Subgroups — #36382056

Order 22599.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419296	C₃	1	1	yes	maximal, characteristic
2	#21419252	C₃	1	6	yes	maximal, complemented
3	#21419251	C₃	1	3	yes	maximal, complemented
4	#21419232	C₃	1	2	yes	maximal
5	C₃₁×#6869	C₃	1	1	yes	maximal, characteristic
6	#14578892	C₃²	1	6	yes	—
7	#14578887	C₃²	1	1	yes	characteristic
8	#14578887	C₃²	1	2	yes	—
9	#14578886	—	3	1	no	—
10	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
11	C₉×C₃₁⋊C₉	—	3	1	no	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃₁×C₂₇⋊C₃	C₃²	1	3	yes	—
14	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
15	C₃×C₉×C₃₁	—	3	1	no	—
16	C₃×C₉×C₃₁	—	3	2	no	—
17	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
18	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
19	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
20	C₃×C₃₁⋊C₉	—	3	1	no	—
21	C₃×C₃₁⋊C₉	—	3	2	no	—
22	C₂₇×C₃₁	—	3	3	no	—
23	C₃₁⋊C₂₇	—	3	6	no	—
24	#10624830	—	31	1	no	maximal, Sylow
25	C₃²×C₃₁	—	3	1	no	—
26	C₃²×C₃₁	—	9	1	no	—
27	C₃²×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
28	C₉×C₃₁	—	3	1	no	—
29	C₉×C₃₁	—	3	1	no	—
30	C₉×C₃₁	#737	1	1	yes	characteristic
31	C₉×C₃₁	#737	1	2	yes	—
32	C₃₁⋊C₉	—	9	1	no	—
33	C₃₁⋊C₉	—	9	2	no	—
34	#6887	—	31	1	no	—
35	#6875	—	31	3	no	—
36	#6875	—	31	6	no	—
37	#6869	—	31	2	no	—
38	#6869	C₃₁⋊C₃	1	1	yes	characteristic
39	C₃×C₃₁	—	3	1	no	—
40	C₃×C₃₁	—	9	1	no	—
41	C₃×C₃₁	#6895	1	1	yes	characteristic
42	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
43	C₂₇⋊C₃	—	31	6	no	—
44	C₂₇⋊C₃	C₃×C₃₁⋊C₃	1	3	yes	—
45	C₉⋊C₉	—	31	1	no	—
46	C₉⋊C₉	—	31	2	no	—
47	C₉⋊C₉	—	93	1	no	—
48	C₃²⋊C₉	—	93	1	no	—
49	C₉²	—	93	1	no	—
50	C₃₁	#10624830	1	1	yes	characteristic, complemented, Sylow
51	C₃³	#11719687	1	1	yes	characteristic
52	C₃×C₉	—	3	1	no	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	93	1	no	—
55	C₃×C₉	—	93	2	no	—
56	C₃×C₉	#11719687	1	1	yes	characteristic
57	C₃×C₉	#11719687	1	1	yes	characteristic
58	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
59	C₂₇	—	3	3	no	—
60	C₂₇	—	93	6	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	9	1	no	—
63	C₃²	C₃×#11719687	1	1	yes	characteristic
64	C₉	—	3	1	no	—
65	C₉	—	3	1	no	—
66	C₉	—	279	1	no	—
67	C₉	—	279	2	no	—
68	C₉	#14578911	1	1	yes	characteristic
69	C₉	#14578913	1	2	yes	—
70	C₃	—	3	1	no	—
71	C₃	—	9	1	no	—
72	C₃	#21419333	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419333
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6869	C₃
socle	C₃×C₃₁	#6895
prime radical	C₃³×C₃₁	3₋¹⁺²
hypercenter	#6869	C₃₁⋊C₃
hypocenter	C₃₁	#10624830
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
2. #21419252	C₃	semidirect
3. #21419251	C₃	semidirect
50. C₃₁	#10624830	semidirect

Maximal subgroups

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

subgroup	index
#21419296	3
#21419252	3
#21419251	3
#21419232	3
C₃₁×#6869	3
#10624830	31