Group #34009666 — subgroups

Subgroups — #34009666

Order 18688.

322 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 156 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	#22626137	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
3	#22624845	C₂	1	4	yes	maximal, complemented
4	M₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₄	1	2	yes	—
6	C₂×C₄×#10596833	C₄	1	2	yes	—
7	#17989476	C₄	1	2	yes	—
8	#17989473	C₂²	1	2	yes	—
9	C₈×#10596833	C₄	1	1	yes	characteristic
10	C₈×#10596833	C₄	1	1	yes	characteristic
11	C₈×#10596833	C₂²	1	2	yes	—
12	C₂×#14556531	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	#17988362	C₄	1	2	yes	—
15	C₁₆×Dic₇₃	C₂²	1	2	yes	—
16	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
17	C₂²×#10596833	C₈	1	4	yes	—
18	#14556665	—	2	4	no	—
19	C₂×#11924031	C₂×C₄	1	4	yes	—
20	C₄×#10596833	C₈	1	4	yes	—
21	C₄×#10596833	C₂×C₄	1	2	yes	—
22	C₄×#10596833	C₂×C₄	1	2	yes	—
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
25	#14556531	—	2	2	no	—
26	C₂×#11924003	C₂×C₄	1	2	yes	—
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₃	—	2	2	no	—
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	C₂×#10596833	—	2	8	no	—
35	C₂×#10596833	C₂×C₈	1	4	yes	—
36	C₂×#10596833	C₂×C₈	1	4	yes	—
37	#11924031	—	2	4	no	—
38	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₄²	1	1	yes	characteristic
40	C₂×#10596831	M₁₆	1	4	yes	—
41	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
43	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
44	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	#11924003	—	2	2	no	—
46	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
50	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	—	2	2	no	—
52	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#97	1	2	yes	—
55	C₂×C₄×C₇₃	#97	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	4	no	—
58	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
59	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
60	#10596833	—	2	4	no	—
61	#10596833	M₃₂	1	8	yes	complemented
62	C₈×C₇₃	#97	1	2	yes	—
63	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
65	#10596831	#97	1	2	yes	—
66	#10596831	C₂×M₁₆	1	2	yes	—
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	#322	1	2	yes	—
69	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₇₃	#322	1	2	yes	—
71	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
72	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
73	Dic₇₃	—	2	2	no	—
74	Dic₇₃	C₂×M₃₂	1	4	yes	—
75	#11402	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	2	2	no	—
77	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
78	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
79	C₄×M₃₂	—	73	1	no	—
80	C₄×M₃₂	—	73	1	no	—
81	C₂×C₈²	—	73	1	no	—
82	#1312	—	73	4	no	—
83	C₇₃	#11402	1	1	yes	characteristic, complemented, Sylow
84	C₂×M₃₂	—	73	2	no	—
85	C₂×M₃₂	—	73	1	no	—
86	C₂×M₃₂	#64408	1	1	yes	characteristic
87	C₂×C₄×C₈	—	73	2	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	#346	—	73	2	no	—
90	#346	—	73	2	no	—
91	C₄×C₁₆	—	73	2	no	—
92	C₄×C₁₆	—	73	2	no	—
93	C₈²	—	73	2	no	—
94	C₈²	—	73	1	no	—
95	C₈²	—	73	1	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	4	no	—
98	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
99	C₂×C₄²	—	73	1	no	—
100	M₃₂	—	2	2	no	—
101	M₃₂	—	146	4	no	—
102	M₃₂	—	146	2	no	—
103	C₂×C₁₆	—	73	4	no	—
104	C₂×C₁₆	—	73	2	no	—
105	C₂×C₁₆	C₂×#64408	1	2	yes	—
106	C₄×C₈	—	73	2	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	4	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₂²×C₄	—	73	2	no	—
114	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	4	no	—
120	C₂×C₈	—	146	8	no	—
121	C₂×C₈	C₄×#64408	1	1	yes	characteristic
122	C₂×C₈	#11924037	1	2	yes	—
123	C₂×C₈	#11924038	1	2	yes	—
124	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
125	C₄²	—	73	2	no	—
126	C₄²	—	73	1	no	—
127	C₄²	—	73	1	no	—
128	C₁₆	—	2	2	no	—
129	C₁₆	—	146	4	no	—
130	C₁₆	—	146	2	no	—
131	C₂³	C₈×#64408	1	1	yes	characteristic
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	73	2	no	—
134	C₂×C₄	—	146	4	no	—
135	C₂×C₄	C₈×#64408	1	1	yes	characteristic
136	C₂×C₄	#14556645	1	2	yes	—
137	C₂×C₄	#14556646	1	2	yes	—
138	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
139	C₈	—	73	8	no	—
140	C₈	—	73	2	no	—
141	C₈	—	73	2	no	—
142	C₈	—	146	4	no	—
143	C₈	#14556646	1	2	yes	—
144	C₈	C₂×#11924037	1	1	yes	characteristic
145	C₈	C₂×#11924038	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	#17989429	1	2	yes	—
148	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
149	C₄	—	73	4	no	—
150	C₄	—	146	2	no	—
151	C₄	#17989429	1	2	yes	—
152	C₄	C₂×#14556645	1	1	yes	characteristic
153	C₄	C₂×#14556646	1	1	yes	characteristic
154	C₂	—	2	2	no	—
155	C₂	C₂×#17989429	1	1	yes	characteristic
156	C₂	M₃₂×#64408	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626137

