Subgroups — #44524941

Order 30861.

180 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	C₃²⋊#11765917	C₃	1	2	yes	maximal, complemented
3	C₃²⋊C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
5	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
6	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
7	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
8	C₃²×C₁₂₇⋊C₃	—	3	6	no	—
9	C₃²×C₁₂₇⋊C₃	C₉	1	3	yes	complemented
10	C₃×#11765917	—	3	2	no	—
11	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
12	C₃×C₉×C₁₂₇	—	3	1	no	—
13	C₃²×C₁₂₇	—	3	1	no	—
14	C₃²×C₁₂₇	—	3	2	no	—
15	C₃²×C₁₂₇	C₃×C₉	1	1	yes	characteristic
16	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃²×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	—	3	9	no	—
20	C₃×C₁₂₇⋊C₃	—	3	9	no	—
21	C₃×C₁₂₇⋊C₃	—	3	3	no	—
22	C₃×C₁₂₇⋊C₃	—	3	3	no	—
23	C₃×C₁₂₇⋊C₃	—	3	6	no	—
24	C₃×C₁₂₇⋊C₃	—	3	9	no	—
25	C₉×C₁₂₇	—	9	1	no	—
26	#11765917	—	9	2	no	—
27	C₃×C₁₂₇	—	3	1	no	—
28	C₃×C₁₂₇	—	3	1	no	—
29	C₃×C₁₂₇	—	3	1	no	—
30	C₃×C₁₂₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃×C₁₂₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
32	C₃×C₁₂₇	AΣL(1,𝔽₂₇)	1	2	yes	—
33	C₁₂₇⋊C₃	—	3	9	no	—
34	#6866	—	127	1	no	maximal, Sylow
35	C₁₂₇	#6866	1	1	yes	characteristic, complemented, Sylow
36	C₃⁴	—	127	1	no	—
37	C₃²⋊C₉	—	127	2	no	—
38	C₃²⋊C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
39	C₃³	—	381	3	no	—
40	C₃³	—	381	6	no	—
41	C₃³	—	381	1	no	—
42	C₃³	—	381	1	no	—
43	C₃³	—	381	1	no	—
44	C₃³	—	127	3	no	—
45	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	381	2	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	3	2	no	—
50	C₃²	—	381	9	no	—
51	C₃²	—	381	9	no	—
52	C₃²	—	381	9	no	—
53	C₃²	—	381	3	no	—
54	C₃²	—	381	6	no	—
55	C₃²	—	381	3	no	—
56	C₃²	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
57	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
58	C₃²	#16812182	1	1	yes	characteristic
59	C₃²	#16812183	1	1	yes	characteristic
60	C₉	—	9	1	no	—
61	C₉	—	1143	2	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	1	no	—
64	C₃	—	3	1	no	—
65	C₃	—	381	9	no	—
66	C₃	#23779831	1	1	yes	characteristic
67	C₃	#23779841	1	1	yes	characteristic
68	C₃	#23779843	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₂₇⋊C₃)
commutator	C₃²×C₁₂₇	C₃×C₉
Frattini	C₃³	C₃×C₁₂₇⋊C₃
Fitting	C₃²⋊C₉×C₁₂₇	C₃
socle	C₃²×C₁₂₇	H(3,𝔽₃)
prime radical	C₃³×C₁₂₇⋊C₃	C₃
cosocle	C₃³×C₁₂₇	C₃²
hypercenter	C₃²⋊C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6866
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃²⋊#11765917	C₃	semidirect
3. C₃²⋊C₉×C₁₂₇	C₃	semidirect
9. C₃²×C₁₂₇⋊C₃	C₉	semidirect
35. C₁₂₇	#6866	semidirect
38. 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
C₃³×C₁₂₇⋊C₃	3
C₃²⋊#11765917	3
C₃²⋊C₉×C₁₂₇	3
#6866	127