Group #34014231 — subgroups

Subgroups — #34014231

Order 18688.

330 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 116 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	#22625045	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989519	—	2	4	no	—
5	C₂×#14556641	C₂²	1	4	yes	complemented
6	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
7	C₂²×#11924003	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₇₃	—	2	4	no	—
9	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
10	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	#14556641	—	2	4	no	—
13	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
14	C₂²×#10596831	C₈	1	1	yes	characteristic
15	C₂×C₈×D₇₃	—	2	2	no	—
16	C₂×C₈×D₇₃	—	2	2	no	—
17	C₂×C₈×D₇₃	C₈	1	2	yes	—
18	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
19	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924003	D₄	1	4	yes	—
21	C₂×#11924003	C₂³	1	1	yes	characteristic
22	C₁₆×D₇₃	—	2	4	no	—
23	C₁₆×D₇₃	—	2	4	no	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
27	C₂³×D₇₃	C₁₆	1	1	yes	characteristic
28	C₂²×Dic₇₃	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₇₃	—	2	2	no	—
30	C₂×C₄×D₇₃	—	2	2	no	—
31	C₂×C₄×D₇₃	C₁₆	1	2	yes	—
32	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
33	C₂×#10596831	—	2	2	no	—
34	C₂×#10596831	C₂×C₈	1	2	yes	—
35	C₈×D₇₃	—	2	4	no	—
36	C₈×D₇₃	—	2	2	no	—
37	C₈×D₇₃	—	2	2	no	—
38	#11924003	C₂²⋊C₄	1	2	yes	—
39	#11924003	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
44	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₁₆×C₇₃	—	2	2	no	—
46	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	2	no	—
48	C₂²×D₇₃	—	2	4	no	—
49	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
51	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
53	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
56	C₄×D₇₃	—	2	4	no	—
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	2	no	—
59	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
60	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
62	#10596831	—	2	2	no	—
63	C₂²×C₇₃	—	2	2	no	—
64	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
65	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
66	C₂×D₇₃	—	2	8	no	—
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	2	2	no	—
69	C₂×D₇₃	M₆₄	1	4	yes	complemented
70	C₄×C₇₃	C₂²⋊C₁₆	1	2	yes	—
71	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
72	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
73	Dic₇₃	—	2	2	no	—
74	#13553	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₂×C₇₃	C₂×M₆₄	1	2	yes	—
78	D₇₃	—	2	4	no	—
79	C₂³×C₁₆	—	73	1	no	—
80	C₂×M₆₄	—	73	2	no	—
81	C₂²⋊C₃₂	—	73	4	no	—
82	C₇₃	#13553	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₈	—	73	1	no	—
84	C₂²×C₁₆	—	73	1	no	—
85	C₂²×C₁₆	—	73	2	no	—
86	C₂²×C₁₆	—	73	2	no	—
87	C₂²×C₁₆	—	146	4	no	—
88	C₂²×C₁₆	#64408	1	1	yes	characteristic
89	M₆₄	—	146	4	no	—
90	C₂×C₃₂	—	73	4	no	—
91	C₂³×C₄	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂²×C₈	—	146	2	no	—
96	C₂²×C₈	—	146	2	no	—
97	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	—	73	4	no	—
100	C₂×C₁₆	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	146	4	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₂×C₁₆	C₂×#64408	1	2	yes	—
105	C₃₂	—	146	4	no	—
106	C₂⁴	—	73	1	no	—
107	C₂²×C₄	—	73	1	no	—
108	C₂²×C₄	—	73	2	no	—
109	C₂²×C₄	—	73	2	no	—
110	C₂²×C₄	—	146	2	no	—
111	C₂²×C₄	—	146	2	no	—
112	C₂²×C₄	#11924036	1	1	yes	characteristic
113	C₂×C₈	—	73	2	no	—
114	C₂×C₈	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	—	146	4	no	—
117	C₂×C₈	—	146	2	no	—
118	C₂×C₈	C₂²⋊#64408	1	2	yes	—
119	C₂×C₈	#11924036	1	1	yes	characteristic
120	C₂×C₈	#11924038	1	2	yes	—
121	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	—	73	2	no	—
124	C₁₆	—	73	2	no	—
125	C₂³	—	73	2	no	—
126	C₂³	—	146	4	no	—
127	C₂³	—	146	2	no	—
128	C₂³	#14556664	1	1	yes	characteristic
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	4	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	#14556648	1	2	yes	—
135	C₂×C₄	#14556664	1	1	yes	characteristic
136	C₂×C₄	#14556666	1	2	yes	—
137	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
138	C₈	—	146	2	no	—
139	C₈	#14556648	1	2	yes	—
140	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
141	C₈	C₂×#11924038	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	73	4	no	—
144	C₂²	—	146	8	no	—
145	C₂²	—	146	2	no	—
146	C₂²	—	146	2	no	—
147	C₂²	#17989482	1	2	yes	—
148	C₂²	C₂×#14556664	1	1	yes	characteristic
149	C₄	—	146	2	no	—
150	C₄	#17989482	1	2	yes	—
151	C₄	C₂×#14556648	1	1	yes	characteristic
152	C₄	C₂×#14556666	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	146	4	no	—
155	C₂	C₂×#17989482	1	1	yes	characteristic
156	C₂	#22626352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal, complemented

