Group #36385348 — subgroups

Subgroups — C₈₉×#7500 #36385348

Order 22784.

294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 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×#2126	C₂	1	2	yes	maximal
2	C₈₉×#1962	C₂	1	1	yes	maximal, characteristic
3	#532×C₈₉	C₂²	1	1	yes	characteristic
4	#423×C₈₉	C₄	1	1	yes	characteristic, complemented
5	C₂×#104×C₈₉	—	2	2	no	—
6	#410×C₈₉	—	2	2	no	—
7	C₄³×C₈₉	C₄	1	1	yes	characteristic, complemented
8	#354×C₈₉	C₄	1	4	yes	complemented
9	#353×C₈₉	—	2	2	no	—
10	C₂×#41×C₈₉	D₄	1	2	yes	—
11	C₂×M₁₆×C₈₉	—	2	1	no	—
12	C₄⋊Q₈×C₈₉	Q₈	1	1	yes	characteristic
13	#127×C₈₉	D₄	1	1	yes	characteristic
14	#124×C₈₉	C₂×C₄	1	2	yes	—
15	#121×C₈₉	—	4	1	no	—
16	C₄×D₄×C₈₉	—	4	1	no	—
17	#117×C₈₉	—	4	2	no	—
18	C₂×C₄²×C₈₉	—	2	1	no	—
19	C₂×C₄²×C₈₉	—	4	1	no	—
20	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	—	4	1	no	—
22	#104×C₈₉	—	4	4	no	—
23	#99×C₈₉	—	2	4	no	—
24	#99×C₈₉	—	4	2	no	—
25	#41×C₈₉	—	4	2	no	—
26	#41×C₈₉	—	4	2	no	—
27	C₂×Q₈×C₈₉	C₄⋊C₄	1	2	yes	—
28	C₂×D₄×C₈₉	—	2	2	no	—
29	C₂×D₄×C₈₉	—	4	1	no	—
30	C₂×D₄×C₈₉	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₄×C₈₉	—	2	1	no	—
32	C₂²×C₄×C₈₉	—	2	2	no	—
33	C₂²×C₄×C₈₉	—	4	1	no	—
34	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
35	M₁₆×C₈₉	—	4	2	no	—
36	C₂×C₈×C₈₉	—	4	1	no	—
37	C₄⋊C₄×C₈₉	—	4	1	no	—
38	C₄⋊C₄×C₈₉	—	8	2	no	—
39	C₂²⋊C₄×C₈₉	—	4	2	no	—
40	C₂²⋊C₄×C₈₉	—	4	4	no	—
41	C₄²×C₈₉	—	2	1	no	—
42	C₄²×C₈₉	—	2	2	no	—
43	C₄²×C₈₉	—	4	1	no	—
44	C₄²×C₈₉	—	4	4	no	—
45	C₄²×C₈₉	—	8	2	no	—
46	C₄²×C₈₉	C₄²	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂³×C₈₉	—	2	2	no	—
49	C₂³×C₈₉	#99	1	1	yes	characteristic
50	Q₈×C₈₉	—	4	2	no	—
51	D₄×C₈₉	—	4	2	no	—
52	D₄×C₈₉	—	8	2	no	—
53	D₄×C₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	—	4	2	no	—
61	C₂×C₄×C₈₉	—	4	4	no	—
62	C₂×C₄×C₈₉	—	4	2	no	—
63	C₂×C₄×C₈₉	—	4	2	no	—
64	C₂×C₄×C₈₉	—	8	4	no	—
65	C₂×C₄×C₈₉	#95	1	1	yes	characteristic
66	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
67	C₈×C₈₉	—	8	1	no	—
68	C₂²×C₈₉	—	2	1	no	—
69	C₂²×C₈₉	—	4	1	no	—
70	C₂²×C₈₉	—	4	2	no	—
71	C₂²×C₈₉	—	8	2	no	—
72	C₂²×C₈₉	#342	1	1	yes	characteristic
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	—	4	1	no	—
76	C₄×C₈₉	—	4	4	no	—
77	C₄×C₈₉	—	4	2	no	—
78	C₄×C₈₉	—	16	4	no	—
79	C₄×C₈₉	#353	1	1	yes	characteristic
80	C₄×C₈₉	#354	1	1	yes	characteristic
81	#7500	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₈₉	—	2	1	no	—
83	C₂×C₈₉	—	4	1	no	—
84	C₂×C₈₉	—	8	2	no	—
85	C₂×C₈₉	#1393	1	1	yes	characteristic
86	#2126	C₂×C₈₉	1	2	yes	—
87	#1962	C₂×C₈₉	1	1	yes	characteristic
88	C₈₉	#7500	1	1	yes	characteristic, direct product factor, Sylow
89	#532	C₂²×C₈₉	1	1	yes	characteristic
90	#423	C₄×C₈₉	1	1	yes	characteristic, complemented
91	C₂×#104	—	2	2	no	—
92	#410	—	2	2	no	—
93	C₄³	C₄×C₈₉	1	1	yes	characteristic, complemented
94	#354	C₄×C₈₉	1	4	yes	complemented
95	#353	—	2	2	no	—
96	C₂×#41	D₄×C₈₉	1	2	yes	—
97	C₂×M₁₆	—	2	1	no	—
98	C₄⋊Q₈	Q₈×C₈₉	1	1	yes	characteristic
99	#127	D₄×C₈₉	1	1	yes	characteristic
100	#124	C₂×C₄×C₈₉	1	2	yes	—
101	#121	—	4	1	no	—
102	C₄×D₄	—	4	1	no	—
103	#117	—	4	2	no	—
104	C₂×C₄²	—	2	1	no	—
105	C₂×C₄²	—	4	1	no	—
106	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
107	C₄⋊C₈	—	4	1	no	—
108	#104	—	4	4	no	—
109	#99	—	2	4	no	—
110	#99	—	4	2	no	—
111	#41	—	4	2	no	—
112	#41	—	4	2	no	—
113	C₂×Q₈	C₄⋊C₄×C₈₉	1	2	yes	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	C₂²⋊C₄×C₈₉	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₄⋊C₄	—	4	1	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₂²⋊C₄	—	4	2	no	—
126	C₂²⋊C₄	—	4	4	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	4	1	no	—
130	C₄²	—	4	4	no	—
131	C₄²	—	8	2	no	—
132	C₄²	C₄²×C₈₉	1	1	yes	characteristic
133	C₄²	C₄⋊C₄×C₈₉	1	1	yes	characteristic
134	C₂³	—	2	2	no	—
135	C₂³	#99×C₈₉	1	1	yes	characteristic
136	Q₈	—	4	2	no	—
137	D₄	—	4	2	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	4	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	#95×C₈₉	1	1	yes	characteristic
152	C₂×C₄	#99×C₈₉	1	1	yes	characteristic
153	C₈	—	8	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	#342×C₈₉	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	4	no	—
163	C₄	—	4	2	no	—
164	C₄	—	16	4	no	—
165	C₄	#353×C₈₉	1	1	yes	characteristic
166	C₄	#354×C₈₉	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	4	1	no	—
169	C₂	—	8	2	no	—
170	C₂	C₈₉×#1393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#2126

