Finite Group Databasebeta

← #450×D107 #28345640

Subgroups — #450×D107 #28345640

Order 13696.

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

normal subgroups only

This group has 996 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×D4×D107C211yesmaximal, characteristic
2C2×SD16×D107C212yesmaximal, complemented
3C22⋊Q8×D107C211yesmaximal, characteristic, complemented
4#20292780C211yesmaximal, characteristic, complemented
5#20292779C212yesmaximal, complemented
6#20292778C211yesmaximal, characteristic, complemented
7#20292777C211yesmaximal, characteristic, complemented
8#20292776C212yesmaximal, complemented
9#102×D107C212yesmaximal, complemented
10C22⋊C8×D107C211yesmaximal, characteristic, complemented
11#450×C107C211yesmaximal, characteristic, complemented
12C24×D107—21no—
13C2×Q8×D107C2211yescharacteristic, complemented
14C22×D428C2211yescharacteristic
15C22×C22⋊D107—21no—
16C2×D4×D107—28no—
17C2×D4×D107—21no—
18C2×D4×D107—21no—
19C2×D4×D107C2212yes—
20C2×D4×D107C2212yes—
21C22×C4×D107C2211yescharacteristic
22C2×#13133807C2212yescomplemented
23C2×#13133806C2212yescomplemented
24C2×#13133805C2212yescomplemented
25SD16×D107—28no—
26C2×C8×D107C2212yescomplemented
27#16812089C2211yescharacteristic, complemented
28#16812088C2211yescharacteristic, complemented
29#16812087—21no—
30#16812086—21no—
31#16812085C2211yescharacteristic, complemented
32C4⋊C4×D107—21no—
33C4⋊C4×D107C2211yescharacteristic, complemented
34C22⋊C4×D107—21no—
35#16812031C2212yescomplemented
36#16812030C2212yescomplemented
37#16812029C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic, complemented
39(C2×D107)⋊C8C2212yescomplemented
40C22×D4×C107C2211yescharacteristic
41C2×SD16×C107C2212yescomplemented
42C22⋊Q8×C107C2211yescharacteristic, complemented
43#102×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic, complemented
45C23×D107—28no—
46C23×D107—21no—
47C23×D107—22no—
48C23×D107—22no—
49C23×D107—44no—
50C23×D107—44no—
51C23×D107D411yescharacteristic
52C2×Dic214—21no—
53C2×Dic214C2311yescharacteristic, complemented
54Q8×D107—24no—
55C2×D428—21no—
56C2×D428—24no—
57C2×D428C2312yes—
58C2×C22⋊D107—28no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61D4×D107—28no—
62D4×D107—28no—
63D4×D107—24no—
64D4×D107—24no—
65D4×D107—24no—
66D4×D107D418yes—
67C22×Dic107D411yescharacteristic
68C2×C4×D107—24no—
69C2×C4×D107—21no—
70C2×C4×D107—21no—
71C2×C4×D107D411yescharacteristic
72C2×C4×D107D411yescharacteristic
73C2×C4×D107C2311yescharacteristic
74C2×C4×D107C2311yescharacteristic
75#13133807—24no—
76#13133806—24no—
77#13133805—24no—
78C2×C107⋊C8C2312yescomplemented
79C8×D107—24no—
80C4⋊Dic107—21no—
81C4⋊Dic107C2311yescharacteristic, complemented
82Dic107⋊C4—21no—
83Dic107⋊C4—21no—
84Dic107⋊C4—21no—
85C22⋊Dic107—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88C24×C107—21no—
89C2×Q8×C107C2311yescharacteristic, complemented
90C2×D4×C107—24no—
91C2×D4×C107—21no—
92C2×D4×C107C2312yes—
93C22×C4×C107C2311yescharacteristic
94SD16×C107—24no—
95C2×C8×C107C2312yescomplemented
96C4⋊C4×C107—21no—
97C4⋊C4×C107C2311yescharacteristic, complemented
98C22⋊C4×C107—21no—
99C23×C107—24no—
100C23×C107—22no—
101C23×C107—42no—
102C23×C107—42no—
103C23×C107C2×D411yescharacteristic
104C22×D107—22no—
105C22×D107—22no—
