Group #34011723 — subgroups

Subgroups — #34011723

Order 18688.

254 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 108 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	#22625473	C₂	1	2	yes	maximal
2	#22624957	C₂	1	2	yes	maximal, complemented
3	#22624956	C₂	1	1	yes	maximal, characteristic, complemented
4	#22624955	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989588	—	2	2	no	—
7	#17989449	—	2	2	no	—
8	#17989448	C₂²	1	2	yes	—
9	#17989448	C₂²	1	2	yes	—
10	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988363	C₄	1	1	yes	characteristic
14	#17988362	C₂²	1	1	yes	characteristic
15	#17988362	C₂²	1	1	yes	characteristic
16	#346×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924036	—	4	2	no	—
18	#14556658	—	4	2	no	—
19	C₄⋊#10596833	—	2	2	no	—
20	C₄⋊#10596833	—	2	2	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
23	C₄²×D₇₃	D₄	1	1	yes	characteristic
24	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
25	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
26	C₁₆×D₇₃	—	2	2	no	—
27	C₁₆×D₇₃	C₈	1	4	yes	complemented
28	C₄×#10596831	Q₈	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	#11924036	—	8	4	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂×#10596833	—	4	2	no	—
37	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
41	C₈×D₇₃	—	2	2	no	—
42	C₈×D₇₃	C₂×C₈	1	2	yes	—
43	C₈×D₇₃	M₁₆	1	2	yes	—
44	C₄×Dic₇₃	SD₁₆	1	2	yes	—
45	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
46	#11924003	—	2	1	no	—
47	#11924003	C₂×C₈	1	2	yes	complemented
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	2	1	no	—
52	C₁₆×C₇₃	C₂×C₈	1	2	yes	complemented
53	C₂²×D₇₃	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#106	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	#106	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
58	C₄×D₇₃	—	2	4	no	—
59	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
60	C₄×D₇₃	#108	1	2	yes	—
61	#10596833	—	8	2	no	—
62	#10596833	—	8	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	#10596831	—	2	1	no	—
67	#10596831	C₄⋊C₈	1	2	yes	—
68	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
69	C₂×D₇₃	#334	1	2	yes	—
70	C₂×D₇₃	#365	1	4	yes	—
71	C₄×C₇₃	—	2	2	no	—
72	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	#334	1	2	yes	—
76	C₂×#1368	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#365	1	2	yes	—
79	D₇₃	#1368	1	4	yes	complemented
80	C₂×#346	—	73	1	no	—
81	C₂×#334	—	73	2	no	—
82	#1368	—	73	1	no	—
83	#1368	—	73	1	no	—
84	#1368	—	73	2	no	—
85	C₇₃	C₂×#1368	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₁₆	—	73	1	no	—
87	C₂²×C₁₆	—	73	1	no	—
88	C₂×C₄⋊C₈	—	146	2	no	—
89	C₂×C₄×C₈	—	73	1	no	—
90	#346	—	73	1	no	—
91	#346	—	73	1	no	—
92	#346	—	73	1	no	—
93	#346	#64408	1	1	yes	characteristic
94	#334	—	73	2	no	—
95	#334	—	73	2	no	—
96	#334	—	146	2	no	—
97	C₂²×C₈	—	292	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂×C₄²	—	73	1	no	—
101	C₂×C₁₆	—	73	1	no	—
102	C₂×C₁₆	—	73	4	no	—
103	C₂×C₁₆	—	73	1	no	—
104	C₂×C₁₆	—	146	2	no	—
105	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
106	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
107	C₄⋊C₈	—	292	2	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₄⋊C₈	—	146	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	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	2	no	—
116	C₂×C₈	—	292	2	no	—
117	C₂×C₈	—	292	2	no	—
118	C₂×C₈	—	584	4	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	2	no	—
121	C₂×C₈	—	73	2	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
125	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
126	C₄²	—	73	2	no	—
127	C₄²	—	73	1	no	—
128	C₄²	C₄⋊#64408	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	73	2	no	—
131	C₁₆	—	146	1	no	—
132	C₁₆	#11924036	1	2	yes	complemented
133	C₂³	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	73	2	no	—
136	C₂×C₄	—	73	2	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	146	4	no	—
139	C₂×C₄	#14556659	1	2	yes	—
140	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	584	2	no	—
143	C₈	—	584	2	no	—
144	C₈	—	73	2	no	—
145	C₈	—	146	1	no	—
146	C₈	C₂×#11924036	1	1	yes	characteristic
147	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
148	C₂²	—	73	2	no	—
149	C₂²	—	73	4	no	—
150	C₂²	C₂×#14556659	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	73	2	no	—
153	C₄	—	146	2	no	—
154	C₄	#17989588	1	1	yes	characteristic
155	C₄	D₇₃⋊#108	1	1	yes	characteristic
156	C₂	—	73	4	no	—
157	C₂	#22625473	1	1	yes	characteristic
158	C₂	D₇₃⋊#365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625473

