Group #34012314 — subgroups

Subgroups — #34012314

Order 18688.

390 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 148 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	#22626132	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
3	#22624968	C₂	1	4	yes	maximal
4	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556664	C₂²	1	2	yes	—
6	C₄×#11924036	C₄	1	1	yes	characteristic
7	C₂×C₈×#64408	C₄	1	2	yes	—
8	C₂×C₈×#64408	C₂²	1	2	yes	—
9	C₄²×#64408	C₄	1	1	yes	characteristic
10	#17989476	C₄	1	2	yes	complemented
11	#17989475	C₄	1	2	yes	complemented
12	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
13	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17988363	C₄	1	1	yes	characteristic, complemented
15	#17988362	C₄	1	2	yes	complemented
16	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#11924036	C₂×C₄	1	2	yes	—
18	C₂×C₄×#64408	C₈	1	4	yes	—
19	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
20	#14556664	—	2	4	no	—
21	C₂×#11924031	C₂×C₄	1	4	yes	—
22	C₄×#10596833	C₈	1	4	yes	—
23	C₈×#64408	—	2	4	no	—
24	C₈×#64408	—	2	4	no	—
25	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
27	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924003	C₂×C₄	1	2	yes	—
29	C₁₆×D₇₃	—	2	4	no	—
30	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#64408	C₂×C₈	1	4	yes	—
36	#11924036	—	2	2	no	—
37	#11924036	—	2	2	no	—
38	C₂×#10596833	C₂×C₈	1	4	yes	—
39	C₄×#64408	—	2	8	no	—
40	C₄×#64408	—	2	2	no	—
41	C₄×#64408	—	2	2	no	—
42	#11924031	—	2	4	no	—
43	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×#10596831	C₄²	1	1	yes	characteristic
46	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
47	C₈×D₇₃	M₁₆	1	4	yes	—
48	C₈×D₇₃	#41	1	4	yes	—
49	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
50	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
51	#11924003	—	2	2	no	—
52	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₄²	1	1	yes	characteristic
55	C₁₆×C₇₃	—	2	2	no	—
56	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×#64408	—	2	4	no	—
58	C₂×#64408	—	2	4	no	—
59	C₂×#64408	M₃₂	1	8	yes	—
60	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
61	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
63	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
64	C₄×D₇₃	—	2	4	no	—
65	C₄×D₇₃	#117	1	2	yes	—
66	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
67	#10596833	—	2	4	no	—
68	C₈×C₇₃	#97	1	2	yes	—
69	C₈×C₇₃	#117	1	2	yes	—
70	#10596831	#97	1	2	yes	—
71	#10596831	#117	1	2	yes	—
72	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
73	C₂×D₇₃	#431	1	2	yes	—
74	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
75	#64408	—	2	8	no	—
76	C₄×C₇₃	—	2	2	no	—
77	C₄×C₇₃	#376	1	1	yes	characteristic
78	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
79	Dic₇₃	—	2	2	no	—
80	Dic₇₃	#431	1	2	yes	—
81	#12591	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	#1744	1	1	yes	characteristic
83	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
84	D₇₃	#2163	1	4	yes	—
85	C₂×#346	—	73	1	no	—
86	C₂×#346	—	73	1	no	—
87	C₄²×C₈	—	73	1	no	—
88	#1374	—	73	4	no	—
89	C₇₃	#12591	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	73	2	no	—
91	C₂²×C₁₆	—	73	2	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	C₂×C₄×C₈	—	73	2	no	—
95	C₂×C₄×C₈	—	73	2	no	—
96	C₄³	—	73	1	no	—
97	#346	—	73	2	no	—
98	#346	—	73	2	no	—
99	#346	—	73	2	no	—
100	#346	—	73	1	no	—
101	#346	#64408	1	1	yes	characteristic, complemented
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂²×C₈	—	73	2	no	—
105	C₂×C₄²	—	73	1	no	—
106	C₂×C₄²	—	73	2	no	—
107	C₂×C₄²	—	73	4	no	—
108	C₂×C₁₆	—	73	4	no	—
109	C₂×C₁₆	—	73	2	no	—
110	C₂×C₁₆	—	146	4	no	—
111	C₂×C₁₆	—	146	4	no	—
112	C₂×C₁₆	C₂×#64408	1	2	yes	—
113	C₄×C₈	—	73	4	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	146	4	no	—
118	C₄×C₈	—	146	4	no	—
119	C₄×C₈	C₂×#64408	1	1	yes	characteristic
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	4	no	—
122	C₂²×C₄	—	73	2	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	4	no	—
126	C₂×C₈	—	73	4	no	—
127	C₂×C₈	—	73	4	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	C₄×#64408	1	1	yes	characteristic
131	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
132	C₄²	—	73	2	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	146	2	no	—
135	C₄²	—	146	2	no	—
136	C₄²	—	146	8	no	—
137	C₄²	C₄×#64408	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	—	146	4	no	—
140	C₁₆	—	146	2	no	—
141	C₂³	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	73	2	no	—
145	C₂×C₄	—	73	8	no	—
146	C₂×C₄	—	73	2	no	—
147	C₂×C₄	—	146	4	no	—
148	C₂×C₄	—	146	4	no	—
149	C₂×C₄	—	146	4	no	—
150	C₂×C₄	C₈×#64408	1	2	yes	—
151	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
152	C₈	—	73	2	no	—
153	C₈	—	73	2	no	—
154	C₈	—	146	4	no	—
155	C₈	#14556645	1	2	yes	—
156	C₈	#14556670	1	2	yes	—
157	C₂²	—	73	4	no	—
158	C₂²	—	73	2	no	—
159	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	73	2	no	—
162	C₄	—	146	8	no	—
163	C₄	—	146	2	no	—
164	C₄	#17989523	1	1	yes	characteristic
165	C₄	C₂×#14556645	1	1	yes	characteristic
166	C₂	—	73	4	no	—
167	C₂	#22625644	1	1	yes	characteristic
168	C₂	M₃₂×#64408	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626132

