Group #36390161 — subgroups

Subgroups — C₈₉×#12313 #36390161

Order 22784.

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3293	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2214	C₂	1	2	yes	maximal, complemented
3	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1358	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1335	C₂	1	2	yes	maximal, complemented
6	#578×C₈₉	—	2	2	no	—
7	#510×C₈₉	—	2	2	no	—
8	#500×C₈₉	—	2	2	no	—
9	#494×C₈₉	C₂²	1	1	yes	characteristic
10	#494×C₈₉	C₂²	1	1	yes	characteristic
11	#473×C₈₉	—	2	2	no	—
12	#456×C₈₉	—	2	2	no	—
13	#443×C₈₉	C₂²	1	2	yes	—
14	#364×C₈₉	C₂²	1	2	yes	—
15	#362×C₈₉	—	2	2	no	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic
17	2₋¹⁺⁴×C₈₉	—	2	2	no	—
18	#137×C₈₉	—	2	4	no	—
19	#137×C₈₉	—	4	4	no	—
20	#135×C₈₉	—	4	2	no	—
21	#135×C₈₉	D₄	1	2	yes	—
22	C₂×Q₁₆×C₈₉	—	2	2	no	—
23	C₂×Q₁₆×C₈₉	—	2	2	no	—
24	C₂×Q₁₆×C₈₉	—	4	2	no	—
25	#131×C₈₉	—	2	2	no	—
26	C₄⋊Q₈×C₈₉	—	2	1	no	—
27	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
28	Q₃₂×C₈₉	—	4	2	no	—
29	SD₃₂×C₈₉	—	4	2	no	—
30	M₃₂×C₈₉	—	2	2	no	—
31	#108×C₈₉	D₄	1	2	yes	—
32	#107×C₈₉	—	4	4	no	—
33	#104×C₈₉	—	2	2	no	—
34	#101×C₈₉	—	2	2	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	#41×C₈₉	—	2	2	no	—
38	#41×C₈₉	—	4	4	no	—
39	#41×C₈₉	—	8	2	no	—
40	C₂×Q₈×C₈₉	—	2	2	no	—
41	C₂×Q₈×C₈₉	—	4	4	no	—
42	C₂×Q₈×C₈₉	—	4	1	no	—
43	Q₁₆×C₈₉	—	4	4	no	—
44	Q₁₆×C₈₉	—	4	2	no	—
45	Q₁₆×C₈₉	—	4	2	no	—
46	Q₁₆×C₈₉	—	4	2	no	—
47	Q₁₆×C₈₉	—	4	2	no	—
48	Q₁₆×C₈₉	D₈	1	2	yes	—
49	SD₁₆×C₈₉	—	2	2	no	—
50	SD₁₆×C₈₉	—	4	4	no	—
51	D₈×C₈₉	D₈	1	2	yes	—
52	M₁₆×C₈₉	—	2	2	no	—
53	M₁₆×C₈₉	—	4	2	no	—
54	C₂×C₈×C₈₉	—	2	2	no	—
55	C₂×C₈×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
57	C₄⋊C₄×C₈₉	—	4	2	no	—
58	C₄⋊C₄×C₈₉	—	4	2	no	—
59	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₁₆×C₈₉	—	4	2	no	—
61	Q₈×C₈₉	—	2	2	no	—
62	Q₈×C₈₉	—	4	4	no	—
63	Q₈×C₈₉	—	8	2	no	—
64	Q₈×C₈₉	—	8	2	no	—
65	D₄×C₈₉	—	2	2	no	—
66	D₄×C₈₉	—	4	2	no	—
67	D₄×C₈₉	—	8	2	no	—
68	C₂×C₄×C₈₉	—	2	1	no	—
69	C₂×C₄×C₈₉	—	4	2	no	—
70	C₂×C₄×C₈₉	—	4	2	no	—
71	C₂×C₄×C₈₉	—	8	4	no	—
72	C₂×C₄×C₈₉	—	8	1	no	—
73	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
74	C₈×C₈₉	—	2	4	no	—
75	C₈×C₈₉	—	4	2	no	—
76	C₈×C₈₉	C₂×D₈	1	2	yes	—
77	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
78	C₂²×C₈₉	—	4	2	no	—
79	C₂²×C₈₉	#453	1	1	yes	characteristic