C₂

yes

maximal

#22624957

C₂

yes

maximal, complemented

#22624956

C₂

yes

maximal, characteristic, complemented

#22624955

C₂

yes

maximal, characteristic, complemented

#346×D₇₃

C₂

yes

maximal, characteristic

#17989588

—

#17989449

—

#17989448

C₂²

yes

—

#17989448

C₂²

yes

—

C₂×C₁₆×D₇₃

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

characteristic

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic

#17988362

C₂²

yes

characteristic

#17988362

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

C₈

yes

complemented

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

—

#11924003

C₂×C₈

yes

complemented

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

C₂²×D₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#106

yes

characteristic

C₂×Dic₇₃

C₂×SD₁₆

yes

—

C₄×D₇₃

—

C₄×D₇₃

C₄⋊C₈

yes

—

C₄×D₇₃

#108

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

#10596831

C₄⋊C₈

yes

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₂×D₇₃

#334

yes

—

C₂×D₇₃

#365

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×#108

yes

characteristic

Dic₇₃

—

Dic₇₃

#334

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

—

—

C₇₃

C₂×#1368

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

146

—

C₂×C₄×C₈

—

—

—

—

yes

—

—

—

146

—

C₂²×C₈

—

292

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂×C₄²

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

146

—

105

C₂×C₁₆

C₂×#64408

yes

characteristic

106

C₂×C₁₆

C₂×#64408

yes

characteristic

107

C₄⋊C₈

—

292

—

108

C₄⋊C₈

—

146

—

109

C₄⋊C₈

—

146

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

C₂×#64408

yes

characteristic

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

292

—

118

C₂×C₈

—

584

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₂×C₈

C₄⋊#64408

yes

characteristic

125

C₂×C₈

C₂²×#64408

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

C₄⋊#64408

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

146

—

132

C₁₆

#11924036

yes

complemented

133

C₂³

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

146

—

139

C₂×C₄

#14556659

yes

—

140

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

141

C₈

—

142

C₈

—

584

—

143

C₈

—

584

—

144

C₈

—

145

C₈

—

146

—

146

C₈

C₂×#11924036

yes

characteristic

147

C₈

D₇₃⋊M₁₆

yes

characteristic

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×#14556659

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

146

—

154

C₄

#17989588

yes

characteristic

155

C₄

D₇₃⋊#108

yes

characteristic

156

C₂

—

157

C₂

#22625473

yes

characteristic

158

C₂

D₇₃⋊#365

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556659
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂²×D₇₃	#106
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	C₂×#1368
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#14556659

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#346×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #22624957	C₂	semidirect
3. #22624956	C₂	semidirect
4. #22624955	C₂	semidirect
27. C₁₆×D₇₃	C₈	semidirect
47. #11924003	C₂×C₈	semidirect
52. C₁₆×C₇₃	C₂×C₈	semidirect
79. D₇₃	#1368	semidirect
85. C₇₃	C₂×#1368	semidirect
132. C₁₆	#11924036	semidirect

H ≅ G/N

type

2. #22624957

C₂

semidirect

3. #22624956

C₂

semidirect

4. #22624955

C₂

semidirect

27. C₁₆×D₇₃

C₈

semidirect

47. #11924003

C₂×C₈

semidirect

52. C₁₆×C₇₃

C₂×C₈

semidirect

79. D₇₃

#1368

semidirect

85. C₇₃

C₂×#1368

semidirect

132. C₁₆

#11924036

semidirect

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

subgroup	index
#22625473	2
#22624957	2
#22624956	2
#22624955	2
#346×D₇₃	2
C₂×#1368	73

subgroup

index

#346×D₇₃

Subgroups — #34011723

Conjugacy classes of subgroups

Complements

Maximal subgroups