Group #34001072 — subgroups

Subgroups — #34001072

Order 18688.

166 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₂×#17989518	C₂	1	1	yes	maximal, characteristic
2	D₇₃⋊#368	C₂	1	1	yes	maximal, characteristic
3	#368×D₇₃	C₂	1	1	yes	maximal, characteristic
4	D₇₃⋊#108	—	2	1	no	—
5	#17989518	—	2	1	no	—
6	#17989518	C₄	1	2	yes	—
7	C₂×#14556641	—	2	1	no	—
8	#17989516	—	2	1	no	—
9	#17989515	C₄	1	2	yes	—
10	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
11	#108×D₇₃	—	2	1	no	—
12	#17988417	C₄	1	1	yes	characteristic
13	#17988416	—	2	1	no	—
14	#368×C₇₃	C₄	1	1	yes	characteristic
15	D₇₃⋊M₁₆	—	4	1	no	—
16	#14556662	—	2	2	no	—
17	#14556661	—	4	1	no	—
18	#14556641	—	2	2	no	—
19	M₁₆×D₇₃	—	4	1	no	—
20	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
21	C₂×#11924003	D₄	1	1	yes	characteristic
22	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₁₆×D₇₃	D₄	1	1	yes	characteristic
25	C₁₆×D₇₃	Q₈	1	1	yes	characteristic
26	#14556527	—	2	1	no	—
27	#14556526	—	4	1	no	—
28	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#108×C₇₃	—	2	1	no	—
30	#11924038	—	4	2	no	—
31	#11924037	—	8	1	no	—
32	#11924036	—	8	1	no	—
33	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
34	#11924012	—	4	1	no	—
35	#11924011	—	8	1	no	—
36	C₂×#10596831	SD₁₆	1	1	yes	characteristic
37	C₈×D₇₃	—	8	1	no	—
38	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×D₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₈×D₇₃	SD₁₆	1	1	yes	characteristic
41	C₈×D₇₃	Q₁₆	1	1	yes	characteristic
42	#11924003	C₂²⋊C₄	1	1	yes	characteristic
43	#11924003	C₄⋊C₄	1	1	yes	characteristic
44	M₁₆×C₇₃	—	4	1	no	—
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₇₃	C₄²	1	1	yes	characteristic
47	C₁₆×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×D₇₃	Q₃₂	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
50	C₂×Dic₇₃	SD₃₂	1	1	yes	characteristic
51	C₄×D₇₃	#102	1	1	yes	characteristic
52	C₄×D₇₃	#107	1	1	yes	characteristic
53	C₄×D₇₃	D₁₆	1	1	yes	characteristic
54	C₄×D₇₃	SD₃₂	1	1	yes	characteristic
55	#10596833	—	8	2	no	—
56	C₈×C₇₃	—	8	1	no	—
57	C₈×C₇₃	#95	1	1	yes	characteristic
58	C₈×C₇₃	#103	1	1	yes	characteristic
59	#10596831	—	8	1	no	—
60	#10596831	#103	1	1	yes	characteristic
61	#10596831	#106	1	1	yes	characteristic
62	C₂²×C₇₃	#358	1	1	yes	characteristic
63	C₂×D₇₃	—	2	2	no	—
64	C₂×D₇₃	#358	1	1	yes	characteristic
65	C₂×D₇₃	#366	1	1	yes	characteristic
66	C₄×C₇₃	#340	1	1	yes	characteristic
67	C₄×C₇₃	#357	1	1	yes	characteristic
68	Dic₇₃	#357	1	1	yes	characteristic
69	Dic₇₃	#367	1	1	yes	characteristic
70	#7472	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#1380	1	1	yes	characteristic
73	D₇₃	—	2	1	no	—
74	D₇₃	#1425	1	2	yes	—
75	C₂²×C₃₂	—	73	1	no	—
76	C₂×#368	—	73	1	no	—
77	C₂×#368	—	73	1	no	—
78	C₇₃	#7472	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₁₆	—	73	1	no	—
80	C₂×#108	—	146	1	no	—
81	C₂×#108	—	146	1	no	—
82	C₂×C₃₂	—	73	2	no	—
83	C₂×C₃₂	—	146	1	no	—
84	C₂×C₃₂	—	146	1	no	—
85	#368	—	73	2	no	—
86	#368	—	73	1	no	—
87	#368	—	146	1	no	—
88	#368	—	146	1	no	—
89	#368	#64408	1	1	yes	characteristic
90	C₂×M₁₆	—	292	1	no	—
91	C₂×M₁₆	—	292	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₂×C₁₆	—	73	1	no	—
94	C₂×C₁₆	—	73	1	no	—
95	C₂×C₁₆	—	73	1	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
99	#108	—	2	1	no	—
100	#108	—	292	1	no	—
101	#108	—	292	1	no	—
102	#108	—	146	2	no	—
103	#108	—	146	1	no	—
104	C₃₂	—	146	2	no	—
105	C₂²×C₄	—	73	1	no	—
106	M₁₆	—	4	1	no	—
107	M₁₆	—	292	2	no	—
108	M₁₆	—	292	1	no	—
109	M₁₆	—	584	1	no	—
110	M₁₆	—	584	1	no	—
111	C₂×C₈	—	584	1	no	—
112	C₂×C₈	—	584	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	1	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
119	C₁₆	—	73	1	no	—
120	C₁₆	—	73	1	no	—
121	C₁₆	C₄×#64408	1	1	yes	characteristic
122	C₁₆	C₄⋊#64408	1	1	yes	characteristic
123	C₂³	—	73	1	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	#14556653	1	1	yes	characteristic
130	C₈	—	8	1	no	—
131	C₈	—	584	2	no	—
132	C₈	—	584	1	no	—
133	C₈	—	73	1	no	—
134	C₈	—	73	1	no	—
135	C₈	(C₂×#64408)⋊C₄	1	1	yes	characteristic
136	C₈	#14556654	1	1	yes	characteristic
137	C₂²	—	73	1	no	—
138	C₂²	—	73	1	no	—
139	C₂²	—	146	2	no	—
140	C₂²	#17989500	1	1	yes	characteristic
141	C₄	—	73	1	no	—
142	C₄	—	73	1	no	—
143	C₄	#17989460	1	1	yes	characteristic
144	C₄	#17989499	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	73	2	no	—
147	C₂	—	146	1	no	—
148	C₂	#22624988	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989518

