Group #34005434 — subgroups

Subgroups — #34005434

Order 18688.

294 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 100 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	#22625871	C₂	1	1	yes	maximal, characteristic
2	#22625868	C₂	1	1	yes	maximal, characteristic
3	#22624785	C₂	1	4	yes	maximal
4	#22608216	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556662	—	2	1	no	—
6	C₂×#14556661	—	2	1	no	—
7	C₂×#14556660	—	2	1	no	—
8	C₂×#14556659	—	2	1	no	—
9	#17989592	C₂²	1	2	yes	—
10	#17989531	C₂²	1	2	yes	—
11	#17988638	C₂²	1	2	yes	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988308	C₄	1	1	yes	characteristic, complemented
14	C₈×#10596831	C₄	1	2	yes	complemented
15	#322×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#11924038	—	2	2	no	—
17	C₂×C₄⋊#64408	—	2	2	no	—
18	#14556662	—	4	2	no	—
19	#14556661	—	4	2	no	—
20	#14556660	—	4	2	no	—
21	#14556659	—	4	2	no	—
22	#14556658	—	2	4	no	—
23	(C₂×#64408)⋊C₄	—	2	4	no	—
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
26	C₄²×D₇₃	C₂³	1	1	yes	characteristic
27	#14556520	—	2	2	no	—
28	#14556503	—	2	2	no	—
29	C₄×#10596831	C₂×C₄	1	2	yes	—
30	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
32	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×#64408	—	4	2	no	—
36	#11924038	—	4	2	no	—
37	#11924038	—	4	2	no	—
38	C₂×#10596833	—	4	2	no	—
39	C₄⋊#64408	—	4	2	no	—
40	C₄⋊#64408	—	4	2	no	—
41	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×D₇₃	#41	1	2	yes	—
43	C₂×#10596831	—	2	2	no	—
44	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
45	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
46	C₈×D₇₃	D₈	1	2	yes	—
47	C₈×D₇₃	SD₁₆	1	2	yes	—
48	C₈×D₇₃	SD₁₆	1	2	yes	—
49	C₈×D₇₃	Q₁₆	1	2	yes	—
50	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₄×Dic₇₃	#41	1	2	yes	—
52	C₂×C₈×C₇₃	—	2	2	no	—
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₇₃	#124	1	1	yes	characteristic
57	C₂×#64408	—	8	4	no	—
58	C₂×#64408	—	8	2	no	—
59	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#117	1	2	yes	—
61	C₂×Dic₇₃	#124	1	1	yes	characteristic
62	C₂×Dic₇₃	#125	1	2	yes	—
63	C₄×D₇₃	—	4	2	no	—
64	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
65	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
68	#10596833	—	8	2	no	—
69	C₈×C₇₃	—	4	2	no	—
70	C₈×C₇₃	#106	1	2	yes	—
71	C₈×C₇₃	#107	1	2	yes	—
72	#10596831	—	4	2	no	—
73	#10596831	#106	1	2	yes	—
74	#10596831	#107	1	2	yes	—
75	C₂²×C₇₃	#382	1	1	yes	characteristic
76	C₂×D₇₃	—	2	2	no	—
77	C₂×D₇₃	#486	1	1	yes	characteristic
78	C₂×D₇₃	#487	1	1	yes	characteristic
79	#64408	—	16	2	no	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
82	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
83	Dic₇₃	—	2	2	no	—
84	Dic₇₃	#486	1	1	yes	characteristic
85	Dic₇₃	#487	1	1	yes	characteristic
86	#8790	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#1833	1	2	yes	—
88	C₂×C₇₃	#1834	1	1	yes	characteristic
89	D₇₃	—	2	1	no	—
90	D₇₃	—	2	1	no	—
91	#1848	—	73	1	no	—
92	#1846	—	73	1	no	—
93	#1450	—	73	1	no	—
94	#1286	—	73	4	no	—
95	C₇₃	#8790	1	1	yes	characteristic, complemented, Sylow
96	#431	—	73	2	no	—
97	C₂×#108	—	146	1	no	—
98	C₂×#108	—	146	1	no	—
99	C₂×#107	—	146	1	no	—
100	C₂×#106	—	146	1	no	—
101	#423	—	73	2	no	—
102	C₂×C₄×C₈	—	73	1	no	—
103	#382	—	73	2	no	—
104	#322	—	73	1	no	—
105	#322	#64408	1	1	yes	characteristic, complemented
106	C₈²	—	73	2	no	—
107	C₂×M₁₆	—	146	2	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₂×C₄⋊C₄	—	146	2	no	—
111	C₂×C₄²	—	73	1	no	—
112	#108	—	292	2	no	—
113	#108	—	292	2	no	—
114	#107	—	292	2	no	—
115	#106	—	292	2	no	—
116	C₄⋊C₈	—	146	2	no	—
117	C₄⋊C₈	—	146	4	no	—
118	C₂²⋊C₈	—	146	2	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	—	73	2	no	—
123	C₄×C₈	C₂×#64408	1	2	yes	—
124	C₄×C₈	C₂×#64408	1	1	yes	characteristic
125	#95	—	146	4	no	—
126	C₂²×C₄	—	292	2	no	—
127	C₂²×C₄	—	73	2	no	—
128	C₂²×C₄	—	73	1	no	—
129	M₁₆	—	292	2	no	—
130	M₁₆	—	292	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	292	2	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	—	73	2	no	—
137	C₂×C₈	—	73	2	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
141	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
142	C₄⋊C₄	—	292	2	no	—
143	C₄⋊C₄	—	292	2	no	—
144	C₄²	—	73	1	no	—
145	C₄²	—	73	2	no	—
146	C₄²	C₂²×#64408	1	1	yes	characteristic
147	C₂³	—	73	1	no	—
148	C₂×C₄	—	292	2	no	—
149	C₂×C₄	—	584	2	no	—
150	C₂×C₄	—	584	4	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	73	1	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
158	C₂×C₄	#14556670	1	2	yes	—
159	C₈	—	4	2	no	—
160	C₈	—	292	2	no	—
161	C₈	—	584	2	no	—
162	C₈	—	73	2	no	—
163	C₈	—	73	2	no	—
164	C₈	#14556659	1	2	yes	—
165	C₈	#14556660	1	2	yes	—
166	C₂²	—	73	1	no	—
167	C₂²	—	73	1	no	—
168	C₂²	—	146	2	no	—
169	C₂²	#17989531	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	73	1	no	—
172	C₄	—	73	1	no	—
173	C₄	—	1168	2	no	—
174	C₄	—	146	2	no	—
175	C₄	C₂×#14556659	1	1	yes	characteristic
176	C₄	C₂×#14556660	1	1	yes	characteristic
177	C₂	—	146	1	no	—
178	C₂	—	146	1	no	—
179	C₂	#22625851	1	2	yes	—
180	C₂	#22625852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625871

