Finite Group Databasebeta

← #448×D113 #28399759

Subgroups — #448×D113 #28399759

Order 14464.

792 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×Q8×D113C211yesmaximal, characteristic, complemented
2C2×SD16×D113C212yesmaximal, complemented
3#121×D113C211yesmaximal, characteristic
4#20298739C211yesmaximal, characteristic, complemented
5#20298738C212yesmaximal, complemented
6#20298737C211yesmaximal, characteristic, complemented
7#20298736C211yesmaximal, characteristic, complemented
8#20298735C212yesmaximal, complemented
9#103×D113C212yesmaximal, complemented
10C22⋊C8×D113C211yesmaximal, characteristic, complemented
11#448×C113C211yesmaximal, characteristic, complemented
12C22×Dic226—21no—
13C22×Dic226C2211yescharacteristic, complemented
14C2×Q8×D113—28no—
15C2×Q8×D113—21no—
16C2×Q8×D113—21no—
17C2×Q8×D113C2212yes—
18C2×Q8×D113C2212yescomplemented
19C2×D4×D113—21no—
20C2×D4×D113C2211yescharacteristic
21C22×C4×D113—21no—
22C22×C4×D113C2211yescharacteristic
23C2×#14268636C2212yescomplemented
24C2×#14268635C2212yescomplemented
25C2×#14268634C2212yescomplemented
26SD16×D113—28no—
27C2×C8×D113C2212yes—
28#16813597C2211yescharacteristic
29#16813596C2211yescharacteristic
30#16813595—21no—
31#16813594—21no—
32#16813593C2211yescharacteristic
33C4⋊C4×D113C2211yescharacteristic
34C22⋊C4×D113—21no—
35#16813547C2212yescomplemented
36#16813546C2212yescomplemented
37#16813545C2212yescomplemented
38C22⋊#11744751C2211yescharacteristic, complemented
39#16813534C2212yescomplemented
40C22×Q8×C113C2211yescharacteristic, complemented
41C2×SD16×C113C2212yescomplemented
42#121×C113C2211yescharacteristic
43#103×C113C2212yescomplemented
44C22⋊C8×C113C2211yescharacteristic, complemented
45C23×D113—21no—
46C23×D113D411yescharacteristic
47C2×Dic226—28no—
48C2×Dic226—21no—
49C2×Dic226—22no—
50C2×Dic226—22no—
51C2×Dic226—24no—
52C2×Dic226C2312yes—
53Q8×D113—28no—
54Q8×D113—28no—
55Q8×D113—24no—
56Q8×D113—24no—
57Q8×D113—24no—
58Q8×D113D418yescomplemented
59C2×D452—21no—
60C2×D452C2311yescharacteristic
61C2×C22⋊D113—21no—
62C2×C22⋊D113—21no—
63C2×C22⋊D113—21no—
64D4×D113—24no—
65D4×D113—44no—
66C22×Dic113—21no—
67C22×Dic113D411yescharacteristic
68C2×C4×D113—28no—
69C2×C4×D113—22no—
70C2×C4×D113—22no—
71C2×C4×D113—24no—
72C2×C4×D113—21no—
73C2×C4×D113D411yescharacteristic
74C2×C4×D113D411yescharacteristic
75C2×C4×D113C2311yescharacteristic
76C2×C4×D113C2311yescharacteristic
77#14268636—24no—
78#14268635—24no—
79#14268634—24no—
80C2×#11744751C2312yes—
81C8×D113—24no—
82C4⋊Dic113C2311yescharacteristic
83#14268626—21no—
84C22⋊Dic113—21no—
85#14268624—21no—
86#14268624—21no—
87C2×Q8×C113—24no—
88C2×Q8×C113—21no—
89C2×Q8×C113C2312yes—
90C2×D4×C113—21no—
91C2×D4×C113C2311yescharacteristic
92C22×C4×C113—21no—
93C22×C4×C113C2311yescharacteristic
94SD16×C113—24no—
95C2×C8×C113C2312yes—
96C4⋊C4×C113C2311yescharacteristic
97C22⋊C4×C113—21no—
98C23×C113—21no—
99C23×C113C2×D411yescharacteristic
100C22×D113—22no—
101C22×D113—22no—
102C22×D113—21no—
103C22×D113—42no—
104C22×D113—42no—
105C22×D113—42no—
106C22×D113SD1614yescomplemented
107C22×D113C2×D411yescharacteristic
108C22×D113C2×D411yescharacteristic
