Finite Group Databasebeta

← #28399808

Subgroups — #28399808

Order 14464.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#16813653C212yesmaximal, complemented
2#20299109C211yesmaximal, characteristic
3#20298754C214yesmaximal, complemented
4#20298737C214yesmaximal, complemented
5#20298607C212yesmaximal, complemented
6#20298562C211yesmaximal, characteristic, complemented
7C23×Q8×C113C211yesmaximal, characteristic, complemented
8C2×#14268647C2211yescharacteristic
9#16813653—28no—
10C2×#14268637C2214yescomplemented
11C2×#14268634C2214yescomplemented
12C2×#14268631C2212yes—
13#16813605—21no—
14#16813601C2212yes—
15#16813596C2212yes—
16#16813591—21no—
17C4×C22⋊D113—22no—
18#16813581C2211yescharacteristic
19#16813546C2218yescomplemented
20C22⋊#11744751C2214yes—
21C22×Q8×C113—28no—
22C22×Q8×C113—22no—
23C22×Q8×C113—21no—
24C22×Q8×C113C2214yes—
25C22×Q8×C113C2212yes—
26C23×C4×C113—21no—
27C23×C4×C113C2211yescharacteristic
28#14268647—24no—
29C2×Dic226C2311yescharacteristic
30C2×D452C2311yescharacteristic
31C2×C22⋊D113—22no—
32C2×C22⋊D113—21no—
33C2×C4×D113—21no—
34#14268637—28no—
35#14268634—28no—
36#14268631—24no—
37C2×#11744751C2314yes—
38C4⋊Dic113C2312yes—
39#14268626—22no—
40C22⋊Dic113—21no—
41C22⋊Dic113—22no—
42#14268624—22no—
43C4×Dic113—21no—
44C24×C113D411yescharacteristic
45C2×Q8×C113—216no—
46C2×Q8×C113—28no—
47C2×Q8×C113—216no—
48C2×Q8×C113—24no—
49C2×Q8×C113—24no—
50C2×Q8×C113—22no—
51C2×Q8×C113D418yescomplemented
52C2×Q8×C113C2314yes—
53C22×C4×C113—28no—
54C22×C4×C113—24no—
55C22×C4×C113—22no—
56C22×C4×C113—24no—
57C22×C4×C113D412yes—
58C22×C4×C113D411yescharacteristic
59C22×C4×C113C2312yes—
60C22×C4×C113C2311yescharacteristic
61C23×C113—22no—
62C23×C113—24no—
63C23×C113C2×D411yescharacteristic
64C23×C113C2×D412yes—
65C22×D113—21no—
66Q8×C113—216no—
67Q8×C113—28no—
68Q8×C113—24no—
69Q8×C113C2×D418yes—
70C2×C4×C113—216no—
71C2×C4×C113—28no—
72C2×C4×C113—22no—
73C2×C4×C113—24no—
74C2×C4×C113—24no—
75C2×C4×C113—24no—
76C2×C4×C113C2×D412yes—
77C2×C4×C113C2×D411yescharacteristic
78C2×C4×C113C2×D414yes—
79C2×C4×C113C2411yescharacteristic
80C22⋊D113—42no—
81C22⋊D113—44no—
82C2×Dic113—22no—
83C2×Dic113—21no—
84D452—22no—
85C4×D113—42no—
86Dic226—22no—
87#11744751—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—28no—
98C4×C113—22no—
99C4×C113#12012yes—
100C4×C113C22×D412yes—
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×Q8D11311yescharacteristic, complemented
111C2×#137—1132no—
112#525—1131no—
113#451—1134no—
114#448—1134no—
115#419—1132no—
116#407—1131no—
117C2×#41—1131no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D11314yes—
122C22×Q8C2×D11312yes—
123C23×C4—21no—
124C23×C4C2×D11311yescharacteristic
125#137—2268no—
126C2×Q16—1134no—
127C2×SD16—1134no—
128C2×M16—1132no—
129#123—2261no—
130C22⋊Q8—1132no—
131#121—1132no—
132#120—2261no—
133C4×D4—2262no—
134#117—1131no—
135#103—1138no—
136C22⋊C8—1134no—
137C24C22⋊D11311yescharacteristic
138#41—2264no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—1131no—
146C2×Q8C22⋊D11318yescomplemented
147C2×Q8C22×D11314yes—
148C2×D4—1131no—
149C2×D4—2262no—
150C2×D4—2261no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—2261no—
156C22×C4C22⋊D11312yes—
157C22×C4C22⋊D11311yescharacteristic
158C22×C4C22×D11312yes—
159C22×C4C22×D11311yescharacteristic
160Q16—2268no—
161SD16—2268no—
162M16—2264no—
163C2×C8—1134no—
164C4⋊C4—1132no—
165C4⋊C4—2262no—
166C22⋊C4—2261no—
167C22⋊C4—2262no—
168C22⋊C4—2262no—
169C42—2261no—
170C23—22no—
171C23—24no—
172C23—2261no—
173C23C2×C22⋊D11311yescharacteristic
174C23C2×C22⋊D11312yes—
175Q8—216no—
176Q8—28no—
177Q8—24no—
178Q8—2262no—
179Q8C2×C22⋊D11318yes—
180D4—4522no—
181D4—4524no—
182D4—2262no—
183C2×C4—216no—
184C2×C4—28no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—4522no—
190C2×C4—2262no—
191C2×C4—2261no—
192C2×C4C2×C22⋊D11312yes—
193C2×C4C2×C22⋊D11311yescharacteristic
194C2×C4C2×C22⋊D11314yes—
195C2×C4C23×D11311yescharacteristic
196C8—2264no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—4521no—
202C22—4522no—
203C22#1681359112yes—
204C22#1681365314yescomplemented
205C22C22×C22⋊D11311yescharacteristic
206C4—28no—
207C4—22no—
208C4—4521no—
209C4—4522no—
210C4#1681359112yes—
211C4C22×C22⋊D11312yes—
212C2—22no—
213C2—24no—
214C2—9041no—
215C2C2×#1681359111yescharacteristic
216C2C2×#1681365312yes—
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC23×Q8×C113C2
socleC22×C113C22×D4
prime radical#20299109C2
hypercenterC23×Q8D113
hypocenterC113#3004
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16813653C2semidirect
3. #20298754C2semidirect
4. #20298737C2semidirect
5. #20298607C2semidirect
6. #20298562C2semidirect
7. C23×Q8×C113C2semidirect
10. C2×#14268637C22semidirect
11. C2×#14268634C22semidirect
19. #16813546C22semidirect
51. C2×Q8×C113D4semidirect
93. C22×C113#137semidirect
109. C113#3004semidirect
110. C23×Q8D113semidirect
146. C2×Q8C22⋊D113semidirect
204. C22#16813653semidirect

Maximal subgroups

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

subgroupindex
C2×#168136532
#202991092
#202987542
#202987372
#202986072
#202985622
C23×Q8×C1132
#3004113