Group #34014240 — subgroups

Subgroups — #34014240

Order 18688.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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	2	yes	maximal, complemented
2	#22625047	C₂	1	4	yes	maximal, complemented
3	C₂×#17989005	C₂	1	1	yes	maximal, characteristic
4	#17989519	—	2	2	no	—
5	#17989519	C₂²	1	4	yes	—
6	C₂×#14556641	—	2	2	no	—
7	C₂×#14556597	C₄	1	1	yes	characteristic
8	#17989005	—	2	2	no	—
9	#17989005	—	2	2	no	—
10	C₂×#14556530	C₂²	1	2	yes	—
11	C₂²×#11924003	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
13	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
14	#14556641	—	2	4	no	—
15	C₂×#11924028	C₈	1	1	yes	characteristic
16	#14556597	—	2	2	no	—
17	#14556597	—	2	2	no	—
18	C₂×#11924012	C₈	1	1	yes	characteristic
19	C₂×#11924011	C₈	1	2	yes	—
20	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
21	#14556530	—	2	4	no	—
22	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924003	D₄	1	2	yes	—
24	C₂×#11924003	D₄	1	2	yes	—
25	C₂×#11924003	C₂³	1	1	yes	characteristic
26	C₁₆×D₇₃	—	2	2	no	—
27	C₁₆×D₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₇₃	—	2	2	no	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	#11924028	—	2	2	no	—
32	#11924028	—	2	2	no	—
33	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
34	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
35	C₂×C₂²⋊D₇₃	C₁₆	1	2	yes	—
36	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
37	#11924012	—	2	2	no	—
38	#11924011	—	2	4	no	—
39	C₂×#10596831	C₂×C₈	1	2	yes	—
40	C₈×D₇₃	—	2	2	no	—
41	C₈×D₇₃	—	2	2	no	—
42	#11924003	C₂²⋊C₄	1	1	yes	characteristic
43	#11924003	C₂²⋊C₄	1	1	yes	characteristic
44	#11924003	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	2	2	no	—
51	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
56	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂²⋊D₇₃	—	2	4	no	—
58	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
59	D₂₉₂	—	2	2	no	—
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	2	2	no	—
62	Dic₁₄₆	—	2	2	no	—
63	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
64	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	#10596831	—	2	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
69	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₂×D₇₃	—	2	2	no	—
71	C₂×D₇₃	—	4	2	no	—
72	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
73	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
75	Dic₇₃	—	2	2	no	—
76	#13556	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	—	2	2	no	—
79	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
80	D₇₃	—	4	2	no	—
81	C₂×#504	—	73	1	no	—
82	C₂×M₆₄	—	73	2	no	—
83	#1400	—	73	4	no	—
84	C₇₃	#13556	1	1	yes	characteristic, complemented, Sylow
85	C₂×#131	—	73	1	no	—
86	#504	—	146	2	no	—
87	#504	—	146	2	no	—
88	C₂×M₃₂	—	73	2	no	—
89	C₂×M₃₂	#64408	1	1	yes	characteristic
90	C₂²×C₁₆	—	73	1	no	—
91	C₂²×C₁₆	—	73	2	no	—
92	M₆₄	—	73	4	no	—
93	M₆₄	—	146	2	no	—
94	C₂×C₃₂	—	146	2	no	—
95	C₂×#41	—	73	1	no	—
96	#131	—	146	2	no	—
97	#131	—	146	2	no	—
98	C₂×M₁₆	—	73	1	no	—
99	C₂×M₁₆	—	73	2	no	—
100	C₂²×C₈	—	73	2	no	—
101	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	—	146	4	no	—
104	C₂×C₁₆	—	73	2	no	—
105	C₂×C₁₆	—	73	2	no	—
106	C₂×C₁₆	—	73	1	no	—
107	C₂×C₁₆	—	73	1	no	—
108	C₂×C₁₆	—	146	2	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	C₂×#64408	1	2	yes	—
111	C₃₂	—	146	4	no	—
112	#41	—	146	2	no	—
113	#41	—	146	2	no	—
114	C₂×Q₈	—	73	1	no	—
115	C₂×D₄	—	73	2	no	—
116	C₂×D₄	—	73	1	no	—
117	C₂²×C₄	—	73	2	no	—
118	C₂²×C₄	#11924036	1	1	yes	characteristic
119	M₁₆	—	146	2	no	—
120	M₁₆	—	146	4	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	C₂²⋊#64408	1	2	yes	—
125	C₂×C₈	#11924036	1	1	yes	characteristic
126	C₂×C₈	#11924038	1	2	yes	—
127	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	73	1	no	—
130	C₁₆	—	73	1	no	—
131	C₁₆	—	73	2	no	—
132	C₂³	—	73	2	no	—
133	C₂³	#14556664	1	1	yes	characteristic
134	Q₈	—	146	2	no	—
135	D₄	—	146	4	no	—
136	D₄	—	146	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	#14556648	1	2	yes	—
141	C₂×C₄	#14556664	1	1	yes	characteristic
142	C₂×C₄	#14556666	1	2	yes	—
143	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
144	C₈	—	146	2	no	—
145	C₈	#14556648	1	2	yes	—
146	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
147	C₈	C₂×#11924038	1	1	yes	characteristic
148	C₂²	—	4	1	no	—
149	C₂²	—	292	2	no	—
150	C₂²	—	146	2	no	—
151	C₂²	#17989482	1	2	yes	—
152	C₂²	C₂×#14556664	1	1	yes	characteristic
153	C₄	—	146	2	no	—
154	C₄	#17989482	1	2	yes	—
155	C₄	C₂×#14556648	1	1	yes	characteristic
156	C₄	C₂×#14556666	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	292	2	no	—
160	C₂	C₂×#17989482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal, complemented

