Group #36384976 — subgroups

Subgroups — C₈₉×#7128 #36384976

Order 22784.

236 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1374	C₂	1	2	yes	maximal
3	C₂×C₄×Q₈×C₈₉	C₄	1	1	yes	characteristic
4	C₂²×C₁₆×C₈₉	—	2	2	no	—
5	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
7	#336×C₈₉	—	2	2	no	—
8	C₂²×Q₈×C₈₉	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₈₉	—	2	1	no	—
10	C₂²×C₈×C₈₉	D₄	1	2	yes	—
11	C₄×Q₈×C₈₉	—	4	1	no	—
12	C₄×Q₈×C₈₉	—	4	1	no	—
13	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
15	C₂×C₄²×C₈₉	—	2	1	no	—
16	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	—	2	4	no	—
18	C₂×C₁₆×C₈₉	—	4	2	no	—
19	C₂×C₁₆×C₈₉	—	4	2	no	—
20	C₄⋊C₈×C₈₉	—	4	1	no	—
21	C₄×C₈×C₈₉	—	4	1	no	—
22	C₂×Q₈×C₈₉	—	2	1	no	—
23	C₂×Q₈×C₈₉	—	4	1	no	—
24	C₂×Q₈×C₈₉	—	4	1	no	—
25	C₂×Q₈×C₈₉	C₁₆	1	2	yes	complemented
26	C₂²×C₄×C₈₉	—	2	1	no	—
27	C₂²×C₄×C₈₉	—	2	1	no	—
28	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	4	1	no	—
34	C₂×C₈×C₈₉	—	4	2	no	—
35	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
36	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₄⋊C₄×C₈₉	—	4	1	no	—
39	C₄²×C₈₉	—	2	1	no	—
40	C₄²×C₈₉	—	2	1	no	—
41	C₄²×C₈₉	—	4	1	no	—
42	C₄²×C₈₉	—	4	1	no	—
43	C₁₆×C₈₉	—	4	4	no	—
44	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
45	Q₈×C₈₉	—	4	2	no	—
46	Q₈×C₈₉	—	4	2	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	4	1	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	—	4	2	no	—
56	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
58	C₂×C₄×C₈₉	#103	1	2	yes	—
59	C₂×C₄×C₈₉	#104	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₈₉	M₃₂	1	1	yes	characteristic
62	C₈×C₈₉	—	2	2	no	—
63	C₈×C₈₉	—	2	2	no	—
64	C₈×C₈₉	—	8	1	no	—
65	C₂²×C₈₉	#323	1	1	yes	characteristic
66	C₂²×C₈₉	Q₈⋊C₈	1	2	yes	—
67	C₂²×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
68	C₂²×C₈₉	#349	1	1	yes	characteristic
69	C₂²×C₈₉	#350	1	2	yes	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	4	2	no	—
73	C₄×C₈₉	—	4	1	no	—
74	C₄×C₈₉	—	8	1	no	—
75	C₄×C₈₉	#328	1	2	yes	—
76	C₄×C₈₉	#329	1	1	yes	characteristic
77	C₄×C₈₉	#333	1	1	yes	characteristic
78	#7128	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×C₈₉	#1272	1	1	yes	characteristic
80	C₂×C₈₉	#1314	1	1	yes	characteristic
81	C₂×C₈₉	#1315	1	1	yes	characteristic
82	C₂×C₈₉	Q₈⋊C₁₆	1	2	yes	—
83	C₂×C₈₉	#1338	1	2	yes	—
84	#1984	C₂×C₈₉	1	1	yes	characteristic
85	#1374	C₂×C₈₉	1	2	yes	—
86	C₈₉	#7128	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×C₄×Q₈	C₄×C₈₉	1	1	yes	characteristic
88	C₂²×C₁₆	—	2	2	no	—
89	C₂×C₄⋊C₈	C₄×C₈₉	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
91	#336	—	2	2	no	—
92	C₂²×Q₈	C₈×C₈₉	1	1	yes	characteristic
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	D₄×C₈₉	1	2	yes	—
95	C₄×Q₈	—	4	1	no	—
96	C₄×Q₈	—	4	1	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	C₈×C₈₉	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	4	2	no	—
103	C₂×C₁₆	—	4	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	C₁₆×C₈₉	1	2	yes	complemented
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
114	C₂²×C₄	M₁₆×C₈₉	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	SD₁₆×C₈₉	1	2	yes	—
120	C₂×C₈	Q₁₆×C₈₉	1	2	yes	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₁₆	—	4	4	no	—
128	C₂³	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
129	Q₈	—	4	2	no	—
130	Q₈	—	4	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	#99×C₈₉	1	1	yes	characteristic
141	C₂×C₄	#100×C₈₉	1	1	yes	characteristic
142	C₂×C₄	#103×C₈₉	1	2	yes	—
143	C₂×C₄	#104×C₈₉	1	2	yes	—
144	C₂×C₄	C₂×C₁₆×C₈₉	1	1	yes	characteristic
145	C₂×C₄	M₃₂×C₈₉	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	2	no	—
148	C₈	—	8	1	no	—
149	C₂²	#323×C₈₉	1	1	yes	characteristic
150	C₂²	Q₈⋊C₈×C₈₉	1	2	yes	—
151	C₂²	C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
152	C₂²	#349×C₈₉	1	1	yes	characteristic
153	C₂²	#350×C₈₉	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	1	no	—
158	C₄	—	8	1	no	—
159	C₄	#328×C₈₉	1	2	yes	—
160	C₄	#329×C₈₉	1	1	yes	characteristic
161	C₄	#333×C₈₉	1	1	yes	characteristic
162	C₂	C₈₉×#1272	1	1	yes	characteristic
163	C₂	C₈₉×#1314	1	1	yes	characteristic
164	C₂	C₈₉×#1315	1	1	yes	characteristic
165	C₂	C₈₉×Q₈⋊C₁₆	1	2	yes	—
166	C₂	C₈₉×#1338	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1984

