Finite Group Databasebeta

← #448×D107 #28345646

Subgroups — #448×D107 #28345646

Order 13696.

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×D107C211yesmaximal, characteristic, complemented
2C2×SD16×D107C212yesmaximal, complemented
3#121×D107C211yesmaximal, characteristic
4#20292768C211yesmaximal, characteristic, complemented
5#20292767C212yesmaximal, complemented
6#20292766C211yesmaximal, characteristic, complemented
7#20292765C211yesmaximal, characteristic, complemented
8#20292764C212yesmaximal, complemented
9#103×D107C212yesmaximal, complemented
10C22⋊C8×D107C211yesmaximal, characteristic, complemented
11#448×C107C211yesmaximal, characteristic, complemented
12C22×Dic214—21no—
13C22×Dic214C2211yescharacteristic, complemented
14C2×Q8×D107—28no—
15C2×Q8×D107—21no—
16C2×Q8×D107—21no—
17C2×Q8×D107C2212yes—
18C2×Q8×D107C2212yescomplemented
19C2×D4×D107—21no—
20C2×D4×D107C2211yescharacteristic
21C22×C4×D107—21no—
22C22×C4×D107C2211yescharacteristic
23C2×#13133807C2212yescomplemented
24C2×#13133806C2212yescomplemented
25C2×#13133805C2212yescomplemented
26SD16×D107—28no—
27C2×C8×D107C2212yes—
28#16812084C2211yescharacteristic
29#16812083C2211yescharacteristic
30#16812082—21no—
31#16812081—21no—
32#16812080C2211yescharacteristic
33C4⋊C4×D107C2211yescharacteristic
34C22⋊C4×D107—21no—
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812032C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic, complemented
39(C2×D107)⋊C8C2212yescomplemented
40C22×Q8×C107C2211yescharacteristic, complemented
41C2×SD16×C107C2212yescomplemented
42#121×C107C2211yescharacteristic
43#103×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic, complemented
45C23×D107—21no—
46C23×D107D411yescharacteristic
47C2×Dic214—28no—
48C2×Dic214—21no—
49C2×Dic214—22no—
50C2×Dic214—22no—
51C2×Dic214—24no—
52C2×Dic214C2312yes—
53Q8×D107—28no—
54Q8×D107—28no—
55Q8×D107—24no—
56Q8×D107—24no—
57Q8×D107—24no—
58Q8×D107D418yescomplemented
59C2×D428—21no—
60C2×D428C2311yescharacteristic
61C2×C22⋊D107—21no—
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64D4×D107—24no—
65D4×D107—44no—
66C22×Dic107—21no—
67C22×Dic107D411yescharacteristic
68C2×C4×D107—28no—
69C2×C4×D107—22no—
70C2×C4×D107—22no—
71C2×C4×D107—24no—
72C2×C4×D107—21no—
73C2×C4×D107D411yescharacteristic
74C2×C4×D107D411yescharacteristic
75C2×C4×D107C2311yescharacteristic
76C2×C4×D107C2311yescharacteristic
77#13133807—24no—
78#13133806—24no—
79#13133805—24no—
80C2×C107⋊C8C2312yes—
81C8×D107—24no—
82C4⋊Dic107C2311yescharacteristic
83Dic107⋊C4—21no—
84C22⋊Dic107—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C2×Q8×C107—24no—
88C2×Q8×C107—21no—
89C2×Q8×C107C2312yes—
90C2×D4×C107—21no—
91C2×D4×C107C2311yescharacteristic
92C22×C4×C107—21no—
93C22×C4×C107C2311yescharacteristic
94SD16×C107—24no—
95C2×C8×C107C2312yes—
96C4⋊C4×C107C2311yescharacteristic
97C22⋊C4×C107—21no—
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—22no—
101C22×D107—22no—
102C22×D107—21no—
103C22×D107—42no—
104C22×D107—42no—
105C22×D107—42no—
106C22×D107SD1614yescomplemented
107C22×D107C2×D411yescharacteristic
