Finite Group Databasebeta

← #28399811

Subgroups — #28399811

Order 14464.

738 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic226C211yesmaximal, characteristic, complemented
2C2×#16813655C212yesmaximal, complemented
3#20299109C211yesmaximal, characteristic
4#20298756C214yesmaximal, complemented
5#20298739C214yesmaximal, complemented
6#20298608C212yesmaximal, complemented
7#407×C113C211yesmaximal, characteristic, complemented
8C2×#14268647C2211yescharacteristic
9C22×Dic226—28no—
10C22×Dic226—22no—
11C22×Dic226—21no—
12C22×Dic226C2214yes—
13C22×Dic226C2212yes—
14C23×Dic113—21no—
15#16813655—28no—
16C2×Dic452C2214yescomplemented
17C2×#14268636C2214yescomplemented
18#16813605—21no—
19#16813601C2212yes—
20#16813596C2212yes—
21#16813591—21no—
22C4×C22⋊D113—22no—
23#16813581C2211yescharacteristic
24#16813547C2218yescomplemented
25C23×C4×C113C2211yescharacteristic
26C2×M16×C113C2212yes—
27C22⋊C8×C113C2214yes—
28#14268647—24no—
29C2×Dic226—28no—
30C2×Dic226—216no—
31C2×Dic226—24no—
32C2×Dic226—24no—
33C2×Dic226—22no—
34C2×Dic226—216no—
35C2×Dic226D418yescomplemented
36C2×Dic226C2311yescharacteristic
37C2×Dic226C2314yes—
38C2×D452C2311yescharacteristic
39C2×C22⋊D113—22no—
40C2×C22⋊D113—21no—
41C22×Dic113—28no—
42C22×Dic113—24no—
43C22×Dic113—22no—
44C2×C4×D113—21no—
45Dic452—28no—
46#14268636—28no—
47C4⋊Dic113C2312yes—
48#14268626—22no—
49C22⋊Dic113—21no—
50C22⋊Dic113—22no—
51#14268624—22no—
52C4×Dic113—21no—
53C24×C113D411yescharacteristic
54C22×C4×C113—24no—
55C22×C4×C113D412yes—
56C22×C4×C113D411yescharacteristic
57C22×C4×C113C2312yes—
58C22×C4×C113C2311yescharacteristic
59M16×C113—24no—
60C2×C8×C113C2314yes—
61C23×C113—22no—
62C23×C113—24no—
63C23×C113C2×D411yescharacteristic
64C23×C113C2×D412yes—
65C22×D113—21no—
66C2×C4×C113—22no—
67C2×C4×C113—24no—
68C2×C4×C113—24no—
69C2×C4×C113C2×D414yes—
70C2×C4×C113C2×D412yes—
71C2×C4×C113C2×D411yescharacteristic
72C2×C4×C113C2411yescharacteristic
73C22⋊D113—42no—
74C22⋊D113—44no—
75C2×Dic113—216no—
76C2×Dic113—28no—
77C2×Dic113—24no—
78C2×Dic113—22no—
79C2×Dic113—21no—
80D452—22no—
81C4×D113—42no—
82Dic226—216no—
83Dic226—28no—
84Dic226—24no—
85Dic226—22no—
86Dic226C2×D418yes—
87C8×C113—24no—
88C22×C113—28no—
89C22×C113—22no—
90C22×C113—22no—
91C22×C113—22no—
92C22×C113#12012yes—
93C22×C113#13714yescomplemented
94C22×C113C22×D411yescharacteristic
95C2×D113—41no—
96C2×D113—42no—
97C4×C113—22no—
98C4×C113#12012yes—
99C4×C113C22×D412yes—
100Dic113—28no—
101Dic113—41no—
102Dic113—42no—
103C2×C113—22no—
104C2×C113—24no—
105C2×C113C2×#12011yescharacteristic
106C2×C113C2×#13712yes—
107D113—81no—
108#3004—1131nomaximal, Sylow
109C113#300411yescharacteristic, complemented, Sylow
110C23×Q8—1131no—
111C2×#137—1132no—
