Group #34032868 — subgroups

Subgroups — #34032868

Order 18688.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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×#64408⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22627040	C₂	1	1	yes	maximal, characteristic
3	M₃₂×#64408	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₄	1	2	yes	—
5	#17989700	C₄	1	2	yes	—
6	#64408⋊C₁₆	—	2	1	no	—
7	#64408⋊C₁₆	—	2	1	no	—
8	#17989671	—	2	1	no	—
9	C₈×#10596842	—	2	1	no	—
10	#17989643	C₄	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	#17989474	—	2	1	no	—
13	C₁₆×#64408	—	2	1	no	—
14	M₃₂×D₇₃	C₄	1	1	yes	characteristic
15	#14556686	—	2	2	no	—
16	#14556686	C₈	1	4	yes	complemented
17	C₂×#11924040	—	2	2	no	—
18	C₄⋊#10596842	C₈	1	4	yes	—
19	C₂²⋊#10596842	—	2	2	no	—
20	C₄×#10596842	—	2	2	no	—
21	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
23	#14556666	—	2	1	no	—
24	#14556665	C₈	1	2	yes	complemented
25	#14556664	—	2	1	no	—
26	C₈×#64408	D₄	1	1	yes	characteristic
27	C₈×#64408	D₄	1	1	yes	characteristic
28	C₈×#64408	D₄	1	1	yes	characteristic
29	C₈×#64408	Q₈	1	1	yes	characteristic
30	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
31	#14556531	C₈	1	1	yes	characteristic, complemented
32	#14556530	—	2	1	no	—
33	C₁₆×D₇₃	—	2	1	no	—
34	M₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
35	C₂×#10596842	—	2	4	no	—
36	#11924040	—	2	4	no	—
37	C₂²×#64408	M₁₆	1	2	yes	—
38	#11924036	C₂²⋊C₄	1	1	yes	characteristic
39	#11924036	C₄⋊C₄	1	1	yes	characteristic
40	#11924036	C₂×C₈	1	2	yes	—
41	C₂×#10596833	M₁₆	1	2	yes	—
42	C₄×#64408	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×#64408	C₄⋊C₄	1	1	yes	characteristic
44	C₄×#64408	C₂×C₈	1	2	yes	—
45	#11924031	—	2	2	no	—
46	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
47	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
48	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₈×D₇₃	M₁₆	1	1	yes	characteristic
51	C₈×D₇₃	M₁₆	1	1	yes	characteristic
52	#11924003	—	2	1	no	—
53	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
54	C₁₆×C₇₃	—	2	1	no	—
55	C₂²×D₇₃	#97	1	1	yes	characteristic
56	C₂×#64408	—	2	4	no	—
57	C₂×#64408	C₂²⋊C₈	1	2	yes	—
58	C₂×#64408	C₄⋊C₈	1	2	yes	—
59	#10596842	—	4	4	no	—
60	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
61	C₂×Dic₇₃	#97	1	1	yes	characteristic
62	C₄×D₇₃	#95	1	1	yes	characteristic
63	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
64	C₄×D₇₃	C₄×C₈	1	1	yes	characteristic
65	C₄×D₇₃	#97	1	1	yes	characteristic
66	#10596833	C₂²⋊C₈	1	2	yes	—
67	#10596833	C₄⋊C₈	1	2	yes	—
