Finite Group Databasebeta

← #28401441

Subgroups — #28401441

Order 14464.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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#20299162C211yesmaximal, characteristic
2#20298936C212yesmaximal, complemented
3#20298922C212yesmaximal, complemented
4#20298846C212yesmaximal, complemented
5#20298829C212yesmaximal, complemented
6#20298748C212yesmaximal, complemented
7#20298737C212yesmaximal, complemented
8#20298565C211yesmaximal, characteristic, complemented
9#536×C113C211yesmaximal, characteristic, complemented
10C22×D452C2211yescharacteristic
11C2×#14268634C2212yes—
12C2×#14268634C2212yes—
13C2×#14268631C2211yescharacteristic, complemented
14C22×#11744751C2211yescharacteristic, complemented
15#16813618—21no—
16#16813611—21no—
17#16813596C2212yes—
18#16813596C2212yes—
19#16813590—21no—
20#16813590—21no—
21#16813581C2211yescharacteristic
22#16813555C2212yescomplemented
23#16813553C2212yescomplemented
24#16813546C2212yescomplemented
25#16813546C2212yescomplemented
26#16813542C2212yes—
27#16813542C2212yes—
28C22⋊#11744751C2212yes—
29C22⋊#11744751C2212yescomplemented
30C2×#41×C113C2211yescharacteristic
31C22×Q8×C113C2211yescharacteristic, complemented
32C4⋊Q8×C113—21no—
33#124×C113—21no—
34#123×C113—21no—
35#123×C113—21no—
36C22⋊Q8×C113C2212yes—
37C22⋊Q8×C113C2212yescomplemented
38#117×C113C2211yescharacteristic, complemented
39C23×D113—21no—
40C2×D452—21no—
41C2×D452—41no—
42C2×D452—41no—
43C2×D452C2312yes—
44C2×C22⋊D113—22no—
45C2×C22⋊D113—22no—
46#14268634—42no—
47#14268634—42no—
48#14268631—41no—
49C2×#11744751—41no—
50C2×#11744751C2312yes—
51C2×#11744751C2312yes—
52C4⋊Dic113C2312yes—
53C22⋊Dic113—21no—
54#14268624—22no—
55#14268624—22no—
56C4×Dic113—21no—
57#41×C113—41no—
58#41×C113—41no—
59C2×Q8×C113—21no—
60C2×Q8×C113—41no—
61C2×Q8×C113—41no—
62C2×Q8×C113D411yescharacteristic
63C2×Q8×C113C2312yes—
64C2×D4×C113D411yescharacteristic
65C2×D4×C113D411yescharacteristic
66C2×D4×C113D411yescharacteristic
67C22×C4×C113—21no—
68C22×C4×C113C2311yescharacteristic
69C22×C4×C113C2311yescharacteristic
70C22×C4×C113C2311yescharacteristic
71C4⋊C4×C113—22no—
72C4⋊C4×C113—22no—
73C4⋊C4×C113C2312yes—
74C22⋊C4×C113—21no—
75C22⋊C4×C113—22no—
76C22⋊C4×C113—22no—
77C42×C113—21no—
78C23×C113C2×D411yescharacteristic
79C23×C113C2×D411yescharacteristic
80C23×C113C2×D411yescharacteristic
81C22×D113—22no—
82C22×D113—42no—
83C22×D113—82no—
84Q8×C113—22no—
85Q8×C113—22no—
86Q8×C113—41no—
87Q8×C113—42no—
88D4×C113—41no—
89D4×C113—41no—
90D4×C113—41no—
91C2×C4×C113—21no—
92C2×C4×C113—21no—
93C2×C4×C113—21no—
94C2×C4×C113—21no—
95C2×C4×C113—21no—
96C2×C4×C113—21no—
97C2×C4×C113—22no—
98C2×C4×C113—22no—
99C2×C4×C113—42no—
100C2×C4×C113C2×D411yescharacteristic
101C2×C4×C113C2×D411yescharacteristic
102C2×C4×C113C2×D411yescharacteristic
103C2×C4×C113C2411yescharacteristic
104C22⋊D113—82no—
105C22⋊D113—82no—
106C2×Dic113—22no—
107D452—22no—
108D452—22no—
109D452—42no—
110#11744751—22no—
111#11744751—22no—
112C22×C113—21no—
113C22×C113—21no—
114C22×C113—21no—
115C22×C113—22no—
116C22×C113—22no—
117C22×C113—22no—
118C22×C113C22×D411yescharacteristic
119C2×D113—42no—
120C2×D113—42no—
121C2×D113—42no—
122C2×D113—84no—
123C4×C113—21no—
124C4×C113—21no—
125C4×C113—21no—
126C4×C113—42no—
127C4×C113—42no—
128C4×C113H(5,𝔽2)11yescharacteristic
129C4×C113H(5,𝔽2)11yescharacteristic
130Dic113—42no—
131C2×C113—41no—
132C2×C113—41no—
133C2×C113—41no—
134C2×C113#53411yescharacteristic
135C2×C113#57712yes—
136D113—82no—
