Group #34011793 — subgroups

Subgroups — #34011793

Order 18688.

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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	#22625492	C₂	1	1	yes	maximal, characteristic
2	#22625486	C₂	1	1	yes	maximal, characteristic
3	#22624957	C₂	1	4	yes	maximal, complemented
4	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989609	—	2	1	no	—
6	#17989591	—	2	1	no	—
7	#17989450	—	2	1	no	—
8	#17989448	—	2	1	no	—
9	#17989448	C₂²	1	2	yes	—
10	#17989448	C₂²	1	2	yes	—
11	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988363	C₄	1	1	yes	characteristic
14	#17988362	C₂²	1	2	yes	—
15	#346×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	4	1	no	—
17	C₄⋊#10596833	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	C₄⋊#10596833	—	4	1	no	—
20	#14556648	—	4	1	no	—
21	#14556646	—	4	1	no	—
22	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
24	C₄²×D₇₃	D₄	1	1	yes	characteristic
25	C₂×#11924003	C₂×C₄	1	2	yes	—
26	C₁₆×D₇₃	—	2	4	no	—
27	C₄×#10596831	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₃	D₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924038	—	8	2	no	—
33	C₂×#10596833	—	4	2	no	—
34	C₂×#10596833	—	4	2	no	—
35	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
38	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
40	C₈×D₇₃	—	2	2	no	—
41	C₈×D₇₃	—	2	2	no	—
42	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
43	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
45	#11924003	M₁₆	1	4	yes	complemented
46	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₁₆×C₇₃	—	2	2	no	—
50	C₂²×D₇₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	#106	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	4	1	no	—
61	#10596833	—	8	2	no	—
62	#10596833	—	8	2	no	—
63	C₈×C₇₃	#108	1	2	yes	—
64	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
65	#10596831	—	2	1	no	—
66	#10596831	—	2	1	no	—
67	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
68	C₂×D₇₃	—	2	1	no	—
69	C₂×D₇₃	—	2	1	no	—
70	C₂×D₇₃	#365	1	2	yes	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	#423	1	1	yes	characteristic
74	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	—	2	1	no	—
78	#12019	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	#1566	1	1	yes	characteristic
80	C₂×C₇₃	C₂×#365	1	1	yes	characteristic
81	C₂×C₇₃	#2155	1	1	yes	characteristic
82	D₇₃	—	2	2	no	—
83	C₂×#346	—	73	1	no	—
84	#1568	—	73	1	no	—
85	#1566	—	73	1	no	—
86	#1368	—	73	4	no	—
87	C₇₃	#12019	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	73	2	no	—
89	#432	—	146	1	no	—
90	#423	—	146	1	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	#346	—	73	2	no	—
93	#346	—	73	1	no	—
94	#346	#64408	1	1	yes	characteristic
95	#335	—	146	1	no	—
96	#334	—	73	2	no	—
97	#334	—	73	2	no	—
98	#334	—	146	1	no	—
99	C₂×M₁₆	—	292	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂²×C₈	—	73	1	no	—
102	C₂×C₄²	—	73	1	no	—
103	C₂×C₁₆	—	73	2	no	—
104	C₂×C₁₆	—	146	4	no	—
105	C₂×C₁₆	C₂×#64408	1	2	yes	—
106	C₄⋊C₈	—	292	1	no	—
107	C₄⋊C₈	—	146	2	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₂²⋊C₈	—	292	1	no	—
110	#97	—	292	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	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	1	no	—
117	C₂²×C₄	—	73	1	no	—
118	M₁₆	—	584	2	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	292	2	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	—	146	2	no	—
125	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
126	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
127	C₄²	—	73	1	no	—
128	C₄²	—	73	1	no	—
129	C₄²	—	73	1	no	—
130	C₄²	C₄⋊#64408	1	1	yes	characteristic
131	C₁₆	—	2	2	no	—
132	C₁₆	—	73	4	no	—
133	C₂³	—	73	1	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	146	2	no	—
139	C₂×C₄	—	146	1	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	#14556659	1	1	yes	characteristic
142	C₂×C₄	#14556659	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
144	C₈	—	584	2	no	—
145	C₈	—	584	2	no	—
146	C₈	—	146	1	no	—
147	C₈	—	146	1	no	—
148	C₈	#14556662	1	2	yes	—
149	C₈	D₇₃⋊M₁₆	1	2	yes	—
150	C₂²	—	73	2	no	—
151	C₂²	—	146	1	no	—
152	C₂²	—	146	1	no	—
153	C₂²	C₂×#14556659	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	146	1	no	—
157	C₄	—	146	1	no	—
158	C₄	—	146	1	no	—
159	C₄	#17989592	1	1	yes	characteristic
160	C₄	C₂×#14556662	1	1	yes	characteristic
161	C₂	—	146	2	no	—
162	C₂	#22625487	1	1	yes	characteristic
163	C₂	D₇₃⋊#365	1	1	yes	characteristic
164	C₂	#22626279	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625492

C₂

yes

maximal, characteristic

#22625486

C₂

yes

maximal, characteristic

#22624957

C₂

yes

maximal, complemented

#346×D₇₃

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

#17989448

C₂²

yes

—

C₂×C₁₆×D₇₃

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic

#17988362

C₂²

yes

—

#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

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

M₁₆

yes

complemented

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#106

yes

characteristic

C₂×Dic₇₃

C₂×SD₁₆

yes

characteristic

C₂×Dic₇₃

C₂×SD₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

#108

yes

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#365

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

—

C₇₃

#12019

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#432

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

—

—

yes

—

146

—

#334

—

#334

—

#334

—

146

—

C₂×M₁₆

—

292

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂×C₄²

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

146

—

105

C₂×C₁₆

C₂×#64408

yes

—

106

C₄⋊C₈

—

292

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

C₂²⋊C₈

—

292

—

110

#97

—

292

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₂×#64408

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

M₁₆

—

584

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

C₄⋊#64408

yes

characteristic

126

C₂×C₈

C₂²×#64408

yes

characteristic

127

C₄²

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₄⋊#64408

yes

characteristic

131

C₁₆

—

132

C₁₆

—

133

C₂³

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

#14556659

yes

characteristic

142

C₂×C₄

#14556659

yes

characteristic

143

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

144

C₈

—

584

—

145

C₈

—

584

—

146

C₈

—

146

—

147

C₈

—

146

—

148

C₈

#14556662

yes

—

149

C₈

D₇₃⋊M₁₆

yes

—

150

C₂²

—

151

C₂²

—

146

—

152

C₂²

—

146

—

153

C₂²

C₂×#14556659

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

146

—

157

C₄

—

146

—

158

C₄

—

146

—

159

C₄

#17989592

yes

characteristic

160

C₄

C₂×#14556662

yes

characteristic

161

C₂

—

146

—

162

C₂

#22625487

yes

characteristic

163

C₂

D₇₃⋊#365

yes

characteristic

164

C₂

#22626279

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556659
commutator	C₂×C₈×C₇₃	C₂²×C₄
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	#12019
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

C₂×#14556659

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

#346×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
3. #22624957	C₂	semidirect
45. #11924003	M₁₆	semidirect
87. C₇₃	#12019	semidirect

H ≅ G/N

type

3. #22624957

C₂

semidirect

45. #11924003

M₁₆

semidirect

87. C₇₃

#12019

semidirect

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

subgroup	index
#22625492	2
#22625486	2
#22624957	2
#346×D₇₃	2
#12019	73

subgroup

index

#346×D₇₃

Subgroups — #34011793

Conjugacy classes of subgroups

Complements

Maximal subgroups