Group #36385227 — subgroups

Subgroups — C₈₉×D₄⋊C₃₂ #36385227

Order 22784.

138 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₆×C₈₉	—	2	1	no	—
5	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₈₉	—	2	1	no	—
7	C₂×C₃₂×C₈₉	—	2	1	no	—
8	C₄⋊C₁₆×C₈₉	C₄	1	1	yes	characteristic
9	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
10	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₈₉	—	2	1	no	—
12	C₄×D₄×C₈₉	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	—	2	1	no	—
14	C₂×C₁₆×C₈₉	—	4	1	no	—
15	C₂×C₁₆×C₈₉	—	4	1	no	—
16	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
18	C₄⋊C₈×C₈₉	C₈	1	1	yes	characteristic
19	C₂²⋊C₈×C₈₉	—	2	1	no	—
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₈₉	—	2	2	no	—
22	C₃₂×C₈₉	—	4	1	no	—
23	C₂×D₄×C₈₉	C₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	—	2	1	no	—
25	C₂×C₈×C₈₉	—	2	1	no	—
26	C₂×C₈×C₈₉	—	4	1	no	—
27	C₂×C₈×C₈₉	—	4	1	no	—
28	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₁₆	1	1	yes	characteristic
31	C₂²⋊C₄×C₈₉	—	2	1	no	—
32	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₈₉	—	2	1	no	—
34	C₁₆×C₈₉	—	4	1	no	—
35	C₁₆×C₈₉	D₈	1	1	yes	characteristic
36	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
37	C₂³×C₈₉	—	2	1	no	—
38	D₄×C₈₉	—	2	1	no	—
39	D₄×C₈₉	C₃₂	1	2	yes	complemented
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	—	4	1	no	—
43	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
46	C₈×C₈₉	—	2	1	no	—
47	C₈×C₈₉	—	4	1	no	—
48	C₈×C₈₉	#102	1	1	yes	characteristic
49	C₈×C₈₉	#104	1	1	yes	characteristic
50	C₂²×C₈₉	—	2	2	no	—
51	C₂²×C₈₉	—	4	1	no	—
52	C₂²×C₈₉	—	4	1	no	—
53	C₂²×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
54	C₄×C₈₉	—	2	1	no	—
55	C₄×C₈₉	—	4	1	no	—
56	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
57	C₄×C₈₉	#350	1	1	yes	characteristic
58	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
59	C₄×C₈₉	M₆₄	1	1	yes	characteristic
60	D₄⋊C₃₂	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₈₉	—	4	2	no	—
62	C₂×C₈₉	D₄⋊C₁₆	1	1	yes	characteristic
63	C₂×C₈₉	C₂²⋊C₃₂	1	1	yes	characteristic
64	C₂×C₈₉	#1401	1	1	yes	characteristic
65	D₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic
66	C₄⋊C₃₂	C₂×C₈₉	1	1	yes	characteristic, complemented
67	C₄×C₃₂	C₂×C₈₉	1	1	yes	characteristic, complemented
68	C₈₉	D₄⋊C₃₂	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₁₆	—	2	1	no	—
70	C₈×D₄	C₄×C₈₉	1	1	yes	characteristic
71	C₂×C₃₂	—	2	1	no	—
72	C₂×C₃₂	—	2	1	no	—
73	C₄⋊C₁₆	C₄×C₈₉	1	1	yes	characteristic
74	C₂²⋊C₁₆	—	2	1	no	—
75	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
76	C₂²×C₈	—	2	1	no	—
77	C₄×D₄	C₈×C₈₉	1	1	yes	characteristic
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	4	1	no	—
80	C₂×C₁₆	—	4	1	no	—
81	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
82	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
83	C₄⋊C₈	C₈×C₈₉	1	1	yes	characteristic
84	C₂²⋊C₈	—	2	1	no	—
85	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
86	C₃₂	—	2	2	no	—
87	C₃₂	—	4	1	no	—
88	C₂×D₄	C₁₆×C₈₉	1	1	yes	characteristic
89	C₂²×C₄	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
94	C₂×C₈	M₁₆×C₈₉	1	1	yes	characteristic
95	C₄⋊C₄	C₁₆×C₈₉	1	1	yes	characteristic
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
98	C₁₆	—	2	1	no	—
99	C₁₆	—	4	1	no	—
100	C₁₆	D₈×C₈₉	1	1	yes	characteristic
101	C₁₆	SD₁₆×C₈₉	1	1	yes	characteristic
102	C₂³	—	2	1	no	—
103	D₄	—	2	1	no	—
104	D₄	C₃₂×C₈₉	1	2	yes	complemented
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
110	C₂×C₄	M₃₂×C₈₉	1	1	yes	characteristic
111	C₈	—	2	1	no	—
112	C₈	—	4	1	no	—
113	C₈	#102×C₈₉	1	1	yes	characteristic
114	C₈	#104×C₈₉	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	1	no	—
118	C₂²	C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	D₄⋊C₈×C₈₉	1	1	yes	characteristic
122	C₄	#350×C₈₉	1	1	yes	characteristic
123	C₄	C₂×C₃₂×C₈₉	1	1	yes	characteristic
124	C₄	M₆₄×C₈₉	1	1	yes	characteristic
125	C₂	—	4	2	no	—
126	C₂	C₈₉×D₄⋊C₁₆	1	1	yes	characteristic
127	C₂	C₈₉×C₂²⋊C₃₂	1	1	yes	characteristic
128	C₂	C₈₉×#1401	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₈₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₄×C₃₂×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₆×C₈₉

