Finite Group Databasebeta

← #28401347

Subgroups — #28401347

Order 14464.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16813598C211yesmaximal, characteristic
2C22⋊Q8×D113C211yesmaximal, characteristic
3#20298934C211yesmaximal, characteristic, complemented
4#20298932C212yesmaximal, complemented
5#20298931C211yesmaximal, characteristic, complemented
6#20298838C212yesmaximal, complemented
7#20298836C212yesmaximal, complemented
8#20298754C211yesmaximal, characteristic, complemented
9#20298752C212yesmaximal, complemented
10C22⋊C8×D113C211yesmaximal, characteristic
11#451×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113C2211yescharacteristic
13C22×C4×D113C2211yescharacteristic
14C2×#14268637C2212yescomplemented
15C2×C8×D113C2212yes—
16#16813602C2211yescharacteristic
17#16813601C2211yescharacteristic
18#16813600—21no—
19#16813599—21no—
20#16813598—21no—
21#16813598—21no—
22#16813598C2212yes—
23#16813598C2211yescharacteristic
24#16813598C2212yes—
25C2×C4⋊Dic113C2211yescharacteristic
26C2×#14268626—21no—
27C4⋊C4×D113—21no—
28C4⋊C4×D113C2211yescharacteristic
29C2×#14268624—21no—
30C22⋊C4×D113—21no—
31#16813556C2212yescomplemented
32#16813555C2212yescomplemented
33#16813547C2212yescomplemented
34#16813546C2212yescomplemented
35#16813546C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic
37#16813534C2212yes—
38C22×Q8×C113C2211yescharacteristic
39C2×Q16×C113C2212yescomplemented
40C22⋊Q8×C113C2211yescharacteristic
41#103×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113D411yescharacteristic
44C2×Dic226—21no—
45C2×Dic226C2311yescharacteristic
46Q8×D113—42no—
47C22×Dic113—21no—
48C22×Dic113D411yescharacteristic
49C2×C4×D113—21no—
50C2×C4×D113—21no—
51C2×C4×D113—42no—
52C2×C4×D113D411yescharacteristic
53C2×C4×D113D411yescharacteristic
54C2×C4×D113C2311yescharacteristic
55C2×C4×D113C2311yescharacteristic
56#14268637—42no—
57C2×#11744751C2312yes—
58C8×D113—42no—
59C4⋊Dic113—21no—
60C4⋊Dic113—21no—
61C4⋊Dic113C2311yescharacteristic
62C4⋊Dic113C2312yes—
63#14268626—21no—
64#14268626—21no—
65#14268626—21no—
66#14268626—22no—
67#14268626—22no—
68C22⋊Dic113—21no—
69#14268624—21no—
70#14268624—21no—
71#14268624—22no—
72#14268624—22no—
73C2×Q8×C113—21no—
74C2×Q8×C113—42no—
75C2×Q8×C113C2312yes—
76C2×Q8×C113C2311yescharacteristic
77C22×C4×C113—21no—
78C22×C4×C113C2311yescharacteristic
79Q16×C113—42no—
80C2×C8×C113C2312yes—
81C4⋊C4×C113—21no—
82C4⋊C4×C113C2311yescharacteristic
83C22⋊C4×C113—21no—
84C23×C113C2×D411yescharacteristic
85C22×D113—41no—
86C22×D113—41no—
87C22×D113Q1614yescomplemented
88C22×D113C2×D411yescharacteristic
89C22×D113C2×D411yescharacteristic
90Q8×C113—22no—
91Q8×C113—22no—
92Q8×C113—22no—
93Q8×C113—42no—
94C2×C4×C113—22no—
95C2×C4×C113—21no—
96C2×C4×C113—21no—
97C2×C4×C113—22no—
98C2×C4×C113—42no—
99C2×C4×C113C2×D411yescharacteristic
100C2×C4×C113C2411yescharacteristic
101C2×Dic113—22no—
102C2×Dic113—21no—
103C2×Dic113—21no—
104C2×Dic113—41no—
105C2×Dic113—42no—
106C2×Dic113C2×D411yescharacteristic
107C2×Dic113C2×D411yescharacteristic
108C4×D113—22no—
109C4×D113—22no—
110C4×D113—41no—
111C4×D113—41no—
112C4×D113—42no—
113C4×D113—42no—
114Dic226—22no—
115Dic226—42no—
116C8×C113—22no—
117#11744751—22no—
118C22×C113—21no—
119C22×C113—21no—
120C22×C113C2×Q1612yes—
121C22×C113C22×D411yescharacteristic
122C2×D113—21no—
123C2×D113—21no—
124C2×D113—21no—
125C2×D113—21no—
126C2×D113—44no—
127C2×D113C2×Q1612yes—
128C2×D113C2×Q1612yes—
129C4×C113—21no—
130C4×C113—41no—
131C4×C113—42no—
132C4×C113—41no—
133C4×C113H(5,𝔽2)12yes—
134Dic113—21no—
135Dic113—21no—
136Dic113—41no—
137Dic113—41no—
138Dic113—42no—
139C2×C113—22no—
140C2×C113#53411yescharacteristic
141C2×C113C22×Q1611yescharacteristic
