Group #34009652 — subgroups

Subgroups — #34009652

Order 18688.

330 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 172 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	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
2	#22625135	C₂	1	1	yes	maximal, characteristic
3	#22624842	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596833	C₂	1	2	yes	maximal, complemented
5	#22611524	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₄	1	2	yes	—
7	#17989605	C₄	1	2	yes	—
8	#17989473	C₂²	1	1	yes	characteristic
9	#17989473	C₂²	1	1	yes	characteristic
10	C₄×#11924031	C₄	1	2	yes	—
11	#17989428	C₄	1	2	yes	—
12	C₈×#10596833	C₂²	1	2	yes	—
13	C₂×#14556530	C₄	1	2	yes	—
14	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	#17988363	C₄	1	1	yes	characteristic
16	C₁₆×Dic₇₃	C₂²	1	2	yes	—
17	#346×C₇₃	C₄	1	1	yes	characteristic
18	#14556664	C₈	1	8	yes	complemented
19	C₂×#11924031	C₂×C₄	1	2	yes	—
20	C₂×#11924031	C₂×C₄	1	2	yes	—
21	#14556658	C₈	1	4	yes	—
22	#14556648	C₈	1	4	yes	—
23	C₄×#10596833	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
27	#14556530	C₈	1	8	yes	complemented
28	C₂×#11924003	C₂×C₄	1	2	yes	—
29	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#10596833	C₂×C₈	1	8	yes	—
35	#11924031	C₂×C₈	1	4	yes	—
36	#11924031	C₂×C₈	1	4	yes	—
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	C₂×C₈	1	4	yes	—
42	C₈×D₇₃	C₂×C₈	1	4	yes	—
43	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
44	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	#11924003	C₂×C₈	1	4	yes	—
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	C₂×C₈	1	4	yes	—
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	C₄×C₈	1	2	yes	—
57	C₄×D₇₃	C₄×C₈	1	2	yes	—
58	#10596833	—	2	8	no	—
59	C₈×C₇₃	C₄×C₈	1	2	yes	—
60	C₈×C₇₃	C₂²×C₈	1	2	yes	—
61	#10596831	C₄×C₈	1	2	yes	—
62	#10596831	C₂²×C₈	1	2	yes	—
63	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×D₇₃	—	2	2	no	—
65	C₂×D₇₃	C₈²	1	2	yes	—
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
69	Dic₇₃	C₈²	1	2	yes	—
70	Dic₇₃	#504	1	4	yes	—
71	#11399	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	C₂×C₈²	1	1	yes	characteristic
73	C₂×C₇₃	#2108	1	2	yes	—
74	D₇₃	—	2	2	no	—
75	C₄×M₃₂	—	73	1	no	—
76	C₂×C₄×C₁₆	—	73	1	no	—
77	#1450	—	73	1	no	—
78	#1311	—	73	2	no	—
79	C₈×C₁₆	—	73	2	no	—
80	C₇₃	#11399	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	73	2	no	—
82	C₂²×C₁₆	—	73	2	no	—
83	#431	—	73	2	no	—
84	C₂×C₄×C₈	—	73	1	no	—
85	#346	—	73	1	no	—
86	#346	#64408	1	1	yes	characteristic
87	C₄×C₁₆	—	73	1	no	—
88	C₄×C₁₆	—	73	1	no	—
89	C₄×C₁₆	—	73	2	no	—
90	C₄×C₁₆	—	73	2	no	—
91	#322	—	73	2	no	—
92	C₈²	—	73	2	no	—
93	C₂²×C₈	—	73	1	no	—
94	C₂²×C₈	—	73	1	no	—
95	C₂×C₄²	—	73	1	no	—
96	M₃₂	—	73	8	no	—
97	C₂×C₁₆	—	73	2	no	—
98	C₂×C₁₆	—	73	2	no	—
99	C₂×C₁₆	—	73	8	no	—
100	C₂×C₁₆	—	73	2	no	—
101	C₂×C₁₆	C₂×#64408	1	2	yes	—
102	C₄⋊C₈	—	73	4	no	—
103	C₂²⋊C₈	—	73	4	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	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	73	8	no	—
116	C₂×C₈	C₄×#64408	1	1	yes	characteristic
117	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
118	C₄²	—	73	2	no	—
119	C₄²	—	73	1	no	—
120	C₄²	C₄×#64408	1	1	yes	characteristic
121	C₁₆	—	73	4	no	—
122	C₁₆	—	73	4	no	—
123	C₁₆	—	73	4	no	—
124	C₁₆	#11924036	1	4	yes	—
125	C₂³	—	73	1	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	146	4	no	—
131	C₂×C₄	C₈×#64408	1	2	yes	—
132	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
133	C₈	—	73	2	no	—
134	C₈	—	73	2	no	—
135	C₈	—	146	8	no	—
136	C₈	C₈×#64408	1	2	yes	—
137	C₈	C₂×#11924036	1	2	yes	—
138	C₂²	—	73	2	no	—
139	C₂²	—	146	2	no	—
140	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	73	4	no	—
143	C₄	—	73	2	no	—
144	C₄	C₂×C₈×#64408	1	1	yes	characteristic
145	C₄	C₄×#11924036	1	1	yes	characteristic
146	C₂	—	146	2	no	—
147	C₂	C₈×#11924036	1	1	yes	characteristic
148	C₂	#22626143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556664