68	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
69	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
70	#10596831	C₂²⋊C₈	1	1	yes	characteristic
71	#10596831	C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₇₃	#322	1	1	yes	characteristic
73	C₂×D₇₃	—	2	2	no	—
74	C₂×D₇₃	#322	1	1	yes	characteristic
75	C₂×D₇₃	#336	1	1	yes	characteristic
76	#64408	—	2	2	no	—
77	#64408	#350	1	4	yes	—
78	C₄×C₇₃	C₈²	1	1	yes	characteristic
79	C₄×C₇₃	#336	1	1	yes	characteristic
80	Dic₇₃	#322	1	1	yes	characteristic
81	Dic₇₃	#336	1	1	yes	characteristic
82	#7141	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	1	no	—
84	C₂×C₇₃	#1273	1	1	yes	characteristic
85	D₇₃	—	2	1	no	—
86	D₇₃	#1379	1	2	yes	—
87	C₄×M₃₂	—	73	1	no	—
88	C₂×C₄×C₁₆	—	73	1	no	—
89	#1450	—	73	1	no	—
90	C₇₃	#7141	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	73	1	no	—
92	C₂×M₃₂	—	73	1	no	—
93	C₂²×C₁₆	—	73	2	no	—
94	#431	—	73	2	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	#346	—	146	1	no	—
97	C₄×C₁₆	—	146	1	no	—
98	C₄×C₁₆	—	146	1	no	—
99	C₄×C₁₆	—	146	1	no	—
100	#322	—	146	1	no	—
101	C₈²	—	146	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	M₃₂	—	73	2	no	—
106	M₃₂	—	73	1	no	—
107	M₃₂	—	146	1	no	—
108	M₃₂	—	146	1	no	—
109	M₃₂	#10596842	1	1	yes	characteristic, complemented
110	C₂×C₁₆	—	73	4	no	—
111	C₂×C₁₆	—	146	1	no	—
112	C₂×C₁₆	—	146	2	no	—
113	C₂×C₁₆	—	146	2	no	—
114	C₂×C₁₆	—	146	1	no	—
115	C₄⋊C₈	—	73	4	no	—
116	C₂²⋊C₈	—	146	2	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	146	2	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	2	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	146	4	no	—
134	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	2	no	—
138	C₁₆	—	2	1	no	—
139	C₁₆	—	146	2	no	—
140	C₁₆	—	146	4	no	—
141	C₁₆	—	146	1	no	—
142	C₂³	—	73	1	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	2	no	—
150	C₂×C₄	—	146	4	no	—
151	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
152	C₈	—	292	4	no	—
153	C₈	—	73	2	no	—
154	C₈	—	73	2	no	—
155	C₈	—	73	1	no	—
156	C₈	—	73	1	no	—
157	C₈	C₂²⋊#10596842	1	1	yes	characteristic
158	C₈	C₄⋊#10596842	1	1	yes	characteristic
159	C₂²	—	73	1	no	—
160	C₂²	—	73	1	no	—
161	C₂²	—	146	2	no	—
162	C₂²	#17989671	1	1	yes	characteristic
163	C₄	—	73	4	no	—
164	C₄	—	73	1	no	—
165	C₄	—	73	1	no	—
166	C₄	—	146	2	no	—
167	C₄	C₈×#10596842	1	1	yes	characteristic
168	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	73	2	no	—
171	C₂	—	146	1	no	—
172	C₂	(C₂×#10596842)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64408⋊C₁₆

