Group #34000445 — subgroups

Subgroups — #34000445

Order 18688.

192 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 44 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	#22625862	C₂	1	1	yes	maximal, characteristic
2	#22625155	C₂	1	1	yes	maximal, characteristic
3	#22611741	C₂	1	1	yes	maximal, characteristic
4	C₄×#11924038	—	2	1	no	—
5	#17989469	—	2	1	no	—
6	#17989458	—	2	1	no	—
7	#17989456	C₄	1	1	yes	characteristic
8	#17989455	C₄	1	1	yes	characteristic
9	#17989429	—	2	1	no	—
10	#17989428	C₄	1	1	yes	characteristic, complemented
11	C₈×#10596833	C₄	1	1	yes	characteristic, complemented
12	#17988539	C₂²	1	1	yes	characteristic
13	#17988407	—	2	1	no	—
14	#17988368	—	2	1	no	—
15	#17988365	C₄	1	1	yes	characteristic, complemented
16	#347×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂×#11924038	—	2	2	no	—
18	D₇₃⋊M₁₆	—	2	1	no	—
19	#14556670	—	2	1	no	—
20	#14556646	—	2	2	no	—
21	C₄×#10596833	—	2	1	no	—
22	C₄×#10596833	—	2	1	no	—
23	C₂×#11924011	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924011	D₄	1	1	yes	characteristic
25	C₄²×D₇₃	D₄	1	1	yes	characteristic
26	#14556531	—	2	1	no	—
27	#14556502	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	M₃₂×C₇₃	—	2	1	no	—
31	#97×C₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924038	—	4	4	no	—
33	#11924038	—	4	1	no	—
34	#11924037	—	4	1	no	—
35	#11924036	—	4	1	no	—
36	C₂×#10596833	—	2	2	no	—
37	C₂×#10596833	—	2	2	no	—
38	C₄⋊#64408	—	4	1	no	—
39	C₂²⋊#64408	—	4	1	no	—
40	C₄×#64408	—	4	1	no	—
41	C₂×C₄×D₇₃	—	2	1	no	—
42	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	#11924011	—	4	1	no	—
44	#11924011	—	4	1	no	—
45	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
46	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
47	C₄×Dic₇₃	—	2	1	no	—
48	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	#11924003	—	4	1	no	—
50	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₁₆×C₇₃	—	4	1	no	—
54	C₂²×D₇₃	#104	1	1	yes	characteristic
55	C₂×#64408	—	8	1	no	—
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	#104	1	1	yes	characteristic
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	4	2	no	—
63	#10596833	—	2	4	no	—
64	#10596833	—	4	2	no	—
65	#10596833	—	8	1	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	—	2	1	no	—
68	#10596831	—	2	1	no	—
69	#10596831	—	2	1	no	—
70	C₂²×C₇₃	#339	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	4	1	no	—
73	#64408	—	16	1	no	—
74	C₄×C₇₃	—	4	1	no	—
75	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
76	C₄×C₇₃	#344	1	1	yes	characteristic
77	Dic₇₃	—	2	2	no	—
78	Dic₇₃	—	2	1	no	—
79	#7225	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	#1288	1	1	yes	characteristic
82	D₇₃	—	4	1	no	—
83	#2165	—	73	1	no	—
84	#1841	—	73	1	no	—
85	#1455	—	73	1	no	—
86	C₇₃	#7225	1	1	yes	characteristic, complemented, Sylow
87	C₄×M₁₆	—	73	1	no	—
88	C₄×M₁₆	—	146	1	no	—
89	#364	—	146	1	no	—
90	#349	—	146	1	no	—
91	#347	—	73	1	no	—
92	#347	#64408	1	1	yes	characteristic, complemented
93	#344	—	146	1	no	—
94	#339	—	146	1	no	—
95	#338	—	73	1	no	—
96	H(3,ℤ/4)	—	73	1	no	—
97	#322	—	73	1	no	—
98	#322	—	146	1	no	—
99	C₈²	—	73	1	no	—
100	C₂×M₁₆	—	73	1	no	—
101	C₂×M₁₆	—	73	1	no	—
102	C₂×M₁₆	—	146	1	no	—
103	C₂×M₁₆	—	146	2	no	—
104	#117	—	146	1	no	—
105	C₂×C₄²	—	73	1	no	—
106	M₃₂	—	2	1	no	—
107	M₃₂	—	146	1	no	—
108	#97	—	73	1	no	—
109	#97	—	146	2	no	—
110	#97	C₂×#64408	1	1	yes	characteristic
111	C₄×C₈	—	73	1	no	—
112	C₄×C₈	—	73	1	no	—
113	C₄×C₈	—	146	1	no	—
114	C₄×C₈	—	146	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	146	1	no	—
117	M₁₆	—	292	1	no	—
118	M₁₆	—	292	1	no	—
119	M₁₆	—	292	4	no	—
120	M₁₆	—	292	1	no	—
121	M₁₆	—	292	1	no	—
122	C₂×C₈	—	292	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	C₄×#64408	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
129	C₄⋊C₄	—	292	1	no	—
130	C₂²⋊C₄	—	292	1	no	—
131	C₄²	—	292	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	—	146	1	no	—
134	C₄²	C₄⋊#64408	1	1	yes	characteristic
135	C₁₆	—	4	1	no	—
136	C₁₆	—	292	1	no	—
137	C₂³	—	73	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	584	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	292	2	no	—
149	C₈	—	584	1	no	—
150	C₈	—	146	4	no	—
151	C₈	—	146	1	no	—
152	C₈	—	146	1	no	—
153	C₂²	—	292	1	no	—
154	C₂²	—	146	1	no	—
155	C₂²	#17989459	1	1	yes	characteristic
156	C₄	—	4	1	no	—
157	C₄	—	1168	1	no	—
158	C₄	—	146	2	no	—
159	C₄	—	146	1	no	—
160	C₄	#17989454	1	1	yes	characteristic
161	C₄	#17989470	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	292	1	no	—
164	C₂	#22624792	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625862

