Group #34032900 — subgroups

Subgroups — #34032900

Order 18688.

210 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 73 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	#22627125	C₂	1	2	yes	maximal
2	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic
3	#17989706	—	2	2	no	—
4	#17989692	C₄	1	4	yes	—
5	#17989691	—	2	2	no	—
6	C₂×#14556664	C₄	1	1	yes	characteristic
7	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
8	C₁₆×#64408	—	2	2	no	—
9	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
10	#14556688	—	2	4	no	—
11	#14556687	—	4	2	no	—
12	#14556686	—	4	2	no	—
13	C₂×#11924036	D₄	1	1	yes	characteristic
14	C₂×C₄×#64408	Q₈	1	1	yes	characteristic
15	#14556664	—	2	1	no	—
16	#14556664	C₈	1	2	yes	—
17	C₂×#11924031	—	2	1	no	—
18	C₈×#64408	C₂×C₄	1	2	yes	—
19	C₈×#64408	D₄	1	2	yes	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	C₈	1	1	yes	characteristic
22	C₁₆×D₇₃	—	2	1	no	—
23	C₁₆×D₇₃	—	2	1	no	—
24	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
25	#11924040	—	4	4	no	—
26	C₂²×#64408	—	2	1	no	—
27	#11924036	C₂²⋊C₄	1	2	yes	—
28	#11924036	D₈	1	1	yes	characteristic
29	#11924036	Q₁₆	1	1	yes	characteristic
30	C₂×#10596833	SD₁₆	1	2	yes	—
31	C₄×#64408	—	2	1	no	—
32	C₄×#64408	C₄⋊C₄	1	2	yes	—
33	#11924031	—	2	2	no	—
34	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#10596831	M₁₆	1	1	yes	characteristic
36	C₈×D₇₃	C₄²	1	1	yes	characteristic
37	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₈×D₇₃	M₁₆	1	1	yes	characteristic
40	#11924003	—	2	1	no	—
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₁₆×C₇₃	—	2	1	no	—
43	C₂²×D₇₃	#108	1	1	yes	characteristic
44	C₂×#64408	—	2	4	no	—
45	C₂×#64408	—	2	2	no	—
46	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×Dic₇₃	#106	1	1	yes	characteristic
48	C₄×D₇₃	#95	1	1	yes	characteristic
49	C₄×D₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₄×D₇₃	#107	1	1	yes	characteristic
51	C₄×D₇₃	#108	1	1	yes	characteristic
52	#10596833	#102	1	2	yes	—
53	#10596833	#103	1	2	yes	—
54	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
55	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
56	#10596831	#97	1	1	yes	characteristic
57	#10596831	C₂²⋊C₈	1	1	yes	characteristic
58	C₂²×C₇₃	#334	1	1	yes	characteristic
59	C₂×D₇₃	—	2	2	no	—
60	C₂×D₇₃	#335	1	1	yes	characteristic
61	C₂×D₇₃	#341	1	1	yes	characteristic
62	#64408	—	2	4	no	—
63	C₄×C₇₃	#335	1	1	yes	characteristic
64	C₄×C₇₃	#336	1	1	yes	characteristic
65	Dic₇₃	#334	1	1	yes	characteristic
66	Dic₇₃	#340	1	1	yes	characteristic
67	#7160	—	73	1	no	maximal, Sylow
68	C₂×C₇₃	—	2	1	no	—
69	C₂×C₇₃	#1277	1	1	yes	characteristic
70	D₇₃	—	2	1	no	—
71	D₇₃	#1373	1	2	yes	—
72	C₂×#364	—	73	2	no	—
73	C₂×C₄×C₁₆	—	73	1	no	—
74	C₇₃	#7160	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₃₂	—	146	2	no	—
76	C₂²×C₁₆	—	73	1	no	—
77	C₂²×C₁₆	—	73	1	no	—
78	C₂×C₄×C₈	—	73	1	no	—
79	#364	—	73	4	no	—
80	#364	—	146	2	no	—
81	C₄×C₁₆	—	146	2	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂×C₄²	—	73	1	no	—
85	M₃₂	—	292	2	no	—
86	M₃₂	—	146	4	no	—
87	C₂×C₁₆	—	292	2	no	—
88	C₂×C₁₆	—	73	2	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	146	1	no	—
91	C₂×C₁₆	—	146	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	C₂×C₁₆	—	146	1	no	—
94	C₂×C₁₆	#10596842	1	1	yes	characteristic
95	C₄×C₈	—	73	2	no	—
96	C₄×C₈	—	73	2	no	—
97	C₂²×C₄	—	73	1	no	—
98	C₂²×C₄	—	146	1	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	73	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
109	C₄²	—	73	2	no	—
110	C₄²	—	146	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	—	292	4	no	—
113	C₁₆	—	146	2	no	—
114	C₁₆	—	146	1	no	—
115	C₂³	—	73	1	no	—
116	C₂×C₄	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	—	146	4	no	—
123	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
124	C₈	—	73	2	no	—
125	C₈	—	73	2	no	—
126	C₈	—	73	1	no	—
127	C₈	—	73	1	no	—
128	C₈	C₄×#10596842	1	1	yes	characteristic
129	C₈	C₂²⋊#10596842	1	1	yes	characteristic
130	C₂²	—	73	1	no	—
131	C₂²	—	73	1	no	—
132	C₂²	—	146	2	no	—
133	C₂²	#17989676	1	1	yes	characteristic
134	C₄	—	73	1	no	—
135	C₄	—	73	1	no	—
136	C₄	—	146	4	no	—
137	C₄	#17989677	1	1	yes	characteristic
138	C₄	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	73	2	no	—
141	C₂	—	146	1	no	—
142	C₂	#22626937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627125

