Group #34000431 — subgroups

Subgroups — #34000431

Order 18688.

206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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	#22626295	C₂	1	1	yes	maximal, characteristic
2	#22625853	C₂	1	1	yes	maximal, characteristic
3	#22608685	C₂	1	1	yes	maximal, characteristic
4	#17989531	—	2	1	no	—
5	C₄⋊#11924031	—	2	1	no	—
6	#17989482	—	2	1	no	—
7	#17989476	C₄	1	2	yes	complemented
8	#17989461	—	2	1	no	—
9	#17989461	C₄	1	2	yes	—
10	#17988609	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic, complemented
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#64408	—	2	2	no	—
16	C₂×#11924031	—	2	2	no	—
17	(C₂×#64408)⋊C₄	—	4	1	no	—
18	(C₂×#64408)⋊C₄	—	4	1	no	—
19	C₂×#11924011	—	4	1	no	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
22	C₄²×D₇₃	D₄	1	1	yes	characteristic
23	C₄⋊#10596831	—	2	1	no	—
24	#14556520	—	4	1	no	—
25	C₄×#10596831	D₄	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#64408	—	4	2	no	—
31	C₄⋊#64408	—	4	2	no	—
32	C₄⋊#64408	—	4	2	no	—
33	#11924031	—	4	2	no	—
34	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
36	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
37	#11924011	—	8	2	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₈×D₇₃	—	4	1	no	—
42	C₈×D₇₃	—	4	1	no	—
43	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
45	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×D₇₃	#102	1	1	yes	characteristic
51	C₂×#64408	—	8	2	no	—
52	C₂×#64408	—	8	2	no	—
53	C₂×#64408	—	8	2	no	—
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
57	C₂×Dic₇₃	#103	1	1	yes	characteristic
58	C₂×Dic₇₃	#106	1	1	yes	characteristic
59	C₂×Dic₇₃	#107	1	1	yes	characteristic
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	—	4	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	—	8	1	no	—
67	#10596831	—	2	1	no	—
68	#10596831	—	2	1	no	—
69	#10596831	—	8	1	no	—
70	C₂²×C₇₃	#340	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	#360	1	2	yes	—
74	#64408	—	16	2	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
78	C₄×C₇₃	#344	1	1	yes	characteristic
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	#7221	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	#1286	1	1	yes	characteristic
84	C₂×C₇₃	#1386	1	1	yes	characteristic
85	C₂×C₇₃	#1389	1	1	yes	characteristic
86	D₇₃	—	2	2	no	—
87	#2163	—	73	1	no	—
88	#1835	—	73	1	no	—
89	#1566	—	73	1	no	—
90	C₇₃	#7221	1	1	yes	characteristic, complemented, Sylow
91	#423	—	146	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	#382	—	146	1	no	—
94	C₄⋊C₁₆	—	146	1	no	—
95	C₂²⋊C₁₆	—	146	1	no	—
96	#346	—	73	2	no	—
97	#340	—	73	2	no	—
98	#340	—	146	1	no	—
99	#334	—	73	1	no	—
100	#334	—	146	1	no	—
101	#334	#64408	1	1	yes	characteristic, complemented
102	C₂×M₁₆	—	292	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂×C₄⋊C₄	—	146	2	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₂×C₁₆	—	146	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	292	1	no	—
110	C₄⋊C₈	—	146	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	C₂×#64408	1	1	yes	characteristic
115	#95	—	292	1	no	—
116	#95	—	292	1	no	—
117	C₂²×C₄	—	292	2	no	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	M₁₆	—	584	2	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	292	1	no	—
124	C₂×C₈	—	292	1	no	—
125	C₂×C₈	—	292	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	C₄×#64408	1	1	yes	characteristic
129	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
130	C₄⋊C₄	—	292	2	no	—
131	C₄⋊C₄	—	292	2	no	—
132	C₄²	—	73	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	C₂²⋊#64408	1	1	yes	characteristic
136	C₁₆	—	292	2	no	—
137	C₂³	—	73	1	no	—
138	C₂×C₄	—	292	1	no	—
139	C₂×C₄	—	584	2	no	—
140	C₂×C₄	—	584	2	no	—
141	C₂×C₄	—	584	2	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#14556653	1	1	yes	characteristic
150	C₂×C₄	#14556654	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	8	1	no	—
154	C₈	—	584	1	no	—
155	C₈	—	146	1	no	—
156	C₈	—	146	1	no	—
157	C₂²	—	73	2	no	—
158	C₂²	—	146	1	no	—
159	C₂²	—	146	1	no	—
160	C₂²	#17989460	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	1168	2	no	—
164	C₄	—	146	1	no	—
165	C₄	—	146	1	no	—
166	C₄	—	146	1	no	—
167	C₄	#17989455	1	1	yes	characteristic
168	C₄	#17989470	1	1	yes	characteristic
169	C₂	—	146	2	no	—
170	C₂	#22624786	1	1	yes	characteristic
171	C₂	#22625004	1	1	yes	characteristic
172	C₂	#22625011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626295

