Group #34032870 — subgroups

Subgroups — #34032870

Order 18688.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22627130	C₂	1	2	yes	maximal
2	#22625644	C₂	1	1	yes	maximal, characteristic
3	#17989690	—	2	2	no	—
4	#17989684	—	2	2	no	—
5	C₄×#11924040	C₄	1	4	yes	complemented
6	C₄×#11924036	C₂²	1	1	yes	characteristic
7	C₄²×#64408	C₄	1	1	yes	characteristic
8	#17989452	—	2	2	no	—
9	C₄×C₈×D₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924040	—	2	4	no	—
11	C₂×#11924036	D₄	1	2	yes	—
12	C₂×C₄×#64408	—	2	1	no	—
13	C₂×C₄×#64408	—	2	1	no	—
14	C₂×C₄×#64408	C₈	1	2	yes	—
15	C₄×#10596833	C₂×C₄	1	2	yes	—
16	C₄×#10596833	D₄	1	1	yes	characteristic
17	C₄×#10596833	Q₈	1	1	yes	characteristic
18	C₂×C₈×D₇₃	—	2	1	no	—
19	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	C₈	1	1	yes	characteristic
21	C₈×Dic₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
23	#11924040	—	4	4	no	—
24	C₂²×#64408	—	2	2	no	—
25	#11924036	—	4	2	no	—
26	C₂×#10596833	C₂²⋊C₄	1	2	yes	—
27	C₂×#10596833	C₄⋊C₄	1	2	yes	—
28	C₄×#64408	—	2	2	no	—
29	C₄×#64408	—	4	2	no	—
30	C₄×#64408	—	4	2	no	—
31	C₄×#64408	C₁₆	1	4	yes	complemented
32	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
35	C₂×#10596831	—	2	1	no	—
36	C₈×D₇₃	—	4	2	no	—
37	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
38	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
43	C₂×#64408	—	2	4	no	—
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	4	2	no	—
46	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×Dic₇₃	#95	1	1	yes	characteristic
50	C₂×Dic₇₃	#97	1	1	yes	characteristic
51	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
55	C₄×D₇₃	M₃₂	1	2	yes	—
56	#10596833	—	2	2	no	—
57	#10596833	—	2	2	no	—
58	C₈×C₇₃	—	4	1	no	—
59	#10596831	—	4	1	no	—
60	C₂²×C₇₃	#336	1	1	yes	characteristic
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	C₂²⋊C₁₆	1	2	yes	—
64	#64408	—	4	4	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
68	C₄×C₇₃	#346	1	1	yes	characteristic
69	Dic₇₃	H(3,ℤ/4)	1	2	yes	—
70	Dic₇₃	#347	1	2	yes	—
71	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
72	#7142	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1274	1	1	yes	characteristic
74	C₂×C₇₃	#1374	1	1	yes	characteristic
75	C₂×C₇₃	#1375	1	1	yes	characteristic
76	D₇₃	—	2	2	no	—
77	#2162	—	73	2	no	—
78	#1744	—	73	1	no	—
79	C₇₃	#7142	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×C₈	—	73	1	no	—
81	C₂×C₄×C₈	—	73	1	no	—
82	C₄³	—	73	1	no	—
83	C₄⋊C₁₆	—	146	2	no	—
84	C₂²⋊C₁₆	—	146	2	no	—
85	C₄×C₁₆	—	73	4	no	—
86	#336	—	146	2	no	—
87	C₂²×C₈	—	73	2	no	—
88	C₂²×C₈	—	146	1	no	—
89	C₂×C₄²	—	73	1	no	—
90	C₂×C₄²	—	73	2	no	—
91	C₂×C₄²	—	146	1	no	—
92	C₂×C₄²	—	146	1	no	—
93	C₂×C₁₆	—	146	4	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	—	146	1	no	—
99	C₄×C₈	#10596842	1	1	yes	characteristic
100	C₂²×C₄	—	73	1	no	—
101	C₂²×C₄	—	73	1	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	146	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	292	2	no	—
106	C₂×C₈	—	292	2	no	—
107	C₂×C₈	—	73	2	no	—
108	C₂×C₈	—	73	2	no	—
109	C₂×C₈	—	146	1	no	—
110	C₄²	—	292	2	no	—
111	C₄²	—	292	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₄²	—	73	4	no	—
116	C₄²	—	146	2	no	—
117	C₄²	C₂×#10596842	1	1	yes	characteristic
118	C₁₆	—	292	4	no	—
119	C₂³	—	73	1	no	—
120	C₂×C₄	—	292	2	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	2	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	4	no	—
130	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
132	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
133	C₈	—	4	1	no	—
134	C₈	—	292	1	no	—
135	C₈	—	146	2	no	—
136	C₈	—	146	2	no	—
137	C₂²	—	73	2	no	—
138	C₂²	—	146	1	no	—
139	C₂²	—	146	1	no	—
140	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	292	4	no	—
144	C₄	—	73	2	no	—
145	C₄	—	73	2	no	—
146	C₄	—	73	2	no	—
147	C₄	#64408⋊C₁₆	1	1	yes	characteristic
148	C₄	#17989682	1	1	yes	characteristic
149	C₂	—	146	2	no	—
150	C₂	#22626934	1	1	yes	characteristic
151	C₂	#22627002	1	1	yes	characteristic
152	C₂	#22627004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627130