#22625047

C₂

yes

maximal, complemented

C₂×#17989005

C₂

yes

maximal, characteristic

#17989519

—

#17989519

C₂²

yes

—

C₂×#14556641

—

C₂×#14556597

C₄

yes

—

—

C₂²

yes

—

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

—

C₂×M₃₂×C₇₃

C₄

yes

characteristic

#14556641

—

C₂×#11924028

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

characteristic

#11924003

C₂²⋊C₄

yes

characteristic

#11924003

C₂×D₄

yes

—

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

—

#13556

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

D₇₃

—

C₂×#504

—

C₂×M₆₄

—

#1400

—

C₇₃

#13556

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

146

—

#504

—

146

—

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

146

—

C₂×#41

—

#131

—

146

—

#131

—

146

—

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×#64408

yes

characteristic

102

M₃₂

—

103

M₃₂

—

146

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

146

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

C₂×#64408

yes

—

111

C₃₂

—

146

—

112

#41

—

146

—

113

#41

—

146

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

#11924036

yes

characteristic

119

M₁₆

—

146

—

120

M₁₆

—

146

—

121

C₂×C₈

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

C₂²⋊#64408

yes

—

125

C₂×C₈

#11924036

yes

characteristic

126

C₂×C₈

#11924038

yes

—

127

C₂×C₈

C₂²×#64408

yes

characteristic

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

—

133

C₂³

#14556664

yes

characteristic

134

Q₈

—

146

—

135

D₄

—

146

—

136

D₄

—

146

—

137

C₂×C₄

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

#14556648

yes

—

141

C₂×C₄

#14556664

yes

characteristic

142

C₂×C₄

#14556666

yes

—

143

C₂×C₄

C₂×#11924036

yes

characteristic

144

C₈

—

146

—

145

C₈

#14556648

yes

—

146

C₈

C₂×C₂²⋊#64408

yes

characteristic

147

C₈

C₂×#11924038

yes

characteristic

148

C₂²

—

149

C₂²

—

292

—

150

C₂²

—

146

—

151

C₂²

#17989482

yes

—

152

C₂²

C₂×#14556664

yes

characteristic

153

C₄

—

146

—

154

C₄

#17989482

yes

—

155

C₄

C₂×#14556648

yes

characteristic

156

C₄

C₂×#14556666

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

292

—

160

C₂

C₂×#17989482

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#11924028	C₈
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#13556
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₂²⋊#64408

commutator

C₂²×C₇₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

socle

C₂×C₇₃

C₂×C₂²⋊C₁₆

C₈

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#17989519	C₂	semidirect
2. #22625047	C₂	semidirect
84. C₇₃	#13556	semidirect

H ≅ G/N

type

1. C₂×#17989519

C₂

semidirect

2. #22625047

C₂

semidirect

84. C₇₃

#13556

semidirect

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

subgroup	index
C₂×#17989519	2
#22625047	2
C₂×#17989005	2
#13556	73

subgroup

index

Subgroups — #34014240

Conjugacy classes of subgroups

Complements

Maximal subgroups