Finite Group Databasebeta

← #28402879

Subgroups — #28402879

Order 14464.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20299252C211yesmaximal, characteristic, complemented
2#20298948C211yesmaximal, characteristic, complemented
3#20298931C212yesmaximal, complemented
4#20298878C211yesmaximal, characteristic, complemented
5#20298846C212yesmaximal, complemented
6#20298800C211yesmaximal, characteristic, complemented
7#20298743C212yesmaximal, complemented
8C4×#14268632C211yesmaximal, characteristic, complemented
9#20298676C211yesmaximal, characteristic, complemented
10#20298599C212yesmaximal, complemented
11#547×C113C211yesmaximal, characteristic, complemented
12C2×#14268646—21no—
13C2×#14268634C2212yes—
14C2×#14268632C2211yescharacteristic
15C2×#14268631C2212yes—
16#16813622C2211yescharacteristic
17#16813606—22no—
18#16813602—21no—
19#16813596C2212yes—
20C4×D452C2211yescharacteristic
21C2×C4⋊Dic113C2212yes—
22#16813555C2212yes—
23#16813555C2211yescharacteristic
24C4⋊#11744751C2211yescharacteristic
25#16813546C2212yes—
26#16813546C2212yes—
27#16813546C2212yes—
28#16813543C2211yescharacteristic
29#16813543C2212yes—
30#16813542C2211yescharacteristic
31C22⋊#11744751C2212yes—
32C4×#11744751C2211yescharacteristic
33C2×#41×C113C2212yes—
34C4⋊Q8×C113C2211yescharacteristic
35#123×C113—22no—
36C22⋊Q8×C113C2212yes—
37#121×C113—21no—
38C4×D4×C113C2211yescharacteristic
39C4×D4×C113C2211yescharacteristic
40C2×C4⋊C4×C113—21no—
41#14268646—44no—
42C2×Dic226—21no—
43C2×D452C2311yescharacteristic
44C2×C22⋊D113—22no—
45C22×Dic113—22no—
46C2×C4×D113—21no—
47#14268634—42no—
48#14268632—22no—
49#14268631—42no—
50C2×#11744751C2311yescharacteristic
51C2×#11744751C2312yes—
52C2×#11744751C2311yescharacteristic
53C4⋊Dic113—22no—
54C4⋊Dic113—21no—
55C4⋊Dic113C2311yescharacteristic
56C4⋊Dic113C2312yes—
57#14268624—21no—
58#14268624—22no—
59#41×C113—24no—
60#41×C113—42no—
61C2×Q8×C113C2312yes—
62C2×D4×C113—22no—
63C2×D4×C113D411yescharacteristic
64C2×D4×C113C2311yescharacteristic
65C22×C4×C113—21no—
66C22×C4×C113—22no—
67C22×C4×C113C2312yes—
68C4⋊C4×C113—21no—
69C4⋊C4×C113—22no—
70C4⋊C4×C113—21no—
71C4⋊C4×C113D411yescharacteristic
72C4⋊C4×C113C2311yescharacteristic
73C22⋊C4×C113—22no—
74C22⋊C4×C113—21no—
75C22⋊C4×C113D412yes—
76C42×C113C2311yescharacteristic
77C23×C113—21no—
78C23×C113C2×D412yes—
79C22×D113—21no—
80Q8×C113—22no—
81Q8×C113—22no—
82D4×C113—22no—
83D4×C113—21no—
84D4×C113—44no—
85D4×C113#4112yescomplemented
86C2×C4×C113—22no—
87C2×C4×C113—22no—
88C2×C4×C113—21no—
89C2×C4×C113—22no—
90C2×C4×C113—44no—
91C2×C4×C113—42no—
92C2×C4×C113C2×D411yescharacteristic
93C2×C4×C113C2×D412yes—
94C2×C4×C113C2×D411yescharacteristic
95C2×C4×C113C2411yescharacteristic
96C22⋊D113—44no—
97C2×Dic113—21no—
98C2×Dic113—22no—
99C2×Dic113—44no—
100D452—21no—
101D452—21no—
102C4×D113—42no—
103Dic226—42no—
104#11744751—21no—
105#11744751—21no—
106#11744751—22no—
107C22×C113—22no—
108C22×C113—22no—
109C22×C113—22no—
110C22×C113—22no—
111C22×C113—41no—
112C22×C113—41no—
113C22×C113C22×D411yescharacteristic
114C2×D113—41no—
115C2×D113—41no—
116C2×D113—41no—
117C4×C113—22no—
118C4×C113—21no—
119C4×C113—42no—
120C4×C113—41no—
121C4×C113AGL(1,ℤ/8)12yes—
122C4×C113C2×#4111yescharacteristic
123C4×C113H(5,𝔽2)11yescharacteristic
124Dic113—42no—
125Dic113—41no—
126C2×C113—42no—
127C2×C113—42no—
128C2×C113#54611yescharacteristic
