Group #34009667 — subgroups

Subgroups — #34009667

Order 18688.

322 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 144 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	#22626138	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924036	C₂	1	1	yes	maximal, characteristic
3	#22624845	C₂	1	4	yes	maximal, complemented
4	#22611524	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556666	C₄	1	2	yes	—
6	C₄×#11924036	C₄	1	2	yes	—
7	#17989475	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×#14556530	C₄	1	2	yes	—
13	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17988363	C₄	1	1	yes	characteristic
15	C₁₆×Dic₇₃	C₂²	1	2	yes	—
16	#346×C₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924036	C₈	1	4	yes	—
18	#14556666	—	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	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
26	#14556530	—	2	4	no	—
27	C₂×#11924003	C₂×C₄	1	2	yes	—
28	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924036	—	2	8	no	—
34	C₂×#10596833	C₂×C₈	1	4	yes	—
35	C₂×#10596833	C₂×C₈	1	4	yes	—
36	#11924031	—	2	4	no	—
37	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
39	C₂×#10596831	C₄²	1	1	yes	characteristic
40	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
41	C₈×D₇₃	M₁₆	1	4	yes	—
42	C₈×D₇₃	M₁₆	1	4	yes	—
43	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
44	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	#11924003	—	2	2	no	—
46	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₄²	1	1	yes	characteristic
49	C₁₆×C₇₃	—	2	2	no	—
50	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
53	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
55	C₄×D₇₃	—	2	4	no	—
56	C₄×D₇₃	#97	1	2	yes	—
57	C₄×D₇₃	#97	1	2	yes	—
58	#10596833	—	2	4	no	—
59	#10596833	M₃₂	1	8	yes	complemented
60	C₈×C₇₃	#97	1	2	yes	—
61	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
62	#10596831	#97	1	2	yes	—
63	#10596831	C₂×M₁₆	1	2	yes	—
64	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂×D₇₃	—	2	2	no	—
66	C₂×D₇₃	#322	1	2	yes	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
69	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
70	Dic₇₃	#322	1	2	yes	—
71	Dic₇₃	C₂×M₃₂	1	4	yes	—
72	#11402	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	C₂×#322	1	1	yes	characteristic
74	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
75	D₇₃	—	2	2	no	—
76	C₄×M₃₂	—	73	1	no	—
77	C₄×M₃₂	—	73	1	no	—
78	C₂×C₈²	—	73	1	no	—
79	#1312	—	73	4	no	—
80	C₇₃	#11402	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	73	2	no	—
82	C₂×M₃₂	—	73	2	no	—
83	C₂×C₄×C₈	—	73	2	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	#346	—	73	2	no	—
86	#346	—	73	1	no	—
87	#346	#64408	1	1	yes	characteristic
88	C₄×C₁₆	—	73	2	no	—
89	C₄×C₁₆	—	73	2	no	—
90	C₈²	—	73	2	no	—
91	C₈²	—	73	1	no	—
92	C₈²	—	73	1	no	—
93	C₂²×C₈	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂²×C₈	—	73	4	no	—
96	C₂×C₄²	—	73	1	no	—
97	M₃₂	—	146	4	no	—
98	M₃₂	—	146	4	no	—
99	C₂×C₁₆	—	73	4	no	—
100	C₂×C₁₆	—	73	2	no	—
101	C₂×C₁₆	C₂×#64408	1	2	yes	—
102	C₄×C₈	—	73	2	no	—
103	C₄×C₈	—	73	2	no	—
104	C₄×C₈	—	73	4	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₄×C₈	C₂×#64408	1	1	yes	characteristic
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂×C₈	—	73	4	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	—	146	8	no	—
118	C₂×C₈	C₄×#64408	1	1	yes	characteristic
119	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
120	C₄²	—	73	2	no	—
121	C₄²	—	73	1	no	—
122	C₄²	C₄×#64408	1	1	yes	characteristic
123	C₁₆	—	2	2	no	—
124	C₁₆	—	146	4	no	—
125	C₁₆	—	146	2	no	—
126	C₂³	—	73	1	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	73	2	no	—
131	C₂×C₄	—	146	4	no	—
132	C₂×C₄	C₈×#64408	1	2	yes	—
133	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
134	C₈	—	73	8	no	—
135	C₈	—	73	2	no	—
136	C₈	—	73	2	no	—
137	C₈	—	146	4	no	—
138	C₈	#14556645	1	2	yes	—
139	C₈	D₇₃⋊M₁₆	1	2	yes	—
140	C₂²	—	73	2	no	—
141	C₂²	—	146	2	no	—
142	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	73	4	no	—
145	C₄	—	73	2	no	—
146	C₄	C₂×#14556645	1	1	yes	characteristic
147	C₄	#17989546	1	1	yes	characteristic
148	C₂	—	146	2	no	—
149	C₂	#22625127	1	1	yes	characteristic
150	C₂	M₃₂×#64408	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626138

