Group #22608796 — subgroups

Subgroups — #22608796

Order 9344.

182 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 71 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	#17988665	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988338	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988309	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	—	2	1	no	—
9	C₂²×#10596831	—	2	1	no	—
10	C₄⋊#10596831	C₂²	1	1	yes	characteristic
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₂²⋊#10596831	—	2	1	no	—
13	C₂²⋊#10596831	—	2	1	no	—
14	C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₃	C₄	1	1	yes	characteristic
17	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
19	#107×C₇₃	C₄	1	1	yes	characteristic
20	#102×C₇₃	C₄	1	1	yes	characteristic
21	#102×C₇₃	C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	#11924012	—	4	2	no	—
24	C₂×#10596831	—	2	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	4	2	no	—
28	C₂×#10596831	D₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	—	2	1	no	—
33	C₂²×C₄×C₇₃	—	2	1	no	—
34	D₈×C₇₃	—	2	2	no	—
35	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₃	—	2	1	no	—
40	C₂²⋊C₄×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	2	1	no	—
43	C₂³×C₇₃	—	2	1	no	—
44	D₄×C₇₃	—	2	1	no	—
45	D₄×C₇₃	—	2	1	no	—
46	D₄×C₇₃	—	2	1	no	—
47	D₄×C₇₃	M₁₆	1	2	yes	complemented
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
57	C₈×C₇₃	—	2	1	no	—
58	C₈×C₇₃	—	2	1	no	—
59	#10596831	—	2	2	no	—
60	#10596831	—	2	1	no	—
61	#10596831	—	2	1	no	—
62	#10596831	—	4	1	no	—
63	#10596831	—	4	1	no	—
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	—	4	1	no	—
66	C₂²×C₇₃	—	4	1	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	1	no	—
70	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	1	no	—
73	C₄×C₇₃	—	4	1	no	—
74	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₄×C₇₃	#131	1	1	yes	characteristic
76	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₇₃	—	4	2	no	—
79	C₂×C₇₃	—	8	1	no	—
80	C₂×C₇₃	#435	1	1	yes	characteristic
81	C₂×C₇₃	#442	1	1	yes	characteristic
82	C₂×C₇₃	#443	1	1	yes	characteristic
83	#1586	—	73	1	no	maximal, Sylow
84	C₇₃	#1586	1	1	yes	characteristic, complemented, Sylow
85	C₄×D₈	D₇₃	1	1	yes	characteristic
86	#436	—	73	1	no	—
87	C₈×D₄	—	73	1	no	—
88	#334	—	73	1	no	—
89	D₄⋊C₈	—	73	1	no	—
90	D₄⋊C₈	—	73	1	no	—
91	#322	—	73	1	no	—
92	C₂×D₈	Dic₇₃	1	1	yes	characteristic
93	C₂×M₁₆	—	146	1	no	—
94	C₂²×C₈	—	146	1	no	—
95	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
96	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
97	#107	Dic₇₃	1	1	yes	characteristic
98	C₄⋊C₈	—	73	1	no	—
99	C₄⋊C₈	—	73	1	no	—
100	#102	Dic₇₃	1	1	yes	characteristic
101	#102	Dic₇₃	1	1	yes	characteristic
102	C₂²⋊C₈	—	146	1	no	—
103	C₂²⋊C₈	—	146	1	no	—
104	C₄×C₈	—	73	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
107	C₂×D₄	C₂×Dic₇₃	1	1	yes	characteristic
108	C₂×D₄	C₂×Dic₇₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	D₈	—	2	2	no	—
112	M₁₆	—	292	2	no	—
113	C₂×C₈	—	292	2	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	146	1	no	—
117	C₂×C₈	—	146	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
120	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
121	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
122	C₄⋊C₄	C₂×Dic₇₃	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	2	1	no	—
125	C₄²	C₂²×D₇₃	1	1	yes	characteristic
126	C₂³	—	2	1	no	—
127	C₂³	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	2	1	no	—
130	D₄	—	2	1	no	—
131	D₄	#11924012	1	2	yes	complemented
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
139	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
140	C₂×C₄	#11924027	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	292	1	no	—
144	C₈	—	292	1	no	—
145	C₈	—	146	2	no	—
146	C₈	—	146	1	no	—
147	C₈	—	146	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	1	no	—
152	C₂²	—	4	1	no	—
153	C₂²	—	4	1	no	—
154	C₂²	D₄×Dic₇₃	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	4	1	no	—
157	C₄	—	4	1	no	—
158	C₄	C₂×#11924012	1	1	yes	characteristic
159	C₄	#14556598	1	1	yes	characteristic
160	C₄	#14556615	1	1	yes	characteristic
161	C₄	#14556615	1	1	yes	characteristic
162	C₂	—	4	2	no	—
163	C₂	—	8	1	no	—
164	C₂	#17988659	1	1	yes	characteristic
165	C₂	#17988699	1	1	yes	characteristic
166	C₂	#17988704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988665

