Finite Group Databasebeta

← #28345689

Subgroups — #28345689

Order 13696.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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, complemented
2C2×#16812136C212yesmaximal, complemented
3#20293140C211yesmaximal, characteristic, complemented
4#20292777C211yesmaximal, characteristic, complemented
5#20292776C212yesmaximal, complemented
6#20292763C211yesmaximal, characteristic, complemented
7#20292762C212yesmaximal, complemented
8#20292761C211yesmaximal, characteristic, complemented
9#20292630C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#450×C107C211yesmaximal, characteristic, complemented
12C24×D107—21no—
13C2×#13133819C2211yescharacteristic, 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
22#16812136—28no—
23C2×#13133805C2212yescomplemented
24C2×D856C2212yescomplemented
25C2×#13133803C2212yescomplemented
26C2×#13133801C2212yescomplemented
27#16812094—21no—
28#16812085C2211yescharacteristic, complemented
29#16812084C2211yescharacteristic, complemented
30#16812083C2211yescharacteristic, complemented
31#16812079—21no—
32#16812071—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic, complemented
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
52#13133819—24no—
53C2×D428—21no—
54C2×D428—24no—
55C2×D428—21no—
56C2×D428—21no—
57C2×D428C2312yes—
58C2×D428C2311yescharacteristic, complemented
59C2×C22⋊D107—28no—
60C2×C22⋊D107—21no—
61C2×C22⋊D107—22no—
62C2×C22⋊D107—22no—
63D4×D107—28no—
64D4×D107—28no—
65D4×D107—24no—
66D4×D107—24no—
67D4×D107—24no—
68D4×D107D418yescomplemented
69C22×Dic107D411yescharacteristic
70C2×C4×D107—24no—
71C2×C4×D107—21no—
72C2×C4×D107D411yescharacteristic
73C2×C4×D107D411yescharacteristic
74C2×C4×D107C2311yescharacteristic
75C2×C4×D107C2311yescharacteristic
76#13133805—24no—
77D856—24no—
78#13133803—24no—
79#13133801—24no—
80C2×C107⋊C8C2312yescomplemented
81C4⋊Dic107C2311yescharacteristic, complemented
82Dic107⋊C4—21no—
83(C2×D107)⋊C4—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86(C2×D107)⋊C4—21no—
87C4×Dic107—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—22no—
107C22×D107—28no—
108C22×D107—24no—
109C22×D107—22no—
110C22×D107—28no—
111C22×D107—21no—
112C22×D107—432no—
113C22×D107—44no—
114C22×D107—44no—
115C22×D107—44no—
116C22×D107—42no—
117C22×D107—42no—
118C22×D107—44no—
119C22×D107C2×D411yescharacteristic
120C22×D107C2×D411yescharacteristic
121Q8×C107—22no—
122D4×C107—24no—
123D4×C107—22no—
124D4×C107C2×D414yes—
125C2×C4×C107—22no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—28no—
131C22⋊D107—28no—
132C22⋊D107—42no—
133C2×Dic107—22no—
134C2×Dic107—21no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—24no—
138D428—22no—
139D428—22no—
140D428—42no—
141D428—42no—
142D428C2×D414yes—
143C4×D107—22no—
144C4×D107—22no—
145C4×D107—22no—
146C4×D107—42no—
147C4×D107C2×D414yes—
148C8×C107—22no—
149C107⋊C8—22no—
150C22×C107—21no—
151C22×C107—21no—
152C22×C107—24no—
153C22×C107—48no—
154C22×C107—42no—
155C22×C107—42no—
156C22×C107AGL(1,ℤ/8)12yescomplemented
157C22×C107C22×D411yescharacteristic
158C2×D107—28no—
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—21no—
162C2×D107—21no—
163C2×D107—24no—
164C2×D107—416no—
165C2×D107—416no—
166C2×D107—48no—
167C2×D107—42no—
168C2×D107—42no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107#12012yes—
173C2×D107AGL(1,ℤ/8)12yescomplemented
174C4×C107—21no—
175C4×C107—41no—
176C4×C107—41no—
177C4×C107C22×D412yes—
178Dic107—21no—
179Dic107—41no—
180Dic107#12012yes—
181C2×C107—22no—
182C2×C107—44no—
183C2×C107C2×#12011yescharacteristic
184C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
186D107—22no—
187D107—22no—
188D107—44no—
189D107—81no—
190#3003—1071nomaximal, Sylow
191C107#300311yescharacteristic, complemented, Sylow
192C23×D4—1071no—
193C2×AGL(1,ℤ/8)—1072no—
194#525—1071no—
195#450—1072no—
196#450—1071no—
197#450D10711yescharacteristic, complemented
198#447—1071no—
199#447—1072no—
200#447—1071no—
201#418—1072no—
202#407—1071no—
203C25—2141no—
