Subgroups — C₁₁²×(C₃⋊D₃)⋊D₇ #44479223

Order 30492.

616 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₁²×D₂₁	C₂	1	2	yes	maximal, complemented
2	C₇×C₁₁²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₁₁²×D₇	—	3	2	no	maximal
4	C₃²×C₇×C₁₁²	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₁₁²×D₂₁	D₃	1	2	yes	complemented
6	D₃×C₇×C₁₁²	—	3	2	no	—
7	D₃×C₇×C₁₁²	—	6	1	no	—
8	C₃×C₁₁²×D₇	—	3	2	no	—
9	D₃²×C₁₁²	—	7	1	no	maximal, Hall
10	C₂×C₁₁²×D₇	—	9	1	no	Hall
11	C₁₁×(C₃⋊D₃)⋊D₇	C₁₁	1	12	yes	maximal, direct product factor
12	C₃×C₇×C₁₁²	—	2	1	no	—
13	C₃×C₇×C₁₁²	C₂×D₃	1	2	yes	complemented
14	C₁₁²×C₃⋊D₃	D₇	1	1	yes	characteristic, complemented
15	C₃×D₃×C₁₁²	—	7	2	no	—
16	C₂×C₇×C₁₁²	—	9	1	no	—
17	C₁₁²×D₇	—	3	2	no	—
18	C₂×D₃×C₁₁²	—	21	2	no	—
19	C₇×C₁₁×C₃⋊D₃	C₂×C₁₁	1	12	yes	complemented
20	C₃×C₁₁×D₂₁	C₂×C₁₁	1	24	yes	complemented
21	C₃²×C₁₁²	C₂×D₇	1	1	yes	characteristic, complemented, Hall
22	D₃×C₁₁×D₇	—	3	24	no	—
23	C₇×C₁₁²	D₃²	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₁₁²	—	21	2	no	—
25	D₃×C₁₁²	—	3	2	no	—
26	D₃×C₁₁²	—	6	1	no	—
27	D₃×C₁₁²	—	7	2	no	—
28	C₃²×C₇×C₁₁	C₂²×C₁₁	1	12	yes	complemented
29	C₂²×C₁₁²	—	63	1	no	Hall
30	C₁₁×D₂₁	D₃×C₁₁	1	24	yes	complemented
31	D₃×C₇×C₁₁	—	3	24	no	—
32	D₃×C₇×C₁₁	—	6	12	no	—
33	C₃×C₁₁×D₇	—	3	24	no	—
34	D₃²×C₁₁	—	7	12	no	—
35	C₃×C₁₁²	—	2	1	no	—
36	C₃×C₁₁²	D₃×D₇	1	2	yes	complemented
37	C₂×C₁₁×D₇	—	9	12	no	—
38	(C₃⋊D₃)⋊D₇	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₁₁²	—	9	1	no	—
40	C₂×C₁₁²	—	21	2	no	—
41	C₃×C₇×C₁₁	—	2	12	no	—
42	C₃×C₇×C₁₁	C₂×D₃×C₁₁	1	24	yes	complemented
43	C₁₁×C₃⋊D₃	C₁₁×D₇	1	12	yes	complemented
44	C₃×D₃×C₁₁	—	7	24	no	—
45	C₂×C₇×C₁₁	—	9	12	no	—
46	C₁₁×D₇	—	3	24	no	—
47	C₂×D₃×C₁₁	—	21	24	no	—
48	C₇×C₃⋊D₃	C₂×C₁₁²	1	1	yes	characteristic, complemented
49	C₃×D₂₁	C₂×C₁₁²	1	2	yes	complemented
50	C₁₁²	(C₃⋊D₃)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
51	C₃²×C₁₁	C₂×C₁₁×D₇	1	12	yes	complemented
52	D₃×D₇	—	3	2	no	—
53	C₇×C₁₁	D₃²×C₁₁	1	12	yes	complemented
54	C₂×C₃×C₁₁	—	21	24	no	—
55	D₃×C₁₁	—	3	24	no	—
56	D₃×C₁₁	—	6	12	no	—
57	D₃×C₁₁	—	7	24	no	—
58	C₃²×C₇	C₂²×C₁₁²	1	1	yes	characteristic, complemented, Hall
59	C₂²×C₁₁	—	63	12	no	—
60	D₂₁	D₃×C₁₁²	1	2	yes	complemented
61	C₃×D₇	—	3	2	no	—
62	D₃×C₇	—	3	2	no	—
63	D₃×C₇	—	6	1	no	—
64	D₃²	—	7	1	no	Hall
65	C₃×C₁₁	—	2	12	no	—
66	C₃×C₁₁	D₃×C₁₁×D₇	1	24	yes	complemented
67	C₂×D₇	—	9	1	no	Hall
68	C₂×C₁₁	—	9	12	no	—
69	C₂×C₁₁	—	21	24	no	—
70	C₃×C₇	—	2	1	no	—
71	C₃×C₇	C₂×D₃×C₁₁²	1	2	yes	complemented
72	C₃⋊D₃	C₁₁²×D₇	1	1	yes	characteristic, complemented
73	C₃×D₃	—	7	2	no	—
74	C₂×C₇	—	9	1	no	—
75	D₇	—	3	2	no	—
76	C₂×D₃	—	21	2	no	—
77	C₁₁	C₁₁×(C₃⋊D₃)⋊D₇	1	12	yes	direct product factor
78	C₃²	C₂×C₁₁²×D₇	1	1	yes	characteristic, complemented, Sylow
79	C₇	D₃²×C₁₁²	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₃	—	21	2	no	—
81	D₃	—	3	2	no	—
82	D₃	—	6	1	no	—
83	D₃	—	7	2	no	—
84	C₂²	—	63	1	no	Sylow
85	C₃	—	2	1	no	—
86	C₃	D₃×C₁₁²×D₇	1	2	yes	complemented
87	C₂	—	9	1	no	—
88	C₂	—	21	2	no	—

