Group #34014234 — subgroups

Subgroups — #34014234

Order 18688.

298 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 112 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×#17989519	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556641	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625045	C₂	1	4	yes	maximal, complemented
4	C₂×#17989005	C₂	1	1	yes	maximal, characteristic
5	#17989519	—	2	2	no	—
6	C₂×#14556641	—	2	2	no	—
7	C₂×#14556641	C₂²	1	2	yes	—
8	C₂×#14556641	C₂²	1	2	yes	—
9	C₂×#14556597	C₄	1	1	yes	characteristic
10	#17989005	—	2	2	no	—
11	#17989005	—	2	2	no	—
12	C₂×#14556530	C₄	1	2	yes	—
13	C₂²×#11924003	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
15	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
16	#14556641	—	2	2	no	—
17	#14556641	—	2	2	no	—
18	C₂×#11924028	C₈	1	1	yes	characteristic
19	#14556597	—	2	2	no	—
20	#14556597	—	2	2	no	—
21	C₂×#11924012	C₈	1	1	yes	characteristic
22	C₂×#11924011	C₈	1	2	yes	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	#14556530	—	2	4	no	—
25	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924003	D₄	1	2	yes	—
27	C₂×#11924003	D₄	1	2	yes	—
28	C₂×#11924003	C₂³	1	1	yes	characteristic
29	C₁₆×D₇₃	—	2	2	no	—
30	C₁₆×D₇₃	—	2	2	no	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₃	—	2	2	no	—
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	#11924028	—	2	2	no	—
35	#11924028	—	2	2	no	—
36	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
37	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
38	C₂×C₂²⋊D₇₃	C₁₆	1	2	yes	—
39	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
40	#11924012	—	2	2	no	—
41	#11924011	—	2	4	no	—
42	C₂×#10596831	C₂×C₈	1	2	yes	—
43	C₈×D₇₃	—	2	2	no	—
44	C₈×D₇₃	—	2	2	no	—
45	#11924003	C₂²⋊C₄	1	2	yes	—
46	#11924003	C₂×D₄	1	1	yes	characteristic
47	#11924003	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₇₃	—	2	2	no	—
54	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
58	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
60	C₂²⋊D₇₃	—	2	4	no	—
61	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
62	D₂₉₂	—	2	2	no	—
63	C₄×D₇₃	—	2	2	no	—
64	C₄×D₇₃	—	2	2	no	—
65	Dic₁₄₆	—	2	2	no	—
66	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
67	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
69	#10596831	—	2	2	no	—
70	C₂²×C₇₃	—	2	2	no	—
71	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
72	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	—	2	4	no	—
75	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
76	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
78	Dic₇₃	—	2	2	no	—
79	#13554	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	2	no	—
81	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
82	C₂×C₇₃	#2258	1	2	yes	—
83	D₇₃	—	4	2	no	—
84	C₂×#504	—	73	1	no	—
85	C₂×M₆₄	—	73	1	no	—
86	C₂²×C₃₂	—	73	1	no	—
87	C₂²⋊C₃₂	—	73	4	no	—
88	C₇₃	#13554	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	—	73	1	no	—
90	#504	—	146	2	no	—
91	#504	—	146	2	no	—
92	C₂×M₃₂	—	73	2	no	—
93	C₂×M₃₂	#64408	1	1	yes	characteristic
94	C₂²×C₁₆	—	73	1	no	—
95	C₂²×C₁₆	—	73	2	no	—
96	M₆₄	—	146	2	no	—
97	C₂×C₃₂	—	73	2	no	—
98	C₂×C₃₂	—	73	2	no	—
99	C₂×C₃₂	—	146	2	no	—
100	C₂×#41	—	73	1	no	—
101	#131	—	146	2	no	—
102	#131	—	146	2	no	—
103	C₂×M₁₆	—	73	1	no	—
104	C₂×M₁₆	—	73	2	no	—
105	C₂²×C₈	—	73	2	no	—
106	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	146	4	no	—
109	C₂×C₁₆	—	73	2	no	—
110	C₂×C₁₆	—	73	2	no	—
111	C₂×C₁₆	—	73	1	no	—
112	C₂×C₁₆	—	73	1	no	—
113	C₂×C₁₆	—	146	2	no	—
114	C₂×C₁₆	—	146	2	no	—
115	C₂×C₁₆	C₂×#64408	1	2	yes	—
116	C₃₂	—	146	2	no	—
117	C₃₂	—	146	2	no	—
118	#41	—	146	2	no	—
119	#41	—	146	2	no	—
120	C₂×Q₈	—	73	1	no	—
121	C₂×D₄	—	73	2	no	—
122	C₂×D₄	—	73	1	no	—
123	C₂²×C₄	—	73	2	no	—
124	C₂²×C₄	#11924036	1	1	yes	characteristic
125	M₁₆	—	146	2	no	—
126	M₁₆	—	146	4	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	C₂²⋊#64408	1	2	yes	—
131	C₂×C₈	#11924036	1	1	yes	characteristic
132	C₂×C₈	#11924038	1	2	yes	—
133	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	—	73	2	no	—
136	C₁₆	—	73	1	no	—
137	C₁₆	—	73	1	no	—
138	C₂³	—	73	2	no	—
139	C₂³	#14556664	1	1	yes	characteristic
140	Q₈	—	146	2	no	—
141	D₄	—	146	4	no	—
142	D₄	—	146	2	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	#14556648	1	2	yes	—
147	C₂×C₄	#14556664	1	1	yes	characteristic
148	C₂×C₄	#14556666	1	2	yes	—
149	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
150	C₈	—	146	2	no	—
151	C₈	#14556648	1	2	yes	—
152	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
153	C₈	C₂×#11924038	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	146	4	no	—
156	C₂²	—	146	2	no	—
157	C₂²	#17989482	1	2	yes	—
158	C₂²	C₂×#14556664	1	1	yes	characteristic
159	C₄	—	146	2	no	—
160	C₄	#17989482	1	2	yes	—
161	C₄	C₂×#14556648	1	1	yes	characteristic
162	C₄	C₂×#14556666	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	292	2	no	—
165	C₂	C₂×#17989482	1	1	yes	characteristic
166	C₂	#22626355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal, characteristic, complemented

