Group #22626104 — subgroups

Subgroups — #22626104

Order 9344.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556646	C₂	1	1	yes	maximal, characteristic
3	#17989453	C₂	1	2	yes	maximal
4	#17989453	C₂	1	2	yes	maximal
5	C₂×#14556556	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596833	C₂²	1	2	yes	—
7	C₂²×#10596833	C₂²	1	2	yes	—
8	C₄⋊#10596833	—	2	2	no	—
9	#14556646	—	2	2	no	—
10	C₂²×Dic₁₄₆	C₄	1	1	yes	characteristic
11	#14556556	—	2	2	no	—
12	#14556556	—	2	2	no	—
13	C₂×#11924007	C₄	1	1	yes	characteristic
14	C₂×#11924007	C₄	1	1	yes	characteristic
15	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
19	C₂×#10596833	—	2	2	no	—
20	C₂×#10596833	—	2	2	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×#10596833	—	2	4	no	—
23	C₂×#10596833	Q₈	1	4	yes	complemented
24	C₂×Dic₁₄₆	—	2	2	no	—
25	C₂×Dic₁₄₆	—	2	2	no	—
26	C₂×Dic₁₄₆	—	2	2	no	—
27	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
31	#11924007	—	2	2	no	—
32	#11924007	—	2	2	no	—
33	C₄×Dic₇₃	—	2	1	no	—
34	C₄×Dic₇₃	—	2	1	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₄×Dic₇₃	—	2	1	no	—
37	C₄×Dic₇₃	D₄	1	2	yes	—
38	C₄×Dic₇₃	D₄	1	2	yes	—
39	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	2	2	no	—
43	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	1	no	—
53	C₂×Dic₇₃	—	2	1	no	—
54	C₂×Dic₇₃	—	2	1	no	—
55	C₂×Dic₇₃	—	2	1	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
58	C₂×Dic₇₃	M₁₆	1	2	yes	—
59	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₇₃	C₂×Q₈	1	2	yes	—
62	C₂×Dic₇₃	#41	1	2	yes	—
63	Dic₁₄₆	—	2	8	no	—
64	#10596833	—	2	4	no	—
65	#10596833	—	4	2	no	—
66	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₇₃	C₄×Q₈	1	2	yes	—
68	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
69	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
70	C₂²×C₇₃	#131	1	2	yes	—
71	C₄×C₇₃	—	2	2	no	—
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	—	4	1	no	—
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	—	2	2	no	—
76	Dic₇₃	—	4	1	no	—
77	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
78	Dic₇₃	#124	1	1	yes	characteristic
79	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₇₃	#391	1	1	yes	characteristic
81	C₂×C₇₃	#407	1	1	yes	characteristic
82	C₂×C₇₃	#408	1	1	yes	characteristic
83	C₂×C₇₃	#446	1	2	yes	—
84	C₂×C₇₃	#446	1	2	yes	—
85	#1985	—	73	1	no	maximal, Sylow
86	C₇₃	#1985	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×Q₈	—	73	1	no	—
88	C₂×C₄⋊C₈	—	73	1	no	—
89	C₂×#97	—	73	1	no	—
90	#336	—	73	2	no	—
91	#336	—	73	2	no	—
92	C₂²×Q₈	—	73	1	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₄×Q₈	—	146	2	no	—
96	C₄×Q₈	—	146	2	no	—
97	C₂×C₄⋊C₄	—	73	1	no	—
98	C₂×C₄⋊C₄	—	73	1	no	—
99	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
100	C₂×C₄²	—	73	1	no	—
101	C₂×C₄²	—	73	1	no	—
102	C₂×C₄²	—	73	1	no	—
103	C₄⋊C₈	—	146	2	no	—
104	#97	—	146	2	no	—
105	C₂×Q₈	—	146	2	no	—
106	C₂×Q₈	—	146	2	no	—
107	C₂×Q₈	—	146	2	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
113	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
114	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	146	2	no	—
117	C₂×C₈	—	146	4	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	146	2	no	—
122	C₄⋊C₄	—	146	2	no	—
123	C₄²	—	73	2	no	—
124	C₄²	—	73	2	no	—
125	C₄²	—	146	1	no	—
126	C₄²	—	146	1	no	—
127	C₄²	—	146	1	no	—
128	C₄²	—	146	1	no	—
129	C₂³	C₂²×#64408	1	1	yes	characteristic
130	Q₈	—	146	8	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	C₂²⋊#64408	1	2	yes	—
149	C₂×C₄	#11924038	1	2	yes	—
150	C₈	—	292	2	no	—
151	C₈	—	146	4	no	—
152	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
153	C₂²	Q₈×#64408	1	2	yes	—
154	C₂²	D₇₃⋊M₁₆	1	1	yes	characteristic
155	C₂²	C₂×#11924038	1	1	yes	characteristic
156	C₂²	#14556678	1	2	yes	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	1	no	—
160	C₄	—	292	1	no	—
161	C₄	—	73	1	no	—
162	C₄	—	73	2	no	—
163	C₄	—	73	1	no	—
164	C₄	—	146	2	no	—
165	C₄	—	146	2	no	—
166	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
167	C₂	#17989558	1	1	yes	characteristic
168	C₂	#17989560	1	1	yes	characteristic
169	C₂	#17989637	1	2	yes	—
170	C₂	#17989639	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10596833