106C22×D107—28no—
107C22×D107—24no—
108C22×D107—22no—
109C22×D107—28no—
110C22×D107—44no—
111C22×D107—432no—
112C22×D107—42no—
113C22×D107—42no—
114C22×D107—44no—
115C22×D107—44no—
116C22×D107—44no—
117C22×D107SD1614yescomplemented
118C22×D107C2×D411yescharacteristic
119C22×D107C2×D411yescharacteristic
120Q8×C107—22no—
121D4×C107—24no—
122D4×C107—22no—
123D4×C107C2×D414yes—
124C2×C4×C107—22no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—28no—
130C22⋊D107—28no—
131C2×Dic107—22no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136D428—24no—
137D428—22no—
138D428C2×D414yes—
139C4×D107—22no—
140C4×D107—22no—
141C4×D107—22no—
142C4×D107—42no—
143C4×D107—42no—
144C4×D107C2×D414yes—
145Dic214—22no—
146Dic214—42no—
147C8×C107—22no—
148C107⋊C8—22no—
149C22×C107—21no—
150C22×C107—21no—
151C22×C107—24no—
152C22×C107—48no—
153C22×C107—42no—
154C22×C107—42no—
155C22×C107C2×SD1612yescomplemented
156C22×C107C22×D411yescharacteristic
157C2×D107—21no—
158C2×D107—21no—
159C2×D107—28no—
160C2×D107—24no—
161C2×D107—416no—
162C2×D107—48no—
163C2×D107—42no—
164C2×D107—42no—
165C2×D107—416no—
166C2×D107#12012yes—
167C2×D107C2×SD1612yes—
168C2×D107C2×SD1612yescomplemented
169C2×D107AGL(1,ℤ/8)12yes—
170C4×C107—21no—
171C4×C107—41no—
172C4×C107—41no—
173C4×C107C22×D412yes—
174Dic107—21no—
175Dic107—41no—
176Dic107—41no—
177Dic107#12012yes—
178C2×C107—22no—
179C2×C107—44no—
180C2×C107C2×#12011yescharacteristic
181C2×C107C22×SD1611yescharacteristic
182C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
183D107—22no—
184D107—44no—
185D107#45014yesdirect product factor
186C2×#450—1071nomaximal, Sylow
187C107C2×#45011yescharacteristic, complemented, Sylow
188C23×D4—1071no—
189C22×SD16—1072no—
190C2×C22⋊Q8—1071no—
191#450—1071no—
192#450—1072no—
193#450—1071no—
194#450—1072no—
195#450—1071no—
196#450D10711yescharacteristic, direct product factor
197C2×#102—1072no—
198C2×C22⋊C8—1071no—
199C25—2141no—
200C22×Q8—1071no—
201C22×D4—1072no—
202C22×D4—1072no—
203C22×D4—1071no—
204C22×D4—2141no—
205C22×D4—2141no—
206C22×D4—2141no—
207C22×D4—2148no—
208C22×D4C2×D10711yescharacteristic
209C23×C4—1071no—
210C2×SD16—1072no—
211C2×SD16—1072no—
212C2×SD16—1072no—
213C2×SD16—2148no—
214C2×SD16C2×D10712yescomplemented
215C22×C8—1072no—
216C22⋊Q8—1071no—
217C22⋊Q8—1071no—
218C22⋊Q8—1071no—
219C22⋊Q8—2141no—
220C22⋊Q8—2141no—
221C22⋊Q8C2×D10711yescharacteristic, complemented
222C2×C4⋊C4—1071no—
223C2×C4⋊C4—2141no—
224C2×C22⋊C4—2141no—
225#102—1072no—
226#102—1072no—
227#102—1072no—
228#102C2×D10712yescomplemented
229C22⋊C8—1071no—
230C22⋊C8—1072no—
231C22⋊C8C2×D10711yescharacteristic, complemented
232C24—21no—
233C24—1071no—
234C24—4284no—
235C24—4284no—
236C24—2142no—
237C24—2142no—
238C24—2148no—
239C24—2141no—
240C2×Q8—1071no—
241C2×Q8—2141no—
242C2×Q8—2144no—
243C2×Q8C22×D10711yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—1078no—
247C2×D4—1072no—
248C2×D4—2144no—
249C2×D4—2144no—
250C2×D4—2142no—
251C2×D4—2142no—
252C2×D4—2144no—
253C2×D4—2148no—
254C2×D4—2148no—
255C2×D4—2148no—
256C2×D4—2144no—
257C2×D4—2141no—
258C2×D4C22×D10712yes—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—1071no—
262C22×C4—1071no—
263C22×C4—1071no—
264C22×C4—2144no—
265C22×C4—2141no—
266C22×C4—2141no—