C₂

yes

maximal, characteristic, complemented

D₄×#10596831

C₂

yes

maximal, characteristic, complemented

#17988338

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596831

C₂

yes

maximal, characteristic, complemented

D₄⋊#10596831

C₂

yes

maximal, characteristic, complemented

#17988309

C₂

yes

maximal, characteristic, complemented

C₄×D₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

C₂²×#10596831

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₄

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₄

yes

characteristic

#102×C₇₃

C₄

yes

characteristic

#102×C₇₃

C₄

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

M₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₄×D₈

D₇₃

yes

characteristic

#436

—

C₈×D₄

—

#334

—

D₄⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×D₈

Dic₇₃

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

146

—

C₄×D₄

C₂×D₇₃

yes

characteristic

C₄×D₄

C₂×D₇₃

yes

characteristic

#107

Dic₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

#102

Dic₇₃

yes

characteristic

101

#102

Dic₇₃

yes

characteristic

102

C₂²⋊C₈

—

146

—

103

C₂²⋊C₈

—

146

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×D₇₃

yes

characteristic

107

C₂×D₄

C₂×Dic₇₃

yes

characteristic

108

C₂×D₄

C₂×Dic₇₃

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

D₈

—

112

M₁₆

—

292

—

113

C₂×C₈

—

292

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₂×C₈

C₂×Dic₇₃

yes

characteristic

120

C₂×C₈

C₂×Dic₇₃

yes

characteristic

121

C₄⋊C₄

C₂×Dic₇₃

yes

characteristic

122

C₄⋊C₄

C₂×Dic₇₃

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

C₂²×D₇₃

yes

characteristic

126

C₂³

—

127

C₂³

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

#11924012

yes

complemented

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

139

C₂×C₄

D₄×D₇₃

yes

characteristic

140

C₂×C₄

#11924027

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

292

—

144

C₈

—

292

—

145

C₈

—

146

—

146

C₈

—

146

—

147

C₈

—

146

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

D₄×Dic₇₃

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂×#11924012

yes

characteristic

159

C₄

#14556598

yes

characteristic

160

C₄

#14556615

yes

characteristic

161

C₄

#14556615

yes

characteristic

162

C₂

—

163

C₂

—

164

C₂

#17988659

yes

characteristic

165

C₂

#17988699

yes

characteristic

166

C₂

#17988704

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₄×D₈×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₄
hypercenter	C₄×D₈	D₇₃
hypocenter	C₇₃	#1586
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₄×D₈×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₄

hypercenter

C₄×D₈

D₇₃

hypocenter

C₇₃

#1586

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #17988665	C₂	semidirect
2. D₄×#10596831	C₂	semidirect
3. #17988338	C₂	semidirect
4. D₄⋊#10596831	C₂	semidirect
5. D₄⋊#10596831	C₂	semidirect
6. #17988309	C₂	semidirect
47. D₄×C₇₃	M₁₆	semidirect
84. C₇₃	#1586	semidirect
131. D₄	#11924012	semidirect

H ≅ G/N

type

1. #17988665

C₂

semidirect

2. D₄×#10596831

C₂

semidirect

3. #17988338

C₂

semidirect

4. D₄⋊#10596831

C₂

semidirect

5. D₄⋊#10596831

C₂

semidirect

6. #17988309

C₂

semidirect

47. D₄×C₇₃

M₁₆

semidirect

84. C₇₃

#1586

semidirect

131. D₄

#11924012

semidirect

Subgroups — #22608796

Conjugacy classes of subgroups

Complements