Subgroups — C₃₁⋊C₃×C₈₁⋊C₃ #36382062

Order 22599.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₇×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#21419258	C₃	1	2	yes	maximal, complemented
3	#21419257	C₃	1	3	yes	maximal, complemented
4	#21419256	C₃	1	3	yes	maximal, complemented
5	C₈₁×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
6	C₃₁×C₈₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₁⋊C₃	C₉	1	1	yes	characteristic
8	C₃×C₃₁⋊C₂₇	C₉	1	2	yes	—
9	C₂₇×C₃₁⋊C₃	C₉	1	2	yes	—
10	C₂₇×C₃₁⋊C₃	C₉	1	2	yes	—
11	C₂₇×C₃₁⋊C₃	C₉	1	2	yes	—
12	C₂₇×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
13	C₂₇×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
14	C₂₇×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
15	C₃₁⋊C₈₁	C₃²	1	6	yes	complemented
16	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
17	C₃₁×C₈₁	C₃²	1	3	yes	complemented
18	C₃²×C₃₁⋊C₃	C₂₇	1	1	yes	characteristic
19	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
20	C₃×C₃₁⋊C₉	C₂₇	1	2	yes	—
21	C₉×C₃₁⋊C₃	C₂₇	1	2	yes	—
22	C₉×C₃₁⋊C₃	C₂₇	1	2	yes	—
23	C₉×C₃₁⋊C₃	C₂₇	1	2	yes	—
24	C₉×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
25	C₉×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
26	C₉×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
27	C₂₇×C₃₁	C₃×C₉	1	2	yes	—
28	C₂₇×C₃₁	C₃³	1	1	yes	characteristic
29	C₃₁⋊C₂₇	C₃×C₉	1	2	yes	—
30	C₃₁⋊C₂₇	C₃×C₉	1	2	yes	—
31	C₃₁⋊C₂₇	C₃×C₉	1	2	yes	—
32	C₃×C₈₁⋊C₃	—	31	1	no	maximal, Sylow
33	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
34	C₃×C₃₁⋊C₃	—	3	3	no	—
35	C₃×C₃₁⋊C₃	C₃×C₂₇	1	1	yes	characteristic
36	C₃×C₃₁⋊C₃	C₃×C₂₇	1	1	yes	characteristic
37	C₃×C₃₁⋊C₃	C₃×C₂₇	1	1	yes	characteristic
38	C₉×C₃₁	C₃×C₂₇	1	2	yes	—
39	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
40	C₃₁⋊C₉	C₃×C₂₇	1	2	yes	—
41	C₃₁⋊C₉	C₃×C₂₇	1	2	yes	—
42	C₃₁⋊C₉	C₃×C₂₇	1	2	yes	—
43	C₃²×C₂₇	—	31	1	no	—
44	C₈₁⋊C₃	—	31	2	no	—
45	C₈₁⋊C₃	—	31	3	no	—
46	C₈₁⋊C₃	—	31	3	no	—
47	C₈₁⋊C₃	C₃₁⋊C₃	1	1	yes	characteristic, direct product factor
48	C₃×C₈₁	—	31	3	no	—
49	C₃×C₃₁	—	3	1	no	—
50	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
51	C₃₁⋊C₃	—	3	1	no	—
52	C₃₁⋊C₃	—	3	1	no	—
53	C₃₁⋊C₃	C₈₁⋊C₃	1	3	yes	direct product factor
54	C₃²×C₉	—	31	1	no	—
55	C₃×C₂₇	—	31	2	no	—
56	C₃×C₂₇	—	31	2	no	—
57	C₃×C₂₇	—	31	2	no	—
58	C₃×C₂₇	—	31	2	no	—
59	C₃×C₂₇	—	31	1	no	—
60	C₃×C₂₇	—	31	1	no	—
61	C₃×C₂₇	—	31	1	no	—
62	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
63	C₈₁	—	31	6	no	—
64	C₈₁	C₃×C₃₁⋊C₃	1	3	yes	complemented
65	C₃₁	C₃×C₈₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
66	C₃³	—	31	1	no	—
67	C₃×C₉	—	31	2	no	—
68	C₃×C₉	—	31	2	no	—
69	C₃×C₉	—	31	2	no	—
70	C₃×C₉	—	31	2	no	—
71	C₃×C₉	—	31	1	no	—
72	C₃×C₉	—	31	1	no	—
73	C₃×C₉	—	31	1	no	—
74	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
75	C₂₇	—	31	2	no	—
76	C₂₇	—	31	2	no	—
77	C₂₇	—	31	2	no	—
78	C₂₇	C₉×C₃₁⋊C₃	1	2	yes	—
79	C₂₇	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
80	C₃²	—	31	1	no	—
81	C₃²	—	31	1	no	—
82	C₃²	—	31	1	no	—
83	C₃²	—	93	3	no	—
84	C₃²	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
85	C₉	—	31	2	no	—
86	C₉	—	31	2	no	—
87	C₉	—	31	2	no	—
88	C₉	C₂₇×C₃₁⋊C₃	1	2	yes	—
89	C₉	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
90	C₃	—	3	1	no	—
91	C₃	—	31	3	no	—
92	C₃	—	93	1	no	—
93	C₃	—	93	1	no	—
94	C₃	C₃×C₂₇×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂₇	C₃²×C₃₁⋊C₃
commutator, socle	C₃×C₃₁	C₃²×C₂₇
Fitting	C₃₁×C₈₁⋊C₃	C₃
prime radical	C₃²×C₃₁⋊C₃	C₂₇
cosocle	C₂₇×C₃₁	C₃³
hypercenter	C₈₁⋊C₃	C₃₁⋊C₃
hypocenter	C₃₁	C₃×C₈₁⋊C₃
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂₇ ◁ C₈₁⋊C₃

Complements

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

N	H ≅ G/N	type
2. #21419258	C₃	semidirect
3. #21419257	C₃	semidirect
4. #21419256	C₃	semidirect
5. C₈₁×C₃₁⋊C₃	C₃	semidirect
6. C₃₁×C₈₁⋊C₃	C₃	semidirect
15. C₃₁⋊C₈₁	C₃²	semidirect
17. C₃₁×C₈₁	C₃²	semidirect
47. C₈₁⋊C₃	C₃₁⋊C₃	direct (N × H)
53. C₃₁⋊C₃	C₈₁⋊C₃	direct (N × H)
64. C₈₁	C₃×C₃₁⋊C₃	semidirect
65. C₃₁	C₃×C₈₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂₇×C₃₁⋊C₃	3
#21419258	3
#21419257	3
#21419256	3
C₈₁×C₃₁⋊C₃	3
C₃₁×C₈₁⋊C₃	3
C₃×C₈₁⋊C₃	31

Subgroups — C31⋊C3×C81⋊C3 #36382062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁⋊C₃×C₈₁⋊C₃ #36382062