267C22×C4C22×D10711yescharacteristic
268SD16—24no—
269SD16—2144no—
270SD16—2144no—
271SD16—2144no—
272C2×C8—1072no—
273C2×C8—2144no—
274C2×C8C22×D10712yescomplemented
275C4⋊C4—21no—
276C4⋊C4—1071no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22×D10711yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—1071no—
291C23—1074no—
292C23—1071no—
293C23—4282no—
294C23—4284no—
295C23—4284no—
296C23—4284no—
297C23—42832no—
298C23—4284no—
299C23—4282no—
300C23—2148no—
301C23—2142no—
302C23—2142no—
303C23—2144no—
304C23—2148no—
305C23—2142no—
306C23D4×D10711yescharacteristic
307Q8—22no—
308Q8—4282no—
309Q8—2142no—
310D4—24no—
311D4—22no—
312D4—1074no—
313D4—2144no—
314D4—2148no—
315D4—2148no—
316D4—2142no—
317D4D4×D10714yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—1071no—
322C2×C4—1074no—
323C2×C4—1071no—
324C2×C4—4282no—
325C2×C4—4282no—
326C2×C4—2142no—
327C2×C4—2142no—
328C2×C4—2142no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2142no—
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—22no—
335C8—2142no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—1072no—
343C22—1072no—
344C22—1072no—
345C22—1072no—
346C22—4288no—
347C22—42816no—
348C22—42816no—
349C22—4282no—
350C22—4282no—
351C22—2148no—
352C22—2144no—
353C22—2141no—
354C22—2141no—
355C22SD16×D10712yescomplemented
356C22C2×D4×D10711yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—1072no—
361C4—4281no—
362C4—4281no—
363C4—2141no—
364C4C2×D4×D10712yes—
365C2—22no—
366C2—44no—
367C2—1074no—
368C2—4284no—
369C2—2142no—
370C2#120×D10711yescharacteristic
371C2C2×SD16×D10711yescharacteristic
372C2AGL(1,ℤ/8)×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalC22×D4×D107C2
hypercenter#450D107
hypocenterC107C2×#450
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D107C2semidirect
3. C22⋊Q8×D107C2semidirect
4. #20292780C2semidirect
5. #20292779C2semidirect
6. #20292778C2semidirect
7. #20292777C2semidirect
8. #20292776C2semidirect
9. #102×D107C2semidirect
10. C22⋊C8×D107C2semidirect
11. #450×C107C2semidirect
13. C2×Q8×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133805C22semidirect
26. C2×C8×D107C22semidirect
27. #16812089C22semidirect
28. #16812088C22semidirect
31. #16812085C22semidirect
33. C4⋊C4×D107C22semidirect
35. #16812031C22semidirect
36. #16812030C22semidirect
37. #16812029C22semidirect
38. C22⋊(C107⋊C8)C22semidirect
39. (C2×D107)⋊C8C22semidirect
41. C2×SD16×C107C22semidirect
42. C22⋊Q8×C107C22semidirect
43. #102×C107C22semidirect
44. C22⋊C8×C107C22semidirect
53. C2×Dic214C23semidirect
78. C2×C107⋊C8C23semidirect
81. C4⋊Dic107C23semidirect
89. C2×Q8×C107C23semidirect
95. C2×C8×C107C23semidirect
97. C4⋊C4×C107C23semidirect
117. C22×D107SD16semidirect
155. C22×C107C2×SD16semidirect
168. C2×D107C2×SD16semidirect
185. D107#450direct (N × H)
187. C107C2×#450semidirect
196. #450D107direct (N × H)
214. C2×SD16C2×D107semidirect
221. C22⋊Q8C2×D107semidirect
228. #102C2×D107semidirect
231. C22⋊C8C2×D107semidirect
243. C2×Q8C22×D107semidirect
274. C2×C8C22×D107semidirect
281. C4⋊C4C22×D107semidirect
355. C22SD16×D107semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×SD16×D1072
C22⋊Q8×D1072
#202927802
#202927792
#202927782
#202927772
#202927762
#102×D1072
C22⋊C8×D1072
#450×C1072
C2×#450107