C₂

yes

maximal, characteristic

C₄×C₈×#64408

C₂

yes

maximal, characteristic

#22624968

C₂

yes

maximal

#346×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

C₂²

yes

—

C₄×#11924036

C₄

yes

characteristic

C₂×C₈×#64408

C₄

yes

—

C₂×C₈×#64408

C₂²

yes

—

C₄²×#64408

C₄

yes

characteristic

#17989476

C₄

yes

complemented

#17989475

C₄

yes

complemented

C₂×C₁₆×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic, complemented

#17988362

C₄

yes

complemented

#346×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924036

C₂×C₄

yes

—

C₂×C₄×#64408

C₈

yes

—

C₂×C₄×#64408

C₂×C₄

yes

—

#14556664

—

C₂×#11924031

C₂×C₄

yes

—

C₄×#10596833

C₈

yes

—

C₈×#64408

—

C₈×#64408

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂³

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

C₂×C₈

yes

—

#11924036

—

#11924036

—

C₂×#10596833

C₂×C₈

yes

—

—

—

—

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

M₁₆

yes

—

C₈×D₇₃

#41

yes

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

—

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₂²×C₈

yes

—

—

M₃₂

yes

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂²×C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#117

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

#10596833

—

C₈×C₇₃

#97

yes

—

C₈×C₇₃

#117

yes

—

#10596831

#97

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

#431

yes

—

C₂×D₇₃

C₂×M₃₂

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

Dic₇₃

—

Dic₇₃

#431

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×M₃₂

yes

—

D₇₃

#2163

yes

—

C₂×#346

—

C₂×#346

—

C₄²×C₈

—

#1374

—

C₇₃

#12591

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

#346

—

#346

—

#346

—

100

#346

—

101

#346

#64408

yes

characteristic, complemented

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

—

146

—

111

C₂×C₁₆

—

146

—

112

C₂×C₁₆

C₂×#64408

yes

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

146

—

118

C₄×C₈

—

146

—

119

C₄×C₈

C₂×#64408

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

C₄×#64408

yes

characteristic

131

C₂×C₈

C₂²×#64408

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

146

—

135

C₄²

—

146

—

136

C₄²

—

146

—

137

C₄²

C₄×#64408

yes

characteristic

138

C₁₆

—

139

C₁₆

—

146

—

140

C₁₆

—

146

—

141

C₂³

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

146

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

C₈×#64408

yes

—

151

C₂×C₄

C₂×C₄×#64408

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

146

—

155

C₈

#14556645

yes

—

156

C₈

#14556670

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×C₈×#64408

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

146

—

163

C₄

—

146

—

164

C₄

#17989523

yes

characteristic

165

C₄

C₂×#14556645

yes

characteristic

166

C₂

—

167

C₂

#22625644

yes

characteristic

168

C₂

M₃₂×#64408

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#346×C₇₃	C₄
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12591
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

#346×C₇₃

C₄

prime radical

C₂²×D₇₃

C₂²×C₈

cosocle

C₂×C₈×D₇₃

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
10. #17989476	C₄	semidirect
11. #17989475	C₄	semidirect
14. #17988363	C₄	semidirect
15. #17988362	C₄	semidirect
16. #346×C₇₃	C₄	semidirect
89. C₇₃	#12591	semidirect
101. #346	#64408	semidirect

H ≅ G/N

type

10. #17989476

C₄

semidirect

11. #17989475

C₄

semidirect

14. #17988363

C₄

semidirect

15. #17988362

C₄

semidirect

16. #346×C₇₃

C₄

semidirect

89. C₇₃

#12591

semidirect

101. #346

#64408

semidirect

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

subgroup	index
#22626132	2
C₄×C₈×#64408	2
#22624968	2
#346×D₇₃	2
#12591	73

subgroup

index

#22626132

C₄×C₈×#64408

#22624968

#346×D₇₃

#12591

Subgroups — #34012314

Conjugacy classes of subgroups

Complements

Maximal subgroups