129C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
130C2×C113#57811yescharacteristic
131D113—81no—
132#3330—1131nomaximal, Sylow
133C113#333011yescharacteristic, complemented, Sylow
134#547—1131no—
135#547D11311yescharacteristic, complemented
136#487—1131no—
137#484—1132no—
138#474—1131no—
139#468—1132no—
140#461—1131no—
141#449—1132no—
142C4×D8—1131no—
143#436—1131no—
144#417—1132no—
145C2×#41—2261no—
146C2×#41C2×D11312yes—
147C2×SD16—1132no—
148C2×D8—1131no—
149C2×M16—1132no—
150C4⋊Q8—1131no—
151C4⋊Q8C2×D11311yescharacteristic
152#123—22no—
153#123—2262no—
154C22⋊Q8—2261no—
155C22⋊Q8C2×D11312yes—
156#121—21no—
157#121—1132no—
158C4×D4—1131no—
159C4×D4C2×D11311yescharacteristic
160C4×D4C2×D11311yescharacteristic
161C2×C4⋊C4—21no—
162C2×C4⋊C4—1132no—
163#107—1131no—
164#107—1132no—
165C4⋊C8—1131no—
166#103—1132no—
167#103—1132no—
168#103—1132no—
169#102—1131no—
170#102—1132no—
171#102—1131no—
172C22⋊C8—1132no—
173C4×C8—1131no—
174#41—24no—
175#41—42no—
176#41—4524no—
177C2×Q8—2261no—
178C2×Q8C22×D11312yes—
179C2×D4—22no—
180C2×D4—1131no—
181C2×D4—2262no—
182C2×D4C22⋊D11311yescharacteristic
183C2×D4C22×D11311yescharacteristic
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—2262no—
187C22×C4—2261no—
188C22×C4C22×D11312yes—
189SD16—4522no—
190D8—2262no—
191M16—4522no—
192C2×C8—1131no—
193C2×C8—1131no—
194C2×C8—1132no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—21no—
198C4⋊C4—1131no—
199C4⋊C4—1132no—
200C4⋊C4—2262no—
201C4⋊C4—2261no—
202C4⋊C4C22⋊D11311yescharacteristic
203C4⋊C4C22×D11311yescharacteristic
204C22⋊C4—22no—
205C22⋊C4—21no—
206C22⋊C4—2261no—
207C22⋊C4—2262no—
208C22⋊C4C22⋊D11312yes—
209C42C22×D11311yescharacteristic
210C23—21no—
211C23—2261no—
212C23C2×C22⋊D11312yes—
213Q8—22no—
214Q8—22no—
215Q8—4522no—
216D4—22no—
217D4—21no—
218D4—44no—
219D4—4524no—
220D4—2261no—
221D4—2261no—
222D4#1426864612yescomplemented
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—4524no—
230C2×C4—4522no—
231C2×C4—2261no—
232C2×C4—2262no—
233C2×C4C2×C22⋊D11311yescharacteristic
234C2×C4C2×C22⋊D11312yes—
235C2×C4C2×C22⋊D11311yescharacteristic
236C2×C4C23×D11311yescharacteristic
237C8—2261no—
238C8—2261no—
239C8—2262no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—4521no—
247C22—4521no—
248C22—4521no—
249C22C22×C22⋊D11311yescharacteristic
250C4—21no—
251C4—22no—
252C4—41no—
253C4—42no—
254C4—4521no—
255C4—4522no—
256C4#1681364712yes—
257C4C2×#1426864611yescharacteristic
258C4#1681366711yescharacteristic
259C2—42no—
260C2—42no—
261C2—9041no—
262C2#2029924211yescharacteristic
263C2C2×#1681364711yescharacteristic
264C2#2029941311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#547×C113C2
socleC22×C113C22×D4
hypercenter#547D113
hypocenterC113#3330
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20299252C2semidirect
2. #20298948C2semidirect
3. #20298931C2semidirect
4. #20298878C2semidirect
5. #20298846C2semidirect
6. #20298800C2semidirect
7. #20298743C2semidirect
8. C4×#14268632C2semidirect
9. #20298676C2semidirect
10. #20298599C2semidirect
11. #547×C113C2semidirect
85. D4×C113#41semidirect
133. C113#3330semidirect
135. #547D113semidirect
222. D4#14268646semidirect

Maximal subgroups

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

subgroupindex
#202992522
#202989482
#202989312
#202988782
#202988462
#202988002
#202987432
C4×#142686322
#202986762
#202985992
#547×C1132
#3330113