C₂

yes

maximal, characteristic

D₇₃⋊#368

C₂

yes

maximal, characteristic

#368×D₇₃

C₂

yes

maximal, characteristic

D₇₃⋊#108

—

#17989518

—

#17989518

C₄

yes

—

C₂×#14556641

—

#17989516

—

#17989515

C₄

yes

—

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

#108×D₇₃

—

#17988417

C₄

yes

characteristic

#17988416

—

#368×C₇₃

C₄

yes

characteristic

D₇₃⋊M₁₆

—

#14556662

—

#14556661

—

#14556641

—

M₁₆×D₇₃

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×#11924003

D₄

yes

characteristic

C₁₆×D₇₃

C₂×C₄

yes

characteristic

C₁₆×D₇₃

C₂×C₄

yes

characteristic

C₁₆×D₇₃

D₄

yes

characteristic

C₁₆×D₇₃

Q₈

yes

characteristic

#14556527

—

#14556526

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#108×C₇₃

—

#11924038

—

#11924037

—

#11924036

—

C₂×C₄×D₇₃

D₈

yes

—

—

SD₁₆

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

C₈×D₇₃

C₄⋊C₄

yes

characteristic

C₈×D₇₃

SD₁₆

yes

characteristic

C₈×D₇₃

Q₁₆

yes

characteristic

#11924003

C₂²⋊C₄

yes

characteristic

#11924003

C₄⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×D₇₃

Q₃₂

yes

characteristic

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×Dic₇₃

SD₃₂

yes

C₄×D₇₃

yes

C₄×D₇₃

yes

C₄×D₇₃

D₁₆

yes

characteristic

C₄×D₇₃

SD₃₂

yes

characteristic

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

yes

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1380

yes

characteristic

D₇₃

—

D₇₃

#1425

yes

—

C₂²×C₃₂

—

C₂×#368

—

C₂×#368

—

C₇₃

#7472

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

146

—

C₂×#108

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

C₂×C₃₂

—

146

—

#368

—

#368

—

#368

—

146

—

#368

—

146

—

#368

#64408

yes

characteristic

C₂×M₁₆

—

292

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#108

—

100

#108

—

292

—

101

#108

—

292

—

102

#108

—

146

—

103

#108

—

146

—

104

C₃₂

—

146

—

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

292

—

108

M₁₆

—

292

—

109

M₁₆

—

584

—

110

M₁₆

—

584

—

111

C₂×C₈

—

584

—

112

C₂×C₈

—

584

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₂²⋊#64408

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

C₄×#64408

yes

characteristic

122

C₁₆

C₄⋊#64408

yes

characteristic

123

C₂³

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#14556653

yes

characteristic

130

C₈

—

131

C₈

—

584

—

132

C₈

—

584

—

133

C₈

—

134

C₈

—

135

C₈

(C₂×#64408)⋊C₄

yes

characteristic

136

C₈

#14556654

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

146

—

140

C₂²

#17989500

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#17989460

yes

characteristic

144

C₄

#17989499

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

146

—

148

C₂

#22624988

yes

characteristic

	subgroup	G/H
center	C₄	#17989499
commutator	C₁₆×C₇₃	C₄²
Frattini	C₂×C₁₆	C₂×#64408
Fitting	#368×C₇₃	C₄
socle	C₂×C₇₃	#1380
prime radical	C₂²×D₇₃	Q₃₂
cosocle	C₂×C₁₆×D₇₃	C₂²
hypercenter	#368	#64408
hypocenter	C₇₃	#7472
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

subgroup

G/H

center

C₄

#17989499

commutator

C₁₆×C₇₃

C₄²

Frattini

C₂×C₁₆

C₂×#64408

Fitting

#368×C₇₃

C₄

socle

C₂×C₇₃

#1380

prime radical

C₂²×D₇₃

Q₃₂

cosocle

C₂×C₁₆×D₇₃

C₂²

C₇₃

G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

N	H ≅ G/N	type
78. C₇₃	#7472	semidirect

H ≅ G/N

type

78. C₇₃

#7472

semidirect

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

subgroup	index
C₂×#17989518	2
D₇₃⋊#368	2
#368×D₇₃	2
#7472	73

subgroup

index

C₂×#17989518

D₇₃⋊#368

#368×D₇₃

#7472

Subgroups — #34001072

Conjugacy classes of subgroups

Complements

Maximal subgroups