C₂

yes

maximal

#22625644

C₂

yes

maximal, characteristic

#17989690

—

#17989684

—

C₄×#11924040

C₄

yes

complemented

C₄×#11924036

C₂²

yes

characteristic

C₄²×#64408

C₄

yes

characteristic

#17989452

—

C₄×C₈×D₇₃

C₄

yes

characteristic

C₂×#11924040

—

C₂×#11924036

D₄

yes

—

C₂×C₄×#64408

—

C₂×C₄×#64408

—

C₂×C₄×#64408

C₈

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₂×C₈×D₇₃

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄×#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₄×C₈×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

—

C₂×#10596833

C₂²⋊C₄

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

—

—

—

C₁₆

yes

complemented

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²⋊C₈

yes

—

—

—

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊C₁₆

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₁₆

yes

C₄×C₇₃

yes

Dic₇₃

yes

—

Dic₇₃

#347

yes

—

Dic₇₃

C₄⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2162

—

#1744

—

C₇₃

#7142

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

#336

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

C₂×C₁₆

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

146

—

C₄×C₈

#10596842

yes

characteristic

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

146

—

104

C₂×C₈

—

105

C₂×C₈

—

292

—

106

C₂×C₈

—

292

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₄²

—

292

—

111

C₄²

—

292

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

146

—

117

C₄²

C₂×#10596842

yes

characteristic

118

C₁₆

—

292

—

119

C₂³

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

C₄×#10596842

yes

characteristic

131

C₂×C₄

C₂²⋊#10596842

yes

characteristic

132

C₂×C₄

C₄⋊#10596842

yes

characteristic

133

C₈

—

134

C₈

—

292

—

135

C₈

—

146

—

136

C₈

—

146

—

137

C₂²

—

138

C₂²

—

146

—

139

C₂²

—

146

—

140

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

292

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#64408⋊C₁₆

yes

characteristic

148

C₄

#17989682

yes

characteristic

149

C₂

—

146

—

150

C₂

#22626934

yes

characteristic

151

C₂

#22627002

yes

characteristic

152

C₂

#22627004

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₂²⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7142
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₄

C₄×#10596842

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₄²

C₂×#10596842

Fitting

C₄×C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₂²⋊C₈

C₂²

C₄×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
5. C₄×#11924040	C₄	semidirect
31. C₄×#64408	C₁₆	semidirect
79. C₇₃	#7142	semidirect

H ≅ G/N

type

5. C₄×#11924040

C₄

semidirect

31. C₄×#64408

C₁₆

semidirect

79. C₇₃

#7142

semidirect

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

subgroup	index
#22627130	2
#22625644	2
#7142	73

subgroup

index

#22627130

#22625644

#7142

Subgroups — #34032870

Conjugacy classes of subgroups

Complements

Maximal subgroups