C₂

yes

maximal, characteristic

C₂×#14556646

C₂

yes

maximal, characteristic

#17989453

C₂

yes

maximal

#17989453

C₂

yes

maximal

C₂×#14556556

C₂

yes

maximal, characteristic

C₂²×#10596833

C₂²

yes

—

C₂²×#10596833

C₂²

yes

—

C₄⋊#10596833

—

#14556646

—

C₂²×Dic₁₄₆

C₄

yes

—

—

C₄

yes

characteristic

C₂×#11924007

C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₄

yes

—

—

—

—

Q₈

yes

complemented

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

characteristic

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

—

C₄×Dic₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

C₂×Dic₇₃

M₁₆

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

#41

yes

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

—

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₂²×C₇₃

#131

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²⋊Q₈

yes

—

Dic₇₃

#124

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#446

yes

—

#1985

—

maximal, Sylow

C₇₃

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#64408

yes

characteristic

100

C₂×C₄²

—

101

C₂×C₄²

—

102

C₂×C₄²

—

103

C₄⋊C₈

—

146

—

104

#97

—

146

—

105

C₂×Q₈

—

146

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×#64408

yes

characteristic

113

C₂²×C₄

C₂×#64408

yes

characteristic

114

C₂²×C₄

C₂×#64408

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

146

—

126

C₄²

—

146

—

127

C₄²

—

146

—

128

C₄²

—

146

—

129

C₂³

C₂²×#64408

yes

characteristic

130

Q₈

—

146

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

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

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

C₂²⋊#64408

yes

—

149

C₂×C₄

#11924038

yes

—

150

C₈

—

292

—

151

C₈

—

146

—

152

C₂²

C₂×C₂²⋊#64408

yes

characteristic

153

C₂²

Q₈×#64408

yes

—

154

C₂²

D₇₃⋊M₁₆

yes

characteristic

155

C₂²

C₂×#11924038

yes

characteristic

156

C₂²

#14556678

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

292

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

146

—

165

C₄

—

146

—

166

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

167

C₂

#17989558

yes

characteristic

168

C₂

#17989560

yes

characteristic

169

C₂

#17989637

yes

—

170

C₂

#17989639

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#64408
commutator, socle	C₂³×C₇₃	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₇₃	C₄
hypercenter	C₂×C₄⋊C₄	#64408
hypocenter	C₇₃	#1985
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×#64408

commutator, socle

C₂³×C₇₃

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₇₃

C₄

C₂×C₄⋊C₄

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
23. C₂×#10596833	Q₈	semidirect
86. C₇₃	#1985	semidirect

H ≅ G/N

type

23. C₂×#10596833

Q₈

semidirect

86. C₇₃

#1985

semidirect

Subgroups — #22626104

Conjugacy classes of subgroups

Complements