C₂

yes

maximal, characteristic

C₂×C₈×#10596833

C₂

yes

maximal, characteristic

#22624845

C₂

yes

maximal, complemented

M₃₂×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#14556665

C₄

yes

—

C₂×C₄×#10596833

C₄

yes

—

#17989476

C₄

yes

—

#17989473

C₂²

yes

—

C₈×#10596833

C₄

yes

characteristic

C₈×#10596833

C₄

yes

characteristic

C₈×#10596833

C₂²

yes

—

C₂×#14556531

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988362

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂×M₃₂×C₇₃

C₄

yes

characteristic

C₂²×#10596833

C₈

yes

—

#14556665

—

C₂×#11924031

C₂×C₄

yes

—

C₄×#10596833

C₈

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

#14556531

—

C₂×#11924003

C₂×C₄

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×#10596833

—

C₂×#10596833

C₂×C₈

yes

—

C₂×#10596833

C₂×C₈

yes

—

#11924031

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

M₁₆

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

#11924003

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂²×C₈

yes

—

#10596833

—

#10596833

M₃₂

yes

complemented

C₈×C₇₃

#97

yes

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

#97

yes

—

#10596831

C₂×M₁₆

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

#322

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

#322

yes

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

Dic₇₃

—

Dic₇₃

C₂×M₃₂

yes

—

#11402

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#322

yes

characteristic

C₂×C₇₃

C₄×M₃₂

yes

—

C₄×M₃₂

—

C₄×M₃₂

—

C₂×C₈²

—

#1312

—

C₇₃

#11402

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

100

M₃₂

—

101

M₃₂

—

146

—

102

M₃₂

—

146

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

C₂×#64408

yes

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₄×#64408

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

146

—

121

C₂×C₈

C₄×#64408

yes

characteristic

122

C₂×C₈

#11924037

yes

—

123

C₂×C₈

#11924038

yes

—

124

C₂×C₈

C₂²×#64408

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₁₆

—

129

C₁₆

—

146

—

130

C₁₆

—

146

—

131

C₂³

C₈×#64408

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

146

—

135

C₂×C₄

C₈×#64408

yes

characteristic

136

C₂×C₄

#14556645

yes

—

137

C₂×C₄

#14556646

yes

—

138

C₂×C₄

C₂×C₄×#64408

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

146

—

143

C₈

#14556646

yes

—

144

C₈

C₂×#11924037

yes

characteristic

145

C₈

C₂×#11924038

yes

characteristic

146

C₂²

—

147

C₂²

#17989429

yes

—

148

C₂²

C₂×C₈×#64408

yes

characteristic

149

C₄

—

150

C₄

—

146

—

151

C₄

#17989429

yes

—

152

C₄

C₂×#14556645

yes

characteristic

153

C₄

C₂×#14556646

yes

characteristic

154

C₂

—

155

C₂

C₂×#17989429

yes

characteristic

156

C₂

M₃₂×#64408

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#11402
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₇₃

C₄

prime radical

C₂³×C₇₃

C₄×C₈

cosocle

C₈×Dic₇₃

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #22624845	C₂	semidirect
61. #10596833	M₃₂	semidirect
83. C₇₃	#11402	semidirect

H ≅ G/N

type

3. #22624845

C₂

semidirect

61. #10596833

M₃₂

semidirect

83. C₇₃

#11402

semidirect

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

subgroup	index
#22626137	2
C₂×C₈×#10596833	2
#22624845	2
M₃₂×Dic₇₃	2
#11402	73

subgroup

index

M₃₂×Dic₇₃

Subgroups — #34009666

Conjugacy classes of subgroups

Complements

Maximal subgroups