Group #34032842 — subgroups

Subgroups — #34032842

Order 18688.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22627001	C₂	1	2	yes	maximal
2	#22626101	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924040	—	2	2	no	—
4	C₂×C₄⋊#10596833	C₄	1	1	yes	characteristic
5	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
6	#17989453	—	2	2	no	—
7	C₂×#14556556	C₄	1	1	yes	characteristic
8	C₂×#11924040	—	2	4	no	—
9	C₂×#11924040	—	4	2	no	—
10	C₂×#11924040	—	4	2	no	—
11	C₂²×#10596833	—	2	1	no	—
12	C₂²×#10596833	D₄	1	2	yes	—
13	C₄⋊#10596833	—	4	1	no	—
14	C₄×#10596833	—	4	1	no	—
15	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
16	#14556556	—	4	1	no	—
17	#14556556	—	4	1	no	—
18	C₂×#11924007	—	2	1	no	—
19	C₂×C₄×Dic₇₃	—	2	1	no	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
22	#11924040	—	4	4	no	—
23	C₂×#10596833	—	2	2	no	—
24	C₂×#10596833	—	2	2	no	—
25	C₂×#10596833	—	4	1	no	—
26	C₂×#10596833	—	4	2	no	—
27	C₂×#10596833	SD₁₆	1	2	yes	—
28	C₂×#10596833	Q₁₆	1	2	yes	—
29	C₂×Dic₁₄₆	—	2	1	no	—
30	C₂×Dic₁₄₆	—	4	1	no	—
31	C₂×Dic₁₄₆	—	4	1	no	—
32	C₂×Dic₁₄₆	C₁₆	1	2	yes	complemented
33	C₂²×Dic₇₃	—	2	1	no	—
34	C₂²×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
36	#11924007	—	4	2	no	—
37	C₄×Dic₇₃	—	2	1	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	4	1	no	—
40	C₄×Dic₇₃	—	4	1	no	—
41	C₂²×C₄×C₇₃	—	2	1	no	—
42	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₄⋊C₄×C₇₃	—	4	1	no	—
44	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	4	1	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	#99	1	1	yes	characteristic
57	C₂×Dic₇₃	#100	1	1	yes	characteristic
58	C₂×Dic₇₃	#103	1	2	yes	—
59	C₂×Dic₇₃	#104	1	2	yes	—
60	Dic₁₄₆	—	4	2	no	—
61	Dic₁₄₆	—	4	2	no	—
62	#10596833	—	2	2	no	—
63	#10596833	—	2	2	no	—
64	#10596833	—	8	1	no	—
65	C₂²×C₇₃	#323	1	1	yes	characteristic
66	C₂²×C₇₃	Q₈⋊C₈	1	2	yes	—
67	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
68	C₂²×C₇₃	#349	1	1	yes	characteristic
69	C₂²×C₇₃	#350	1	2	yes	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	8	1	no	—
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	—	4	1	no	—
75	Dic₇₃	#328	1	2	yes	—
76	Dic₇₃	#329	1	1	yes	characteristic
77	Dic₇₃	#333	1	1	yes	characteristic
78	#7128	—	73	1	no	maximal, Sylow
79	C₂×C₇₃	#1272	1	1	yes	characteristic
80	C₂×C₇₃	#1314	1	1	yes	characteristic
81	C₂×C₇₃	#1315	1	1	yes	characteristic
82	C₂×C₇₃	Q₈⋊C₁₆	1	2	yes	—
83	C₂×C₇₃	#1338	1	2	yes	—
84	#1984	—	73	1	no	—
85	#1374	—	73	2	no	—
86	C₇₃	#7128	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₄×Q₈	—	73	1	no	—
88	C₂²×C₁₆	—	146	2	no	—
89	C₂×C₄⋊C₈	—	73	1	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	#336	—	146	2	no	—
92	C₂²×Q₈	—	73	1	no	—
93	C₂²×C₈	—	73	2	no	—
94	C₂²×C₈	—	146	1	no	—
95	C₄×Q₈	—	292	1	no	—
96	C₄×Q₈	—	292	1	no	—
97	C₂×C₄⋊C₄	—	146	1	no	—
98	C₂×C₄⋊C₄	#10596842	1	1	yes	characteristic
99	C₂×C₄²	—	73	1	no	—
100	C₂×C₄²	—	146	1	no	—
101	C₂×C₁₆	—	292	2	no	—
102	C₂×C₁₆	—	292	2	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₄⋊C₈	—	292	1	no	—
105	C₄×C₈	—	292	1	no	—
106	C₂×Q₈	—	292	1	no	—
107	C₂×Q₈	—	292	1	no	—
108	C₂×Q₈	—	73	2	no	—
109	C₂×Q₈	—	146	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	146	1	no	—
114	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
115	C₂×C₈	—	292	1	no	—
116	C₂×C₈	—	292	2	no	—
117	C₂×C₈	—	73	2	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	—	146	2	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	292	2	no	—
123	C₄²	—	292	1	no	—
124	C₄²	—	292	1	no	—
125	C₄²	—	146	1	no	—
126	C₄²	—	146	1	no	—
127	C₁₆	—	292	4	no	—
128	C₂³	C₂²⋊#10596842	1	1	yes	characteristic
129	Q₈	—	292	2	no	—
130	Q₈	—	292	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	292	1	no	—
136	C₂×C₄	—	292	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	#14556686	1	1	yes	characteristic
145	C₂×C₄	#14556688	1	1	yes	characteristic
146	C₈	—	584	1	no	—
147	C₈	—	146	2	no	—
148	C₈	—	146	2	no	—
149	C₂²	#17989672	1	1	yes	characteristic
150	C₂²	Q₈⋊#10596842	1	2	yes	—
151	C₂²	#17989684	1	1	yes	characteristic
152	C₂²	#17989686	1	1	yes	characteristic
153	C₂²	#17989687	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	292	2	no	—
158	C₄	—	292	1	no	—
159	C₄	—	73	1	no	—
160	C₄	—	73	2	no	—
161	C₄	—	73	1	no	—
162	C₂	#22626932	1	1	yes	characteristic
163	C₂	#22626953	1	1	yes	characteristic
164	C₂	#22626954	1	1	yes	characteristic
165	C₂	#22626986	1	2	yes	—
166	C₂	#22626987	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22627001

