Group #36474919 — subgroups

Subgroups — #36474919

Order 22784.

198 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 99 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	M₆₄×D₈₉	C₂	1	2	yes	maximal
2	#26021847	C₂	1	1	yes	maximal, characteristic
3	#26018450	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018449	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667951	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
8	#19151552	C₄	1	1	yes	characteristic
9	#19151432	C₂²	1	2	yes	—
10	#19151431	—	2	2	no	—
11	C₃₂×D₈₉	—	2	2	no	—
12	#19151374	C₄	1	1	yes	characteristic
13	#19151373	C₄	1	1	yes	characteristic
14	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
15	M₆₄×C₈₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083557	C₈	1	1	yes	characteristic
18	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
19	#15667969	C₈	1	1	yes	characteristic
20	#15667951	—	2	2	no	—
21	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
22	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₈₉	D₄	1	2	yes	—
24	C₁₆×D₈₉	Q₈	1	2	yes	—
25	#15667923	C₈	1	1	yes	characteristic
26	#15667922	C₈	1	1	yes	characteristic
27	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
28	#15667918	—	2	2	no	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₃₂×C₈₉	—	2	2	no	—
33	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
34	#13083557	C₁₆	1	4	yes	—
35	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
36	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
37	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
38	C₈×D₈₉	M₁₆	1	2	yes	—
39	#13083553	—	2	1	no	—
40	#13083551	—	2	1	no	—
41	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
42	#13083549	—	2	1	no	—
43	#13083549	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₁₆×C₈₉	—	2	1	no	—
48	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	C₂²×D₈₉	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₈₉	—	2	1	no	—
54	C₂×Dic₈₉	M₃₂	1	1	yes	characteristic
55	C₄×D₈₉	C₄⋊C₈	1	2	yes	—
56	C₄×D₈₉	C₂×C₁₆	1	2	yes	—
57	C₈×C₈₉	C₂×C₁₆	1	2	yes	—
58	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
60	#10623494	C₄⋊C₈	1	2	yes	—
61	#10623494	C₂×C₁₆	1	2	yes	—
62	C₂²×C₈₉	C₂×M₃₂	1	1	yes	characteristic
63	C₂×D₈₉	—	4	1	no	—
64	C₂×D₈₉	C₄⋊C₁₆	1	2	yes	—
65	C₄×C₈₉	—	4	1	no	—
66	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
68	Dic₈₉	—	4	1	no	—
69	Dic₈₉	C₄⋊C₁₆	1	2	yes	—
70	#13630	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
73	D₈₉	—	2	2	no	—
74	C₂×M₆₄	—	89	2	no	—
75	#2108	—	89	1	no	—
76	#1422	—	89	2	no	—
77	#1422	—	89	1	no	—
78	#1422	D₈₉	1	1	yes	characteristic, complemented
79	C₈₉	#13630	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₃₂	—	89	1	no	—
81	C₂²×C₁₆	—	89	1	no	—
82	#405	—	89	1	no	—
83	M₆₄	—	89	2	no	—
84	M₆₄	—	178	2	no	—
85	M₆₄	C₂×D₈₉	1	2	yes	—
86	C₂×C₃₂	—	178	2	no	—
87	#346	—	89	1	no	—
88	#346	—	89	1	no	—
89	C₄×C₁₆	—	89	1	no	—
90	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
91	C₂×M₁₆	—	89	1	no	—
92	C₂²×C₈	—	89	1	no	—
93	#117	—	89	1	no	—
94	M₃₂	—	178	2	no	—
95	C₂×C₁₆	—	89	1	no	—
96	C₂×C₁₆	—	89	2	no	—
97	C₂×C₁₆	—	89	2	no	—
98	C₂×C₁₆	—	89	1	no	—
99	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
100	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
101	#97	—	89	1	no	—
102	#97	—	89	1	no	—
103	C₄×C₈	—	89	1	no	—
104	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
105	C₃₂	—	2	2	no	—
106	C₃₂	—	178	2	no	—
107	C₂²×C₄	—	89	1	no	—
108	M₁₆	—	89	4	no	—
109	C₂×C₈	—	89	1	no	—
110	C₂×C₈	—	89	1	no	—
111	C₂×C₈	—	89	2	no	—
112	C₂×C₈	—	89	2	no	—
113	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₄⋊C₄	—	178	1	no	—
116	C₂²⋊C₄	—	178	1	no	—
117	C₄²	—	89	1	no	—
118	C₄²	C₈×D₈₉	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	89	2	no	—
121	C₁₆	—	178	1	no	—
122	C₁₆	D₄×D₈₉	1	1	yes	characteristic
123	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
124	C₂³	—	89	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	89	1	no	—
127	C₂×C₄	—	89	2	no	—
128	C₂×C₄	—	89	2	no	—
129	C₂×C₄	—	178	1	no	—
130	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
131	C₈	—	89	2	no	—
132	C₈	—	89	2	no	—
133	C₈	C₁₆×D₈₉	1	2	yes	—
134	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
135	C₈	M₁₆×D₈₉	1	1	yes	characteristic
136	C₂²	—	89	2	no	—
137	C₂²	—	356	1	no	—
138	C₂²	M₃₂×D₈₉	1	1	yes	characteristic
139	C₄	—	4	1	no	—
140	C₄	—	89	2	no	—
141	C₄	—	356	1	no	—
142	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
143	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	178	2	no	—
146	C₂	C₄⋊C₁₆×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₆₄×D₈₉