109Q8×C113—24no—
110Q8×C113—22no—
111Q8×C113C2×D414yescomplemented
112D4×C113—22no—
113D4×C113—42no—
114C2×C4×C113—24no—
115C2×C4×C113—22no—
116C2×C4×C113—22no—
117C2×C4×C113—21no—
118C2×C4×C113C2×D411yescharacteristic
119C2×C4×C113C2411yescharacteristic
120C22⋊D113—42no—
121C22⋊D113—42no—
122C22⋊D113—42no—
123C2×Dic113—24no—
124C2×Dic113—22no—
125C2×Dic113—22no—
126C2×Dic113—21no—
127C2×Dic113C2×D411yescharacteristic
128C2×Dic113C2×D411yescharacteristic
129D452—22no—
130D452—42no—
131C4×D113—28no—
132C4×D113—28no—
133C4×D113—22no—
134C4×D113—22no—
135C4×D113—22no—
136C4×D113—42no—
137C4×D113C2×D414yes—
138Dic226—28no—
139Dic226—24no—
140Dic226—22no—
141Dic226—28no—
142Dic226C2×D414yescomplemented
143C8×C113—22no—
144#11744751—22no—
145C22×C113—21no—
146C22×C113—21no—
147C22×C113—41no—
148C22×C113—41no—
149C22×C113—41no—
150C22×C113C2×SD1612yescomplemented
151C22×C113C22×D411yescharacteristic
152C2×D113—21no—
153C2×D113—21no—
154C2×D113—28no—
155C2×D113—41no—
156C2×D113—41no—
157C2×D113—41no—
158C2×D113—84no—
159C2×D113#12012yes—
160C2×D113C2×SD1612yes—
161C2×D113C2×SD1612yescomplemented
162C2×D113#13712yes—
163C4×C113—24no—
164C4×C113—21no—
165C4×C113—41no—
166C4×C113C22×D412yes—
167Dic113—21no—
168Dic113—24no—
169Dic113—41no—
170Dic113#12012yes—
171C2×C113—22no—
172C2×C113—81no—
173C2×C113C2×#12011yescharacteristic
174C2×C113C22×SD1611yescharacteristic
175C2×C113C2×#13711yescharacteristic
176D113—22no—
177D113—81no—
178D113#44814yesdirect product factor
179C2×#448—1131nomaximal, Sylow
180C113C2×#44811yescharacteristic, complemented, Sylow
181C23×Q8—1131no—
182C22×SD16—1132no—
183C2×#121—1131no—
184#448—1131no—
185#448—1132no—
186#448—1131no—
187#448—1132no—
188#448—1131no—
189#448D11311yescharacteristic, direct product factor
190C2×#103—1132no—
191C2×C22⋊C8—1131no—
192C22×Q8—1132no—
193C22×Q8—1132no—
194C22×Q8—1131no—
195C22×Q8—2261no—
196C22×Q8—2261no—
197C22×Q8—2261no—
198C22×Q8—2268no—
199C22×Q8C2×D11311yescharacteristic, complemented
200C22×D4—1131no—
201C22×D4—2261no—
202C23×C4—1131no—
203C23×C4—2261no—
204C2×SD16—1132no—
205C2×SD16—1132no—
206C2×SD16—1132no—
207C2×SD16—2268no—
208C2×SD16C2×D11312yescomplemented
209C22×C8—1132no—
210#121—1131no—
211#121—1131no—
212#121—1131no—
213#121—2261no—
214#121—2261no—
215#121C2×D11311yescharacteristic
216C2×C4⋊C4—1131no—
217C2×C22⋊C4—2261no—
218#103—1132no—
219#103—1132no—
220#103—1132no—
221#103C2×D11312yescomplemented
222C22⋊C8—1131no—
223C22⋊C8—1132no—
224C22⋊C8C2×D11311yescharacteristic, complemented
225C24—1131no—
226C24—2261no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—1138no—
230C2×Q8—1132no—
231C2×Q8—2262no—
232C2×Q8—2262no—
233C2×Q8—2264no—
234C2×Q8—2268no—
235C2×Q8—2264no—
236C2×Q8—2264no—
237C2×Q8—2268no—
238C2×Q8—2268no—
239C2×Q8—2264no—
240C2×Q8—2261no—
241C2×Q8C22×D11312yes—
242C2×D4—21no—
243C2×D4—1131no—
244C2×D4—4524no—
245C2×D4—2261no—
246C2×D4—2264no—
247C2×D4—2261no—
248C2×D4—2261no—
249C2×D4—2261no—
250C2×D4C22×D11311yescharacteristic
251C22×C4—21no—
252C22×C4—1131no—
253C22×C4—1131no—
254C22×C4—1131no—
255C22×C4—1131no—
256C22×C4—1131no—
257C22×C4—2264no—
258C22×C4—2261no—
259C22×C4—2262no—