C₂

yes

maximal

#22626101

C₂

yes

maximal, characteristic

C₂²×#11924040

—

C₂×C₄⋊#10596833

C₄

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

characteristic

#17989453

—

C₂×#14556556

C₄

yes

—

—

—

—

D₄

yes

—

C₄⋊#10596833

—

C₄×#10596833

—

C₂²×Dic₁₄₆

C₈

yes

—

—

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₈

yes

—

—

—

—

—

SD₁₆

yes

—

C₂×#10596833

Q₁₆

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₁₆

yes

complemented

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₇₃

M₁₆

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

C₂×Dic₇₃

#104

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

#10596833

—

C₂²×C₇₃

#323

yes

characteristic

C₂²×C₇₃

Q₈⋊C₈

yes

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#328

yes

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

Q₈⋊C₁₆

yes

—

C₂×C₇₃

#1338

yes

—

#1984

—

#1374

—

C₇₃

#7128

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₂²×C₁₆

—

146

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#336

—

146

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₄×Q₈

—

292

—

C₄×Q₈

—

292

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

#10596842

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

—

146

—

101

C₂×C₁₆

—

292

—

102

C₂×C₁₆

—

292

—

103

C₂×C₁₆

—

146

—

104

C₄⋊C₈

—

292

—

105

C₄×C₈

—

292

—

106

C₂×Q₈

—

292

—

107

C₂×Q₈

—

292

—

108

C₂×Q₈

—

109

C₂×Q₈

—

146

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

146

—

114

C₂²×C₄

C₂×#10596842

yes

characteristic

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

146

—

120

C₂×C₈

—

146

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

292

—

123

C₄²

—

292

—

124

C₄²

—

292

—

125

C₄²

—

146

—

126

C₄²

—

146

—

127

C₁₆

—

292

—

128

C₂³

C₂²⋊#10596842

yes

characteristic

129

Q₈

—

292

—

130

Q₈

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

292

—

136

C₂×C₄

—

292

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

#14556686

yes

characteristic

145

C₂×C₄

#14556688

yes

characteristic

146

C₈

—

584

—

147

C₈

—

146

—

148

C₈

—

146

—

149

C₂²

#17989672

yes

characteristic

150

C₂²

Q₈⋊#10596842

yes

—

151

C₂²

#17989684

yes

characteristic

152

C₂²

#17989686

yes

characteristic

153

C₂²

#17989687

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

292

—

158

C₄

—

292

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₂

#22626932

yes

characteristic

163

C₂

#22626953

yes

characteristic

164

C₂

#22626954

yes

characteristic

165

C₂

#22626986

yes

—

166

C₂

#22626987

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊#10596842
commutator	C₂×C₄×C₇₃	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂×C₄⋊C₄×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄×#10596833	C₂²
hypercenter	C₂×C₄⋊C₄	#10596842
hypocenter	C₇₃	#7128
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#10596842

commutator

C₂×C₄×C₇₃

C₂×C₁₆

Frattini

C₂²×C₄

C₂×#10596842

Fitting

C₂×C₄⋊C₄×C₇₃

C₈

socle, prime radical

C₂³×C₇₃

C₂²⋊C₈

C₂²

C₂×C₄⋊C₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
32. C₂×Dic₁₄₆	C₁₆	semidirect
86. C₇₃	#7128	semidirect

H ≅ G/N

type

32. C₂×Dic₁₄₆

C₁₆

semidirect

86. C₇₃

#7128

semidirect

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

subgroup	index
#22627001	2
#22626101	2
#7128	73

subgroup

index

#22627001

#22626101

#7128

Subgroups — #34032842

Conjugacy classes of subgroups

Complements

Maximal subgroups