C₂

yes

maximal, characteristic

C₈×#11924036

C₂

yes

maximal, characteristic

#22624845

C₂

yes

maximal, complemented

#22611524

C₂

yes

maximal, characteristic

C₂×#14556666

C₄

yes

—

C₄×#11924036

C₄

yes

—

#17989475

C₄

yes

—

#17989473

C₂²

yes

—

C₈×#10596833

C₄

yes

characteristic

C₈×#10596833

C₄

yes

characteristic

C₈×#10596833

C₂²

yes

—

C₂×#14556530

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

—

#346×C₇₃

C₄

yes

characteristic

C₂×#11924036

C₈

yes

—

#14556666

—

C₂×#11924031

C₂×C₄

yes

—

C₄×#10596833

C₈

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

#14556530

—

C₂×#11924003

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924036

—

C₂×#10596833

C₂×C₈

yes

—

C₂×#10596833

C₂×C₈

yes

—

#11924031

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

characteristic

C₈×D₇₃

M₁₆

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

characteristic

C₂×Dic₇₃

C₂²×C₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

#97

yes

—

C₄×D₇₃

#97

yes

—

#10596833

—

#10596833

M₃₂

yes

complemented

C₈×C₇₃

#97

yes

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

#97

yes

—

#10596831

C₂×M₁₆

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#322

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×#97

yes

characteristic

C₄×C₇₃

C₂×#97

yes

characteristic

Dic₇₃

#322

yes

—

Dic₇₃

C₂×M₃₂

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×M₃₂

yes

—

D₇₃

—

C₄×M₃₂

—

C₄×M₃₂

—

C₂×C₈²

—

#1312

—

C₇₃

#11402

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂×#64408

yes

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#64408

yes

characteristic

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

146

—

118

C₂×C₈

C₄×#64408

yes

characteristic

119

C₂×C₈

C₂²×#64408

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

C₄×#64408

yes

characteristic

123

C₁₆

—

124

C₁₆

—

146

—

125

C₁₆

—

146

—

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

146

—

132

C₂×C₄

C₈×#64408

yes

—

133

C₂×C₄

C₂×C₄×#64408

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

146

—

138

C₈

#14556645

yes

—

139

C₈

D₇₃⋊M₁₆

yes

—

140

C₂²

—

141

C₂²

—

146

—

142

C₂²

C₂×C₈×#64408

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₂×#14556645

yes

characteristic

147

C₄

#17989546

yes

characteristic

148

C₂

—

146

—

149

C₂

#22625127

yes

characteristic

150

C₂

M₃₂×#64408

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#346×C₇₃	C₄
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#11402
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

#346×C₇₃

C₄

prime radical

C₂²×D₇₃

C₄×C₈

cosocle

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #22624845	C₂	semidirect
59. #10596833	M₃₂	semidirect
80. C₇₃	#11402	semidirect

H ≅ G/N

type

3. #22624845

C₂

semidirect

59. #10596833

M₃₂

semidirect

80. C₇₃

#11402

semidirect

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

subgroup	index
#22626138	2
C₈×#11924036	2
#22624845	2
#22611524	2
#11402	73

subgroup

index

Subgroups — #34009667

Conjugacy classes of subgroups

Complements

Maximal subgroups