Group #36474974 — subgroups

Subgroups — #36474974

Order 22784.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₃₂×D₈₉	C₂	1	2	yes	maximal, complemented
2	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26018389	C₂	1	2	yes	maximal, complemented
4	C₄×#15667918	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
7	C₈×D₃₅₆	C₄	1	1	yes	characteristic
8	C₂×#15667918	C₂²	1	1	yes	characteristic
9	C₂×#15667918	C₂²	1	1	yes	characteristic
10	C₃₂×D₈₉	—	2	4	no	—
11	C₄⋊#13083549	C₄	1	1	yes	characteristic
12	#19151377	C₄	1	2	yes	—
13	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
16	C₄×D₃₅₆	C₈	1	1	yes	characteristic
17	C₂×#13083549	C₂×C₄	1	2	yes	—
18	C₁₆×D₈₉	—	2	2	no	—
19	C₁₆×D₈₉	—	2	2	no	—
20	C₄⋊#10623494	C₈	1	1	yes	characteristic
21	#15667924	C₈	1	2	yes	—
22	#15667918	—	2	1	no	—
23	#15667918	D₄	1	2	yes	complemented
24	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₃₂×C₈₉	—	2	2	no	—
28	C₂×D₃₅₆	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
30	C₂×#10623494	C₂×C₈	1	2	yes	—
31	C₈×D₈₉	—	2	2	no	—
32	C₈×D₈₉	—	2	2	no	—
33	C₄⋊Dic₈₉	C₁₆	1	1	yes	characteristic
34	#13083551	C₁₆	1	2	yes	—
35	#13083549	—	2	2	no	—
36	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₈₉	—	2	1	no	—
40	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	#41	1	1	yes	characteristic
42	C₂²×D₈₉	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
47	D₃₅₆	C₃₂	1	4	yes	complemented
48	C₄×D₈₉	—	2	2	no	—
49	C₄×D₈₉	—	2	2	no	—
50	C₈×C₈₉	—	2	1	no	—
51	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
52	C₈×C₈₉	#131	1	1	yes	characteristic
53	#10623494	—	2	2	no	—
54	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×D₈₉	—	2	2	no	—
56	C₂×D₈₉	—	2	2	no	—
57	C₂×D₈₉	C₂×C₃₂	1	4	yes	complemented
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	C₂×C₃₂	1	2	yes	complemented
60	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
61	C₄×C₈₉	#504	1	1	yes	characteristic
62	Dic₈₉	—	2	2	no	—
63	D₄×C₃₂	—	89	1	no	maximal, Sylow
64	C₂×C₈₉	D₄×C₁₆	1	1	yes	characteristic
65	C₂×C₈₉	C₂²×C₃₂	1	1	yes	characteristic
66	C₂×C₈₉	#2258	1	1	yes	characteristic
67	D₈₉	—	2	4	no	—
68	C₂²×C₃₂	—	89	2	no	—
69	D₄×C₁₆	—	89	1	no	—
70	C₄⋊C₃₂	D₈₉	1	1	yes	characteristic, complemented
71	C₂²⋊C₃₂	—	89	2	no	—
72	C₄×C₃₂	—	89	1	no	—
73	C₈₉	D₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	89	2	no	—
75	C₈×D₄	—	89	1	no	—
76	C₂×C₃₂	—	89	1	no	—
77	C₂×C₃₂	—	89	1	no	—
78	C₂×C₃₂	—	178	4	no	—
79	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
80	C₄⋊C₁₆	—	89	1	no	—
81	C₂²⋊C₁₆	—	89	2	no	—
82	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
83	C₂²×C₈	—	89	2	no	—
84	C₄×D₄	—	89	1	no	—
85	C₂×C₁₆	—	89	2	no	—
86	C₂×C₁₆	—	178	2	no	—
87	C₂×C₁₆	—	178	2	no	—
88	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
90	C₄⋊C₈	—	89	1	no	—
91	C₂²⋊C₈	—	89	2	no	—
92	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
93	C₃₂	—	2	2	no	—
94	C₃₂	—	89	2	no	—
95	C₃₂	—	178	1	no	—
96	C₂×D₄	—	89	1	no	—
97	C₂²×C₄	—	89	2	no	—
98	C₂×C₈	—	89	2	no	—
99	C₂×C₈	—	178	2	no	—
100	C₂×C₈	—	178	2	no	—
101	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
102	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
103	C₄⋊C₄	—	89	1	no	—
104	C₂²⋊C₄	—	89	2	no	—
105	C₄²	C₈×D₈₉	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	178	2	no	—
108	C₁₆	D₄×D₈₉	1	1	yes	characteristic
109	C₁₆	#13083575	1	1	yes	characteristic
110	C₂³	—	89	2	no	—
111	D₄	—	89	4	no	—
112	C₂×C₄	—	89	2	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	—	178	2	no	—
115	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
116	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
117	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	178	2	no	—
120	C₈	#15667974	1	1	yes	characteristic
121	C₈	#15668020	1	1	yes	characteristic
122	C₂²	—	89	4	no	—
123	C₂²	—	178	2	no	—
124	C₂²	—	178	2	no	—
125	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	178	2	no	—
128	C₄	C₃₂×D₈₉	1	2	yes	complemented
129	C₄	#19151663	1	1	yes	characteristic
130	C₄	#19152016	1	1	yes	characteristic
131	C₂	—	178	4	no	—
132	C₂	#26022071	1	1	yes	characteristic
133	C₂	C₂×C₃₂×D₈₉	1	1	yes	characteristic
134	C₂	#26022547	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₈₉