C₂

yes

maximal, characteristic

#22625135

C₂

yes

maximal, characteristic

#22624842

C₂

yes

maximal, complemented

C₁₆×#10596833

C₂

yes

maximal, complemented

#22611524

C₂

yes

maximal, characteristic

C₂×#14556664

C₄

yes

—

#17989605

C₄

yes

—

#17989473

C₂²

yes

characteristic

#17989473

C₂²

yes

characteristic

C₄×#11924031

C₄

yes

—

#17989428

C₄

yes

—

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

#14556664

C₈

yes

complemented

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

#14556658

C₈

yes

—

#14556648

C₈

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

C₈

yes

complemented

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

C₂×#10596833

C₂×C₈

yes

—

#11924031

C₂×C₈

yes

—

#11924031

C₂×C₈

yes

—

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

C₂×C₈

yes

—

C₈×D₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

characteristic

#11924003

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

—

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

C₄×C₈

yes

—

C₄×D₇₃

C₄×C₈

yes

—

#10596833

—

C₈×C₇₃

C₄×C₈

yes

—

C₈×C₇₃

C₂²×C₈

yes

—

#10596831

C₄×C₈

yes

—

#10596831

C₂²×C₈

yes

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₈²

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

Dic₇₃

C₈²

yes

—

Dic₇₃

#504

yes

—

#11399

—

maximal, Sylow

C₂×C₇₃

C₂×C₈²

yes

characteristic

C₂×C₇₃

#2108

yes

—

D₇₃

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

#1311

—

C₈×C₁₆

—

C₇₃

#11399

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#322

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂×#64408

yes

—

102

C₄⋊C₈

—

103

C₂²⋊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

C₄×#64408

yes

characteristic

117

C₂×C₈

C₂²×#64408

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×#64408

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

#11924036

yes

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

C₈×#64408

yes

—

132

C₂×C₄

C₂×C₄×#64408

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

146

—

136

C₈

C₈×#64408

yes

—

137

C₈

C₂×#11924036

yes

—

138

C₂²

—

139

C₂²

—

146

—

140

C₂²

C₂×C₈×#64408

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×C₈×#64408

yes

characteristic

145

C₄

C₄×#11924036

yes

characteristic

146

C₂

—

146

—

147

C₂

C₈×#11924036

yes

characteristic

148

C₂

#22626143

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₈²
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₄×C₈
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#11399
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator

C₂×C₇₃

C₂×C₈²

Fitting

#346×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄×C₈

prime radical

C₂²×D₇₃

C₄×C₈

cosocle

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #22624842	C₂	semidirect
4. C₁₆×#10596833	C₂	semidirect
18. #14556664	C₈	semidirect
27. #14556530	C₈	semidirect
80. C₇₃	#11399	semidirect

H ≅ G/N

type

3. #22624842

C₂

semidirect

4. C₁₆×#10596833

C₂

semidirect

18. #14556664

C₈

semidirect

27. #14556530

C₈

semidirect

80. C₇₃

#11399

semidirect

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

subgroup	index
C₄×#14556664	2
#22625135	2
#22624842	2
C₁₆×#10596833	2
#22611524	2
#11399	73

subgroup

index

C₁₆×#10596833

Subgroups — #34009652

Conjugacy classes of subgroups

Complements

Maximal subgroups