C₂

yes

maximal, characteristic

C₈₉×#1374

C₂

yes

maximal

C₂×C₄×Q₈×C₈₉

C₄

yes

characteristic

C₂²×C₁₆×C₈₉

—

C₂×C₄⋊C₈×C₈₉

C₄

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×Q₈×C₈₉

C₈

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄×Q₈×C₈₉

—

C₄×Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₈

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

C₁₆

yes

complemented

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂²×C₄×C₈₉

M₁₆

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂²⋊C₈

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#104

yes

—

C₂×C₄×C₈₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₈₉

M₃₂

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

#323

yes

characteristic

C₂²×C₈₉

Q₈⋊C₈

yes

—

C₂²×C₈₉

C₂²⋊C₁₆

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#328

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

Q₈⋊C₁₆

yes

—

C₂×C₈₉

#1338

yes

—

#1984

C₂×C₈₉

yes

characteristic

#1374

C₂×C₈₉

yes

—

C₈₉

#7128

yes

characteristic, direct product factor, Sylow

C₂×C₄×Q₈

C₄×C₈₉

yes

characteristic

C₂²×C₁₆

—

C₂×C₄⋊C₈

C₄×C₈₉

yes

characteristic

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

#336

—

C₂²×Q₈

C₈×C₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₄×C₈₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₈×C₈₉

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₁₆×C₈₉

yes

complemented

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

113

C₂²×C₄

C₂×C₈×C₈₉

yes

characteristic

114

C₂²×C₄

M₁₆×C₈₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

SD₁₆×C₈₉

yes

—

120

C₂×C₈

Q₁₆×C₈₉

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₁₆

—

128

C₂³

C₂²⋊C₈×C₈₉

yes

characteristic

129

Q₈

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#99×C₈₉

yes

characteristic

141

C₂×C₄

#100×C₈₉

yes

characteristic

142

C₂×C₄

#103×C₈₉

yes

—

143

C₂×C₄

#104×C₈₉

yes

—

144

C₂×C₄

C₂×C₁₆×C₈₉

yes

characteristic

145

C₂×C₄

M₃₂×C₈₉

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₂²

#323×C₈₉

yes

characteristic

150

C₂²

Q₈⋊C₈×C₈₉

yes

—

151

C₂²

C₂²⋊C₁₆×C₈₉

yes

characteristic

152

C₂²

#349×C₈₉

yes

characteristic

153

C₂²

#350×C₈₉

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#328×C₈₉

yes

—

160

C₄

#329×C₈₉

yes

characteristic

161

C₄

#333×C₈₉

yes

characteristic

162

C₂

C₈₉×#1272

yes

characteristic

163

C₂

C₈₉×#1314

yes

characteristic

164

C₂

C₈₉×#1315

yes

characteristic

165

C₂

C₈₉×Q₈⋊C₁₆

yes

—

166

C₂

C₈₉×#1338

yes

—

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₁₆×C₈₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₈₉
socle, prime radical	C₂³×C₈₉	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

subgroup

G/H

center

C₂²×C₄×C₈₉

C₂²⋊C₄

commutator

C₂×C₄

C₂×C₁₆×C₈₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₈₉

socle, prime radical

C₂³×C₈₉

C₂²⋊C₈

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
25. C₂×Q₈×C₈₉	C₁₆	semidirect
78. #7128	C₈₉	direct (N × H)
86. C₈₉	#7128	direct (N × H)
109. C₂×Q₈	C₁₆×C₈₉	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₈₉

C₁₆

semidirect

78. #7128

C₈₉

direct (N × H)

86. C₈₉

#7128

direct (N × H)

109. C₂×Q₈

C₁₆×C₈₉

semidirect

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

subgroup	index
C₈₉×#1984	2
C₈₉×#1374	2
#7128	89

subgroup

index

C₈₉×#1984

C₈₉×#1374

#7128

Subgroups — C₈₉×#7128 #36384976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7128 #36384976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7128 #36384976