C₂

yes

maximal, characteristic

#22625155

C₂

yes

maximal, characteristic

#22611741

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

characteristic

#17989455

C₄

yes

characteristic

#17989429

—

#17989428

C₄

yes

characteristic, complemented

C₈×#10596833

C₄

yes

characteristic, complemented

#17988539

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#347×C₇₃

C₄

yes

characteristic, complemented

C₂×#11924038

—

D₇₃⋊M₁₆

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924011

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

#14556531

—

#14556502

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

M₃₂×C₇₃

—

#97×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

—

—

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

#104

yes

characteristic

C₂×#64408

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#104

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#339

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7225

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1288

yes

characteristic

D₇₃

—

#2165

—

#1841

—

#1455

—

C₇₃

#7225

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

C₄×M₁₆

—

146

—

#364

—

146

—

#349

—

146

—

#347

—

#347

#64408

yes

characteristic, complemented

#344

—

146

—

#339

—

146

—

—

—

—

—

146

—

C₈²

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

146

—

103

C₂×M₁₆

—

146

—

104

#117

—

146

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

146

—

108

#97

—

109

#97

—

146

—

110

#97

C₂×#64408

yes

characteristic

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

146

—

114

C₄×C₈

—

146

—

115

C₂²×C₄

—

116

C₂²×C₄

—

146

—

117

M₁₆

—

292

—

118

M₁₆

—

292

—

119

M₁₆

—

292

—

120

M₁₆

—

292

—

121

M₁₆

—

292

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₂×C₈

C₄×#64408

yes

characteristic

128

C₂×C₈

C₂²⋊#64408

yes

characteristic

129

C₄⋊C₄

—

292

—

130

C₂²⋊C₄

—

292

—

131

C₄²

—

292

—

132

C₄²

—

133

C₄²

—

146

—

134

C₄²

C₄⋊#64408

yes

characteristic

135

C₁₆

—

136

C₁₆

—

292

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

292

—

140

C₂×C₄

—

584

—

141

C₂×C₄

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

292

—

149

C₈

—

584

—

150

C₈

—

146

—

151

C₈

—

146

—

152

C₈

—

146

—

153

C₂²

—

292

—

154

C₂²

—

146

—

155

C₂²

#17989459

yes

characteristic

156

C₄

—

157

C₄

—

1168

—

158

C₄

—

146

—

159

C₄

—

146

—

160

C₄

#17989454

yes

characteristic

161

C₄

#17989470

yes

characteristic

162

C₂

—

163

C₂

—

292

—

164

C₂

#22624792

yes

characteristic

	subgroup	G/H
center	C₄	#17989454
commutator	C₂×C₈×C₇₃	C₄²
Frattini	#97	C₂×#64408
Fitting	#347×C₇₃	C₄
socle	C₂×C₇₃	#1288
prime radical	C₂²×D₇₃	#104
cosocle	#17988539	C₂²
hypercenter	#347	#64408
hypocenter	C₇₃	#7225
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

subgroup

G/H

center

C₄

#17989454

commutator

C₂×C₈×C₇₃

C₄²

#347×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #347

N	H ≅ G/N	type
10. #17989428	C₄	semidirect
11. C₈×#10596833	C₄	semidirect
15. #17988365	C₄	semidirect
16. #347×C₇₃	C₄	semidirect
86. C₇₃	#7225	semidirect
92. #347	#64408	semidirect

H ≅ G/N

type

10. #17989428

C₄

semidirect

11. C₈×#10596833

C₄

semidirect

15. #17988365

C₄

semidirect

16. #347×C₇₃

C₄

semidirect

86. C₇₃

#7225

semidirect

92. #347

#64408

semidirect

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

subgroup	index
#22625862	2
#22625155	2
#22611741	2
#7225	73

subgroup

index

Subgroups — #34000445

Conjugacy classes of subgroups

Complements

Maximal subgroups