80	C₄×C₈₉	—	2	2	no	—
81	C₄×C₈₉	—	4	2	no	—
82	C₄×C₈₉	—	8	2	no	—
83	C₄×C₈₉	—	16	1	no	—
84	C₄×C₈₉	#447	1	2	yes	—
85	#12313	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₈₉	—	2	1	no	—
87	C₂×C₈₉	—	8	2	no	—
88	C₂×C₈₉	#1619	1	1	yes	characteristic
89	#3293	C₂×C₈₉	1	1	yes	characteristic
90	#2214	C₂×C₈₉	1	2	yes	complemented
91	#1713	C₂×C₈₉	1	1	yes	characteristic, complemented
92	#1358	C₂×C₈₉	1	1	yes	characteristic, complemented
93	#1335	C₂×C₈₉	1	2	yes	complemented
94	C₈₉	#12313	1	1	yes	characteristic, direct product factor, Sylow
95	#578	—	2	2	no	—
96	#510	—	2	2	no	—
97	#500	—	2	2	no	—
98	#494	C₂²×C₈₉	1	1	yes	characteristic
99	#494	C₂²×C₈₉	1	1	yes	characteristic
100	#473	—	2	2	no	—
101	#456	—	2	2	no	—
102	#443	C₂²×C₈₉	1	2	yes	—
103	#364	C₂²×C₈₉	1	2	yes	—
104	#362	—	2	2	no	—
105	C₈²	C₂²×C₈₉	1	1	yes	characteristic
106	2₋¹⁺⁴	—	2	2	no	—
107	#137	—	2	4	no	—
108	#137	—	4	4	no	—
109	#135	—	4	2	no	—
110	#135	D₄×C₈₉	1	2	yes	—
111	C₂×Q₁₆	—	2	2	no	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×Q₁₆	—	4	2	no	—
114	#131	—	2	2	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	D₄×C₈₉	1	2	yes	—
117	Q₃₂	—	4	2	no	—
118	SD₃₂	—	4	2	no	—
119	M₃₂	—	2	2	no	—
120	#108	D₄×C₈₉	1	2	yes	—
121	#107	—	4	4	no	—
122	#104	—	2	2	no	—
123	#101	—	2	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
126	#41	—	2	2	no	—
127	#41	—	4	4	no	—
128	#41	—	8	2	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂×Q₈	—	4	1	no	—
132	Q₁₆	—	4	4	no	—
133	Q₁₆	—	4	2	no	—
134	Q₁₆	—	4	2	no	—
135	Q₁₆	—	4	2	no	—
136	Q₁₆	—	4	2	no	—
137	Q₁₆	D₈×C₈₉	1	2	yes	—
138	SD₁₆	—	2	2	no	—
139	SD₁₆	—	4	4	no	—
140	D₈	D₈×C₈₉	1	2	yes	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	4	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
149	C₁₆	—	4	2	no	—
150	Q₈	—	2	2	no	—
151	Q₈	—	4	4	no	—
152	Q₈	—	8	2	no	—
153	Q₈	—	8	2	no	—
154	D₄	—	2	2	no	—
155	D₄	—	4	2	no	—
156	D₄	—	8	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	4	2	no	—
165	C₈	C₂×D₈×C₈₉	1	2	yes	—
166	C₈	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
167	C₂²	—	4	2	no	—
168	C₂²	#453×C₈₉	1	1	yes	characteristic
169	C₄	—	2	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	8	2	no	—
172	C₄	—	16	1	no	—
173	C₄	#447×C₈₉	1	2	yes	—
174	C₂	—	2	1	no	—
175	C₂	—	8	2	no	—
176	C₂	C₈₉×#1619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#3293

