Subgroups — C₉₇×#6894 #37555734

Order 23571.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₉₇	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₉₇	C₃	1	1	yes	maximal, characteristic
3	C₉⋊C₉×C₉₇	C₃	1	6	yes	maximal, complemented
4	C₉⋊C₉×C₉₇	C₃	1	3	yes	maximal, complemented
5	C₃²⋊C₉×C₉₇	C₃	1	2	yes	maximal
6	C₃³×C₉₇	C₃²	1	1	yes	characteristic
7	3₋¹⁺²×C₉₇	—	3	3	no	—
8	C₃×C₉×C₉₇	—	3	3	no	—
9	C₃×C₉×C₉₇	C₃²	1	6	yes	—
10	C₃×C₉×C₉₇	C₃²	1	2	yes	—
11	C₃×C₉×C₉₇	C₃²	1	3	yes	—
12	C₃×C₉×C₉₇	C₃²	1	1	yes	characteristic
13	C₃²×C₉₇	—	3	1	no	—
14	C₃²×C₉₇	—	3	2	no	—
15	C₃²×C₉₇	3₋¹⁺²	1	3	yes	—
16	C₃²×C₉₇	C₃³	1	1	yes	characteristic
17	C₉×C₉₇	—	3	2	no	—
18	C₉×C₉₇	—	3	3	no	—
19	C₉×C₉₇	—	3	6	no	—
20	C₉×C₉₇	3₋¹⁺²	1	3	yes	complemented
21	C₃×C₉₇	—	3	3	no	—
22	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₉₇	C₃×3₋¹⁺²	1	1	yes	characteristic
24	C₃×C₉₇	#737	1	2	yes	—
25	#6894	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₉₇	#6894	1	1	yes	characteristic, direct product factor, Sylow
27	C₃×3₋¹⁺²	C₃×C₉₇	1	1	yes	characteristic
28	C₃²×C₉	C₃×C₉₇	1	1	yes	characteristic
29	C₉⋊C₉	C₃×C₉₇	1	6	yes	complemented
30	C₉⋊C₉	C₃×C₉₇	1	3	yes	complemented
31	C₃²⋊C₉	C₃×C₉₇	1	2	yes	—
32	C₃³	C₃²×C₉₇	1	1	yes	characteristic
33	3₋¹⁺²	—	3	3	no	—
34	C₃×C₉	—	3	3	no	—
35	C₃×C₉	C₃²×C₉₇	1	6	yes	—
36	C₃×C₉	C₃²×C₉₇	1	2	yes	—
37	C₃×C₉	C₃²×C₉₇	1	3	yes	—
38	C₃×C₉	C₃²×C₉₇	1	1	yes	characteristic
39	C₃²	—	3	1	no	—
40	C₃²	—	3	2	no	—
41	C₃²	3₋¹⁺²×C₉₇	1	3	yes	—
42	C₃²	C₃³×C₉₇	1	1	yes	characteristic
43	C₉	—	3	2	no	—
44	C₉	—	3	3	no	—
45	C₉	—	3	6	no	—
46	C₉	3₋¹⁺²×C₉₇	1	3	yes	complemented
47	C₃	—	3	3	no	—
48	C₃	C₃×3₋¹⁺²×C₉₇	1	1	yes	characteristic
49	C₃	C₃×3₋¹⁺²×C₉₇	1	1	yes	characteristic
50	C₃	#737×C₉₇	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₉₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃³×C₉₇
prime radical	C₃³×C₉₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₉₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₉⋊C₉×C₉₇	C₃	semidirect
4. C₉⋊C₉×C₉₇	C₃	semidirect
20. C₉×C₉₇	3₋¹⁺²	semidirect
25. #6894	C₉₇	direct (N × H)
26. C₉₇	#6894	direct (N × H)
29. C₉⋊C₉	C₃×C₉₇	semidirect
30. C₉⋊C₉	C₃×C₉₇	semidirect
46. C₉	3₋¹⁺²×C₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₉₇	3
C₃²×C₉×C₉₇	3
C₉⋊C₉×C₉₇	3
C₃²⋊C₉×C₉₇	3
#6894	97

Subgroups — C97×#6894 #37555734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#6894 #37555734