Subgroups — C₄⋊(C₃⋊(C₃⋊C₃₂)) #11773009

Order 1152.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₃⋊C₃₂)	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊C₃₂)	—	3	4	no	maximal
4	C₂×C₃²×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₃⋊(C₃⋊C₃₂)	—	2	2	no	—
8	C₃×C₄×C₁₆	D₃	1	4	yes	—
9	C₂×C₃⋊C₃₂	—	3	8	no	—
10	C₂×C₃²×C₈	C₈	1	1	yes	characteristic
11	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄²	C₈	1	1	yes	characteristic
13	C₃²×C₁₆	—	2	1	no	—
14	C₃²×C₁₆	D₄	1	1	yes	characteristic
15	C₃²×C₁₆	Q₈	1	1	yes	characteristic
16	C₄⋊C₃₂	—	9	1	no	Sylow
17	C₂×C₃×C₁₆	Dic₃	1	4	yes	—
18	C₂×C₃×C₁₆	C₂×D₃	1	4	yes	—
19	C₃×C₄×C₈	Dic₃	1	4	yes	—
20	C₃⋊C₃₂	—	6	8	no	—
21	C₂×C₃²×C₄	C₁₆	1	1	yes	characteristic
22	C₂×C₃²×C₄	C₁₆	1	1	yes	characteristic
23	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
24	C₃²×C₈	—	2	1	no	—
25	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
26	C₃²×C₈	M₁₆	1	1	yes	characteristic
27	C₂×C₃₂	—	9	2	no	—
28	C₄×C₁₆	C₃⋊D₃	1	1	yes	characteristic
29	C₂×C₃×C₈	C₃⋊C₈	1	4	yes	—
30	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
31	C₃×C₄²	C₃⋊C₈	1	4	yes	—
32	C₃×C₁₆	—	2	4	no	—
33	C₃×C₁₆	Dic₆	1	4	yes	—
34	C₃×C₁₆	D₁₂	1	4	yes	—
35	C₂²×C₃²	C₂×C₁₆	1	1	yes	characteristic
36	C₃²×C₄	—	2	1	no	—
37	C₃²×C₄	C₃₂	1	2	yes	complemented
38	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
39	C₃²×C₄	M₃₂	1	1	yes	characteristic
40	C₂×C₁₆	C₃⋊Dic₃	1	1	yes	characteristic
41	C₂×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
42	C₄×C₈	C₃⋊Dic₃	1	1	yes	characteristic
43	C₃₂	—	18	2	no	—
44	C₂×C₃×C₄	C₃⋊C₁₆	1	4	yes	—
45	C₂×C₃×C₄	C₃⋊C₁₆	1	4	yes	—
46	C₂×C₃×C₄	C₂×C₃⋊C₈	1	4	yes	—
47	C₃×C₈	—	2	4	no	—
48	C₃×C₈	#208	1	4	yes	—
49	C₃×C₈	C₄⋊Dic₃	1	4	yes	—
50	C₂×C₃²	C₄⋊C₁₆	1	1	yes	characteristic
51	C₂×C₃²	C₂×C₃₂	1	1	yes	characteristic
52	C₂×C₃²	M₆₄	1	1	yes	characteristic
53	C₂×C₈	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
54	C₂×C₈	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₄²	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	#634	1	1	yes	characteristic
58	C₁₆	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
59	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	—
60	C₃×C₄	—	2	4	no	—
61	C₃×C₄	C₃⋊C₃₂	1	8	yes	complemented
62	C₃×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
63	C₃×C₄	#812	1	4	yes	—
64	C₃²	C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄	C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
66	C₂×C₄	C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
67	C₂×C₄	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	#3748	1	1	yes	characteristic
70	C₈	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
71	C₂×C₃	C₄⋊(C₃⋊C₁₆)	1	4	yes	—
72	C₂×C₃	C₂×C₃⋊C₃₂	1	4	yes	—
73	C₂×C₃	#4499	1	4	yes	—
74	C₂²	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₃⋊(C₃⋊C₃₂)	1	2	yes	complemented
77	C₄	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
78	C₄	#63639	1	1	yes	characteristic
79	C₃	C₄⋊(C₃⋊C₃₂)	1	4	yes	complemented
80	C₂	C₄⋊(C₃⋊(C₃⋊C₁₆))	1	1	yes	characteristic
81	C₂	C₂×C₃⋊(C₃⋊C₃₂)	1	1	yes	characteristic
82	C₂	#10589365	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₃₂
Fitting	C₃²×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃²	C₂×C₁₆
cosocle	C₂×C₃²×C₁₆	C₂²
hypercenter	C₄×C₁₆	C₃⋊D₃
hypocenter	C₃²	C₄⋊C₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
37. C₃²×C₄	C₃₂	semidirect
61. C₃×C₄	C₃⋊C₃₂	semidirect
64. C₃²	C₄⋊C₃₂	semidirect
76. C₄	C₃⋊(C₃⋊C₃₂)	semidirect
79. C₃	C₄⋊(C₃⋊C₃₂)	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×C₁₆	2
C₂×C₃⋊(C₃⋊C₃₂)	2
C₄⋊(C₃⋊C₃₂)	3

Subgroups — C4⋊(C3⋊(C3⋊C32)) #11773009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊(C₃⋊(C₃⋊C₃₂)) #11773009