—

C₈×D₄×C₈₉

C₄

yes

characteristic

C₂×C₃₂×C₈₉

—

C₂×C₃₂×C₈₉

—

C₄⋊C₁₆×C₈₉

C₄

yes

characteristic

C₂²⋊C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

—

C₄×D₄×C₈₉

C₈

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

C₈

yes

characteristic

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₃₂×C₈₉

—

C₂×D₄×C₈₉

C₁₆

yes

characteristic

C₂²×C₄×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

characteristic

C₄⋊C₄×C₈₉

C₁₆

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

characteristic

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

C₃₂

yes

complemented

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

M₃₂

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²⋊C₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

D₄⋊C₈

yes

characteristic

C₄×C₈₉

#350

yes

characteristic

C₄×C₈₉

C₂×C₃₂

yes

characteristic

C₄×C₈₉

M₆₄

yes

characteristic

D₄⋊C₃₂

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

D₄⋊C₁₆

yes

characteristic

C₂×C₈₉

C₂²⋊C₃₂

yes

characteristic

C₂×C₈₉

#1401

yes

characteristic

D₄×C₁₆

C₂×C₈₉

yes

characteristic

C₄⋊C₃₂

C₂×C₈₉

yes

characteristic, complemented

C₄×C₃₂

C₂×C₈₉

yes

characteristic, complemented

C₈₉

D₄⋊C₃₂

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

—

C₈×D₄

C₄×C₈₉

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

C₄×C₈₉

yes

characteristic

C₂²⋊C₁₆

—

C₄×C₁₆

C₂²×C₈₉

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₈×C₈₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₄⋊C₈

C₈×C₈₉

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

C₂×C₄×C₈₉

yes

characteristic

C₃₂

—

C₃₂

—

C₂×D₄

C₁₆×C₈₉

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₈₉

yes

characteristic

C₂×C₈

M₁₆×C₈₉

yes

characteristic

C₄⋊C₄

C₁₆×C₈₉

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂×C₈×C₈₉

yes

characteristic

C₁₆

—

C₁₆

—

100

C₁₆

D₈×C₈₉

yes

characteristic

101

C₁₆

SD₁₆×C₈₉

yes

characteristic

102

C₂³

—

103

D₄

—

104

D₄

C₃₂×C₈₉

yes

complemented

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

109

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

110

C₂×C₄

M₃₂×C₈₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

#102×C₈₉

yes

characteristic

114

C₈

#104×C₈₉

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²⋊C₁₆×C₈₉

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

D₄⋊C₈×C₈₉

yes

characteristic

122

C₄

#350×C₈₉

yes

characteristic

123

C₄

C₂×C₃₂×C₈₉

yes

characteristic

124

C₄

M₆₄×C₈₉

yes

characteristic

125

C₂

—

126

C₂

C₈₉×D₄⋊C₁₆

yes

characteristic

127

C₂

C₈₉×C₂²⋊C₃₂

yes

characteristic

128

C₂

C₈₉×#1401

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆×C₈₉	D₄
commutator	C₄	C₂×C₃₂×C₈₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₉
socle	C₂²×C₈₉	C₂²⋊C₁₆
prime radical	C₂×D₄×C₈₉	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

subgroup

G/H

center

C₂×C₁₆×C₈₉

D₄

commutator

C₄

C₂×C₃₂×C₈₉

Frattini, cosocle

C₄×C₁₆

C₂²×C₈₉

socle

C₂²×C₈₉

C₂²⋊C₁₆

prime radical

C₂×D₄×C₈₉

C₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

N	H ≅ G/N	type
2. C₈₉×C₄⋊C₃₂	C₂	semidirect
3. C₄×C₃₂×C₈₉	C₂	semidirect
39. D₄×C₈₉	C₃₂	semidirect
60. D₄⋊C₃₂	C₈₉	direct (N × H)
66. C₄⋊C₃₂	C₂×C₈₉	semidirect
67. C₄×C₃₂	C₂×C₈₉	semidirect
68. C₈₉	D₄⋊C₃₂	direct (N × H)
104. D₄	C₃₂×C₈₉	semidirect

H ≅ G/N

type

2. C₈₉×C₄⋊C₃₂

C₂

semidirect

3. C₄×C₃₂×C₈₉

C₂

semidirect

39. D₄×C₈₉

C₃₂

semidirect

60. D₄⋊C₃₂

C₈₉

direct (N × H)

66. C₄⋊C₃₂

C₂×C₈₉

semidirect

67. C₄×C₃₂

C₂×C₈₉

semidirect

68. C₈₉

D₄⋊C₃₂

direct (N × H)

104. D₄

C₃₂×C₈₉

semidirect

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

subgroup	index
D₄×C₁₆×C₈₉	2
C₈₉×C₄⋊C₃₂	2
C₄×C₃₂×C₈₉	2
D₄⋊C₃₂	89

subgroup

index

D₄×C₁₆×C₈₉

C₈₉×C₄⋊C₃₂

C₄×C₃₂×C₈₉

D₄⋊C₃₂

Subgroups — C₈₉×D₄⋊C₃₂ #36385227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×D4⋊C32 #36385227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×D₄⋊C₃₂ #36385227