Group #22625427 — subgroups

Subgroups — D₇₃⋊#332 #22625427

Order 9344.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 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	#17989588	C₂	1	1	yes	maximal, characteristic
2	#17989588	C₂	1	1	yes	maximal, characteristic
3	#17989446	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	—	2	1	no	—
6	C₂×#11924036	—	2	1	no	—
7	#14556658	C₂²	1	2	yes	—
8	#14556658	C₂²	1	2	yes	—
9	C₄⋊#10596833	—	2	1	no	—
10	C₄⋊#10596833	—	2	1	no	—
11	C₂×Q₈×D₇₃	—	2	1	no	—
12	#14556591	C₄	1	1	yes	characteristic
13	#14556590	—	2	1	no	—
14	#14556590	C₂²	1	2	yes	—
15	#14556589	—	2	1	no	—
16	C₄⋊C₄×D₇₃	—	2	1	no	—
17	C₄⋊C₄×D₇₃	C₄	1	2	yes	—
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic
20	#11924036	—	4	2	no	—
21	#11924036	—	4	2	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₂×#10596833	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×Dic₁₄₆	—	2	2	no	—
26	Q₈×D₇₃	—	4	2	no	—
27	Q₈×D₇₃	—	4	2	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
34	#11924007	—	2	2	no	—
35	#11924007	—	2	2	no	—
36	C₄×Dic₇₃	D₄	1	1	yes	characteristic
37	C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
42	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	—	4	4	no	—
55	C₄×D₇₃	D₈	1	2	yes	—
56	C₄×D₇₃	SD₁₆	1	2	yes	—
57	C₄×D₇₃	SD₁₆	1	2	yes	—
58	C₄×D₇₃	Q₁₆	1	2	yes	—
59	Dic₁₄₆	—	4	2	no	—
60	Dic₁₄₆	—	4	2	no	—
61	Dic₁₄₆	—	4	2	no	—
62	#10596833	—	4	2	no	—
63	#10596833	—	4	2	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₃	#101	1	2	yes	—
66	C₂×D₇₃	#102	1	2	yes	—
67	C₂×D₇₃	#103	1	2	yes	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	4	2	no	—
70	C₄×C₇₃	#102	1	2	yes	—
71	C₄×C₇₃	#103	1	2	yes	—
72	Dic₇₃	—	2	1	no	—
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	C₂×D₈	1	1	yes	characteristic
75	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
76	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
77	Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
79	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
80	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
81	D₇₃	#332	1	4	yes	complemented
82	C₂×#332	—	73	1	no	maximal, Sylow
83	C₇₃	C₂×#332	1	1	yes	characteristic, complemented, Sylow
84	C₂×C₄⋊Q₈	—	73	1	no	—
85	C₂×C₄⋊C₈	—	73	1	no	—
86	C₂×C₄⋊C₈	—	73	1	no	—
87	#332	—	73	4	no	—
88	C₂²×Q₈	—	146	1	no	—
89	C₂²×C₈	—	146	1	no	—
90	C₂²×C₈	—	146	1	no	—
91	C₄⋊Q₈	—	73	2	no	—
92	C₄⋊Q₈	—	73	1	no	—
93	C₄⋊Q₈	—	146	1	no	—
94	C₄⋊Q₈	—	146	1	no	—
95	C₄⋊Q₈	#64408	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	73	2	no	—
97	C₂×C₄⋊C₄	—	146	1	no	—
98	C₂×C₄²	—	73	1	no	—
99	C₄⋊C₈	—	73	2	no	—
100	C₄⋊C₈	—	73	2	no	—
101	C₄⋊C₈	—	146	1	no	—
102	C₄⋊C₈	—	146	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	292	2	no	—
105	C₂×Q₈	—	292	2	no	—
106	C₂×Q₈	—	146	2	no	—
107	C₂×Q₈	—	146	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	146	2	no	—
112	C₂×C₈	—	292	2	no	—
113	C₂×C₈	—	292	2	no	—
114	C₂×C₈	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	73	2	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₄⋊C₄	—	146	2	no	—
120	C₄⋊C₄	—	146	1	no	—
121	C₄⋊C₄	C₂×#64408	1	2	yes	—
122	C₄²	—	73	1	no	—
123	C₄²	—	73	1	no	—
124	C₄²	—	73	1	no	—
125	C₄²	C₂×#64408	1	1	yes	characteristic
126	C₂³	—	73	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	292	2	no	—
129	Q₈	—	292	2	no	—
130	Q₈	—	292	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	292	4	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
144	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
145	C₈	—	292	2	no	—
146	C₈	—	292	2	no	—
147	C₂²	—	73	2	no	—
148	C₂²	—	73	2	no	—
149	C₂²	—	73	2	no	—
150	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	292	2	no	—
154	C₄	—	73	1	no	—
155	C₄	—	73	1	no	—
156	C₄	—	73	1	no	—
157	C₄	—	73	1	no	—
158	C₄	—	146	1	no	—
159	C₄	#14556653	1	2	yes	—
160	C₄	#14556654	1	2	yes	—
161	C₂	—	73	4	no	—
162	C₂	D₇₃⋊#101	1	1	yes	characteristic
163	C₂	C₂×#14556653	1	1	yes	characteristic
164	C₂	C₂×#14556654	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989588

C₂

yes

maximal, characteristic

#17989588

C₂

yes

maximal, characteristic

#17989446

C₂

yes

maximal, complemented

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

—

C₂×#11924036

—

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

C₄⋊#10596833

—

C₄⋊#10596833

—

C₂×Q₈×D₇₃

—

#14556591

C₄

yes

characteristic

#14556590

—

#14556590

C₂²

yes

—

#14556589

—

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₄

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

—

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

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

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

D₈

yes

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

#101

yes

—

C₂×D₇₃

#102

yes

—

C₂×D₇₃

#103

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#102

yes

—

C₄×C₇₃

#103

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×D₈

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×Q₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#332

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

#64408

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

146

—

102

C₄⋊C₈

—

146

—

103

C₂×Q₈

—

104

C₂×Q₈

—

292

—

105

C₂×Q₈

—

292

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

146

—

112

C₂×C₈

—

292

—

113

C₂×C₈

—

292

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

146

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

C₂×#64408

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂×#64408

yes

characteristic

126

C₂³

—

127

Q₈

—

128

Q₈

—

292

—

129

Q₈

—

292

—

130

Q₈

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

C₂²⋊#64408

yes

characteristic

143

C₂×C₄

C₂²⋊#64408

yes

characteristic

144

C₂×C₄

C₂²×#64408

yes

characteristic

145

C₈

—

292

—

146

C₈

—

292

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊#64408

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

292

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

146

—

159

C₄

#14556653

yes

—

160

C₄

#14556654

yes

—

161

C₂

—

162

C₂

D₇₃⋊#101

yes

characteristic

163

C₂

C₂×#14556653

yes

characteristic

164

C₂

C₂×#14556654

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	C₂×#332
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
3. #17989446	C₂	semidirect
81. D₇₃	#332	semidirect
83. C₇₃	C₂×#332	semidirect

H ≅ G/N

type

3. #17989446

C₂

semidirect

81. D₇₃

#332

semidirect

83. C₇₃

C₂×#332

semidirect

Subgroups — D₇₃⋊#332 #22625427

Conjugacy classes of subgroups

Complements

Subgroups — D73⋊#332 #22625427

Conjugacy classes of subgroups

Complements

Subgroups — D₇₃⋊#332 #22625427