Group #22625362 — subgroups

Subgroups — #22625362

Order 9344.

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

normal subgroups only

This group has 264 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	#17989605	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989531	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989439	C₂	1	4	yes	maximal, complemented
4	#17989138	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#64408	—	2	1	no	—
6	#14556658	—	2	1	no	—
7	#14556648	C₂²	1	2	yes	—
8	C₄×#10596833	—	2	1	no	—
9	(C₂×#64408)⋊C₄	—	2	1	no	—
10	(C₂×#64408)⋊C₄	C₂²	1	2	yes	—
11	C₂×#11924029	—	2	1	no	—
12	#14556588	C₄	1	1	yes	characteristic, complemented
13	#14556580	—	2	1	no	—
14	#14556566	—	2	1	no	—
15	#14556566	C₂²	1	2	yes	—
16	#14556553	—	2	1	no	—
17	#14556553	C₄	1	2	yes	complemented
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	—	2	2	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₄⋊#64408	—	4	2	no	—
23	#11924029	—	4	2	no	—
24	#11924029	—	4	2	no	—
25	C₂×D₂₉₂	—	2	2	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	—	2	1	no	—
28	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
33	#11924005	—	2	2	no	—
34	#11924005	—	2	2	no	—
35	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₇₃	D₄	1	1	yes	characteristic
37	C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₂×Q₈×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
41	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×D₇₃	—	2	2	no	—
43	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	4	2	no	—
46	C₂×#64408	—	4	2	no	—
47	Q₈×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
55	D₂₉₂	—	4	2	no	—
56	D₂₉₂	—	4	2	no	—
57	D₂₉₂	—	4	2	no	—
58	D₂₉₂	—	4	2	no	—
59	D₂₉₂	—	4	2	no	—
60	C₄×D₇₃	—	4	1	no	—
61	C₄×D₇₃	—	4	1	no	—
62	C₄×D₇₃	—	4	4	no	—
63	C₄×D₇₃	D₈	1	2	yes	—
64	C₄×D₇₃	SD₁₆	1	2	yes	—
65	#10596833	—	4	2	no	—
66	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×D₇₃	—	2	1	no	—
68	C₂×D₇₃	—	2	1	no	—
69	C₂×D₇₃	—	4	2	no	—
70	C₂×D₇₃	—	4	2	no	—
71	C₂×D₇₃	—	4	2	no	—
72	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
73	C₂×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
74	#64408	—	8	2	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	4	2	no	—
78	C₄×C₇₃	#102	1	2	yes	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	#102	1	2	yes	—
82	C₂×C₇₃	#410	1	1	yes	characteristic
83	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
84	C₂×C₇₃	#421	1	1	yes	characteristic
85	D₇₃	—	2	1	no	—
86	D₇₃	—	2	1	no	—
87	D₇₃	—	8	2	no	—
88	#1517	—	73	1	no	maximal, Sylow
89	C₇₃	#1517	1	1	yes	characteristic, complemented, Sylow
90	#532	—	73	1	no	—
91	#431	—	73	1	no	—
92	#382	—	73	1	no	—
93	#327	—	73	4	no	—
94	C₂×#41	—	146	1	no	—
95	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
96	#127	—	73	1	no	—
97	#124	—	146	1	no	—
98	#121	—	73	2	no	—
99	#121	—	146	1	no	—
100	C₄×D₄	—	73	2	no	—
101	C₄×D₄	—	146	1	no	—
102	C₂×C₄⋊C₄	—	146	1	no	—
103	C₂×C₄²	—	73	1	no	—
104	C₄⋊C₈	—	146	1	no	—
105	C₂²⋊C₈	—	73	2	no	—
106	C₄×C₈	—	146	1	no	—
107	#95	—	73	2	no	—
108	#95	—	146	1	no	—
109	#41	—	292	2	no	—
110	#41	—	292	2	no	—
111	C₂×Q₈	—	2	1	no	—
112	C₂×D₄	—	73	2	no	—
113	C₂×D₄	—	146	2	no	—
114	C₂×D₄	—	146	1	no	—
115	C₂×D₄	—	146	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	146	2	no	—
120	C₂²×C₄	—	146	2	no	—
121	C₂×C₈	—	146	2	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	292	2	no	—
124	C₄⋊C₄	C₂×#64408	1	2	yes	—
125	C₂²⋊C₄	—	146	2	no	—
126	C₂²⋊C₄	—	146	2	no	—
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	C₂×#64408	1	1	yes	characteristic
131	C₂³	—	73	1	no	—
132	C₂³	—	146	2	no	—
133	Q₈	—	4	2	no	—
134	D₄	—	292	2	no	—
135	D₄	—	292	2	no	—
136	D₄	—	292	2	no	—
137	D₄	—	292	2	no	—
138	D₄	—	292	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	292	4	no	—
143	C₂×C₄	—	292	2	no	—
144	C₂×C₄	—	292	2	no	—
145	C₂×C₄	—	292	2	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	1	no	—
151	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
152	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
153	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
154	C₈	—	292	2	no	—
155	C₂²	—	292	2	no	—
156	C₂²	—	292	2	no	—
157	C₂²	—	292	2	no	—
158	C₂²	—	73	1	no	—
159	C₂²	—	73	1	no	—
160	C₂²	—	146	1	no	—
161	C₂²	—	146	1	no	—
162	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	4	2	no	—
166	C₄	—	584	2	no	—
167	C₄	—	73	2	no	—
168	C₄	—	146	1	no	—
169	C₄	—	146	1	no	—
170	C₄	#14556653	1	2	yes	—
171	C₂	—	584	2	no	—
172	C₂	—	146	1	no	—
173	C₂	—	146	1	no	—
174	C₂	#17989568	1	1	yes	characteristic
175	C₂	C₂×#14556653	1	1	yes	characteristic
176	C₂	#17989586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989605

