Group #36467829 — subgroups

Subgroups — #36467829

Order 22784.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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	#26032051	C₂	1	1	yes	maximal, characteristic
2	#26022330	C₂	1	2	yes	maximal, complemented
3	#26018289	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018226	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152399	—	2	2	no	—
7	#19152039	—	2	2	no	—
8	#19151865	—	2	2	no	—
9	#19151765	—	2	2	no	—
10	#19151716	C₂²	1	2	yes	—
11	#19151419	C₂²	1	2	yes	complemented
12	#19151414	—	2	2	no	—
13	#494×C₈₉	C₂²	1	1	yes	characteristic, complemented
14	#493×C₈₉	C₂²	1	1	yes	characteristic
15	#486×C₈₉	—	2	2	no	—
16	C₈²×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	#15668057	—	2	2	no	—
18	#15668038	—	2	4	no	—
19	#15668036	—	4	4	no	—
20	#15668034	D₄	1	2	yes	—
21	#15668031	—	4	2	no	—
22	D₈×D₈₉	—	4	2	no	—
23	#15668018	—	2	2	no	—
24	#15667958	—	4	2	no	—
25	#15667955	—	4	2	no	—
26	#15667952	—	2	2	no	—
27	#15667948	D₄	1	2	yes	—
28	#15667940	—	2	2	no	—
29	#15667929	—	2	2	no	—
30	C₂×Q₁₆×C₈₉	—	2	2	no	—
31	C₂×D₈×C₈₉	—	2	2	no	—
32	C₄⋊Q₈×C₈₉	—	2	1	no	—
33	#127×C₈₉	D₄	1	2	yes	—
34	#102×C₈₉	—	4	4	no	—
35	C₄×C₈×C₈₉	—	2	1	no	—
36	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
37	#13083574	—	2	2	no	—
38	#13083573	—	4	4	no	—
39	C₂×C₂²⋊D₈₉	—	8	2	no	—
40	D₄×D₈₉	—	4	4	no	—
41	Dic₃₅₆	D₈	1	2	yes	—
42	#13083563	—	2	2	no	—
43	#13083562	—	4	4	no	—
44	D₇₁₂	D₈	1	2	yes	—
45	#13083559	—	4	4	no	—
46	#13083558	—	2	2	no	—
47	#13083557	—	4	2	no	—
48	C₈×D₈₉	—	4	2	no	—
49	#13083549	—	4	2	no	—
50	C₂×Q₈×C₈₉	—	4	1	no	—
51	C₂×D₄×C₈₉	—	2	2	no	—
52	C₂×D₄×C₈₉	—	4	2	no	—
53	Q₁₆×C₈₉	—	4	2	no	—
54	Q₁₆×C₈₉	—	4	2	no	—
55	D₈×C₈₉	—	4	2	no	—
56	D₈×C₈₉	—	4	2	no	—
57	C₂×C₈×C₈₉	—	2	2	no	—
58	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₈₉	—	4	2	no	—
60	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂³×C₈₉	—	4	2	no	—
62	C₂²×D₈₉	—	8	2	no	—
63	Q₈×C₈₉	—	8	2	no	—
64	D₄×C₈₉	—	4	4	no	—
65	D₄×C₈₉	—	8	4	no	—
66	C₂×C₄×C₈₉	—	2	1	no	—
67	C₂×C₄×C₈₉	—	8	1	no	—
68	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
69	C₂²⋊D₈₉	—	4	2	no	—
70	C₂²⋊D₈₉	—	8	4	no	—
71	C₂×Dic₈₉	—	8	2	no	—
72	D₃₅₆	—	2	2	no	—
73	C₄×D₈₉	—	4	2	no	—
74	Dic₁₇₈	—	2	2	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	#10623494	—	4	2	no	—
79	C₂²×C₈₉	—	8	2	no	—
80	C₂²×C₈₉	—	16	2	no	—
81	C₂²×C₈₉	#453	1	1	yes	characteristic
82	C₂×D₈₉	—	4	2	no	—
83	C₂×D₈₉	—	32	2	no	—
84	C₄×C₈₉	—	2	2	no	—
85	C₄×C₈₉	—	16	1	no	—