142C2×C113#57511yescharacteristic
143D113—22no—
144D113—22no—
145#3189—1131nomaximal, Sylow
146C113#318911yescharacteristic, complemented, Sylow
147C2×C22⋊Q8—1131no—
148C2×C22⋊Q8—1131no—
149#484—1131no—
150#484—1132no—
151#484—1131no—
152#467—1132no—
153#467—1132no—
154#451—1131no—
155#451—1132no—
156#451D11311yescharacteristic, complemented
157C2×C22⋊C8—1131no—
158C22×Q8—1131no—
159C22×Q8C2×D11311yescharacteristic
160C23×C4—1131no—
161C2×Q16—1132no—
162C2×Q16C2×D11312yescomplemented
163C22×C8—1132no—
164C22⋊Q8—1132no—
165C22⋊Q8—1131no—
166C22⋊Q8—1132no—
167C22⋊Q8—1131no—
168C22⋊Q8—1131no—
169C22⋊Q8—2261no—
170C22⋊Q8—2261no—
171C22⋊Q8—2261no—
172C22⋊Q8—2261no—
173C22⋊Q8C2×D11311yescharacteristic
174C2×C4⋊C4—1131no—
175C2×C4⋊C4—1131no—
176C2×C4⋊C4—2261no—
177C2×C4⋊C4—2261no—
178C2×C22⋊C4—2261no—
179C2×C22⋊C4—2261no—
180#107—1132no—
181#107—1132no—
182#103—1132no—
183#103—1132no—
184#103—1132no—
185#103C2×D11312yescomplemented
186C22⋊C8—1131no—
187C22⋊C8—1132no—
188C22⋊C8C2×D11311yescharacteristic
189C24—1131no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—1131no—
193C2×Q8—4522no—
194C2×Q8—2261no—
195C2×Q8C22×D11311yescharacteristic
196C2×Q8C22×D11312yes—
197C22×C4—21no—
198C22×C4—1131no—
199C22×C4—1131no—
200C22×C4—1131no—
201C22×C4—1131no—
202C22×C4—1131no—
203C22×C4—4522no—
204C22×C4—2261no—
205C22×C4—2261no—
206C22×C4—2261no—
207C22×C4C22×D11311yescharacteristic
208Q16—42no—
209Q16—4522no—
210C2×C8—1132no—
211C2×C8—4522no—
212C2×C8C22×D11312yes—
213C4⋊C4—21no—
214C4⋊C4—1131no—
215C4⋊C4—1132no—
216C4⋊C4—2261no—
217C4⋊C4—2261no—
218C4⋊C4—2262no—
219C4⋊C4—2262no—
220C4⋊C4—2261no—
221C4⋊C4—2261no—
222C4⋊C4—2261no—
223C4⋊C4C22×D11311yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—2261no—
226C22⋊C4—2261no—
227C22⋊C4—2262no—
228C22⋊C4—2262no—
229C22⋊C4—2261no—
230C23—1134no—
231C23—1131no—
232C23—1131no—
233C23—4521no—
234C23—4521no—
235C23D4×D11311yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—4522no—
241Q8—2262no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—1131no—
248C2×C4—1131no—
249C2×C4—4521no—
250C2×C4—4521no—
251C2×C4—4522no—
252C2×C4—4522no—
253C2×C4—4522no—
254C2×C4—4521no—
255C2×C4—2262no—
256C2×C4—2261no—
257C2×C4—2261no—
258C2×C4—2262no—
259C2×C4—2262no—
260C2×C4D4×D11311yescharacteristic
261C2×C4C23×D11311yescharacteristic
262C8—22no—
263C8—2262no—
264C22—21no—
265C22—21no—
266C22—1132no—
267C22—1132no—
268C22—4524no—
269C22—2261no—
270C22—2261no—
271C22—2261no—
272C22—2261no—
273C22Q16×D11312yes—
274C22C2×D4×D11311yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—4521no—
280C4—4522no—
281C4—4521no—
282C4—2261no—
283C4—2261no—
284C4#1681366712yes—
285C2—22no—
286C2—2262no—
287C2—2262no—
288C2#2029915811yescharacteristic
289C2C2×Q16×D11311yescharacteristic
290C2#2029939711yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#451×C113C2
socleC22×C113C22×D4
prime radicalC23×D113D4
hypercenter#451D113
hypocenterC113#3189
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #20298934C2semidirect
4. #20298932C2semidirect
5. #20298931C2semidirect
6. #20298838C2semidirect
7. #20298836C2semidirect
8. #20298754C2semidirect
9. #20298752C2semidirect
11. #451×C113C2semidirect
14. C2×#14268637C22semidirect
31. #16813556C22semidirect
32. #16813555C22semidirect
33. #16813547C22semidirect
34. #16813546C22semidirect
35. #16813546C22semidirect
39. C2×Q16×C113C22semidirect
41. #103×C113C22semidirect
87. C22×D113Q16semidirect
146. C113#3189semidirect
156. #451D113semidirect
162. C2×Q16C2×D113semidirect
185. #103C2×D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168135982
C22⋊Q8×D1132
#202989342
#202989322
#202989312
#202988382
#202988362
#202987542
#202987522
C22⋊C8×D1132
#451×C1132
#3189113