Group #34005499 — subgroups

Subgroups — #34005499

Order 18688.

354 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 140 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	#22625872	C₂	1	2	yes	maximal
2	#22625487	C₂	1	2	yes	maximal
3	#22625486	C₂	1	2	yes	maximal
4	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556662	—	2	4	no	—
6	#17989591	C₂²	1	4	yes	—
7	#17989449	—	2	4	no	—
8	#17989448	—	2	4	no	—
9	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₄	1	1	yes	characteristic
13	C₈×#10596831	C₄	1	1	yes	characteristic
14	C₈×#10596831	C₄	1	1	yes	characteristic
15	C₈²×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	2	4	no	—
17	#14556662	—	4	8	no	—
18	C₄⋊#10596833	—	2	4	no	—
19	C₄⋊#10596833	—	2	4	no	—
20	C₂×C₈×D₇₃	D₄	1	2	yes	—
21	C₂×C₈×D₇₃	Q₈	1	2	yes	—
22	C₂×C₈×D₇₃	Q₈	1	2	yes	—
23	C₄²×D₇₃	C₂³	1	1	yes	characteristic
24	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	2	yes	—
28	C₈×Dic₇₃	D₄	1	2	yes	—
29	C₈×Dic₇₃	Q₈	1	2	yes	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924038	—	4	8	no	—
34	C₂×#10596833	—	4	4	no	—
35	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×#10596831	C₄⋊C₄	1	2	yes	—
39	C₂×#10596831	C₄⋊C₄	1	2	yes	—
40	C₂×#10596831	C₄⋊C₄	1	2	yes	—
41	C₈×D₇₃	—	2	4	no	—
42	C₈×D₇₃	—	2	4	no	—
43	C₈×D₇₃	SD₁₆	1	8	yes	—
44	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
45	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	#127	1	1	yes	characteristic
56	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×D₇₃	—	2	2	no	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	C₂×SD₁₆	1	4	yes	—
61	#10596833	—	8	4	no	—
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	#106	1	4	yes	—
64	C₈×C₇₃	#108	1	4	yes	complemented
65	#10596831	—	2	2	no	—
66	#10596831	#106	1	4	yes	—
67	#10596831	#108	1	4	yes	complemented
68	C₂²×C₇₃	#384	1	1	yes	characteristic
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	#498	1	2	yes	—
72	C₄×C₇₃	C₂×#106	1	2	yes	—
73	C₄×C₇₃	C₂×#108	1	2	yes	—
74	C₄×C₇₃	C₂×#108	1	2	yes	—
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	#492	1	2	yes	—
78	#8822	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	#1844	1	1	yes	characteristic
80	C₂×C₇₃	#1848	1	1	yes	characteristic
81	C₂×C₇₃	#1848	1	1	yes	characteristic
82	D₇₃	—	2	2	no	—
83	#1848	—	73	2	no	—
84	#1566	—	73	2	no	—
85	#1566	—	73	2	no	—
86	C₂×C₈²	—	73	1	no	—
87	C₇₃	#8822	1	1	yes	characteristic, complemented, Sylow
88	C₂×#108	—	146	4	no	—
89	#423	—	73	4	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	#334	—	146	4	no	—
94	#334	—	146	4	no	—
95	C₈²	—	73	1	no	—
96	C₈²	—	73	1	no	—
97	C₈²	—	73	1	no	—
98	C₈²	#64408	1	1	yes	characteristic
99	C₂×M₁₆	—	146	4	no	—
100	C₂²×C₈	—	73	2	no	—
101	C₂²×C₈	—	73	2	no	—
102	C₂²×C₈	—	73	2	no	—
103	C₂×C₄²	—	73	1	no	—
104	#108	—	292	8	no	—
105	C₄⋊C₈	—	146	4	no	—
106	C₄⋊C₈	—	146	4	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	—	73	2	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	C₂×#64408	1	1	yes	characteristic
114	C₄×C₈	C₂×#64408	1	1	yes	characteristic
115	C₄×C₈	C₂×#64408	1	1	yes	characteristic
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	M₁₆	—	292	8	no	—
120	C₂×C₈	—	292	4	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	73	2	no	—
123	C₂×C₈	—	73	8	no	—
124	C₂×C₈	—	73	2	no	—
125	C₂×C₈	—	146	4	no	—
126	C₂×C₈	—	146	4	no	—
127	C₂×C₈	C₄⋊#64408	1	2	yes	—
128	C₂×C₈	C₄⋊#64408	1	2	yes	—
129	C₂×C₈	C₄⋊#64408	1	2	yes	—
130	C₄²	—	73	1	no	—
131	C₄²	—	73	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	C₂²×#64408	1	1	yes	characteristic
134	C₂³	—	73	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	4	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	584	4	no	—
146	C₈	—	73	4	no	—
147	C₈	—	73	4	no	—
148	C₈	—	146	2	no	—
149	C₈	#14556659	1	4	yes	—
150	C₈	#14556662	1	4	yes	complemented
151	C₂²	—	73	2	no	—
152	C₂²	—	146	1	no	—
153	C₂²	—	146	1	no	—
154	C₂²	#17989533	1	1	yes	characteristic
155	C₄	—	73	2	no	—
156	C₄	—	146	1	no	—
157	C₄	—	146	1	no	—
158	C₄	C₂×#14556659	1	2	yes	—
159	C₄	D₇₃⋊#108	1	2	yes	—
160	C₄	C₂×#14556662	1	2	yes	—
161	C₂	—	146	2	no	—
162	C₂	#22625866	1	1	yes	characteristic
163	C₂	#22625871	1	1	yes	characteristic
164	C₂	#22625872	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625872