86	C₄×C₈₉	#447	1	2	yes	—
87	Dic₈₉	—	4	2	no	—
88	#12312	—	89	1	no	maximal, Sylow
89	C₂×C₈₉	—	2	1	no	—
90	C₂×C₈₉	—	16	2	no	—
91	C₂×C₈₉	#1619	1	1	yes	characteristic
92	D₈₉	—	8	2	no	—
93	#3292	—	89	1	no	—
94	#2214	—	89	2	no	—
95	#1709	D₈₉	1	1	yes	characteristic, complemented
96	#1357	—	89	1	no	—
97	#1335	—	89	2	no	—
98	C₈₉	#12312	1	1	yes	characteristic, complemented, Sylow
99	#576	—	178	2	no	—
100	#510	—	178	2	no	—
101	#494	C₂×D₈₉	1	1	yes	characteristic, complemented
102	#493	C₂×D₈₉	1	1	yes	characteristic
103	#486	—	2	2	no	—
104	#472	—	178	2	no	—
105	#453	—	178	2	no	—
106	#443	—	89	2	no	—
107	#364	—	89	2	no	—
108	#362	—	178	2	no	—
109	C₈²	C₂×D₈₉	1	1	yes	characteristic, complemented
110	H(5,𝔽₂)	—	178	2	no	—
111	AGL(1,ℤ/8)	—	356	4	no	—
112	AGL(1,ℤ/8)	—	178	4	no	—
113	#135	—	89	2	no	—
114	#135	—	356	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	—	356	2	no	—
118	#131	—	178	2	no	—
119	C₄⋊Q₈	—	2	1	no	—
120	#127	C₂²⋊D₈₉	1	2	yes	—
121	Q₃₂	—	356	2	no	—
122	SD₃₂	—	356	2	no	—
123	M₃₂	—	178	2	no	—
124	#108	—	89	2	no	—
125	#104	—	178	2	no	—
126	#102	—	4	4	no	—
127	#100	—	178	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
130	#41	—	356	4	no	—
131	#41	—	178	2	no	—
132	C₂×Q₈	—	4	1	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	4	2	no	—
135	C₂×D₄	—	356	4	no	—
136	C₂×D₄	—	712	2	no	—
137	Q₁₆	—	4	2	no	—
138	Q₁₆	—	4	2	no	—
139	Q₁₆	—	89	2	no	—
140	SD₁₆	—	356	4	no	—
141	SD₁₆	—	178	2	no	—
142	D₈	—	4	2	no	—
143	D₈	—	4	2	no	—
144	D₈	—	89	2	no	—
145	D₈	—	356	4	no	—
146	M₁₆	—	356	2	no	—
147	M₁₆	—	178	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	356	2	no	—
150	C₂×C₈	D₄×D₈₉	1	2	yes	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₁₆	—	356	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	712	2	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	178	2	no	—
158	D₄	—	4	4	no	—
159	D₄	—	8	4	no	—
160	D₄	—	356	2	no	—
161	D₄	—	178	2	no	—
162	D₄	—	712	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	356	2	no	—
166	C₂×C₄	—	712	2	no	—
167	C₂×C₄	#15667980	1	1	yes	characteristic
168	C₈	—	4	2	no	—
169	C₈	—	356	2	no	—
170	C₈	D₈×D₈₉	1	2	yes	—
171	C₈	#15668036	1	2	yes	—
172	C₂²	—	8	2	no	—
173	C₂²	—	16	2	no	—
174	C₂²	—	2848	2	no	—
175	C₂²	—	356	2	no	—
176	C₂²	#19151765	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	16	1	no	—
179	C₄	—	356	2	no	—
180	C₄	#19151730	1	2	yes	—
181	C₂	—	2	1	no	—
182	C₂	—	16	2	no	—
183	C₂	—	712	2	no	—
184	C₂	#26019304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26032051

