Subgroups — #37555933

Order 23571.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21463377	C₃	1	1	yes	maximal, characteristic
2	#21463357	C₃	1	2	yes	maximal, complemented
3	#21463356	C₃	1	3	yes	maximal, complemented
4	#21463355	C₃	1	3	yes	maximal, complemented
5	C₂₇×C₉₇⋊C₃	C₃	1	3	yes	maximal, complemented
6	C₃×C₂₇×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
7	#14579757	C₉	1	2	yes	—
8	#14579756	C₉	1	3	yes	—
9	#14579755	C₉	1	3	yes	—
10	C₃²⋊(C₉₇⋊C₃)	C₉	1	1	yes	characteristic
11	C₉×C₉₇⋊C₃	C₃²	1	3	yes	—
12	C₉₇⋊C₂₇	C₃²	1	6	yes	—
13	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
14	C₂₇×C₉₇	C₃²	1	1	yes	characteristic
15	C₂₇×C₉₇	C₃²	1	1	yes	characteristic
16	C₂₇×C₉₇	C₃²	1	1	yes	characteristic
17	C₃²×C₉₇	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉₇⋊C₃	C₃×C₉	1	3	yes	—
19	C₉×C₉₇	C₃×C₉	1	1	yes	characteristic
20	C₉×C₉₇	C₃×C₉	1	1	yes	characteristic
21	C₉×C₉₇	C₃³	1	1	yes	characteristic
22	C₉₇⋊C₉	C₃×C₉	1	6	yes	—
23	C₃×C₉₇	—	3	1	no	—
24	C₃×C₉₇	C₃²×C₉	1	1	yes	characteristic
25	C₉₇⋊C₃	—	3	3	no	—
26	#6903	—	97	1	no	maximal, Sylow
27	C₉₇	#6903	1	1	yes	characteristic, complemented, Sylow
28	#737	—	97	1	no	—
29	C₂₇⋊C₃	—	97	2	no	—
30	C₂₇⋊C₃	—	97	3	no	—
31	C₂₇⋊C₃	—	97	3	no	—
32	C₃×C₂₇	—	97	3	no	—
33	C₃×C₂₇	C₉₇⋊C₃	1	1	yes	characteristic, complemented
34	3₋¹⁺²	—	97	2	no	—
35	3₋¹⁺²	—	97	3	no	—
36	3₋¹⁺²	—	97	3	no	—
37	H(3,𝔽₃)	—	97	1	no	—
38	C₃×C₉	—	97	3	no	—
39	C₃×C₉	C₃×C₉₇⋊C₃	1	1	yes	characteristic
40	C₂₇	—	97	6	no	—
41	C₂₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic
42	C₂₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic
43	C₂₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic
44	C₃²	—	97	3	no	—
45	C₃²	C₉×C₉₇⋊C₃	1	1	yes	characteristic
46	C₉	—	97	6	no	—
47	C₉	C₉×C₉₇⋊C₃	1	1	yes	characteristic
48	C₉	C₉×C₉₇⋊C₃	1	1	yes	characteristic
49	C₉	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
50	C₃	—	3	1	no	—
51	C₃	—	291	3	no	—
52	C₃	C₃×C₉×C₉₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂₇	C₃×C₉₇⋊C₃
commutator, socle	C₃×C₉₇	C₃²×C₉
Frattini	C₉	C₃²×C₉₇⋊C₃
Fitting	C₃×C₂₇×C₉₇	C₃
prime radical	C₃²⋊(C₉₇⋊C₃)	C₉
cosocle	C₉×C₉₇	C₃³
hypercenter	C₃×C₂₇	C₉₇⋊C₃
hypocenter	C₉₇	#6903
LCS	G ▷ C₃×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇

Complements

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

N	H ≅ G/N	type
2. #21463357	C₃	semidirect
3. #21463356	C₃	semidirect
4. #21463355	C₃	semidirect
5. C₂₇×C₉₇⋊C₃	C₃	semidirect
6. C₃×C₂₇×C₉₇	C₃	semidirect
27. C₉₇	#6903	semidirect
33. C₃×C₂₇	C₉₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#21463377	3
#21463357	3
#21463356	3
#21463355	3
C₂₇×C₉₇⋊C₃	3
C₃×C₂₇×C₉₇	3
#6903	97