Group #34032889 — subgroups

Subgroups — #34032889

Order 18688.

230 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22627121	C₂	1	2	yes	maximal
2	#22625127	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556688	—	2	2	no	—
4	#17989690	—	2	2	no	—
5	#17989690	C₄	1	4	yes	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₄×#11924036	C₂²	1	1	yes	characteristic
8	#17989429	—	2	2	no	—
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	#14556688	—	4	2	no	—
11	#14556688	—	4	2	no	—
12	C₂×#11924040	—	2	4	no	—
13	C₂×#11924036	C₈	1	2	yes	—
14	C₂×#11924036	D₄	1	1	yes	characteristic
15	C₂×#11924036	Q₈	1	1	yes	characteristic
16	C₄×#10596833	—	2	1	no	—
17	C₄×#10596833	—	2	1	no	—
18	C₄×#10596833	C₂×C₄	1	2	yes	—
19	C₄×#10596833	D₄	1	2	yes	—
20	C₂×C₈×D₇₃	—	2	1	no	—
21	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#10596831	C₈	1	1	yes	characteristic
23	C₈×Dic₇₃	—	2	1	no	—
24	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
25	#11924040	—	4	4	no	—
26	#11924036	—	2	1	no	—
27	#11924036	—	2	1	no	—
28	#11924036	—	4	2	no	—
29	#11924036	D₈	1	1	yes	characteristic
30	#11924036	SD₁₆	1	2	yes	—
31	#11924036	Q₁₆	1	1	yes	characteristic
32	C₂×#10596833	—	2	2	no	—
33	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
34	C₂×#10596833	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	2	1	no	—
39	C₈×D₇₃	—	4	2	no	—
40	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
41	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	#95	1	1	yes	characteristic
50	C₂×Dic₇₃	#97	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	#106	1	1	yes	characteristic
55	C₄×D₇₃	#107	1	1	yes	characteristic
56	C₄×D₇₃	#108	1	1	yes	characteristic
57	C₄×D₇₃	#108	1	1	yes	characteristic
58	#10596833	—	2	2	no	—
59	#10596833	—	4	2	no	—
60	#10596833	#102	1	2	yes	—
61	#10596833	#103	1	2	yes	—
62	C₈×C₇₃	—	4	1	no	—
63	#10596831	—	4	1	no	—
64	C₂²×C₇₃	#336	1	1	yes	characteristic
65	C₂×D₇₃	—	2	1	no	—
66	C₂×D₇₃	—	2	1	no	—
67	C₂×D₇₃	#334	1	1	yes	characteristic
68	C₂×D₇₃	#335	1	1	yes	characteristic
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	#334	1	1	yes	characteristic
72	C₄×C₇₃	#335	1	1	yes	characteristic
73	Dic₇₃	#340	1	1	yes	characteristic
74	Dic₇₃	#341	1	1	yes	characteristic
75	Dic₇₃	#347	1	2	yes	—
76	Dic₇₃	#349	1	2	yes	—
77	#7152	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	#1277	1	1	yes	characteristic
79	C₂×C₇₃	#1375	1	1	yes	characteristic
80	C₂×C₇₃	#1377	1	1	yes	characteristic
81	D₇₃	—	2	1	no	—
82	D₇₃	—	2	1	no	—
83	#2150	—	73	2	no	—
84	C₂×#322	—	73	1	no	—
85	C₇₃	#7152	1	1	yes	characteristic, complemented, Sylow
86	C₂×M₃₂	—	146	2	no	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	C₂×C₄×C₈	—	73	1	no	—
90	C₄⋊C₁₆	—	73	4	no	—
91	C₄⋊C₁₆	—	146	2	no	—
92	#322	—	146	2	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	146	1	no	—
97	C₂×C₄²	—	73	1	no	—
98	M₃₂	—	292	2	no	—
99	M₃₂	—	292	2	no	—
100	C₂×C₁₆	—	146	4	no	—
101	C₄×C₈	—	73	2	no	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	—	146	1	no	—
105	C₄×C₈	—	146	1	no	—
106	C₄×C₈	—	146	1	no	—
107	C₄×C₈	#10596842	1	1	yes	characteristic
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	292	2	no	—
113	C₂×C₈	—	292	2	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₄²	—	73	1	no	—
124	C₄²	—	73	1	no	—
125	C₄²	—	73	1	no	—
126	C₄²	C₂×#10596842	1	1	yes	characteristic
127	C₁₆	—	292	4	no	—
128	C₂³	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	—	146	2	no	—
138	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
140	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	292	1	no	—
143	C₈	—	292	2	no	—
144	C₈	—	73	2	no	—
145	C₈	—	73	2	no	—
146	C₈	—	146	2	no	—
147	C₂²	—	73	1	no	—
148	C₂²	—	73	1	no	—
149	C₂²	—	146	1	no	—
150	C₂²	—	146	1	no	—
151	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	73	1	no	—
155	C₄	—	73	1	no	—
156	C₄	—	73	2	no	—
157	C₄	—	73	2	no	—
158	C₄	#17989676	1	1	yes	characteristic
159	C₄	#17989677	1	1	yes	characteristic
160	C₂	—	146	1	no	—
161	C₂	—	146	1	no	—
162	C₂	#22626937	1	1	yes	characteristic
163	C₂	#22627003	1	1	yes	characteristic
164	C₂	#22627008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627121

C₂

yes

maximal

#22625127

C₂

yes

maximal, characteristic

C₂×#14556688

—

#17989690

—

#17989690

C₄

yes

—

C₄×#11924036

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

characteristic

#17989429

—

C₄×C₈×D₇₃

C₄

yes

—

—

—

C₈

yes

—

C₂×#11924036

D₄

yes

characteristic

C₂×#11924036

Q₈

yes

—

—

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

—

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

—

—

—

—

D₈

yes

characteristic

#11924036

SD₁₆

yes

—

#11924036

Q₁₆

yes

characteristic

C₂×#10596833

—

C₂×#10596833

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

—

—

yes

—

#10596833

#103

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

Dic₇₃

#349

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#2150

—

C₂×#322

—

C₇₃

#7152

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

#322

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

M₃₂

—

292

—

M₃₂

—

292

—

100

C₂×C₁₆

—

146

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

146

—

105

C₄×C₈

—

146

—

106

C₄×C₈

—

146

—

107

C₄×C₈

#10596842

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂×C₈

—

112

C₂×C₈

—

292

—

113

C₂×C₈

—

292

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×#10596842

yes

characteristic

127

C₁₆

—

292

—

128

C₂³

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

C₄×#10596842

yes

characteristic

139

C₂×C₄

C₂²⋊#10596842

yes

characteristic

140

C₂×C₄

C₄⋊#10596842

yes

characteristic

141

C₈

—

142

C₈

—

292

—

143

C₈

—

292

—

144

C₈

—

145

C₈

—

146

C₈

—

146

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

146

—

150

C₂²

—

146

—

151

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#17989676

yes

characteristic

159

C₄

#17989677

yes

characteristic

160

C₂

—

146

—

161

C₂

—

146

—

162

C₂

#22626937

yes

characteristic

163

C₂

#22627003

yes

characteristic

164

C₂

#22627008

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7152
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

(C₂×#10596842)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₄⋊C₈

C₂²

C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
85. C₇₃	#7152	semidirect

H ≅ G/N

type

85. C₇₃

#7152

semidirect

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

subgroup	index
#22627121	2
#22625127	2
#7152	73

subgroup

index

#22627121

#22625127

#7152

Subgroups — #34032889

Conjugacy classes of subgroups

Complements

Maximal subgroups