C₂

yes

maximal, characteristic

#22625868

C₂

yes

maximal, characteristic

#22624785

C₂

yes

maximal

#22608216

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#17989531

C₂²

yes

—

#17988638

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988308

C₄

yes

characteristic, complemented

C₈×#10596831

C₄

yes

complemented

#322×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924038

—

C₂×C₄⋊#64408

—

—

—

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

#41

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

SD₁₆

yes

—

C₈×D₇₃

SD₁₆

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

C₂²×D₇₃

yes

—

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×Dic₇₃

#124

yes

characteristic

C₂×Dic₇₃

#125

yes

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

characteristic

C₄×D₇₃

C₂×SD₁₆

yes

characteristic

C₄×D₇₃

C₂×SD₁₆

yes

characteristic

C₄×D₇₃

C₂×Q₁₆

yes

characteristic

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

#107

yes

—

#10596831

—

#10596831

#106

yes

—

#10596831

#107

yes

—

C₂²×C₇₃

#382

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

—

C₂×C₇₃

#1834

yes

characteristic

D₇₃

—

D₇₃

—

—

—

—

—

C₇₃

#8790

yes

characteristic, complemented, Sylow

#431

—

C₂×#108

—

146

—

C₂×#108

—

146

—

C₂×#107

—

146

—

100

C₂×#106

—

146

—

101

#423

—

102

C₂×C₄×C₈

—

103

#382

—

104

#322

—

105

#322

#64408

yes

characteristic, complemented

106

C₈²

—

107

C₂×M₁₆

—

146

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂×C₄⋊C₄

—

146

—

111

C₂×C₄²

—

112

#108

—

292

—

113

#108

—

292

—

114

#107

—

292

—

115

#106

—

292

—

116

C₄⋊C₈

—

146

—

117

C₄⋊C₈

—

146

—

118

C₂²⋊C₈

—

146

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

C₂×#64408

yes

—

124

C₄×C₈

C₂×#64408

yes

characteristic

125

#95

—

146

—

126

C₂²×C₄

—

292

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

M₁₆

—

292

—

130

M₁₆

—

292

—

131

C₂×C₈

—

132

C₂×C₈

—

292

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

146

—

140

C₂×C₈

C₄⋊#64408

yes

characteristic

141

C₂×C₈

C₄⋊#64408

yes

characteristic

142

C₄⋊C₄

—

292

—

143

C₄⋊C₄

—

292

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂²×#64408

yes

characteristic

147

C₂³

—

148

C₂×C₄

—

292

—

149

C₂×C₄

—

584

—

150

C₂×C₄

—

584

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

158

C₂×C₄

#14556670

yes

—

159

C₈

—

160

C₈

—

292

—

161

C₈

—

584

—

162

C₈

—

163

C₈

—

164

C₈

#14556659

yes

—

165

C₈

#14556660

yes

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

146

—

169

C₂²

#17989531

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

1168

—

174

C₄

—

146

—

175

C₄

C₂×#14556659

yes

characteristic

176

C₄

C₂×#14556660

yes

characteristic

177

C₂

—

146

—

178

C₂

—

146

—

179

C₂

#22625851

yes

—

180

C₂

#22625852

yes

characteristic

	subgroup	G/H
center	C₂²	#17989531
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	#382
prime radical	C₂²×D₇₃	#124
cosocle	C₄²×D₇₃	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	#8790
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17989531

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

#322×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
13. #17988308	C₄	semidirect
14. C₈×#10596831	C₄	semidirect
15. #322×C₇₃	C₄	semidirect
95. C₇₃	#8790	semidirect
105. #322	#64408	semidirect

H ≅ G/N

type

13. #17988308

C₄

semidirect

14. C₈×#10596831

C₄

semidirect

15. #322×C₇₃

C₄

semidirect

95. C₇₃

#8790

semidirect

105. #322

#64408

semidirect

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

subgroup	index
#22625871	2
#22625868	2
#22624785	2
#22608216	2
#8790	73

subgroup

index

Subgroups — #34005434

Conjugacy classes of subgroups

Complements

Maximal subgroups