Subgroups — C₂³×C₃⋊#157 #11724749

Order 864.

804 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊#157	C₂	1	14	yes	maximal, direct product factor
3	C₂³×#157	—	3	1	no	maximal
4	C₂³×C₃³	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₃×C₃⋊D₃	C₄	1	7	yes	complemented
6	C₂²×C₃×C₃⋊D₃	C₂²	1	7	yes	—
7	C₂×C₃⋊#157	C₂²	1	28	yes	direct product factor
8	C₂³×C₃⋊D₃	D₃	1	1	yes	characteristic
9	C₂³×C₃×D₃	—	6	2	no	—
10	C₂²×#157	—	3	14	no	—
11	C₂²×C₃³	C₂×C₄	1	7	yes	complemented
12	C₂×C₃×C₃⋊D₃	C₂×C₄	1	21	yes	complemented
13	C₂×C₃×C₃⋊D₃	C₂³	1	7	yes	—
14	C₃⋊#157	C₂³	1	8	yes	direct product factor
15	C₂³×Dic₃	—	9	1	no	maximal
16	C₂³×C₃²	—	2	2	no	—
17	C₂³×C₃²	—	4	2	no	—
18	C₂³×C₃²	Dic₃	1	1	yes	characteristic, complemented
19	C₂²×C₃⋊D₃	Dic₃	1	7	yes	complemented
20	C₂²×C₃⋊D₃	C₂×D₃	1	7	yes	—
21	C₂²×C₃×D₃	—	6	14	no	—
22	C₂²×C₃×D₃	—	6	14	no	—
23	C₂×#157	—	3	28	no	—
24	C₂×C₃³	C₂²×C₄	1	7	yes	complemented
25	C₃×C₃⋊D₃	C₂²×C₄	1	7	yes	complemented
26	C₃×C₃⋊D₃	C₂⁴	1	1	yes	characteristic
27	C₂⁴×C₃	—	9	1	no	—
28	C₂³×D₃	—	6	2	no	—
29	C₂²×Dic₃	—	9	14	no	—
30	C₂²×C₃²	—	2	14	no	—
31	C₂²×C₃²	—	4	14	no	—
32	C₂²×C₃²	C₂×Dic₃	1	7	yes	complemented
33	C₂×C₃⋊D₃	C₂×Dic₃	1	21	yes	complemented
34	C₂×C₃⋊D₃	C₂²×D₃	1	7	yes	—
35	C₂×C₃×D₃	—	6	42	no	—
36	C₂×C₃×D₃	—	6	14	no	—
37	#157	—	3	8	no	—
38	C₂³×C₄	—	27	1	no	Sylow
39	C₃³	C₂³×C₄	1	1	yes	characteristic, complemented, Sylow
40	C₂³×C₃	—	2	2	no	—
41	C₂³×C₃	—	4	2	no	—
42	C₂³×C₃	—	9	7	no	—
43	C₂³×C₃	—	9	7	no	—
44	C₂³×C₃	#157	1	1	yes	characteristic, complemented
45	C₂²×D₃	—	6	14	no	—
46	C₂²×D₃	—	6	14	no	—
47	C₂×Dic₃	—	9	28	no	—
48	C₂×C₃²	—	2	14	no	—
49	C₂×C₃²	—	4	14	no	—
50	C₂×C₃²	C₂²×Dic₃	1	7	yes	complemented
51	C₃⋊D₃	C₂²×Dic₃	1	7	yes	complemented
52	C₃⋊D₃	C₂³×D₃	1	1	yes	characteristic
53	C₃×D₃	—	6	14	no	—
54	C₃×D₃	—	6	2	no	—
55	C₂⁴	—	9	1	no	—
56	C₂²×C₄	—	27	14	no	—
57	C₂²×C₃	—	2	14	no	—
58	C₂²×C₃	—	4	14	no	—
59	C₂²×C₃	—	9	7	no	—
60	C₂²×C₃	—	9	21	no	—
61	C₂²×C₃	C₂×#157	1	7	yes	complemented
62	C₂×D₃	—	6	42	no	—
63	C₂×D₃	—	6	14	no	—
64	Dic₃	—	9	8	no	—
65	C₃²	—	2	2	no	—
66	C₃²	—	4	2	no	—
67	C₃²	C₂³×Dic₃	1	1	yes	characteristic, complemented
68	C₂³	—	9	7	no	—
69	C₂³	—	9	7	no	—
70	C₂³	C₃⋊#157	1	1	yes	characteristic, direct product factor
71	C₂×C₄	—	27	28	no	—
72	C₂×C₃	—	2	14	no	—
73	C₂×C₃	—	4	14	no	—
74	C₂×C₃	—	9	1	no	—
75	C₂×C₃	—	9	7	no	—
76	C₂×C₃	C₂²×#157	1	7	yes	complemented
77	D₃	—	6	14	no	—
78	D₃	—	6	2	no	—
79	C₂²	—	9	7	no	—
80	C₂²	—	9	21	no	—
81	C₂²	C₂×C₃⋊#157	1	7	yes	direct product factor
82	C₄	—	27	8	no	—
83	C₃	—	2	2	no	—
84	C₃	—	4	2	no	—
85	C₃	C₂³×#157	1	1	yes	characteristic, complemented
86	C₂	—	9	1	no	—
87	C₂	—	9	7	no	—
88	C₂	C₂²×C₃⋊#157	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	C₃⋊#157
commutator, hypocenter	C₃³	C₂³×C₄
Frattini	C₁	G
Fitting, socle	C₂³×C₃³	C₄
prime radical	C₂³×C₃×C₃⋊D₃	C₂
cosocle	C₃×C₃⋊D₃	C₂⁴
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊#157	C₂	direct (N × H)
4. C₂³×C₃³	C₄	semidirect
5. C₂²×C₃×C₃⋊D₃	C₄	semidirect
7. C₂×C₃⋊#157	C₂²	direct (N × H)
11. C₂²×C₃³	C₂×C₄	semidirect
12. C₂×C₃×C₃⋊D₃	C₂×C₄	semidirect
14. C₃⋊#157	C₂³	direct (N × H)
18. C₂³×C₃²	Dic₃	semidirect
19. C₂²×C₃⋊D₃	Dic₃	semidirect
24. C₂×C₃³	C₂²×C₄	semidirect
25. C₃×C₃⋊D₃	C₂²×C₄	semidirect
32. C₂²×C₃²	C₂×Dic₃	semidirect
33. C₂×C₃⋊D₃	C₂×Dic₃	semidirect
39. C₃³	C₂³×C₄	semidirect
44. C₂³×C₃	#157	semidirect
50. C₂×C₃²	C₂²×Dic₃	semidirect
51. C₃⋊D₃	C₂²×Dic₃	semidirect
61. C₂²×C₃	C₂×#157	semidirect
67. C₃²	C₂³×Dic₃	semidirect
70. C₂³	C₃⋊#157	direct (N × H)
76. C₂×C₃	C₂²×#157	semidirect
81. C₂²	C₂×C₃⋊#157	direct (N × H)
85. C₃	C₂³×#157	semidirect
88. C₂	C₂²×C₃⋊#157	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₃⋊D₃	2
C₂²×C₃⋊#157	2
C₂³×#157	3
C₂³×Dic₃	9

Subgroups — C23×C3⋊#157 #11724749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃⋊#157 #11724749