Group #34034077 — subgroups

Subgroups — #34034077

Order 18688.

358 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 133 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	#22627087	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556686	C₂	1	1	yes	maximal, characteristic
3	#22627002	C₂	1	4	yes	maximal
4	#22625589	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556686	C₂²	1	2	yes	—
6	C₂×#14556686	C₂²	1	2	yes	—
7	#17989681	C₄	1	4	yes	complemented
8	C₄×#11924040	C₄	1	4	yes	complemented
9	C₂×#14556645	C₄	1	2	yes	—
10	C₄×#11924036	C₂²	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	2	yes	—
12	#17989523	C₄	1	1	yes	characteristic
13	#97×D₇₃	C₄	1	1	yes	characteristic
14	#14556686	—	2	4	no	—
15	#14556686	—	2	4	no	—
16	C₂×#11924040	C₂×C₄	1	4	yes	—
17	C₂×#11924040	C₂×C₄	1	4	yes	—
18	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924036	C₂³	1	1	yes	characteristic
20	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
21	#14556645	—	2	4	no	—
22	C₄×#10596833	C₂×C₄	1	2	yes	—
23	C₄×#10596833	C₂×C₄	1	2	yes	—
24	C₈×#64408	—	2	4	no	—
25	(C₂×#64408)⋊C₄	C₈	1	4	yes	—
26	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
27	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
28	#14556502	C₈	1	1	yes	characteristic
29	#14556501	C₈	1	2	yes	—
30	#97×C₇₃	C₈	1	1	yes	characteristic
31	#11924040	—	2	4	no	—
32	#11924040	—	2	4	no	—
33	C₂²×#64408	C₂×C₈	1	4	yes	—
34	#11924036	M₁₆	1	4	yes	—
35	#11924036	#41	1	4	yes	—
36	C₂×#10596833	C₄²	1	2	yes	—
37	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
38	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
39	C₄×#64408	—	2	2	no	—
40	C₄×#64408	—	2	2	no	—
41	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×#10596831	C₂×C₈	1	2	yes	—
44	C₈×D₇₃	—	2	2	no	—
45	C₈×D₇₃	—	2	2	no	—
46	C₄×Dic₇₃	C₄²	1	2	yes	—
47	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
49	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×#64408	—	2	4	no	—
52	C₂×#64408	—	2	8	no	—
53	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
56	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
57	C₄×D₇₃	—	2	4	no	—
58	C₄×D₇₃	#117	1	2	yes	—
59	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
60	#10596833	#97	1	4	yes	—
61	#10596833	#117	1	2	yes	—
62	#10596833	#117	1	2	yes	—
63	C₈×C₇₃	—	2	2	no	—
64	#10596831	—	2	2	no	—
65	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×D₇₃	#431	1	2	yes	—
67	C₂×D₇₃	#504	1	4	yes	—
68	#64408	—	4	4	no	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	#431	1	2	yes	—
71	Dic₇₃	—	2	2	no	—
72	Dic₇₃	#376	1	1	yes	characteristic
73	Dic₇₃	C₂×#97	1	1	yes	characteristic
74	#12592	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	#1744	1	1	yes	characteristic
76	C₂×C₇₃	#2108	1	2	yes	—
77	D₇₃	#2164	1	4	yes	—
78	C₂×#346	—	73	1	no	—
79	C₂×C₄×C₁₆	—	73	1	no	—
80	#1726	—	73	1	no	—
81	#1374	—	73	4	no	—
82	C₇₃	#12592	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	73	2	no	—
84	C₂²×C₁₆	—	73	2	no	—
85	C₂×#97	—	73	1	no	—
86	C₂×#97	—	73	2	no	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₂×C₄×C₈	—	73	2	no	—
89	#376	—	73	1	no	—
90	#346	—	73	4	no	—
91	C₄×C₁₆	—	73	4	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₂²×C₈	—	73	1	no	—
94	C₂²×C₈	—	73	2	no	—
95	C₂×C₄²	—	73	1	no	—
96	C₂×C₄²	—	73	2	no	—
97	C₂×C₁₆	—	73	4	no	—
98	C₂×C₁₆	—	73	4	no	—
99	C₂×C₁₆	—	146	4	no	—
100	C₂×C₁₆	—	146	4	no	—
101	#97	—	73	2	no	—
102	#97	—	73	1	no	—
103	#97	—	146	4	no	—
104	#97	#10596842	1	1	yes	characteristic
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	73	2	no	—
107	C₄×C₈	—	146	4	no	—
108	#95	—	73	4	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	2	no	—
111	C₂²×C₄	—	73	4	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	4	no	—
116	C₂×C₈	—	73	4	no	—
117	C₂×C₈	—	73	2	no	—
118	C₂×C₈	—	146	2	no	—
119	C₂×C₈	—	146	2	no	—
120	C₂×C₈	C₂×#10596842	1	2	yes	—
121	C₄²	—	73	2	no	—
122	C₄²	—	73	1	no	—
123	C₄²	—	146	2	no	—
124	C₄²	—	146	2	no	—
125	C₄²	C₂×#10596842	1	1	yes	characteristic
126	C₁₆	—	146	4	no	—
127	C₁₆	—	146	4	no	—
128	C₂³	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	2	no	—
131	C₂×C₄	—	73	2	no	—
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	146	4	no	—
134	C₂×C₄	—	146	4	no	—
135	C₂×C₄	—	146	8	no	—
136	C₂×C₄	C₄×#10596842	1	2	yes	—
137	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	73	2	no	—
140	C₈	—	73	2	no	—
141	C₈	—	73	4	no	—
142	C₈	—	146	2	no	—
143	C₂²	—	73	4	no	—
144	C₂²	—	73	2	no	—
145	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	292	4	no	—
148	C₄	—	73	1	no	—
149	C₄	—	73	1	no	—
150	C₄	—	146	2	no	—
151	C₄	#17989700	1	2	yes	—
152	C₂	—	73	4	no	—
153	C₂	#22627065	1	1	yes	characteristic
154	C₂	#22627094	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22627087