C₂

yes

maximal, complemented

C₁₆×D₃₅₆

C₂

yes

maximal, characteristic

#26018389

C₂

yes

maximal, complemented

C₄×#15667918

C₂

yes

maximal, characteristic, complemented

C₈₉×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₈₉

C₂²

yes

—

C₈×D₃₅₆

C₄

yes

characteristic

C₂×#15667918

C₂²

yes

characteristic

C₂×#15667918

C₂²

yes

characteristic

C₃₂×D₈₉

—

C₄⋊#13083549

C₄

yes

characteristic

#19151377

C₄

yes

—

C₂×C₃₂×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₈×D₈₉

C₂×C₄

yes

—

C₄×D₃₅₆

C₈

yes

characteristic

C₂×#13083549

C₂×C₄

yes

—

C₁₆×D₈₉

—

C₁₆×D₈₉

—

C₄⋊#10623494

C₈

yes

characteristic

#15667924

C₈

yes

—

#15667918

—

#15667918

D₄

yes

complemented

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₂×D₃₅₆

C₁₆

yes

characteristic

C₂×C₄×D₈₉

C₂×C₈

yes

—

C₂×#10623494

C₂×C₈

yes

—

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

C₁₆

yes

characteristic

#13083551

C₁₆

yes

—

#13083549

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

C₂²×D₈₉

C₂×C₁₆

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

C₂×C₁₆

yes

—

D₃₅₆

C₃₂

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂²×C₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×C₃₂

yes

complemented

C₄×C₈₉

—

C₄×C₈₉

C₂×C₃₂

yes

complemented

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#504

yes

characteristic

Dic₈₉

—

D₄×C₃₂

—

maximal, Sylow

C₂×C₈₉

D₄×C₁₆

yes

characteristic

C₂×C₈₉

C₂²×C₃₂

yes

characteristic

C₂×C₈₉

#2258

yes

characteristic

D₈₉

—

C₂²×C₃₂

—

D₄×C₁₆

—

C₄⋊C₃₂

D₈₉

yes

characteristic, complemented

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₈₉

D₄×C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₈×D₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

178

—

C₂×C₃₂

C₂×D₈₉

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

178

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₂×C₈

C₈×D₈₉

yes

characteristic

102

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₈×D₈₉

yes

characteristic

106

C₁₆

—

107

C₁₆

—

178

—

108

C₁₆

D₄×D₈₉

yes

characteristic

109

C₁₆

#13083575

yes

characteristic

110

C₂³

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

178

—

114

C₂×C₄

—

178

—

115

C₂×C₄

C₁₆×D₈₉

yes

characteristic

116

C₂×C₄

C₁₆×D₈₉

yes

characteristic

117

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

118

C₈

—

119

C₈

—

178

—

120

C₈

#15667974

yes

characteristic

121

C₈

#15668020

yes

characteristic

122

C₂²

—

123

C₂²

—

178

—

124

C₂²

—

178

—

125

C₂²

C₂×C₁₆×D₈₉

yes

characteristic

126

C₄

—

127

C₄

—

178

—

128

C₄

C₃₂×D₈₉

yes

complemented

129

C₄

#19151663

yes

characteristic

130

C₄

#19152016

yes

characteristic

131

C₂

—

178

—

132

C₂

#26022071

yes

characteristic

133

C₂

C₂×C₃₂×D₈₉

yes

characteristic

134

C₂

#26022547

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₃₂
Fitting	C₈₉×C₄⋊C₃₂	C₂
socle	C₂²×C₈₉	C₂²×C₁₆
prime radical	C₂×D₃₅₆	C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₄⋊C₃₂	D₈₉
hypocenter	C₈₉	D₄×C₃₂
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₈₉

commutator

C₂×C₈₉

C₂²×C₃₂

Fitting

C₈₉×C₄⋊C₃₂

C₂

socle

C₂²×C₈₉

C₂²×C₁₆

prime radical

C₂×D₃₅₆

C₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₄⋊C₃₂

D₈₉

hypocenter

C₈₉

D₄×C₃₂

LCS

G ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
1. C₂×C₃₂×D₈₉	C₂	semidirect
3. #26018389	C₂	semidirect
4. C₄×#15667918	C₂	semidirect
5. C₈₉×C₄⋊C₃₂	C₂	semidirect
23. #15667918	D₄	semidirect
47. D₃₅₆	C₃₂	semidirect
57. C₂×D₈₉	C₂×C₃₂	semidirect
59. C₄×C₈₉	C₂×C₃₂	semidirect
70. C₄⋊C₃₂	D₈₉	semidirect
73. C₈₉	D₄×C₃₂	semidirect
128. C₄	C₃₂×D₈₉	semidirect

H ≅ G/N

type

1. C₂×C₃₂×D₈₉

C₂

semidirect

3. #26018389

C₂

semidirect

4. C₄×#15667918

C₂

semidirect

5. C₈₉×C₄⋊C₃₂

C₂

semidirect

23. #15667918

D₄

semidirect

47. D₃₅₆

C₃₂

semidirect

57. C₂×D₈₉

C₂×C₃₂

semidirect

59. C₄×C₈₉

C₂×C₃₂

semidirect

70. C₄⋊C₃₂

D₈₉

semidirect

73. C₈₉

D₄×C₃₂

semidirect

128. C₄

C₃₂×D₈₉

semidirect

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

subgroup	index
C₂×C₃₂×D₈₉	2
C₁₆×D₃₅₆	2
#26018389	2
C₄×#15667918	2
C₈₉×C₄⋊C₃₂	2
D₄×C₃₂	89

subgroup

index

C₂×C₃₂×D₈₉

C₁₆×D₃₅₆

#26018389

C₄×#15667918

C₈₉×C₄⋊C₃₂

D₄×C₃₂

Subgroups — #36474974

Conjugacy classes of subgroups

Complements

Maximal subgroups