C₂

yes

maximal, characteristic, complemented

#17989531

C₂

yes

maximal, characteristic, complemented

#17989439

C₂

yes

maximal, complemented

#17989138

C₂

yes

maximal, characteristic

C₂×C₄⋊#64408

—

#14556658

—

#14556648

C₂²

yes

—

C₄×#10596833

—

(C₂×#64408)⋊C₄

—

(C₂×#64408)⋊C₄

C₂²

yes

—

C₂×#11924029

—

#14556588

C₄

yes

characteristic, complemented

#14556580

—

#14556566

—

#14556566

C₂²

yes

—

#14556553

—

#14556553

C₄

yes

complemented

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924005

—

#11924005

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂×D₄

yes

—

—

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

D₈

yes

—

C₄×D₇₃

SD₁₆

yes

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

C₂×SD₁₆

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#102

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

D₇₃

—

#1517

—

maximal, Sylow

C₇₃

#1517

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#41

—

146

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#127

—

#124

—

146

—

#121

—

#121

—

146

—

100

C₄×D₄

—

101

C₄×D₄

—

146

—

102

C₂×C₄⋊C₄

—

146

—

103

C₂×C₄²

—

104

C₄⋊C₈

—

146

—

105

C₂²⋊C₈

—

106

C₄×C₈

—

146

—

107

#95

—

108

#95

—

146

—

109

#41

—

292

—

110

#41

—

292

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

146

—

114

C₂×D₄

—

146

—

115

C₂×D₄

—

146

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

146

—

120

C₂²×C₄

—

146

—

121

C₂×C₈

—

146

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

292

—

124

C₄⋊C₄

C₂×#64408

yes

—

125

C₂²⋊C₄

—

146

—

126

C₂²⋊C₄

—

146

—

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂×#64408

yes

characteristic

131

C₂³

—

132

C₂³

—

146

—

133

Q₈

—

134

D₄

—

292

—

135

D₄

—

292

—

136

D₄

—

292

—

137

D₄

—

292

—

138

D₄

—

292

—

139

C₂×C₄

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

292

—

143

C₂×C₄

—

292

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

292

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

C₂²⋊#64408

yes

characteristic

152

C₂×C₄

C₂²⋊#64408

yes

characteristic

153

C₂×C₄

C₂²×#64408

yes

characteristic

154

C₈

—

292

—

155

C₂²

—

292

—

156

C₂²

—

292

—

157

C₂²

—

292

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

146

—

161

C₂²

—

146

—

162

C₂²

C₂×C₂²⋊#64408

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

584

—

167

C₄

—

168

C₄

—

146

—

169

C₄

—

146

—

170

C₄

#14556653

yes

—

171

C₂

—

584

—

172

C₂

—

146

—

173

C₂

—

146

—

174

C₂

#17989568

yes

characteristic

175

C₂

C₂×#14556653

yes

characteristic

176

C₂

#17989586

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1517
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #17989605	C₂	semidirect
2. #17989531	C₂	semidirect
3. #17989439	C₂	semidirect
12. #14556588	C₄	semidirect
17. #14556553	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
89. C₇₃	#1517	semidirect
95. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

1. #17989605

C₂

semidirect

2. #17989531

C₂

semidirect

3. #17989439

C₂

semidirect

12. #14556588

C₄

semidirect

17. #14556553

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

89. C₇₃

#1517

semidirect

95. C₄⋊Q₈

#64408

semidirect

Subgroups — #22625362

Conjugacy classes of subgroups

Complements