Subgroups — #37555950

Order 23571.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21463378	C₃	1	2	yes	maximal, complemented
2	#21463376	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₉₇⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	#21463366	C₃	1	2	yes	maximal, complemented
5	#21463364	C₃	1	1	yes	maximal, characteristic, complemented
6	#21463363	C₃	1	2	yes	maximal, complemented
7	#21463361	C₃	1	1	yes	maximal, characteristic, complemented
8	#21463359	C₃	1	2	yes	maximal, complemented
9	AΣL(1,𝔽₂₇)×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
10	#14579757	—	3	2	no	—
11	#14579757	C₃²	1	2	yes	—
12	#14579757	C₃²	1	2	yes	—
13	#14579756	—	3	2	no	—
14	#14579755	—	3	2	no	—
15	C₃²⋊(C₉₇⋊C₃)	C₃²	1	2	yes	—
16	C₃×C₉₇⋊C₉	—	3	1	no	—
17	C₃×C₉₇⋊C₉	—	3	2	no	—
18	C₃×C₉₇⋊C₉	—	3	1	no	—
19	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
20	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₉₇⋊C₉	C₃²	1	1	yes	characteristic
22	C₉×C₉₇⋊C₃	—	3	2	no	—
23	C₃³×C₉₇	C₃²	1	1	yes	characteristic
24	3₋¹⁺²×C₉₇	C₃²	1	2	yes	—
25	H(3,𝔽₃)×C₉₇	C₃²	1	1	yes	characteristic
26	C₃²×C₉₇	—	3	3	no	—
27	C₃²×C₉₇	—	3	1	no	—
28	C₃²×C₉₇	—	3	1	no	—
29	C₃²×C₉₇	C₃³	1	1	yes	characteristic
30	C₃×C₉₇⋊C₃	—	3	2	no	—
31	C₉×C₉₇	—	3	2	no	—
32	C₉₇⋊C₉	—	3	2	no	—
33	C₉₇⋊C₉	—	3	1	no	—
34	C₉₇⋊C₉	—	3	1	no	—
35	C₉₇⋊C₉	—	3	2	no	—
36	C₉₇⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
37	C₉₇⋊C₉	H(3,𝔽₃)	1	2	yes	—
38	C₃×C₉₇	—	3	3	no	—
39	C₃×C₉₇	—	3	1	no	—
40	C₃×C₉₇	—	9	1	no	—
41	C₃×C₉₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	C₉₇⋊C₃	—	9	2	no	—
43	#6908	—	97	1	no	maximal, Sylow
44	C₉₇	#6908	1	1	yes	characteristic, complemented, Sylow
45	#737	—	97	2	no	—
46	#737	—	97	1	no	—
47	C₃²×C₉	—	97	1	no	—
48	#733	—	97	2	no	—
49	#732	—	97	1	no	—
50	#731	—	97	2	no	—
51	#731	—	97	1	no	—
52	AΣL(1,𝔽₂₇)	—	97	2	no	—
53	AΣL(1,𝔽₂₇)	C₉₇⋊C₃	1	1	yes	characteristic, complemented
54	C₃³	C₃×C₉₇⋊C₃	1	1	yes	characteristic
55	3₋¹⁺²	—	291	2	no	—
56	3₋¹⁺²	—	291	2	no	—
57	3₋¹⁺²	—	291	2	no	—
58	3₋¹⁺²	—	97	2	no	—
59	3₋¹⁺²	—	97	2	no	—
60	3₋¹⁺²	C₃×C₉₇⋊C₃	1	2	yes	—
61	H(3,𝔽₃)	—	97	2	no	—
62	H(3,𝔽₃)	C₃×C₉₇⋊C₃	1	1	yes	characteristic
63	C₃×C₉	—	291	2	no	—
64	C₃×C₉	—	291	1	no	—
65	C₃×C₉	—	291	2	no	—
66	C₃×C₉	—	291	1	no	—
67	C₃×C₉	—	97	1	no	—
68	C₃×C₉	—	97	1	no	—
69	C₃×C₉	—	97	1	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	291	2	no	—
74	C₃²	C₃²×C₉₇⋊C₃	1	1	yes	characteristic
75	C₉	—	3	2	no	—
76	C₉	—	291	1	no	—
77	C₉	—	291	1	no	—
78	C₉	—	291	2	no	—
79	C₉	—	291	2	no	—
80	C₉	—	97	1	no	—
81	C₉	—	97	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	3	no	—
84	C₃	—	9	1	no	—
85	C₃	—	873	2	no	—
86	C₃	H(3,𝔽₃)×C₉₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₉₇⋊C₃
commutator, cosocle	C₃²×C₉₇	C₃³
Frattini	C₃²	C₃²×C₉₇⋊C₃
Fitting	AΣL(1,𝔽₂₇)×C₉₇	C₃
socle	C₃×C₉₇	C₃×H(3,𝔽₃)
hypercenter	AΣL(1,𝔽₂₇)	C₉₇⋊C₃
hypocenter	C₉₇	#6908
LCS	G ▷ C₃²×C₉₇ ▷ C₃×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. #21463378	C₃	semidirect
2. #21463376	C₃	semidirect
3. C₃²×C₉₇⋊C₉	C₃	semidirect
4. #21463366	C₃	semidirect
5. #21463364	C₃	semidirect
6. #21463363	C₃	semidirect
7. #21463361	C₃	semidirect
8. #21463359	C₃	semidirect
9. AΣL(1,𝔽₂₇)×C₉₇	C₃	semidirect
44. C₉₇	#6908	semidirect
53. AΣL(1,𝔽₂₇)	C₉₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#21463378	3
#21463376	3
C₃²×C₉₇⋊C₉	3
#21463366	3
#21463364	3
#21463363	3
#21463361	3
#21463359	3
AΣL(1,𝔽₂₇)×C₉₇	3
#6908	97