C₂²×#14556641

C₂

yes

maximal, characteristic, complemented

#22625045

C₂

yes

maximal, complemented

C₂×#17989005

C₂

yes

maximal, characteristic

#17989519

—

C₂×#14556641

—

C₂×#14556641

C₂²

yes

—

C₂×#14556641

C₂²

yes

—

C₂×#14556597

C₄

yes

—

—

C₄

yes

—

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₂²

yes

—

C₂×M₃₂×C₇₃

C₄

yes

—

—

C₈

yes

—

—

C₈

yes

characteristic

C₂×#11924011

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

#14556530

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

D₄

yes

—

C₂×#11924003

D₄

yes

—

C₂×#11924003

C₂³

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

C₁₆

yes

characteristic

C₂×D₂₉₂

C₁₆

yes

characteristic

C₂×C₂²⋊D₇₃

C₁₆

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

#11924012

—

#11924011

—

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

C₂²⋊C₄

yes

—

#11924003

C₂×D₄

yes

characteristic

#11924003

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂²⋊C₈

yes

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

#13554

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₃

#2258

yes

—

D₇₃

—

C₂×#504

—

C₂×M₆₄

—

C₂²×C₃₂

—

C₂²⋊C₃₂

—

C₇₃

#13554

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

146

—

#504

—

146

—

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

100

C₂×#41

—

101

#131

—

146

—

102

#131

—

146

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂×#64408

yes

characteristic

107

M₃₂

—

108

M₃₂

—

146

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

146

—

114

C₂×C₁₆

—

146

—

115

C₂×C₁₆

C₂×#64408

yes

—

116

C₃₂

—

146

—

117

C₃₂

—

146

—

118

#41

—

146

—

119

#41

—

146

—

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

#11924036

yes

characteristic

125

M₁₆

—

146

—

126

M₁₆

—

146

—

127

C₂×C₈

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

C₂²⋊#64408

yes

—

131

C₂×C₈

#11924036

yes

characteristic

132

C₂×C₈

#11924038

yes

—

133

C₂×C₈

C₂²×#64408

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₁₆

—

137

C₁₆

—

138

C₂³

—

139

C₂³

#14556664

yes

characteristic

140

Q₈

—

146

—

141

D₄

—

146

—

142

D₄

—

146

—

143

C₂×C₄

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

#14556648

yes

—

147

C₂×C₄

#14556664

yes

characteristic

148

C₂×C₄

#14556666

yes

—

149

C₂×C₄

C₂×#11924036

yes

characteristic

150

C₈

—

146

—

151

C₈

#14556648

yes

—

152

C₈

C₂×C₂²⋊#64408

yes

characteristic

153

C₈

C₂×#11924038

yes

characteristic

154

C₂²

—

155

C₂²

—

146

—

156

C₂²

—

146

—

157

C₂²

#17989482

yes

—

158

C₂²

C₂×#14556664

yes

characteristic

159

C₄

—

146

—

160

C₄

#17989482

yes

—

161

C₄

C₂×#14556648

yes

characteristic

162

C₄

C₂×#14556666

yes

characteristic

163

C₂

—

164

C₂

—

292

—

165

C₂

C₂×#17989482

yes

characteristic

166

C₂

#22626355

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂×#11924028	C₈
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#13554
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂×M₃₂×C₇₃

C₄

C₈

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#17989519	C₂	semidirect
2. C₂²×#14556641	C₂	semidirect
3. #22625045	C₂	semidirect
88. C₇₃	#13554	semidirect

H ≅ G/N

type

1. C₂×#17989519

C₂

semidirect

2. C₂²×#14556641

C₂

semidirect

3. #22625045

C₂

semidirect

88. C₇₃

#13554

semidirect

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

subgroup	index
C₂×#17989519	2
C₂²×#14556641	2
#22625045	2
C₂×#17989005	2
#13554	73

subgroup

index

Subgroups — #34014234

Conjugacy classes of subgroups

Complements

Maximal subgroups