Group #36385191 — subgroups

Subgroups — C₈₉×#7343 #36385191

Order 22784.

324 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 46 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×#1784	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1320	C₂	1	2	yes	maximal
3	C₂³×Q₈×C₈₉	C₄	1	1	yes	characteristic
4	C₂×#101×C₈₉	—	2	2	no	—
5	#407×C₈₉	C₄	1	1	yes	characteristic
6	#407×C₈₉	C₂²	1	1	yes	characteristic
7	#349×C₈₉	—	2	2	no	—
8	C₂²×Q₈×C₈₉	—	2	1	no	—
9	C₂²×Q₈×C₈₉	—	4	1	no	—
10	C₂²×Q₈×C₈₉	—	4	1	no	—
11	C₂²×Q₈×C₈₉	—	4	4	no	—
12	C₂²×Q₈×C₈₉	C₈	1	2	yes	—
13	C₂³×C₄×C₈₉	—	2	1	no	—
14	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₈₉	—	2	1	no	—
16	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
17	M₃₂×C₈₉	—	4	2	no	—
18	#101×C₈₉	—	4	4	no	—
19	C₂²⋊C₈×C₈₉	—	4	1	no	—
20	C₂²⋊C₈×C₈₉	—	4	1	no	—
21	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
22	C₂×Q₈×C₈₉	—	2	4	no	—
23	C₂×Q₈×C₈₉	—	4	2	no	—
24	C₂×Q₈×C₈₉	—	4	4	no	—
25	C₂×Q₈×C₈₉	—	4	2	no	—
26	C₂×Q₈×C₈₉	—	4	8	no	—
27	C₂×Q₈×C₈₉	—	4	4	no	—
28	C₂×Q₈×C₈₉	—	4	4	no	—
29	C₂×Q₈×C₈₉	—	4	1	no	—
30	C₂×Q₈×C₈₉	—	4	1	no	—
31	C₂²×C₄×C₈₉	—	2	2	no	—
32	C₂²×C₄×C₈₉	—	2	1	no	—
33	C₂²×C₄×C₈₉	—	2	1	no	—
34	C₂²×C₄×C₈₉	—	4	4	no	—
35	C₂²×C₄×C₈₉	—	4	2	no	—
36	C₂²×C₄×C₈₉	—	4	2	no	—
37	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
39	M₁₆×C₈₉	—	4	2	no	—
40	M₁₆×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	—	4	1	no	—
43	C₁₆×C₈₉	—	8	2	no	—
44	C₂³×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	—	4	1	no	—
46	C₂³×C₈₉	—	4	2	no	—
47	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
48	Q₈×C₈₉	—	4	4	no	—
49	Q₈×C₈₉	—	4	4	no	—
50	Q₈×C₈₉	—	4	4	no	—
51	Q₈×C₈₉	—	4	4	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	—	4	2	no	—
56	C₂×C₄×C₈₉	—	4	2	no	—
57	C₂×C₄×C₈₉	—	4	8	no	—
58	C₂×C₄×C₈₉	—	4	2	no	—
59	C₂×C₄×C₈₉	—	4	2	no	—
60	C₂×C₄×C₈₉	—	4	1	no	—
61	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
62	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
63	C₈×C₈₉	—	4	2	no	—
64	C₈×C₈₉	—	8	1	no	—
65	C₂²×C₈₉	—	2	2	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	—	4	1	no	—
68	C₂²×C₈₉	—	4	1	no	—
69	C₂²×C₈₉	—	4	2	no	—
70	C₂²×C₈₉	—	4	1	no	—
71	C₂²×C₈₉	—	4	2	no	—
72	C₂²×C₈₉	#323	1	1	yes	characteristic
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	4	4	no	—
75	C₄×C₈₉	—	4	1	no	—
76	C₄×C₈₉	AΣL(1,𝔽₁₆)	1	2	yes	—
77	#7343	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂×C₈₉	—	2	1	no	—
79	C₂×C₈₉	—	4	2	no	—
80	C₂×C₈₉	—	4	1	no	—
81	C₂×C₈₉	#1316	1	1	yes	characteristic
82	#1784	C₂×C₈₉	1	1	yes	characteristic
83	#1320	C₂×C₈₉	1	2	yes	—
84	C₈₉	#7343	1	1	yes	characteristic, direct product factor, Sylow
85	C₂³×Q₈	C₄×C₈₉	1	1	yes	characteristic
86	C₂×#101	—	2	2	no	—
87	#407	C₄×C₈₉	1	1	yes	characteristic
88	#407	C₂²×C₈₉	1	1	yes	characteristic
89	#349	—	2	2	no	—
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	—	4	1	no	—
92	C₂²×Q₈	—	4	1	no	—
93	C₂²×Q₈	—	4	4	no	—
94	C₂²×Q₈	C₈×C₈₉	1	2	yes	—
95	C₂³×C₄	—	2	1	no	—
96	C₂³×C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
99	M₃₂	—	4	2	no	—
100	#101	—	4	4	no	—
101	C₂²⋊C₈	—	4	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	C₂⁴	M₁₆×C₈₉	1	1	yes	characteristic
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	4	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂×Q₈	—	4	8	no	—
110	C₂×Q₈	—	4	4	no	—
111	C₂×Q₈	—	4	1	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	4	4	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
120	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	1	no	—
125	C₁₆	—	8	2	no	—
126	C₂³	—	2	1	no	—
127	C₂³	—	4	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
130	Q₈	—	4	4	no	—
131	Q₈	—	4	4	no	—
132	Q₈	—	4	4	no	—
133	Q₈	—	4	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	8	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	#99×C₈₉	1	1	yes	characteristic
144	C₂×C₄	#100×C₈₉	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₈	—	8	1	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	#323×C₈₉	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	4	4	no	—
157	C₄	—	4	1	no	—
158	C₄	AΣL(1,𝔽₁₆)×C₈₉	1	2	yes	—
159	C₂	—	2	1	no	—
160	C₂	—	4	2	no	—
161	C₂	—	4	1	no	—
162	C₂	C₈₉×#1316	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#1784

