Group #34013297 — subgroups

Subgroups — #34013297

Order 18688.

282 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 108 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	#22625872	C₂	1	2	yes	maximal
2	#22624991	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	—	2	2	no	—
5	C₂×#14556662	C₂²	1	4	yes	—
6	#17989591	—	2	2	no	—
7	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₄×#11924003	C₄	1	1	yes	characteristic
10	C₁₆×Dic₇₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	2	4	no	—
13	#14556662	—	2	8	no	—
14	#14556662	—	4	4	no	—
15	C₄⋊#10596833	—	4	2	no	—
16	C₄⋊#10596833	—	4	2	no	—
17	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
18	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
19	C₄²×D₇₃	D₄	1	1	yes	characteristic
20	C₂×#11924003	C₂×C₄	1	2	yes	—
21	C₁₆×D₇₃	—	2	4	no	—
22	C₄×#10596831	Q₈	1	1	yes	characteristic
23	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	#11924038	—	4	4	no	—
28	#11924038	—	4	4	no	—
29	C₂×#10596833	—	4	4	no	—
30	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
32	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
35	C₈×D₇₃	SD₁₆	1	4	yes	—
36	C₈×D₇₃	#41	1	4	yes	—
37	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	D₈	1	1	yes	characteristic
39	C₄×Dic₇₃	D₈	1	1	yes	characteristic
40	#11924003	—	2	2	no	—
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	2	no	—
45	C₂²×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	#107	1	1	yes	characteristic
50	C₂×Dic₇₃	#107	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	—	2	2	no	—
54	C₄×D₇₃	#124	1	2	yes	—
55	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
56	#10596833	—	8	4	no	—
57	C₈×C₇₃	#106	1	2	yes	—
58	C₈×C₇₃	#117	1	2	yes	—
59	#10596831	#106	1	2	yes	—
60	#10596831	#125	1	2	yes	—
61	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	—	2	1	no	—
64	C₂×D₇₃	#487	1	2	yes	—
65	C₄×C₇₃	#368	1	2	yes	—
66	C₄×C₇₃	#368	1	2	yes	—
67	C₄×C₇₃	#382	1	1	yes	characteristic
68	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
69	Dic₇₃	—	2	1	no	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	#486	1	2	yes	—
72	#12891	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#1834	1	1	yes	characteristic
74	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
76	D₇₃	—	2	2	no	—
77	C₂×C₄×C₁₆	—	73	1	no	—
78	#1848	—	73	2	no	—
79	#1381	—	73	4	no	—
80	C₇₃	#12891	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₁₆	—	73	2	no	—
82	C₂×#108	—	73	4	no	—
83	C₂×#108	—	146	2	no	—
84	#423	—	146	2	no	—
85	C₂×C₄×C₈	—	73	1	no	—
86	C₄×C₁₆	—	73	2	no	—
87	C₄×C₁₆	—	73	1	no	—
88	C₄×C₁₆	#64408	1	1	yes	characteristic
89	C₂×M₁₆	—	146	4	no	—
90	C₂²×C₈	—	73	1	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂×C₄²	—	73	1	no	—
93	C₂×C₁₆	—	73	2	no	—
94	C₂×C₁₆	—	146	4	no	—
95	C₂×C₁₆	C₂×#64408	1	2	yes	—
96	#108	—	292	4	no	—
97	#108	—	146	8	no	—
98	C₄⋊C₈	—	292	2	no	—
99	C₄⋊C₈	—	292	2	no	—
100	C₄×C₈	—	73	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	C₂×#64408	1	1	yes	characteristic
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	—	73	1	no	—
107	M₁₆	—	292	4	no	—
108	M₁₆	—	292	4	no	—
109	C₂×C₈	—	292	4	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	4	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
115	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	1	no	—
119	C₄²	C₄⋊#64408	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	146	2	no	—
122	C₂³	—	73	1	no	—
123	C₂×C₄	—	73	1	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	73	2	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	#14556660	1	1	yes	characteristic
131	C₂×C₄	#14556660	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
133	C₈	—	584	4	no	—
134	C₈	—	73	2	no	—
135	C₈	—	73	2	no	—
136	C₈	#14556659	1	2	yes	—
137	C₈	#14556670	1	2	yes	—
138	C₂²	—	73	2	no	—
139	C₂²	—	146	1	no	—
140	C₂²	—	146	1	no	—
141	C₂²	C₂×#14556660	1	1	yes	characteristic
142	C₄	—	73	2	no	—
143	C₄	—	146	1	no	—
144	C₄	—	146	1	no	—
145	C₄	#17989515	1	2	yes	—
146	C₄	#17989516	1	2	yes	—
147	C₄	#17989531	1	1	yes	characteristic
148	C₄	C₂×#14556659	1	1	yes	characteristic
149	C₂	—	146	2	no	—
150	C₂	#22625852	1	1	yes	characteristic
151	C₂	C₂×#17989515	1	1	yes	characteristic
152	C₂	C₂×#17989516	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625872