108C22×D107C2×D411yescharacteristic
109Q8×C107—24no—
110Q8×C107—22no—
111Q8×C107C2×D414yescomplemented
112D4×C107—22no—
113D4×C107—42no—
114C2×C4×C107—24no—
115C2×C4×C107—22no—
116C2×C4×C107—22no—
117C2×C4×C107—21no—
118C2×C4×C107C2×D411yescharacteristic
119C2×C4×C107C2411yescharacteristic
120C22⋊D107—42no—
121C22⋊D107—42no—
122C22⋊D107—42no—
123C2×Dic107—24no—
124C2×Dic107—22no—
125C2×Dic107—22no—
126C2×Dic107—21no—
127C2×Dic107C2×D411yescharacteristic
128C2×Dic107C2×D411yescharacteristic
129D428—22no—
130D428—42no—
131C4×D107—28no—
132C4×D107—28no—
133C4×D107—22no—
134C4×D107—22no—
135C4×D107—22no—
136C4×D107—42no—
137C4×D107C2×D414yes—
138Dic214—28no—
139Dic214—24no—
140Dic214—22no—
141Dic214—28no—
142Dic214C2×D414yescomplemented
143C8×C107—22no—
144C107⋊C8—22no—
145C22×C107—21no—
146C22×C107—21no—
147C22×C107—41no—
148C22×C107—41no—
149C22×C107—41no—
150C22×C107C2×SD1612yescomplemented
151C22×C107C22×D411yescharacteristic
152C2×D107—21no—
153C2×D107—21no—
154C2×D107—28no—
155C2×D107—41no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—84no—
159C2×D107#12012yes—
160C2×D107C2×SD1612yes—
161C2×D107C2×SD1612yescomplemented
162C2×D107#13712yes—
163C4×C107—24no—
164C4×C107—21no—
165C4×C107—41no—
166C4×C107C22×D412yes—
167Dic107—21no—
168Dic107—24no—
169Dic107—41no—
170Dic107#12012yes—
171C2×C107—22no—
172C2×C107—81no—
173C2×C107C2×#12011yescharacteristic
174C2×C107C22×SD1611yescharacteristic
175C2×C107C2×#13711yescharacteristic
176D107—22no—
177D107—81no—
178D107#44814yesdirect product factor
179C2×#448—1071nomaximal, Sylow
180C107C2×#44811yescharacteristic, complemented, Sylow
181C23×Q8—1071no—
182C22×SD16—1072no—
183C2×#121—1071no—
184#448—1071no—
185#448—1072no—
186#448—1071no—
187#448—1072no—
188#448—1071no—
189#448D10711yescharacteristic, direct product factor
190C2×#103—1072no—
191C2×C22⋊C8—1071no—
192C22×Q8—1072no—
193C22×Q8—1072no—
194C22×Q8—1071no—
195C22×Q8—2141no—
196C22×Q8—2141no—
197C22×Q8—2141no—
198C22×Q8—2148no—
199C22×Q8C2×D10711yescharacteristic, complemented
200C22×D4—1071no—
201C22×D4—2141no—
202C23×C4—1071no—
203C23×C4—2141no—
204C2×SD16—1072no—
205C2×SD16—1072no—
206C2×SD16—1072no—
207C2×SD16—2148no—
208C2×SD16C2×D10712yescomplemented
209C22×C8—1072no—
210#121—1071no—
211#121—1071no—
212#121—1071no—
213#121—2141no—
214#121—2141no—
215#121C2×D10711yescharacteristic
216C2×C4⋊C4—1071no—
217C2×C22⋊C4—2141no—
218#103—1072no—
219#103—1072no—
220#103—1072no—
221#103C2×D10712yescomplemented
222C22⋊C8—1071no—
223C22⋊C8—1072no—
224C22⋊C8C2×D10711yescharacteristic, complemented
225C24—1071no—
226C24—2141no—
227C2×Q8—24no—
228C2×Q8—21no—
229C2×Q8—1078no—
230C2×Q8—1072no—
231C2×Q8—2142no—
232C2×Q8—2142no—
233C2×Q8—2144no—
234C2×Q8—2148no—
235C2×Q8—2144no—
236C2×Q8—2144no—
237C2×Q8—2148no—
238C2×Q8—2148no—
239C2×Q8—2144no—
240C2×Q8—2141no—
241C2×Q8C22×D10712yes—
242C2×D4—21no—
243C2×D4—1071no—
244C2×D4—4284no—
245C2×D4—2141no—
246C2×D4—2144no—
247C2×D4—2141no—
248C2×D4—2141no—
249C2×D4—2141no—
250C2×D4C22×D10711yescharacteristic
251C22×C4—21no—
252C22×C4—1071no—
253C22×C4—1071no—
254C22×C4—1071no—
255C22×C4—1071no—