C₂

yes

maximal

#26021847

C₂

yes

maximal, characteristic

#26018450

C₂

yes

maximal, characteristic, complemented

#26018449

C₂

yes

maximal, complemented

C₈₉×#1422

C₂

yes

maximal, characteristic, complemented

C₂×#15667951

C₄

yes

characteristic

C₂×C₁₆×D₈₉

C₂²

yes

characteristic

#19151552

C₄

yes

characteristic

#19151432

C₂²

yes

—

#19151431

—

C₃₂×D₈₉

—

#19151374

C₄

yes

characteristic

#19151373

C₄

yes

characteristic

C₁₆×Dic₈₉

C₂²

yes

characteristic

M₆₄×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#13083557

C₈

yes

characteristic

C₂×C₈×D₈₉

C₂×C₄

yes

characteristic

#15667969

C₈

yes

characteristic

#15667951

—

C₂×#13083549

C₂×C₄

yes

characteristic

C₂×#13083549

C₂×C₄

yes

characteristic

C₁₆×D₈₉

D₄

yes

—

C₁₆×D₈₉

Q₈

yes

—

#15667923

C₈

yes

characteristic

#15667922

C₈

yes

characteristic

C₈×Dic₈₉

C₂×C₄

yes

characteristic

#15667918

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₂×C₄×D₈₉

C₂×C₈

yes

characteristic

#13083557

C₁₆

yes

—

C₂×#10623494

C₂×C₈

yes

characteristic

C₂×#10623494

C₂×C₈

yes

characteristic

C₈×D₈₉

C₄⋊C₄

yes

—

C₈×D₈₉

M₁₆

yes

—

#13083553

—

#13083551

—

C₄×Dic₈₉

C₂×C₈

yes

characteristic

#13083549

—

#13083549

C₄⋊C₄

yes

—

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

C₂×Q₈

yes

characteristic

C₂²×D₈₉

M₃₂

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

M₃₂

yes

characteristic

C₄×D₈₉

C₄⋊C₈

yes

—

C₄×D₈₉

C₂×C₁₆

yes

—

C₈×C₈₉

C₂×C₁₆

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₂×M₁₆

yes

characteristic

#10623494

C₄⋊C₈

yes

—

#10623494

C₂×C₁₆

yes

—

C₂²×C₈₉

C₂×M₃₂

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

C₄⋊C₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₈₉

C₂²×C₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

C₄⋊C₁₆

yes

—

#13630

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊C₁₆

yes

characteristic

D₈₉

—

C₂×M₆₄

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#13630

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

178

—

M₆₄

C₂×D₈₉

yes

—

C₂×C₃₂

—

178

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

100

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₈₉

yes

characteristic

105

C₃₂

—

106

C₃₂

—

178

—

107

C₂²×C₄

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₈×D₈₉

yes

characteristic

114

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

115

C₄⋊C₄

—

178

—

116

C₂²⋊C₄

—

178

—

117

C₄²

—

118

C₄²

C₈×D₈₉

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

178

—

122

C₁₆

D₄×D₈₉

yes

characteristic

123

C₁₆

Q₈×D₈₉

yes

characteristic

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

178

—

130

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

C₁₆×D₈₉

yes

—

134

C₈

C₄⋊C₄×D₈₉

yes

characteristic

135

C₈

M₁₆×D₈₉

yes

characteristic

136

C₂²

—

137

C₂²

—

356

—

138

C₂²

M₃₂×D₈₉

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

356

—

142

C₄

C₄⋊C₈×D₈₉

yes

characteristic

143

C₄

C₂×C₁₆×D₈₉

yes

characteristic

144

C₂

—

145

C₂

—

178

—

146

C₂

C₄⋊C₁₆×D₈₉

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1422	C₂
socle	C₂×C₈₉	C₂×C₄⋊C₁₆
prime radical	C₂²×D₈₉	M₃₂
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1422	D₈₉
hypocenter	C₈₉	#13630
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

subgroup

G/H

center

C₁₆

D₄×D₈₉

commutator

C₄×C₈₉

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#1422

C₂

socle

C₂×C₈₉

C₂×C₄⋊C₁₆

prime radical

C₂²×D₈₉

M₃₂

cosocle

C₂×C₁₆×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

N	H ≅ G/N	type
3. #26018450	C₂	semidirect
4. #26018449	C₂	semidirect
5. C₈₉×#1422	C₂	semidirect
78. #1422	D₈₉	semidirect
79. C₈₉	#13630	semidirect

H ≅ G/N

type

3. #26018450

C₂

semidirect

4. #26018449

C₂

semidirect

5. C₈₉×#1422

C₂

semidirect

78. #1422

D₈₉

semidirect

79. C₈₉

#13630

semidirect

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

subgroup	index
M₆₄×D₈₉	2
#26021847	2
#26018450	2
#26018449	2
C₈₉×#1422	2
#13630	89

subgroup

index

M₆₄×D₈₉

C₈₉×#1422

Subgroups — #36474919

Conjugacy classes of subgroups

Complements

Maximal subgroups