137#3200—1131nomaximal, Sylow
138C113#320011yescharacteristic, complemented, Sylow
139#536D11311yescharacteristic, complemented
140#535—1131no—
141#485—1132no—
142#482—1132no—
143#468—1132no—
144#465—1132no—
145#450—1132no—
146#448—1132no—
147#408—1131no—
148C2×#41C2×D11311yescharacteristic
149C22×Q8C2×D11311yescharacteristic, complemented
150C22×D4—1131no—
151C2×SD16—1132no—
152C2×SD16—1132no—
153C2×M16—1131no—
154C22×C8—1131no—
155C4⋊Q8—21no—
156#127—2261no—
157#124—21no—
158#124—2261no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D11312yes—
162C22⋊Q8C2×D11312yescomplemented
163#121—1132no—
164#121—1132no—
165#120—2261no—
166#120—2261no—
167#117—1131no—
168#117C2×D11311yescharacteristic, complemented
169#107—1132no—
170#106—1132no—
171#103—1132no—
172#103—1132no—
173#102—1132no—
174#102—1132no—
175C22⋊C8—1132no—
176C22⋊C8—1132no—
177C24—2261no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D11311yescharacteristic
184C2×Q8C22×D11312yes—
185C2×D4—1132no—
186C2×D4—4521no—
187C2×D4—4521no—
188C2×D4—2262no—
189C2×D4—2262no—
190C2×D4—2261no—
191C2×D4C22⋊D11311yescharacteristic
192C2×D4C22⋊D11311yescharacteristic
193C2×D4C22⋊D11311yescharacteristic
194C22×C4—21no—
195C22×C4C22×D11311yescharacteristic
196C22×C4C22×D11311yescharacteristic
197C22×C4C22×D11311yescharacteristic
198SD16—4522no—
199SD16—4522no—
200M16—4521no—
201C2×C8—1132no—
202C2×C8—1132no—
203C2×C8—4521no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—1132no—
207C4⋊C4C22×D11312yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—2261no—
212C22⋊C4—2262no—
213C22⋊C4—2262no—
214C42—21no—
215C42—2261no—
216C23—9042no—
217C23—4522no—
218C23—2262no—
219C23C2×C22⋊D11311yescharacteristic
220C23C2×C22⋊D11311yescharacteristic
221C23C2×C22⋊D11311yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—9042no—
230D4—9042no—
231D4—4522no—
232D4—2262no—
233D4—2262no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—2262no—
244C2×C4C2×C22⋊D11311yescharacteristic
245C2×C4C2×C22⋊D11311yescharacteristic
246C2×C4C2×C22⋊D11311yescharacteristic
247C2×C4C23×D11311yescharacteristic
248C8—2262no—
249C8—2262no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—9044no—
257C22—4522no—
258C22—4522no—
259C22—4522no—
260C22C22×C22⋊D11311yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—4522no—
267C4#1681366711yescharacteristic
268C4#1681366711yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—9042no—
273C2#2029915711yescharacteristic
274C2#2029940612yes—
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#536×C113C2
socleC22×C113C22×D4
prime radical#20299162C2
hypercenter#536D113
hypocenterC113#3200
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20298936C2semidirect
3. #20298922C2semidirect
4. #20298846C2semidirect
5. #20298829C2semidirect
6. #20298748C2semidirect
7. #20298737C2semidirect
8. #20298565C2semidirect
9. #536×C113C2semidirect
13. C2×#14268631C22semidirect
14. C22×#11744751C22semidirect
22. #16813555C22semidirect
23. #16813553C22semidirect
24. #16813546C22semidirect
25. #16813546C22semidirect
29. C22⋊#11744751C22semidirect
31. C22×Q8×C113C22semidirect
37. C22⋊Q8×C113C22semidirect
38. #117×C113C22semidirect
138. C113#3200semidirect
139. #536D113semidirect
149. C22×Q8C2×D113semidirect
162. C22⋊Q8C2×D113semidirect
168. #117C2×D113semidirect

Maximal subgroups

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

subgroupindex
#202991622
#202989362
#202989222
#202988462
#202988292
#202987482
#202987372
#202985652
#536×C1132
#3200113