C₂

yes

maximal, characteristic

C₈₉×#1320

C₂

yes

maximal

C₂³×Q₈×C₈₉

C₄

yes

characteristic

C₂×#101×C₈₉

—

#407×C₈₉

C₄

yes

characteristic

#407×C₈₉

C₂²

yes

characteristic

#349×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

C₈

yes

—

C₂³×C₄×C₈₉

—

C₂³×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₉

—

C₂×M₁₆×C₈₉

D₄

yes

—

M₃₂×C₈₉

—

#101×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₂⁴×C₈₉

M₁₆

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₈

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₈

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

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

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#323

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

AΣL(1,𝔽₁₆)

yes

—

#7343

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1316

yes

characteristic

#1784

C₂×C₈₉

yes

characteristic

#1320

C₂×C₈₉

yes

—

C₈₉

#7343

yes

characteristic, direct product factor, Sylow

C₂³×Q₈

C₄×C₈₉

yes

characteristic

C₂×#101

—

#407

C₄×C₈₉

yes

characteristic

#407

C₂²×C₈₉

yes

characteristic

#349

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₈×C₈₉

yes

—

C₂³×C₄

—

C₂³×C₄

C₂×C₄×C₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

D₄×C₈₉

yes

—

M₃₂

—

100

#101

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

C₂⁴

M₁₆×C₈₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

C₂²⋊C₄×C₈₉

yes

characteristic

120

C₂²×C₄

C₂×C₈×C₈₉

yes

characteristic

121

M₁₆

—

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₁₆

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²⋊C₈×C₈₉

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#99×C₈₉

yes

characteristic

144

C₂×C₄

#100×C₈₉

yes

characteristic

145

C₈

—

146

C₈

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

#323×C₈₉

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

AΣL(1,𝔽₁₆)×C₈₉

yes

—

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

C₈₉×#1316

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#1316
commutator	C₂²×C₄	C₂×C₈×C₈₉
Frattini, cosocle	#407	C₂²×C₈₉
prime radical	C₂⁴×C₈₉	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #407×C₈₉ ◁ G

subgroup

G/H

center, socle

C₂×C₈₉

#1316

commutator

C₂²×C₄

C₂×C₈×C₈₉

Frattini, cosocle

#407

C₂²×C₈₉

prime radical

C₂⁴×C₈₉

M₁₆

LCS

G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈₉ ◁ C₂×C₄×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #407×C₈₉ ◁ G

N	H ≅ G/N	type
77. #7343	C₈₉	direct (N × H)
84. C₈₉	#7343	direct (N × H)

H ≅ G/N

type

77. #7343

C₈₉

direct (N × H)

84. C₈₉

#7343

direct (N × H)

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

subgroup	index
C₈₉×#1784	2
C₈₉×#1320	2
#7343	89

subgroup

index

C₈₉×#1784

C₈₉×#1320

#7343

Subgroups — C₈₉×#7343 #36385191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7343 #36385191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7343 #36385191