C₂

yes

maximal

C₂×C₁₆×#64408

C₂

yes

maximal, characteristic

#17989706

—

#17989692

C₄

yes

—

#17989691

—

C₂×#14556664

C₄

yes

characteristic

C₂×C₈×#64408

C₂²

yes

characteristic

C₁₆×#64408

—

C₂×C₁₆×D₇₃

C₄

yes

—

—

—

D₄

yes

characteristic

C₂×C₄×#64408

Q₈

yes

characteristic

#14556664

—

#14556664

C₈

yes

—

C₂×#11924031

—

C₈×#64408

C₂×C₄

yes

—

C₈×#64408

D₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

C₈

yes

characteristic

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₂×C₁₆×C₇₃

C₈

yes

characteristic

#11924040

—

C₂²×#64408

—

#11924036

C₂²⋊C₄

yes

—

#11924036

D₈

yes

characteristic

#11924036

Q₁₆

yes

characteristic

C₂×#10596833

SD₁₆

yes

—

C₄×#64408

—

C₄×#64408

C₄⋊C₄

yes

—

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×#10596831

M₁₆

yes

characteristic

C₈×D₇₃

C₄²

yes

characteristic

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

C₂×C₈

yes

characteristic

C₈×D₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

—

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

C₄⋊C₈

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

yes

—

#10596833

#103

yes

—

C₈×C₇₃

C₄×C₈

yes

characteristic

C₈×C₇₃

C₂²⋊C₈

yes

yes

C₂²⋊C₈

yes

characteristic

C₂²×C₇₃

#334

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1277

yes

characteristic

D₇₃

—

D₇₃

#1373

yes

—

C₂×#364

—

C₂×C₄×C₁₆

—

C₇₃

#7160

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

#364

—

#364

—

146

—

C₄×C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

#10596842

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×#10596842

yes

characteristic

109

C₄²

—

110

C₄²

—

146

—

111

C₁₆

—

112

C₁₆

—

292

—

113

C₁₆

—

146

—

114

C₁₆

—

146

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

C₄⋊#10596842

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

C₄×#10596842

yes

characteristic

129

C₈

C₂²⋊#10596842

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

146

—

133

C₂²

#17989676

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

146

—

137

C₄

#17989677

yes

characteristic

138

C₄

(C₂×#10596842)⋊C₄

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

146

—

142

C₂

#22626937

yes

characteristic

	subgroup	G/H
center	C₄	#17989677
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₈	C₂×#10596842
Fitting	C₂×C₁₆×C₇₃	C₈
socle	C₂×C₇₃	#1277
prime radical	C₂²×D₇₃	#108
cosocle	C₂×C₈×#64408	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7160
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#17989677

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₂×C₈

C₂×#10596842

Fitting

C₂×C₁₆×C₇₃

C₈

C₂×C₇₃

C₂²×D₇₃

C₂×C₈×#64408

C₂²

C₂×C₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
74. C₇₃	#7160	semidirect

H ≅ G/N

type

74. C₇₃

#7160

semidirect

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

subgroup	index
#22627125	2
C₂×C₁₆×#64408	2
#7160	73

subgroup

index

#22627125

C₂×C₁₆×#64408

#7160

Subgroups — #34032900

Conjugacy classes of subgroups

Complements

Maximal subgroups