	subgroup	G/H
center, hypercenter	C₁₁²	(C₃⋊D₃)⋊D₇
commutator, cosocle, hypocenter	C₃²×C₇	C₂²×C₁₁²
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₁₁²	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₁₁²

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₁²×D₂₁	C₂	semidirect
2. C₇×C₁₁²×C₃⋊D₃	C₂	semidirect
4. C₃²×C₇×C₁₁²	C₂²	semidirect
5. C₁₁²×D₂₁	D₃	semidirect
11. C₁₁×(C₃⋊D₃)⋊D₇	C₁₁	direct (N × H)
13. C₃×C₇×C₁₁²	C₂×D₃	semidirect
14. C₁₁²×C₃⋊D₃	D₇	semidirect
19. C₇×C₁₁×C₃⋊D₃	C₂×C₁₁	semidirect
20. C₃×C₁₁×D₂₁	C₂×C₁₁	semidirect
21. C₃²×C₁₁²	C₂×D₇	semidirect
23. C₇×C₁₁²	D₃²	semidirect
28. C₃²×C₇×C₁₁	C₂²×C₁₁	semidirect
30. C₁₁×D₂₁	D₃×C₁₁	semidirect
36. C₃×C₁₁²	D₃×D₇	semidirect
38. (C₃⋊D₃)⋊D₇	C₁₁²	direct (N × H)
42. C₃×C₇×C₁₁	C₂×D₃×C₁₁	semidirect
43. C₁₁×C₃⋊D₃	C₁₁×D₇	semidirect
48. C₇×C₃⋊D₃	C₂×C₁₁²	semidirect
49. C₃×D₂₁	C₂×C₁₁²	semidirect
50. C₁₁²	(C₃⋊D₃)⋊D₇	direct (N × H)
51. C₃²×C₁₁	C₂×C₁₁×D₇	semidirect
53. C₇×C₁₁	D₃²×C₁₁	semidirect
58. C₃²×C₇	C₂²×C₁₁²	semidirect
60. D₂₁	D₃×C₁₁²	semidirect
66. C₃×C₁₁	D₃×C₁₁×D₇	semidirect
71. C₃×C₇	C₂×D₃×C₁₁²	semidirect
72. C₃⋊D₃	C₁₁²×D₇	semidirect
77. C₁₁	C₁₁×(C₃⋊D₃)⋊D₇	direct (N × H)
78. C₃²	C₂×C₁₁²×D₇	semidirect
79. C₇	D₃²×C₁₁²	semidirect
86. C₃	D₃×C₁₁²×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₁₁²×D₂₁	2
C₇×C₁₁²×C₃⋊D₃	2
D₃×C₁₁²×D₇	3
D₃²×C₁₁²	7
C₁₁×(C₃⋊D₃)⋊D₇	11

Subgroups — C112×(C3⋊D3)⋊D7 #44479223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁²×(C₃⋊D₃)⋊D₇ #44479223