C₂

yes

maximal

#22624991

C₂

yes

maximal

C₄×C₁₆×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

—

C₂×#14556662

C₂²

yes

—

#17989591

—

C₂×C₁₆×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄×#11924003

C₄

yes

characteristic

C₁₆×Dic₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

C₄

yes

—

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂³

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

SD₁₆

yes

—

C₈×D₇₃

#41

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₂×Q₁₆

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂×D₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

#124

yes

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

C₈×C₇₃

#106

yes

—

C₈×C₇₃

#117

yes

—

#10596831

#106

yes

—

#10596831

#125

yes

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#487

yes

—

C₄×C₇₃

#368

yes

—

C₄×C₇₃

#368

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#486

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂×C₄×C₁₆

—

#1848

—

#1381

—

C₇₃

#12891

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

—

146

—

#423

—

146

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#64408

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×#64408

yes

—

#108

—

292

—

#108

—

146

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

292

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#64408

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

M₁₆

—

292

—

108

M₁₆

—

292

—

109

C₂×C₈

—

292

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₄⋊#64408

yes

characteristic

115

C₂×C₈

C₂²×#64408

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄⋊#64408

yes

characteristic

120

C₁₆

—

121

C₁₆

—

146

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

#14556660

yes

characteristic

131

C₂×C₄

#14556660

yes

characteristic

132

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

133

C₈

—

584

—

134

C₈

—

135

C₈

—

136

C₈

#14556659

yes

—

137

C₈

#14556670

yes

—

138

C₂²

—

139

C₂²

—

146

—

140

C₂²

—

146

—

141

C₂²

C₂×#14556660

yes

characteristic

142

C₄

—

143

C₄

—

146

—

144

C₄

—

146

—

145

C₄

#17989515

yes

—

146

C₄

#17989516

yes

—

147

C₄

#17989531

yes

characteristic

148

C₄

C₂×#14556659

yes

characteristic

149

C₂

—

146

—

150

C₂

#22625852

yes

characteristic

151

C₂

C₂×#17989515

yes

characteristic

152

C₂

C₂×#17989516

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556660
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂²×D₇₃	C₂×Q₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#12891
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14556660

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₄×C₁₆×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#107

prime radical

C₂²×D₇₃

C₂×Q₁₆

cosocle

C₂×C₈×D₇₃

C₂³

C₄×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
80. C₇₃	#12891	semidirect

H ≅ G/N

type

80. C₇₃

#12891

semidirect

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

subgroup	index
#22625872	2
#22624991	2
C₄×C₁₆×D₇₃	2
#12891	73

subgroup

index

#22625872

#22624991

C₄×C₁₆×D₇₃

#12891

Subgroups — #34013297

Conjugacy classes of subgroups

Complements

Maximal subgroups