112#525—1131no—
113#451—1134no—
114#448—1134no—
115#419—1132no—
116#407D11311yescharacteristic, complemented
117C2×#41—1131no—
118C22×Q8—1132no—
119C22×Q8—1134no—
120C22×Q8—2268no—
121C22×Q8—2262no—
122C22×Q8—2261no—
123C23×C4—2261no—
124C23×C4C2×D11311yescharacteristic
125#137—2268no—
126C2×Q16—1134no—
127C2×SD16—1134no—
128C2×M16C2×D11312yes—
129#123—2261no—
130C22⋊Q8—1132no—
131#121—1132no—
132#120—2261no—
133C4×D4—2262no—
134#117—1131no—
135#103—1138no—
136C22⋊C8C2×D11314yes—
137C24D45211yescharacteristic
138#41—2264no—
139C2×Q8—1138no—
140C2×Q8—1134no—
141C2×Q8—1131no—
142C2×Q8—22616no—
143C2×Q8—2268no—
144C2×Q8—22616no—
145C2×Q8—2264no—
146C2×Q8—2264no—
147C2×Q8—2262no—
148C2×D4—1131no—
149C2×D4—2262no—
150C2×D4—2261no—
151C22×C4—24no—
152C22×C4—2261no—
153C22×C4—2268no—
154C22×C4—2264no—
155C22×C4—2262no—
156C22×C4D45212yes—
157C22×C4D45211yescharacteristic
158C22×C4C22×D11312yes—
159C22×C4C22×D11311yescharacteristic
160Q16—2268no—
161SD16—2268no—
162M16—24no—
163C2×C8C22×D11314yes—
164C4⋊C4—1132no—
165C4⋊C4—2262no—
166C22⋊C4—2261no—
167C22⋊C4—2262no—
168C22⋊C4—2262no—
169C42—2261no—
170C23—22no—
171C23—24no—
172C23—2261no—
173C23C2×D45211yescharacteristic
174C23C2×D45212yes—
175Q8—1138no—
176Q8—22616no—
177Q8—2262no—
178Q8—2268no—
179Q8—2264no—
180D4—4522no—
181D4—4524no—
182D4—2262no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—4522no—
187C2×C4—2262no—
188C2×C4—2261no—
189C2×C4—22616no—
190C2×C4—2268no—
191C2×C4—2264no—
192C2×C4D4×D11314yes—
193C2×C4C2×D45212yes—
194C2×C4C2×D45211yescharacteristic
195C2×C4C23×D11311yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—4521no—
202C22—4522no—
203C22#1681359212yes—
204C22#1681365514yescomplemented
205C22C22×D45211yescharacteristic
206C4—22no—
207C4—4521no—
208C4—4522no—
209C4—2268no—
210C4#1681359212yes—
211C4C2×D4×D11312yes—
212C2—22no—
213C2—24no—
214C2—9041no—
215C2C2×#1681359211yescharacteristic
216C2C2×#1681365512yes—
subgroupG/H
centerC22C22×D452
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#407×C113C2
socleC22×C113C22×D4
prime radical#20299109C2
hypercenter#407D113
hypocenterC113#3004
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×Dic226C2semidirect
2. C2×#16813655C2semidirect
4. #20298756C2semidirect
5. #20298739C2semidirect
6. #20298608C2semidirect
7. #407×C113C2semidirect
16. C2×Dic452C22semidirect
17. C2×#14268636C22semidirect
24. #16813547C22semidirect
35. C2×Dic226D4semidirect
93. C22×C113#137semidirect
109. C113#3004semidirect
116. #407D113semidirect
204. C22#16813655semidirect

Maximal subgroups

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

subgroupindex
C23×Dic2262
C2×#168136552
#202991092
#202987562
#202987392
#202986082
#407×C1132
#3004113