256C22×C4—1071no—
257C22×C4—2144no—
258C22×C4—2141no—
259C22×C4—2142no—
260C22×C4—2142no—
261C22×C4—2148no—
262C22×C4—2141no—
263C22×C4C22×D10711yescharacteristic
264SD16—24no—
265SD16—2144no—
266SD16—2144no—
267SD16—2144no—
268C2×C8—1072no—
269C2×C8—2144no—
270C2×C8C22×D10712yes—
271C4⋊C4—1071no—
272C4⋊C4—2141no—
273C4⋊C4C22×D10711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C23—21no—
279C23—1071no—
280C23—1074no—
281C23—1071no—
282C23—4282no—
283C23—4282no—
284C23—4282no—
285C23—2142no—
286C23—2142no—
287C23—2141no—
288C23D4×D10711yescharacteristic
289Q8—24no—
290Q8—22no—
291Q8—1074no—
292Q8—2148no—
293Q8—2148no—
294Q8—2144no—
295Q8—2142no—
296Q8D4×D10714yescomplemented
297D4—22no—
298D4—42no—
299D4—4282no—
300D4—4282no—
301D4—4282no—
302D4—4282no—
303D4—2142no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—1071no—
309C2×C4—1074no—
310C2×C4—1071no—
311C2×C4—4282no—
312C2×C4—2142no—
313C2×C4—2142no—
314C2×C4—2148no—
315C2×C4—2148no—
316C2×C4—2142no—
317C2×C4—2141no—
318C2×C4—2144no—
319C2×C4—2142no—
320C2×C4—2142no—
321C2×C4D4×D10711yescharacteristic
322C2×C4C23×D10711yescharacteristic
323C8—22no—
324C8—2142no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—8564no—
331C22—1072no—
332C22—1072no—
333C22—1072no—
334C22—1072no—
335C22—4281no—
336C22—4281no—
337C22—4281no—
338C22—2148no—
339C22—2141no—
340C22—2141no—
341C22SD16×D10712yescomplemented
342C22C2×D4×D10711yescharacteristic
343C4—24no—
344C4—21no—
345C4—41no—
346C4—1072no—
347C4—4281no—
348C4—2141no—
349C4—2144no—
350C4C2×D4×D10712yes—
351C2—22no—
352C2—81no—
353C2—8561no—
354C2—1074no—
355C2—2142no—
356C2#120×D10711yescharacteristic
357C2C2×SD16×D10711yescharacteristic
358C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
prime radical#121×D107C2
hypercenter#448D107
hypocenterC107C2×#448
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D107C2semidirect
2. C2×SD16×D107C2semidirect
4. #20292768C2semidirect
5. #20292767C2semidirect
6. #20292766C2semidirect
7. #20292765C2semidirect
8. #20292764C2semidirect
9. #103×D107C2semidirect
10. C22⋊C8×D107C2semidirect
11. #448×C107C2semidirect
13. C22×Dic214C22semidirect
18. C2×Q8×D107C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133806C22semidirect
25. C2×#13133805C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812032C22semidirect
38. C22⋊(C107⋊C8)C22semidirect
39. (C2×D107)⋊C8C22semidirect
40. C22×Q8×C107C22semidirect
41. C2×SD16×C107C22semidirect
43. #103×C107C22semidirect
44. C22⋊C8×C107C22semidirect
58. Q8×D107D4semidirect
106. C22×D107SD16semidirect
111. Q8×C107C2×D4semidirect
142. Dic214C2×D4semidirect
150. C22×C107C2×SD16semidirect
161. C2×D107C2×SD16semidirect
178. D107#448direct (N × H)
180. C107C2×#448semidirect
189. #448D107direct (N × H)
199. C22×Q8C2×D107semidirect
208. C2×SD16C2×D107semidirect
221. #103C2×D107semidirect
224. C22⋊C8C2×D107semidirect
296. Q8D4×D107semidirect
341. C22SD16×D107semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1072
C2×SD16×D1072
#121×D1072
#202927682
#202927672
#202927662
#202927652
#202927642
#103×D1072
C22⋊C8×D1072
#448×C1072
C2×#448107