Finite Group Databasebeta

← #28401348

Subgroups — #28401348

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×#16813602C211yesmaximal, characteristic
2C22⋊Q8×D113C211yesmaximal, characteristic
3#20298935C212yesmaximal, complemented
4#20298933C211yesmaximal, characteristic, complemented
5#20298840C212yesmaximal, complemented
6#20298838C212yesmaximal, complemented
7#20298756C211yesmaximal, characteristic, complemented
8#20298755C212yesmaximal, complemented
9#20298753C211yesmaximal, characteristic, complemented
10C22⋊C8×D113C211yesmaximal, characteristic
11#484×C113C211yesmaximal, characteristic, complemented
12C22×Dic226C2211yescharacteristic
13C2×Q8×D113C2211yescharacteristic
14C22×C4×D113C2211yescharacteristic
15C2×Dic452C2212yescomplemented
16C2×#14268637C2212yescomplemented
17C2×C8×D113C2212yes—
18#16813602—21no—
19#16813602—21no—
20#16813602C2212yes—
21#16813602C2211yescharacteristic
22#16813602C2212yes—
23#16813601C2211yescharacteristic
24#16813600—21no—
25#16813599—21no—
26#16813598C2211yescharacteristic
27C2×C4⋊Dic113—21no—
28C4⋊C4×D113—21no—
29C4⋊C4×D113C2211yescharacteristic
30C2×#14268624—21no—
31C22⋊C4×D113—21no—
32#16813555C2212yescomplemented
33#16813547C2212yescomplemented
34#16813545C2212yescomplemented
35#16813545C2212yescomplemented
36C22⋊#11744751C2211yescharacteristic
37#16813534C2212yes—
38C22⋊Q8×C113C2211yescharacteristic
39C2×C4⋊C4×C113C2211yescharacteristic
40#107×C113C2212yescomplemented
41#103×C113C2212yescomplemented
42C22⋊C8×C113C2211yescharacteristic
43C23×D113D411yescharacteristic
44C2×Dic226—21no—
45C2×Dic226—21no—
46C2×Dic226—42no—
47C2×Dic226C2311yescharacteristic
48C2×Dic226C2312yes—
49Q8×D113—42no—
50C22×Dic113—21no—
51C22×Dic113D411yescharacteristic
52C2×C4×D113—21no—
53C2×C4×D113—21no—
54C2×C4×D113—42no—
55C2×C4×D113D411yescharacteristic
56C2×C4×D113D411yescharacteristic
57C2×C4×D113C2311yescharacteristic
58C2×C4×D113C2311yescharacteristic
59Dic452—42no—
60#14268637—42no—
61C2×#11744751C2312yes—
62C8×D113—42no—
63C4⋊Dic113—21no—
64C4⋊Dic113—22no—
65C4⋊Dic113—22no—
66C4⋊Dic113C2311yescharacteristic
67#14268626—21no—
68#14268626—21no—
69#14268626—21no—
70C22⋊Dic113—21no—
71#14268624—21no—
72#14268624—21no—
73#14268624—22no—
74#14268624—22no—
75C2×Q8×C113C2311yescharacteristic
76C22×C4×C113—21no—
77C22×C4×C113C2311yescharacteristic
78C2×C8×C113C2312yes—
79C4⋊C4×C113—21no—
80C4⋊C4×C113—21no—
81C4⋊C4×C113C2311yescharacteristic
82C4⋊C4×C113C2312yes—
83C22⋊C4×C113—21no—
84C23×C113C2×D411yescharacteristic
85C22×D113—41no—
86C22×D113—41no—
87C22×D113Q1614yescomplemented
88C22×D113C2×D411yescharacteristic
89C22×D113C2×D411yescharacteristic
90Q8×C113—22no—
91C2×C4×C113—22no—
92C2×C4×C113—21no—
93C2×C4×C113—21no—
94C2×C4×C113—22no—
95C2×C4×C113—42no—
96C2×C4×C113C2×D411yescharacteristic
97C2×C4×C113C2411yescharacteristic
98C2×Dic113—22no—
99C2×Dic113—21no—
100C2×Dic113—21no—
101C2×Dic113—41no—
102C2×Dic113—42no—
103C2×Dic113C2×D411yescharacteristic
104C2×Dic113C2×D411yescharacteristic