C₂

yes

maximal

C₈₉×#1962

C₂

yes

maximal, characteristic

#532×C₈₉

C₂²

yes

characteristic

#423×C₈₉

C₄

yes

characteristic, complemented

C₂×#104×C₈₉

—

#410×C₈₉

—

C₄³×C₈₉

C₄

yes

characteristic, complemented

#354×C₈₉

C₄

yes

complemented

#353×C₈₉

—

C₂×#41×C₈₉

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

Q₈

yes

characteristic

#127×C₈₉

D₄

yes

characteristic

#124×C₈₉

C₂×C₄

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

#117×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

#104×C₈₉

—

#99×C₈₉

—

#99×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₄⋊C₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₄²

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

#99

yes

characteristic

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#342

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1393

yes

characteristic

#2126

C₂×C₈₉

yes

—

#1962

C₂×C₈₉

yes

characteristic

C₈₉

#7500

yes

characteristic, direct product factor, Sylow

#532

C₂²×C₈₉

yes

characteristic

#423

C₄×C₈₉

yes

characteristic, complemented

C₂×#104

—

#410

—

C₄³

C₄×C₈₉

yes

characteristic, complemented

#354

C₄×C₈₉

yes

complemented

#353

—

C₂×#41

D₄×C₈₉

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Q₈×C₈₉

yes

characteristic

#127

D₄×C₈₉

yes

characteristic

100

#124

C₂×C₄×C₈₉

yes

—

101

#121

—

102

C₄×D₄

—

103

#117

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

C₂×C₄×C₈₉

yes

characteristic

107

C₄⋊C₈

—

108

#104

—

109

#99

—

110

#99

—

111

#41

—

112

#41

—

113

C₂×Q₈

C₄⋊C₄×C₈₉

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²⋊C₄×C₈₉

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

121

M₁₆

—

122

C₂×C₈

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₄²×C₈₉

yes

characteristic

133

C₄²

C₄⋊C₄×C₈₉

yes

characteristic

134

C₂³

—

135

C₂³

#99×C₈₉

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#95×C₈₉

yes

characteristic

152

C₂×C₄

#99×C₈₉

yes

characteristic

153

C₈

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#342×C₈₉

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#353×C₈₉

yes

characteristic

166

C₄

#354×C₈₉

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

C₈₉×#1393

yes

characteristic

	subgroup	G/H
center	C₄×C₈₉	#354
commutator	C₄²	C₄²×C₈₉
Frattini, cosocle	#532	C₂²×C₈₉
socle	C₂×C₈₉	#1393
prime radical	#532×C₈₉	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #532×C₈₉ ◁ G

subgroup

G/H

center

C₄×C₈₉

#354

commutator

C₄²

C₄²×C₈₉

C₂²×C₈₉

C₂×C₈₉

#532×C₈₉

C₂²

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #532×C₈₉ ◁ G

N	H ≅ G/N	type
4. #423×C₈₉	C₄	semidirect
7. C₄³×C₈₉	C₄	semidirect
8. #354×C₈₉	C₄	semidirect
81. #7500	C₈₉	direct (N × H)
88. C₈₉	#7500	direct (N × H)
90. #423	C₄×C₈₉	semidirect
93. C₄³	C₄×C₈₉	semidirect
94. #354	C₄×C₈₉	semidirect

H ≅ G/N

type

4. #423×C₈₉

C₄

semidirect

7. C₄³×C₈₉

C₄

semidirect

8. #354×C₈₉

C₄

semidirect

81. #7500

C₈₉

direct (N × H)

88. C₈₉

#7500

direct (N × H)

90. #423

C₄×C₈₉

semidirect

93. C₄³

C₄×C₈₉

semidirect

94. #354

C₄×C₈₉

semidirect

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

subgroup	index
C₈₉×#2126	2
C₈₉×#1962	2
#7500	89

subgroup

index

C₈₉×#2126

C₈₉×#1962

#7500

Subgroups — C₈₉×#7500 #36385348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7500 #36385348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7500 #36385348