C₂

yes

maximal, characteristic

C₈₉×#2214

C₂

yes

maximal, complemented

C₈₉×#1713

C₂

yes

maximal, characteristic, complemented

C₈₉×#1358

C₂

yes

maximal, characteristic, complemented

C₈₉×#1335

C₂

yes

maximal, complemented

#578×C₈₉

—

#510×C₈₉

—

#500×C₈₉

—

#494×C₈₉

C₂²

yes

characteristic

#494×C₈₉

C₂²

yes

characteristic

#473×C₈₉

—

#456×C₈₉

—

#443×C₈₉

C₂²

yes

—

#364×C₈₉

C₂²

yes

—

#362×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

2₋¹⁺⁴×C₈₉

—

#137×C₈₉

—

#137×C₈₉

—

#135×C₈₉

—

#135×C₈₉

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

#131×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

D₄

yes

—

Q₃₂×C₈₉

—

SD₃₂×C₈₉

—

M₃₂×C₈₉

—

#108×C₈₉

D₄

yes

—

#107×C₈₉

—

#104×C₈₉

—

#101×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₈

yes

—

SD₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

D₈

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

#453

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

#12313

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

#3293

C₂×C₈₉

yes

characteristic

#2214

C₂×C₈₉

yes

complemented

#1713

C₂×C₈₉

yes

characteristic, complemented

#1358

C₂×C₈₉

yes

characteristic, complemented

#1335

C₂×C₈₉

yes

complemented

C₈₉

#12313

yes

characteristic, direct product factor, Sylow

—

—

—

C₂²×C₈₉

yes

characteristic

#494

C₂²×C₈₉

yes

characteristic

100

#473

—

101

#456

—

102

#443

C₂²×C₈₉

yes

—

103

#364

C₂²×C₈₉

yes

—

104

#362

—

105

C₈²

C₂²×C₈₉

yes

characteristic

106

2₋¹⁺⁴

—

107

#137

—

108

#137

—

109

#135

—

110

#135

D₄×C₈₉

yes

—

111

C₂×Q₁₆

—

112

C₂×Q₁₆

—

113

C₂×Q₁₆

—

114

#131

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

D₄×C₈₉

yes

—

117

Q₃₂

—

118

SD₃₂

—

119

M₃₂

—

120

#108

D₄×C₈₉

yes

—

121

#107

—

122

#104

—

123

#101

—

124

C₄×C₈

—

125

C₄×C₈

C₂³×C₈₉

yes

characteristic

126

#41

—

127

#41

—

128

#41

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

Q₁₆

—

133

Q₁₆

—

134

Q₁₆

—

135

Q₁₆

—

136

Q₁₆

—

137

Q₁₆

D₈×C₈₉

yes

—

138

SD₁₆

—

139

SD₁₆

—

140

D₈

D₈×C₈₉

yes

—

141

M₁₆

—

142

M₁₆

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

C₂×D₄×C₈₉

yes

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄²

C₂×D₄×C₈₉

yes

characteristic

149

C₁₆

—

150

Q₈

—

151

Q₈

—

152

Q₈

—

153

Q₈

—

154

D₄

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

#120×C₈₉

yes

characteristic

163

C₈

—

164

C₈

—

165

C₈

C₂×D₈×C₈₉

yes

—

166

C₈

AGL(1,ℤ/8)×C₈₉

yes

—

167

C₂²

—

168

C₂²

#453×C₈₉

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#447×C₈₉

yes

—

174

C₂

—

175

C₂

—

176

C₂

C₈₉×#1619

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#1619
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₈₉
prime radical	C₈₉×#3293	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₄×C₈×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂×C₈₉

#1619

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₈₉

prime radical

C₈₉×#3293

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
2. C₈₉×#2214	C₂	semidirect
3. C₈₉×#1713	C₂	semidirect
4. C₈₉×#1358	C₂	semidirect
5. C₈₉×#1335	C₂	semidirect
85. #12313	C₈₉	direct (N × H)
90. #2214	C₂×C₈₉	semidirect
91. #1713	C₂×C₈₉	semidirect
92. #1358	C₂×C₈₉	semidirect
93. #1335	C₂×C₈₉	semidirect
94. C₈₉	#12313	direct (N × H)

H ≅ G/N

type

2. C₈₉×#2214

C₂

semidirect

3. C₈₉×#1713

C₂

semidirect

4. C₈₉×#1358

C₂

semidirect

5. C₈₉×#1335

C₂

semidirect

85. #12313

C₈₉

direct (N × H)

90. #2214

C₂×C₈₉

semidirect

91. #1713

C₂×C₈₉

semidirect

92. #1358

C₂×C₈₉

semidirect

93. #1335

C₂×C₈₉

semidirect

94. C₈₉

#12313

direct (N × H)

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

subgroup	index
C₈₉×#3293	2
C₈₉×#2214	2
C₈₉×#1713	2
C₈₉×#1358	2
C₈₉×#1335	2
#12313	89

subgroup

index

C₈₉×#3293

C₈₉×#2214

C₈₉×#1713

C₈₉×#1358

C₈₉×#1335

#12313

Subgroups — C₈₉×#12313 #36390161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#12313 #36390161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12313 #36390161