Group #34005507 — subgroups

Subgroups — #34005507

Order 18688.

338 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 128 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	1	yes	maximal, characteristic
2	#22625870	C₂	1	1	yes	maximal, characteristic
3	#22625487	C₂	1	2	yes	maximal
4	#22625486	C₂	1	2	yes	maximal
5	#322×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556662	—	2	2	no	—
7	C₂×#14556661	—	2	2	no	—
8	#17989591	C₂²	1	2	yes	—
9	#17989591	C₂²	1	1	yes	characteristic
10	#17989591	C₂²	1	1	yes	characteristic
11	#17989449	—	2	4	no	—
12	#17989448	—	2	4	no	—
13	C₄×C₈×D₇₃	C₂²	1	2	yes	—
14	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	#17988309	C₄	1	2	yes	—
16	#17988308	C₄	1	1	yes	characteristic
17	#322×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924038	—	2	2	no	—
19	C₂×#11924038	—	2	1	no	—
20	C₂×#11924038	—	2	1	no	—
21	#14556662	—	4	4	no	—
22	#14556661	—	4	4	no	—
23	C₄⋊#10596833	—	2	2	no	—
24	C₄⋊#10596833	—	2	2	no	—
25	C₄⋊#10596833	—	2	2	no	—
26	C₄⋊#10596833	—	2	2	no	—
27	C₂×C₈×D₇₃	D₄	1	2	yes	—
28	C₂×C₈×D₇₃	Q₈	1	4	yes	—
29	C₄²×D₇₃	C₂³	1	1	yes	characteristic
30	C₄×#10596831	C₂×C₄	1	2	yes	—
31	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₇₃	D₄	1	4	yes	—
33	C₈×Dic₇₃	Q₈	1	2	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	#11924038	—	4	4	no	—
37	#11924038	—	4	2	no	—
38	#11924038	—	4	2	no	—
39	C₂×#10596833	—	4	2	no	—
40	C₂×#10596833	—	4	1	no	—
41	C₂×#10596833	—	4	1	no	—
42	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	4	yes	—
45	C₂×#10596831	C₄⋊C₄	1	2	yes	—
46	C₈×D₇₃	—	4	4	no	—
47	C₈×D₇₃	SD₁₆	1	4	yes	—
48	C₈×D₇₃	SD₁₆	1	4	yes	—
49	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
50	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
52	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
53	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×Dic₇₃	#127	1	1	yes	characteristic
58	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
59	C₄×D₇₃	—	2	4	no	—
60	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
61	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
62	#10596833	—	8	2	no	—
63	#10596833	—	8	1	no	—
64	#10596833	—	8	1	no	—
65	C₈×C₇₃	—	2	4	no	—
66	C₈×C₇₃	#106	1	4	yes	—
67	#10596831	—	2	4	no	—
68	#10596831	#106	1	4	yes	—
69	C₂²×C₇₃	#384	1	1	yes	characteristic
70	C₂×D₇₃	—	2	2	no	—
71	C₂×D₇₃	#498	1	2	yes	—
72	C₄×C₇₃	C₂×#106	1	2	yes	—
73	C₄×C₇₃	#430	1	4	yes	—
74	Dic₇₃	—	2	2	no	—
75	Dic₇₃	#492	1	1	yes	characteristic
76	Dic₇₃	#492	1	1	yes	characteristic
77	#8825	—	73	1	no	maximal, Sylow
78	C₂×C₇₃	#1844	1	1	yes	characteristic
79	C₂×C₇₃	#1849	1	2	yes	—
80	D₇₃	—	2	2	no	—
81	#1848	—	73	1	no	—
82	#1848	—	73	1	no	—
83	#1566	—	73	2	no	—
84	#1566	—	73	2	no	—
85	C₂×#322	—	73	1	no	—
86	C₇₃	#8825	1	1	yes	characteristic, complemented, Sylow
87	C₂×#108	—	146	2	no	—
88	C₂×#108	—	146	2	no	—
89	#423	—	73	1	no	—
90	#423	—	73	2	no	—
91	#423	—	73	1	no	—
92	C₂×C₄×C₈	—	73	2	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	#334	—	146	4	no	—
95	#334	—	146	4	no	—
96	#322	—	73	1	no	—
97	#322	—	73	2	no	—
98	#322	#64408	1	1	yes	characteristic
99	C₂×M₁₆	—	146	2	no	—
100	C₂×M₁₆	—	146	1	no	—
101	C₂×M₁₆	—	146	1	no	—
102	C₂²×C₈	—	73	4	no	—
103	C₂²×C₈	—	73	2	no	—
104	C₂×C₄²	—	73	1	no	—
105	#108	—	292	4	no	—
106	#108	—	292	4	no	—
107	C₄⋊C₈	—	146	2	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₄⋊C₈	—	146	2	no	—
110	C₄⋊C₈	—	146	2	no	—
111	C₄×C₈	—	73	4	no	—
112	C₄×C₈	—	73	2	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	2	no	—
115	C₄×C₈	C₂×#64408	1	2	yes	—
116	C₄×C₈	C₂×#64408	1	1	yes	characteristic
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	2	no	—
119	M₁₆	—	292	4	no	—
120	M₁₆	—	292	2	no	—
121	M₁₆	—	292	2	no	—
122	C₂×C₈	—	292	4	no	—
123	C₂×C₈	—	292	2	no	—
124	C₂×C₈	—	292	1	no	—
125	C₂×C₈	—	292	1	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	73	4	no	—
128	C₂×C₈	—	73	4	no	—
129	C₂×C₈	—	73	4	no	—
130	C₂×C₈	C₄⋊#64408	1	4	yes	—
131	C₂×C₈	C₄⋊#64408	1	2	yes	—
132	C₄²	—	73	2	no	—
133	C₄²	—	73	1	no	—
134	C₄²	C₂²×#64408	1	1	yes	characteristic
135	C₂³	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	146	4	no	—
141	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	584	2	no	—
145	C₈	—	584	1	no	—
146	C₈	—	584	1	no	—
147	C₈	—	73	4	no	—
148	C₈	—	146	4	no	—
149	C₈	#14556659	1	4	yes	—
150	C₂²	—	73	2	no	—
151	C₂²	—	146	2	no	—
152	C₂²	#17989533	1	1	yes	characteristic
153	C₄	—	73	1	no	—
154	C₄	—	73	1	no	—
155	C₄	—	146	2	no	—
156	C₄	C₂×#14556659	1	2	yes	—
157	C₄	#17989602	1	4	yes	—
158	C₂	—	146	2	no	—
159	C₂	#22625866	1	1	yes	characteristic
160	C₂	#22625873	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625872