C₂

yes

maximal, characteristic

#26022330

C₂

yes

maximal, complemented

#26018289

C₂

yes

maximal, characteristic, complemented

#26018226

C₂

yes

maximal, complemented

C₈₉×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

—

#19151419

C₂²

yes

complemented

#19151414

—

#494×C₈₉

C₂²

yes

characteristic, complemented

#493×C₈₉

C₂²

yes

characteristic

#486×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic, complemented

—

—

—

D₄

yes

—

#15668031

—

D₈×D₈₉

—

—

—

—

—

D₄

yes

—

#15667940

—

#15667929

—

C₂×Q₁₆×C₈₉

—

C₂×D₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

#127×C₈₉

D₄

yes

—

#102×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

#13083574

—

#13083573

—

C₂×C₂²⋊D₈₉

—

D₄×D₈₉

—

Dic₃₅₆

D₈

yes

—

#13083563

—

#13083562

—

D₇₁₂

D₈

yes

—

#13083559

—

#13083558

—

#13083557

—

C₈×D₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#120

yes

characteristic

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

AGL(1,ℤ/8)

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#453

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#447

yes

—

Dic₈₉

—

#12312

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1619

yes

characteristic

D₈₉

—

#3292

—

#2214

—

#1709

D₈₉

yes

characteristic, complemented

#1357

—

#1335

—

C₈₉

#12312

yes

characteristic, complemented, Sylow

#576

—

178

—

100

#510

—

178

—

101

#494

C₂×D₈₉

yes

characteristic, complemented

102

#493

C₂×D₈₉

yes

characteristic

103

#486

—

104

#472

—

178

—

105

#453

—

178

—

106

#443

—

107

#364

—

108

#362

—

178

—

109

C₈²

C₂×D₈₉

yes

characteristic, complemented

110

H(5,𝔽₂)

—

178

—

111

AGL(1,ℤ/8)

—

356

—

112

AGL(1,ℤ/8)

—

178

—

113

#135

—

114

#135

—

356

—

115

C₂×Q₁₆

—

116

C₂×D₈

—

117

C₂×D₈

—

356

—

118

#131

—

178

—

119

C₄⋊Q₈

—

120

#127

C₂²⋊D₈₉

yes

—

121

Q₃₂

—

356

—

122

SD₃₂

—

356

—

123

M₃₂

—

178

—

124

#108

—

125

#104

—

178

—

126

#102

—

127

#100

—

178

—

128

C₄×C₈

—

129

C₄×C₈

C₂²×D₈₉

yes

characteristic

130

#41

—

356

—

131

#41

—

178

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

356

—

136

C₂×D₄

—

712

—

137

Q₁₆

—

138

Q₁₆

—

139

Q₁₆

—

140

SD₁₆

—

356

—

141

SD₁₆

—

178

—

142

D₈

—

143

D₈

—

144

D₈

—

145

D₈

—

356

—

146

M₁₆

—

356

—

147

M₁₆

—

178

—

148

C₂×C₈

—

149

C₂×C₈

—

356

—

150

C₂×C₈

D₄×D₈₉

yes

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

153

C₁₆

—

356

—

154

C₂³

—

155

C₂³

—

712

—

156

Q₈

—

157

Q₈

—

178

—

158

D₄

—

159

D₄

—

160

D₄

—

356

—

161

D₄

—

178

—

162

D₄

—

712

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

356

—

166

C₂×C₄

—

712

—

167

C₂×C₄

#15667980

yes

characteristic

168

C₈

—

169

C₈

—

356

—

170

C₈

D₈×D₈₉

yes

—

171

C₈

#15668036

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

2848

—

175

C₂²

—

356

—

176

C₂²

#19151765

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

356

—

180

C₄

#19151730

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

712

—

184

C₂

#26019304

yes

characteristic

	subgroup	G/H
center	C₂	#26019304
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1709	C₂
socle	C₂×C₈₉	#1619
prime radical	#26032051	C₂
hypercenter	#1709	D₈₉
hypocenter	C₈₉	#12312
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

subgroup

G/H

center

C₂

#26019304

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1709

C₂

C₂×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1709

N	H ≅ G/N	type
2. #26022330	C₂	semidirect
3. #26018289	C₂	semidirect
4. #26018226	C₂	semidirect
5. C₈₉×#1709	C₂	semidirect
11. #19151419	C₂²	semidirect
13. #494×C₈₉	C₂²	semidirect
16. C₈²×C₈₉	C₂²	semidirect
95. #1709	D₈₉	semidirect
98. C₈₉	#12312	semidirect
101. #494	C₂×D₈₉	semidirect
109. C₈²	C₂×D₈₉	semidirect

H ≅ G/N

type

2. #26022330

C₂

semidirect

3. #26018289

C₂

semidirect

4. #26018226

C₂

semidirect

5. C₈₉×#1709

C₂

semidirect

11. #19151419

C₂²

semidirect

13. #494×C₈₉

C₂²

semidirect

16. C₈²×C₈₉

C₂²

semidirect

95. #1709

D₈₉

semidirect

98. C₈₉

#12312

semidirect

101. #494

C₂×D₈₉

semidirect

109. C₈²

C₂×D₈₉

semidirect

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

subgroup	index
#26032051	2
#26022330	2
#26018289	2
#26018226	2
C₈₉×#1709	2
#12312	89

subgroup

index

C₈₉×#1709

Subgroups — #36467829

Conjugacy classes of subgroups

Complements

Maximal subgroups