#22625045

C₂

yes

maximal, complemented

C₂²×C₁₆×D₇₃

C₂

yes

maximal, characteristic

#17989519

—

C₂×#14556641

C₂²

yes

complemented

C₂²×C₈×D₇₃

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

—

C₂×C₁₆×D₇₃

C₄

yes

—

C₂×C₁₆×D₇₃

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

#14556641

—

C₂²×C₄×D₇₃

C₈

yes

characteristic

C₂²×#10596831

C₈

yes

characteristic

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

D₄

yes

—

C₂×#11924003

C₂³

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂³×D₇₃

C₁₆

yes

characteristic

C₂²×Dic₇₃

C₁₆

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₁₆

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

—

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

C₂²⋊C₄

yes

—

#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

—

C₂²×D₇₃

—

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

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

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

—

C₂×D₇₃

—

C₂×D₇₃

M₆₄

yes

complemented

C₄×C₇₃

C₂²⋊C₁₆

yes

—

C₄×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

#13553

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₆₄

yes

—

D₇₃

—

C₂³×C₁₆

—

C₂×M₆₄

—

C₂²⋊C₃₂

—

C₇₃

#13553

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

146

—

C₂²×C₁₆

#64408

yes

characteristic

M₆₄

—

146

—

C₂×C₃₂

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂²×C₈

—

146

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

146

—

103

C₂×C₁₆

—

146

—

104

C₂×C₁₆

C₂×#64408

yes

—

105

C₃₂

—

146

—

106

C₂⁴

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

—

146

—

112

C₂²×C₄

#11924036

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₂×C₈

—

146

—

118

C₂×C₈

C₂²⋊#64408

yes

—

119

C₂×C₈

#11924036

yes

characteristic

120

C₂×C₈

#11924038

yes

—

121

C₂×C₈

C₂²×#64408

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

—

126

C₂³

—

146

—

127

C₂³

—

146

—

128

C₂³

#14556664

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

#14556648

yes

—

135

C₂×C₄

#14556664

yes

characteristic

136

C₂×C₄

#14556666

yes

—

137

C₂×C₄

C₂×#11924036

yes

characteristic

138

C₈

—

146

—

139

C₈

#14556648

yes

—

140

C₈

C₂×C₂²⋊#64408

yes

characteristic

141

C₈

C₂×#11924038

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

146

—

145

C₂²

—

146

—

146

C₂²

—

146

—

147

C₂²

#17989482

yes

—

148

C₂²

C₂×#14556664

yes

characteristic

149

C₄

—

146

—

150

C₄

#17989482

yes

—

151

C₄

C₂×#14556648

yes

characteristic

152

C₄

C₂×#14556666

yes

characteristic

153

C₂

—

154

C₂

—

146

—

155

C₂

C₂×#17989482

yes

characteristic

156

C₂

#22626352

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂²×C₁₆×C₇₃	C₄
prime radical	C₂³×D₇₃	C₁₆
cosocle	C₂×#11924003	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#13553
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂²×C₁₆×C₇₃

C₄

prime radical

C₂³×D₇₃

C₁₆

C₂³

C₂²×C₁₆

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×#17989519	C₂	semidirect
2. #22625045	C₂	semidirect
5. C₂×#14556641	C₂²	semidirect
69. C₂×D₇₃	M₆₄	semidirect
82. C₇₃	#13553	semidirect

H ≅ G/N

type

1. C₂×#17989519

C₂

semidirect

2. #22625045

C₂

semidirect

5. C₂×#14556641

C₂²

semidirect

69. C₂×D₇₃

M₆₄

semidirect

82. C₇₃

#13553

semidirect

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

subgroup	index
C₂×#17989519	2
#22625045	2
C₂²×C₁₆×D₇₃	2
#13553	73

subgroup

index

C₂×#17989519

#22625045

C₂²×C₁₆×D₇₃

#13553

Subgroups — #34014231

Conjugacy classes of subgroups

Complements

Maximal subgroups