C₂

yes

maximal, characteristic

#22625853

C₂

yes

maximal, characteristic

#22608685

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

complemented

#17989461

—

#17989461

C₄

yes

—

#17988609

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic, complemented

#17988337

—

#334×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

C₄⋊#64408

—

C₄⋊#64408

—

#11924031

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

—

—

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#360

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

#1835

—

#1566

—

C₇₃

#7221

yes

characteristic, complemented, Sylow

#423

—

146

—

C₂×C₄×C₈

—

#382

—

146

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

146

—

#346

—

#340

—

#340

—

146

—

#334

—

100

#334

—

146

—

101

#334

#64408

yes

characteristic, complemented

102

C₂×M₁₆

—

292

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂×C₄⋊C₄

—

146

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

146

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

292

—

110

C₄⋊C₈

—

146

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#64408

yes

characteristic

115

#95

—

292

—

116

#95

—

292

—

117

C₂²×C₄

—

292

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

M₁₆

—

584

—

122

C₂×C₈

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

C₄×#64408

yes

characteristic

129

C₂×C₈

C₄⋊#64408

yes

characteristic

130

C₄⋊C₄

—

292

—

131

C₄⋊C₄

—

292

—

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#64408

yes

characteristic

136

C₁₆

—

292

—

137

C₂³

—

138

C₂×C₄

—

292

—

139

C₂×C₄

—

584

—

140

C₂×C₄

—

584

—

141

C₂×C₄

—

584

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

149

C₂×C₄

#14556653

yes

characteristic

150

C₂×C₄

#14556654

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

584

—

155

C₈

—

146

—

156

C₈

—

146

—

157

C₂²

—

158

C₂²

—

146

—

159

C₂²

—

146

—

160

C₂²

#17989460

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

1168

—

164

C₄

—

146

—

165

C₄

—

146

—

166

C₄

—

146

—

167

C₄

#17989455

yes

characteristic

168

C₄

#17989470

yes

characteristic

169

C₂

—

146

—

170

C₂

#22624786

yes

characteristic

171

C₂

#22625004

yes

characteristic

172

C₂

#22625011

yes

characteristic

	subgroup	G/H
center	C₂²	#17989460
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7221
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#17989460

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#334×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
7. #17989476	C₄	semidirect
12. #17988338	C₄	semidirect
14. #334×C₇₃	C₄	semidirect
90. C₇₃	#7221	semidirect
101. #334	#64408	semidirect

H ≅ G/N

type

7. #17989476

C₄

semidirect

12. #17988338

C₄

semidirect

14. #334×C₇₃

C₄

semidirect

90. C₇₃

#7221

semidirect

101. #334

#64408

semidirect

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

subgroup	index
#22626295	2
#22625853	2
#22608685	2
#7221	73

subgroup

index

Subgroups — #34000431

Conjugacy classes of subgroups

Complements

Maximal subgroups