Subgroups — #44525027

Order 30861.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×#16812183	C₃	1	3	yes	maximal, complemented
3	C₃²⋊#11765917	C₃	1	2	yes	maximal, complemented
4	#23779831	C₃	1	6	yes	maximal, complemented
5	C₃²⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
7	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
8	C₃²×C₁₂₇⋊C₃	—	3	6	no	—
9	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
10	C₃²×C₁₂₇⋊C₃	C₃²	1	2	yes	—
11	#16812183	—	3	9	no	—
12	C₃×#11765917	C₃²	1	6	yes	complemented
13	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₁₂₇	C₃²	1	3	yes	complemented
15	C₃²×C₁₂₇	—	3	1	no	—
16	C₃²×C₁₂₇	—	3	2	no	—
17	C₃²×C₁₂₇	3₋¹⁺²	1	3	yes	complemented
18	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	—	3	27	no	—
20	C₃×C₁₂₇⋊C₃	—	3	2	no	—
21	C₃×C₁₂₇⋊C₃	—	3	2	no	—
22	C₃×C₁₂₇⋊C₃	—	3	2	no	—
23	C₃×C₁₂₇⋊C₃	—	3	1	no	—
24	C₃×C₁₂₇⋊C₃	—	3	2	no	—
25	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
26	C₃×C₁₂₇⋊C₃	3₋¹⁺²	1	6	yes	complemented
27	C₉×C₁₂₇	—	3	3	no	—
28	#11765917	—	3	6	no	—
29	C₃×C₁₂₇	—	3	3	no	—
30	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₁₂₇	C₃×3₋¹⁺²	1	2	yes	—
32	C₃×C₁₂₇	C₃×3₋¹⁺²	1	1	yes	characteristic
33	C₁₂₇⋊C₃	—	3	3	no	—
34	C₁₂₇⋊C₃	—	3	6	no	—
35	#6891	—	127	1	no	maximal, Sylow
36	C₁₂₇	#6891	1	1	yes	characteristic, complemented, Sylow
37	C₃⁴	—	127	1	no	—
38	C₃×3₋¹⁺²	—	127	3	no	—
39	C₃²⋊C₉	—	127	2	no	—
40	C₃²⋊C₉	—	127	6	no	—
41	C₃²⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
42	C₃³	—	381	3	no	—
43	C₃³	—	381	6	no	—
44	C₃³	—	381	3	no	—
45	C₃³	—	127	2	no	—
46	C₃³	—	127	1	no	—
47	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
48	3₋¹⁺²	—	381	9	no	—
49	C₃×C₉	—	127	6	no	—
50	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
51	C₃²	—	3	1	no	—
52	C₃²	—	3	2	no	—
53	C₃²	—	381	27	no	—
54	C₃²	—	381	2	no	—
55	C₃²	—	381	2	no	—
56	C₃²	—	381	2	no	—
57	C₃²	—	381	2	no	—
58	C₃²	—	381	1	no	—
59	C₃²	—	127	3	no	—
60	C₃²	—	127	6	no	—
61	C₃²	#16812183	1	3	yes	complemented
62	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
63	C₉	—	3	3	no	—
64	C₉	—	381	6	no	—
65	C₃	—	3	3	no	—
66	C₃	—	381	6	no	—
67	C₃	—	381	3	no	—
68	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic
69	C₃	3₋¹⁺²×C₁₂₇⋊C₃	1	2	yes	—
70	C₃	C₃×#16812183	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⋊C₃	C₃
hypercenter	C₃²⋊C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6891
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. C₃×#16812183	C₃	semidirect
3. C₃²⋊#11765917	C₃	semidirect
4. #23779831	C₃	semidirect
5. C₃²⋊C₉×C₁₂₇	C₃	semidirect
12. C₃×#11765917	C₃²	semidirect
14. C₃×C₉×C₁₂₇	C₃²	semidirect
17. C₃²×C₁₂₇	3₋¹⁺²	semidirect
26. C₃×C₁₂₇⋊C₃	3₋¹⁺²	semidirect
36. C₁₂₇	#6891	semidirect
41. C₃²⋊C₉	C₁₂₇⋊C₃	semidirect
50. C₃×C₉	C₃×C₁₂₇⋊C₃	semidirect
61. C₃²	#16812183	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₁₂₇⋊C₃	3
C₃×#16812183	3
C₃²⋊#11765917	3
#23779831	3
C₃²⋊C₉×C₁₂₇	3
#6891	127