	1	2	yes	—
15	C₃×#11765917	C₃²	1	2	yes	—
16	C₃×#11765917	C₃²	1	1	yes	characteristic
17	C₉×C₁₂₇⋊C₃	—	3	3	no	—
18	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
19	C₃×C₉×C₁₂₇	C₃²	1	2	yes	—
20	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
21	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
22	C₃×C₁₂₇⋊C₃	—	3	1	no	—
23	C₃×C₁₂₇⋊C₃	—	3	2	no	—
24	C₃×C₁₂₇⋊C₃	C₃×C₉	1	3	yes	—
25	C₉×C₁₂₇	—	3	1	no	—
26	C₉×C₁₂₇	—	3	2	no	—
27	C₉×C₁₂₇	C₃×C₉	1	3	yes	complemented
28	#11765917	—	3	1	no	—
29	#11765917	—	3	2	no	—
30	#11765917	—	3	2	no	—
31	#11765917	C₃×C₉	1	6	yes	complemented
32	#11765917	3₋¹⁺²	1	3	yes	complemented
33	C₃×C₁₂₇	C₃²×C₉	1	1	yes	characteristic
34	C₃×C₁₂₇	C₃×3₋¹⁺²	1	1	yes	characteristic
35	C₃×C₁₂₇	#737	1	2	yes	—
36	C₁₂₇⋊C₃	—	3	3	no	—
37	C₉×3₋¹⁺²	—	127	1	no	maximal, Sylow
38	C₁₂₇	C₉×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
39	C₃×3₋¹⁺²	—	127	1	no	—
40	C₃²×C₉	—	127	1	no	—
41	C₉⋊C₉	—	127	2	no	—
42	C₉⋊C₉	—	127	2	no	—
43	C₉⋊C₉	—	127	1	no	—
44	C₉⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
45	C₃²⋊C₉	—	127	2	no	—
46	C₉²	—	127	1	no	—
47	C₉²	—	127	2	no	—
48	C₃³	—	127	1	no	—
49	3₋¹⁺²	—	127	9	no	—
50	C₃×C₉	—	381	3	no	—
51	C₃×C₉	—	127	2	no	—
52	C₃×C₉	—	127	2	no	—
53	C₃×C₉	—	127	1	no	—
54	C₃×C₉	—	127	1	no	—
55	C₃×C₉	—	127	2	no	—
56	C₃×C₉	C₃×C₁₂₇⋊C₃	1	2	yes	—
57	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
58	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
59	C₃²	—	381	1	no	—
60	C₃²	—	381	2	no	—
61	C₃²	—	127	3	no	—
62	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
63	C₉	—	3	1	no	—
64	C₉	—	3	2	no	—
65	C₉	—	381	1	no	—
66	C₉	—	381	2	no	—
67	C₉	—	381	2	no	—
68	C₉	—	127	6	no	—
69	C₉	—	127	3	no	—
70	C₉	C₉×C₁₂₇⋊C₃	1	3	yes	complemented
71	C₃	—	381	3	no	—
72	C₃	C₃×C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
73	C₃	3₋¹⁺²×C₁₂₇⋊C₃	1	1	yes	characteristic
74	C₃	#23779861	1	2	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	C₉×3₋¹⁺²
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
3. #23779834	C₃	semidirect
4. #23779833	C₃	semidirect
5. #23779833	C₃	semidirect
7. C₉×#11765917	C₃	semidirect
8. C₉×#11765917	C₃	semidirect
9. C₉⋊C₉×C₁₂₇	C₃	semidirect
11. #16812182	C₉	semidirect
27. C₉×C₁₂₇	C₃×C₉	semidirect
31. #11765917	C₃×C₉	semidirect
32. #11765917	3₋¹⁺²	semidirect
38. C₁₂₇	C₉×3₋¹⁺²	semidirect
44. C₉⋊C₉	C₁₂₇⋊C₃	semidirect
70. 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×#16812182	3
C₃×C₉×C₁₂₇⋊C₃	3
#23779834	3
#23779833	3
#23779831	3
C₉×#11765917	3
C₉⋊C₉×C₁₂₇	3
C₉×3₋¹⁺²	127