C₂

yes

maximal, characteristic

C₄×#14556686

C₂

yes

maximal, characteristic

#22627002

C₂

yes

maximal

#22625589

C₂

yes

maximal, characteristic

C₂×#14556686

C₂²

yes

—

C₂×#14556686

C₂²

yes

—

#17989681

C₄

yes

complemented

C₄×#11924040

C₄

yes

complemented

C₂×#14556645

C₄

yes

—

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₂²

yes

—

#17989523

C₄

yes

characteristic

#97×D₇₃

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11924040

C₂×C₄

yes

—

C₂×#11924036

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

C₂×C₄

yes

—

#14556645

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₈×#64408

—

(C₂×#64408)⋊C₄

C₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

#14556502

C₈

yes

characteristic

#14556501

C₈

yes

—

#97×C₇₃

C₈

yes

characteristic

#11924040

—

#11924040

—

C₂²×#64408

C₂×C₈

yes

—

#11924036

M₁₆

yes

—

#11924036

#41

yes

—

C₂×#10596833

C₄²

yes

—

C₂×#10596833

C₂²×C₄

yes

characteristic

C₂×#10596833

C₂²×C₄

yes

characteristic

C₄×#64408

—

C₄×#64408

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂²×C₈

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄×C₈

yes

—

C₂×Dic₇₃

C₂×C₄²

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

#117

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

#10596833

#97

yes

—

#10596833

#117

yes

—

#10596833

#117

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄×C₈

yes

characteristic

C₂×D₇₃

#431

yes

—

C₂×D₇₃

#504

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#431

yes

—

Dic₇₃

—

Dic₇₃

#376

yes

characteristic

Dic₇₃

C₂×#97

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

#2164

yes

—

C₂×#346

—

C₂×C₄×C₁₆

—

#1726

—

#1374

—

C₇₃

#12592

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#346

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

100

C₂×C₁₆

—

146

—

101

#97

—

102

#97

—

103

#97

—

146

—

104

#97

#10596842

yes

characteristic

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

146

—

108

#95

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

146

—

119

C₂×C₈

—

146

—

120

C₂×C₈

C₂×#10596842

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

146

—

124

C₄²

—

146

—

125

C₄²

C₂×#10596842

yes

characteristic

126

C₁₆

—

146

—

127

C₁₆

—

146

—

128

C₂³

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

C₄×#10596842

yes

—

137

C₂×C₄

C₂²×#10596842

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

146

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂×C₄×#10596842

yes

characteristic

146

C₄

—

147

C₄

—

292

—

148

C₄

—

149

C₄

—

150

C₄

—

146

—

151

C₄

#17989700

yes

—

152

C₂

—

153

C₂

#22627065

yes

characteristic

154

C₂

#22627094

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂×C₄×C₈
Fitting	#97×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	#97	#10596842
hypocenter	C₇₃	#12592
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂×C₄×C₈

Fitting

#97×C₇₃

C₈

prime radical

C₂²×D₇₃

C₂²×C₈

C₂³

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
7. #17989681	C₄	semidirect
8. C₄×#11924040	C₄	semidirect
82. C₇₃	#12592	semidirect

H ≅ G/N

type

7. #17989681

C₄

semidirect

8. C₄×#11924040

C₄

semidirect

82. C₇₃

#12592

semidirect

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

subgroup	index
#22627087	2
C₄×#14556686	2
#22627002	2
#22625589	2
#12592	73

subgroup

index

Subgroups — #34034077

Conjugacy classes of subgroups

Complements

Maximal subgroups