105C4×D113—22no—
106C4×D113—22no—
107C4×D113—41no—
108C4×D113—41no—
109C4×D113—42no—
110C4×D113—42no—
111Dic226—22no—
112Dic226—22no—
113Dic226—22no—
114Dic226—42no—
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—41no—
132C4×C113—42no—
133C4×C113H(5,𝔽2)12yes—
134Dic113—21no—
135Dic113—21no—
136Dic113—41no—
137Dic113—42no—
138Dic113—41no—
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#484D11311yescharacteristic, complemented
152#467—1132no—
153#467—1132no—
154#451—1131no—
155#451—1132no—
156#451—1131no—
157C2×C22⋊C8—1131no—
158C22×Q8—1131no—
159C22×Q8—1131no—
160C23×C4—1131no—
161C2×Q16—1132no—
162C2×Q16—1132no—
163C22×C8—1132no—
164C22⋊Q8—1131no—
165C22⋊Q8—1131no—
166C22⋊Q8—1132no—
167C22⋊Q8—1131no—
168C22⋊Q8—1132no—
169C22⋊Q8—2261no—
170C22⋊Q8—2261no—
171C22⋊Q8—2261no—
172C22⋊Q8—2261no—
173C22⋊Q8C2×D11311yescharacteristic
174C2×C4⋊C4—1131no—
175C2×C4⋊C4—2261no—
176C2×C4⋊C4—2261no—
177C2×C4⋊C4C2×D11311yescharacteristic
178C2×C22⋊C4—2261no—
179C2×C22⋊C4—2261no—
180#107—1132no—
181#107C2×D11312yescomplemented
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—1132no—
191C2×Q8—1131no—
192C2×Q8—4522no—
193C2×Q8—4522no—
194C2×Q8—2261no—
195C2×Q8—2261no—
196C2×Q8C22×D11311yescharacteristic
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—4522no—
209Q16—4522no—
210C2×C8—1132no—
211C2×C8—4522no—
212C2×C8C22×D11312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—1131no—
216C4⋊C4—2261no—
217C4⋊C4—2261no—
218C4⋊C4—2261no—
219C4⋊C4—2262no—
220C4⋊C4—2262no—
221C4⋊C4—2261no—
222C4⋊C4C22×D11311yescharacteristic
223C4⋊C4C22×D11312yes—
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—4522no—
238Q8—4522no—
239Q8—2262no—
240Q8—2262no—
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—2261no—
256C2×C4—2262no—
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—42no—
277C4—41no—
278C4—41no—
279C4—4521no—
280C4—4521no—
281C4—4522no—
282C4—2261no—
283C4—2261no—
284C4#1681366812yes—
285C2—22no—
286C2—2262no—
287C2—2262no—
288C2#2029915911yescharacteristic
289C2C2×Q16×D11311yescharacteristic
290C2#2029939411yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#484×C113C2
socleC22×C113C22×D4
prime radicalC23×D113D4
hypercenter#484D113
hypocenterC113#3189
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20298935C2semidirect
4. #20298933C2semidirect
5. #20298840C2semidirect
6. #20298838C2semidirect
7. #20298756C2semidirect
8. #20298755C2semidirect
9. #20298753C2semidirect
11. #484×C113C2semidirect
15. C2×Dic452C22semidirect
16. C2×#14268637C22semidirect
32. #16813555C22semidirect
33. #16813547C22semidirect
34. #16813545C22semidirect
35. #16813545C22semidirect
40. #107×C113C22semidirect
41. #103×C113C22semidirect
87. C22×D113Q16semidirect
146. C113#3189semidirect
151. #484D113semidirect
181. #107C2×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×#168136022
C22⋊Q8×D1132
#202989352
#202989332
#202988402
#202988382
#202987562
#202987552
#202987532
C22⋊C8×D1132
#484×C1132
#3189113