C₂

yes

maximal, characteristic

#22625870

C₂

yes

maximal, characteristic

#22625487

C₂

yes

maximal

#22625486

C₂

yes

maximal

#322×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

—

C₂×#14556661

—

#17989591

C₂²

yes

—

#17989591

C₂²

yes

characteristic

#17989591

C₂²

yes

characteristic

#17989449

—

#17989448

—

C₄×C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₄

yes

—

#17988308

C₄

yes

characteristic

#322×C₇₃

C₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

SD₁₆

yes

—

C₈×D₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

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

—

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

C₂×SD₁₆

yes

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

#10596831

—

#10596831

#106

yes

—

C₂²×C₇₃

#384

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#498

yes

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

#430

yes

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

—

C₇₃

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

C₂×#108

—

146

—

#423

—

#423

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

146

—

#334

—

146

—

—

—

yes

C₂×M₁₆

—

146

—

100

C₂×M₁₆

—

146

—

101

C₂×M₁₆

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

#108

—

292

—

106

#108

—

292

—

107

C₄⋊C₈

—

146

—

108

C₄⋊C₈

—

146

—

109

C₄⋊C₈

—

146

—

110

C₄⋊C₈

—

146

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

C₂×#64408

yes

—

116

C₄×C₈

C₂×#64408

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

M₁₆

—

292

—

120

M₁₆

—

292

—

121

M₁₆

—

292

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊#64408

yes

—

131

C₂×C₈

C₄⋊#64408

yes

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²×#64408

yes

characteristic

135

C₂³

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

142

C₂×C₄

C₂×C₄⋊#64408

yes

—

143

C₈

—

144

C₈

—

584

—

145

C₈

—

584

—

146

C₈

—

584

—

147

C₈

—

148

C₈

—

146

—

149

C₈

#14556659

yes

—

150

C₂²

—

151

C₂²

—

146

—

152

C₂²

#17989533

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

146

—

156

C₄

C₂×#14556659

yes

—

157

C₄

#17989602

yes

—

158

C₂

—

146

—

159

C₂

#22625866

yes

characteristic

160

C₂

#22625873

yes

—

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	#322×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	C₄⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	#322	#64408
hypocenter	C₇₃	#8825
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17989533

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

#322×C₇₃

C₄

socle

C₂²×C₇₃

#384

prime radical

C₂²×D₇₃

C₄⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
86. C₇₃	#8825	semidirect

H ≅ G/N

type

86. C₇₃

#8825

semidirect

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

subgroup	index
#22625872	2
#22625870	2
#22625487	2
#22625486	2
#322×D₇₃	2
#8825	73

subgroup

index

#322×D₇₃

Subgroups — #34005507

Conjugacy classes of subgroups

Complements

Maximal subgroups