260C22×C4—2262no—
261C22×C4—2268no—
262C22×C4—2261no—
263C22×C4C22×D11311yescharacteristic
264SD16—24no—
265SD16—2264no—
266SD16—2264no—
267SD16—2264no—
268C2×C8—1132no—
269C2×C8—2264no—
270C2×C8C22×D11312yes—
271C4⋊C4—1131no—
272C4⋊C4—2261no—
273C4⋊C4C22×D11311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2261no—
276C22⋊C4—2261no—
277C22⋊C4—2261no—
278C23—21no—
279C23—1131no—
280C23—1134no—
281C23—1131no—
282C23—4522no—
283C23—4522no—
284C23—4522no—
285C23—2262no—
286C23—2262no—
287C23—2261no—
288C23D4×D11311yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—1134no—
292Q8—2268no—
293Q8—2268no—
294Q8—2264no—
295Q8—2262no—
296Q8D4×D11314yescomplemented
297D4—22no—
298D4—42no—
299D4—4522no—
300D4—4522no—
301D4—4522no—
302D4—4522no—
303D4—2262no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—1131no—
309C2×C4—1134no—
310C2×C4—1131no—
311C2×C4—4522no—
312C2×C4—2262no—
313C2×C4—2262no—
314C2×C4—2268no—
315C2×C4—2268no—
316C2×C4—2262no—
317C2×C4—2261no—
318C2×C4—2264no—
319C2×C4—2262no—
320C2×C4—2262no—
321C2×C4D4×D11311yescharacteristic
322C2×C4C23×D11311yescharacteristic
323C8—22no—
324C8—2262no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—1132no—
331C22—1132no—
332C22—1132no—
333C22—1132no—
334C22—9044no—
335C22—4521no—
336C22—4521no—
337C22—4521no—
338C22—2268no—
339C22—2261no—
340C22—2261no—
341C22SD16×D11312yescomplemented
342C22C2×D4×D11311yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—1132no—
347C4—4521no—
348C4—2261no—
349C4—2264no—
350C4C2×D4×D11312yes—
351C2—22no—
352C2—81no—
353C2—1134no—
354C2—9041no—
355C2—2262no—
356C2#120×D11311yescharacteristic
357C2C2×SD16×D11311yescharacteristic
358C2#137×D11311yescharacteristic
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#448×C113C2
socleC22×C113C22×D4
prime radical#121×D113C2
hypercenter#448D113
hypocenterC113C2×#448
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D113C2semidirect
2. C2×SD16×D113C2semidirect
4. #20298739C2semidirect
5. #20298738C2semidirect
6. #20298737C2semidirect
7. #20298736C2semidirect
8. #20298735C2semidirect
9. #103×D113C2semidirect
10. C22⋊C8×D113C2semidirect
11. #448×C113C2semidirect
13. C22×Dic226C22semidirect
18. C2×Q8×D113C22semidirect
23. C2×#14268636C22semidirect
24. C2×#14268635C22semidirect
25. C2×#14268634C22semidirect
35. #16813547C22semidirect
36. #16813546C22semidirect
37. #16813545C22semidirect
38. C22⋊#11744751C22semidirect
39. #16813534C22semidirect
40. C22×Q8×C113C22semidirect
41. C2×SD16×C113C22semidirect
43. #103×C113C22semidirect
44. C22⋊C8×C113C22semidirect
58. Q8×D113D4semidirect
106. C22×D113SD16semidirect
111. Q8×C113C2×D4semidirect
142. Dic226C2×D4semidirect
150. C22×C113C2×SD16semidirect
161. C2×D113C2×SD16semidirect
178. D113#448direct (N × H)
180. C113C2×#448semidirect
189. #448D113direct (N × H)
199. C22×Q8C2×D113semidirect
208. C2×SD16C2×D113semidirect
221. #103C2×D113semidirect
224. C22⋊C8C2×D113semidirect
296. Q8D4×D113semidirect
341. C22SD16×D113semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1132
C2×SD16×D1132
#121×D1132
#202987392
#202987382
#202987372
#202987362
#202987352
#103×D1132
C22⋊C8×D1132
#448×C1132
C2×#448113