C₂

yes

maximal

#22625487

C₂

yes

maximal

#22625486

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

—

#17989591

C₂²

yes

—

#17989449

—

#17989448

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

yes

—

yes

—

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#498

yes

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

C₂×#108

yes

—

C₄×C₇₃

C₂×#108

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#492

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1848

—

#1566

—

#1566

—

C₂×C₈²

—

C₇₃

#8822

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

146

—

#334

—

146

—

C₈²

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂×M₁₆

—

146

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂×C₄²

—

104

#108

—

292

—

105

C₄⋊C₈

—

146

—

106

C₄⋊C₈

—

146

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#64408

yes

characteristic

114

C₄×C₈

C₂×#64408

yes

characteristic

115

C₄×C₈

C₂×#64408

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₂×C₈

C₄⋊#64408

yes

—

128

C₂×C₈

C₄⋊#64408

yes

—

129

C₂×C₈

C₄⋊#64408

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

133

C₄²

C₂²×#64408

yes

characteristic

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

144

C₈

—

145

C₈

—

584

—

146

C₈

—

147

C₈

—

148

C₈

—

146

—

149

C₈

#14556659

yes

—

150

C₈

#14556662

yes

complemented

151

C₂²

—

152

C₂²

—

146

—

153

C₂²

—

146

—

154

C₂²

#17989533

yes

characteristic

155

C₄

—

156

C₄

—

146

—

157

C₄

—

146

—

158

C₄

C₂×#14556659

yes

—

159

C₄

D₇₃⋊#108

yes

—

160

C₄

C₂×#14556662

yes

—

161

C₂

—

146

—

162

C₂

#22625866

yes

characteristic

163

C₂

#22625871

yes

characteristic

164

C₂

#22625872

yes

characteristic

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	C₄⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8822
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989533

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

#384

prime radical

C₂²×D₇₃

C₄⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
64. C₈×C₇₃	#108	semidirect
67. #10596831	#108	semidirect
87. C₇₃	#8822	semidirect
150. C₈	#14556662	semidirect

H ≅ G/N

type

64. C₈×C₇₃

#108

semidirect

67. #10596831

#108

semidirect

87. C₇₃

#8822

semidirect

150. C₈

#14556662

semidirect

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

subgroup	index
#22625872	2
#22625487	2
#22625486	2
C₈²×D₇₃	2
#8822	73

subgroup

index

C₈²×D₇₃

Subgroups — #34005499

Conjugacy classes of subgroups

Complements

Maximal subgroups