C₂

yes

maximal, characteristic

#22627040

C₂

yes

maximal, characteristic

M₃₂×#64408

C₂

yes

maximal, characteristic

C₂×#14556686

C₄

yes

—

#17989700

C₄

yes

—

#64408⋊C₁₆

—

#64408⋊C₁₆

—

#17989671

—

C₈×#10596842

—

#17989643

C₄

yes

characteristic

C₂×C₈×#64408

C₂²

yes

characteristic

#17989474

—

C₁₆×#64408

—

M₃₂×D₇₃

C₄

yes

characteristic

#14556686

—

#14556686

C₈

yes

complemented

C₂×#11924040

—

C₄⋊#10596842

C₈

yes

—

C₂²⋊#10596842

—

C₄×#10596842

—

C₂×#11924036

C₂×C₄

yes

characteristic

C₂×C₄×#64408

C₂×C₄

yes

characteristic

#14556666

—

#14556665

C₈

yes

complemented

#14556664

—

C₈×#64408

D₄

yes

characteristic

C₈×#64408

D₄

yes

characteristic

C₈×#64408

D₄

yes

characteristic

C₈×#64408

Q₈

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556531

C₈

yes

characteristic, complemented

#14556530

—

C₁₆×D₇₃

—

M₃₂×C₇₃

C₈

yes

characteristic, complemented

C₂×#10596842

—

#11924040

—

C₂²×#64408

M₁₆

yes

—

#11924036

C₂²⋊C₄

yes

characteristic

#11924036

C₄⋊C₄

yes

characteristic

#11924036

C₂×C₈

yes

—

C₂×#10596833

M₁₆

yes

—

C₄×#64408

C₂²⋊C₄

yes

characteristic

C₄×#64408

C₄⋊C₄

yes

characteristic

C₄×#64408

C₂×C₈

yes

—

#11924031

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×#10596831

C₂×C₈

yes

characteristic

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

—

C₂²⋊C₈

yes

—

C₂×#64408

C₄⋊C₈

yes

—

#10596842

—

C₂×C₄×C₇₃

C₄×C₈

yes

C₂×Dic₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

C₄×C₈

yes

characteristic

C₄×D₇₃

C₄×C₈

yes

C₄×D₇₃

yes

C₂²⋊C₈

yes

—

#10596833

C₄⋊C₈

yes

—

C₈×C₇₃

C₂²⋊C₈

yes

characteristic

C₈×C₇₃

C₄⋊C₈

yes

characteristic

#10596831

C₂²⋊C₈

yes

characteristic

#10596831

C₄⋊C₈

yes

characteristic

C₂²×C₇₃

#322

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

yes

—

C₄×C₇₃

C₈²

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1273

yes

characteristic

D₇₃

—

D₇₃

#1379

yes

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

C₇₃

#7141

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

#346

—

146

—

C₄×C₁₆

—

146

—

C₄×C₁₆

—

146

—

C₄×C₁₆

—

146

—

100

#322

—

146

—

101

C₈²

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

M₃₂

—

106

M₃₂

—

107

M₃₂

—

146

—

108

M₃₂

—

146

—

109

M₃₂

#10596842

yes

characteristic, complemented

110

C₂×C₁₆

—

111

C₂×C₁₆

—

146

—

112

C₂×C₁₆

—

146

—

113

C₂×C₁₆

—

146

—

114

C₂×C₁₆

—

146

—

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

146

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

146

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

146

—

134

C₂×C₈

C₂×#10596842

yes

characteristic

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

139

C₁₆

—

146

—

140

C₁₆

—

146

—

141

C₁₆

—

146

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

146

—

151

C₂×C₄

C₄×#10596842

yes

characteristic

152

C₈

—

292

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

C₂²⋊#10596842

yes

characteristic

158

C₈

C₄⋊#10596842

yes

characteristic

159

C₂²

—

160

C₂²

—

161

C₂²

—

146

—

162

C₂²

#17989671

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

146

—

167

C₄

C₈×#10596842

yes

characteristic

168

C₄

(C₂×#10596842)⋊C₄

yes

characteristic

169

C₂

—

170

C₂

—

171

C₂

—

146

—

172

C₂

(C₂×#10596842)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10596842
commutator	C₄×C₇₃	C₈²
Frattini	C₂×C₈	C₂×#10596842
Fitting	M₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1273
prime radical	C₂²×D₇₃	#97
cosocle	C₂×C₈×#64408	C₂²
hypercenter	M₃₂	#10596842
hypocenter	C₇₃	#7141
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

subgroup

G/H

center

C₈

C₂²⋊#10596842

commutator

C₄×C₇₃

C₈²

Frattini

C₂×C₈

C₂×#10596842

Fitting

M₃₂×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₂×C₈×#64408

C₂²

M₃₂

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

N	H ≅ G/N	type
16. #14556686	C₈	semidirect
24. #14556665	C₈	semidirect
31. #14556531	C₈	semidirect
34. M₃₂×C₇₃	C₈	semidirect
90. C₇₃	#7141	semidirect
109. M₃₂	#10596842	semidirect

H ≅ G/N

type

16. #14556686

C₈

semidirect

24. #14556665

C₈

semidirect

31. #14556531

C₈

semidirect

34. M₃₂×C₇₃

C₈

semidirect

90. C₇₃

#7141

semidirect

109. M₃₂

#10596842

semidirect

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

subgroup	index
C₂×#64408⋊C₁₆	2
#22627040	2
M₃₂×#64408	2
#7141	73

subgroup

index

C₂×#64408⋊C₁₆

#22627040

M₃₂×#64408

#7141

Subgroups — #34032868

Conjugacy classes of subgroups

Complements

Maximal subgroups