204C2×#41—1071no—
205C22×D4—1072no—
206C22×D4—1072no—
207C22×D4—1071no—
208C22×D4—2141no—
209C22×D4—2141no—
210C22×D4—2141no—
211C22×D4—2148no—
212C22×D4C2×D10711yescharacteristic
213C23×C4—1071no—
214AGL(1,ℤ/8)—2148no—
215C2×SD16—1072no—
216C2×SD16C2×D10712yescomplemented
217C2×D8—1072no—
218C2×D8—1072no—
219C2×M16—1072no—
220#123—2141no—
221C22⋊Q8—1071no—
222C22⋊Q8C2×D10711yescharacteristic, complemented
223#121—1071no—
224#121—1071no—
225#120—2141no—
226C4×D4—2141no—
227C4×D4—2141no—
228#117—1071no—
229#102—1072no—
230#102—1072no—
231#102—1072no—
232#102C2×D10712yescomplemented
233C22⋊C8—1071no—
234C22⋊C8—1072no—
235C22⋊C8C2×D10711yescharacteristic, complemented
236C24—21no—
237C24—1071no—
238C24—4284no—
239C24—4284no—
240C24—2142no—
241C24—2142no—
242C24—2148no—
243C24—2141no—
244#41—2144no—
245C2×Q8C22×D10711yescharacteristic, complemented
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—1078no—
249C2×D4—1072no—
250C2×D4—1071no—
251C2×D4—2144no—
252C2×D4—2141no—
253C2×D4—2141no—
254C2×D4—2144no—
255C2×D4—2142no—
256C2×D4—2142no—
257C2×D4—2144no—
258C2×D4—2148no—
259C2×D4—2148no—
260C2×D4—2141no—
261C2×D4—2148no—
262C2×D4—2144no—
263C2×D4—2141no—
264C2×D4C22×D10712yes—
265C22×C4—1071no—
266C22×C4—1071no—
267C22×C4—1071no—
268C22×C4—1071no—
269C22×C4—1071no—
270C22×C4—2141no—
271C22×C4—2144no—
272C22×C4C22×D10711yescharacteristic
273SD16—24no—
274SD16—2144no—
275D8—2144no—
276D8—2144no—
277M16—2144no—
278C2×C8—1072no—
279C2×C8C22×D10712yescomplemented
280C4⋊C4—21no—
281C4⋊C4—1071no—
282C4⋊C4—2141no—
283C4⋊C4C22×D10711yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—2141no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C42—2141no—
290C23—24no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—1071no—
295C23—1071no—
296C23—4282no—
297C23—4284no—
298C23—42832no—
299C23—4284no—
300C23—4284no—
301C23—4284no—
302C23—4282no—
303C23—2148no—
304C23—2142no—
305C23—2142no—
306C23—2144no—
307C23—2148no—
308C23—2141no—
309C23—2142no—
310C23—2142no—
311C23D4×D10711yescharacteristic
312Q8—22no—
313D4—24no—
314D4—22no—
315D4—1074no—
316D4—4282no—
317D4—4282no—
318D4—4282no—
319D4—2144no—
320D4—2148no—
321D4—2148no—
322D4—2142no—
323D4—2142no—
324D4D4×D10714yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1071no—
329C2×C4—1074no—
330C2×C4—1071no—
331C2×C4—4282no—
332C2×C4—2142no—
333C2×C4—2142no—
334C2×C4—2142no—
335C2×C4—2141no—
336C2×C4—2142no—
337C2×C4D4×D10711yescharacteristic
338C2×C4C23×D10711yescharacteristic
339C8—22no—
340C8—2142no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—1072no—
348C22—1072no—
349C22—4288no—
350C22—42816no—
351C22—42816no—
352C22—4281no—
353C22—4281no—
354C22—4281no—
355C22—4282no—
356C22—4282no—
357C22—2148no—
358C22—2144no—
359C22—2141no—
360C22—2141no—
361C22—2141no—
362C22—2141no—
363C22#1681213612yescomplemented
364C22C2×D4×D10711yescharacteristic
365C4—21no—
366C4—41no—
367C4—41no—
368C4—1072no—
369C4—4281no—
370C4—2141no—
371C4C2×D4×D10712yes—
372C2—22no—
373C2—44no—
374C2—8561no—
375C2—4284no—
376C2—2142no—
377C2—2142no—
378C2#120×D10711yescharacteristic
379C2AGL(1,ℤ/8)×D10711yescharacteristic
380C2C2×#1681213611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
hypercenter#450D107
hypocenterC107#3003
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
1. C22×D4×D107C2semidirect
2. C2×#16812136C2semidirect
3. #20293140C2semidirect
4. #20292777C2semidirect
5. #20292776C2semidirect
6. #20292763C2semidirect
7. #20292762C2semidirect
8. #20292761C2semidirect
9. #20292630C2semidirect
10. #20292589C2semidirect
11. #450×C107C2semidirect
13. C2×#13133819C22semidirect
23. C2×#13133805C22semidirect
24. C2×D856C22semidirect
25. C2×#13133803C22semidirect
26. C2×#13133801C22semidirect
28. #16812085C22semidirect
29. #16812084C22semidirect
30. #16812083C22semidirect
34. #16812067C22semidirect
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
58. C2×D428C23semidirect
68. D4×D107D4semidirect
80. C2×C107⋊C8C23semidirect
81. C4⋊Dic107C23semidirect
89. C2×Q8×C107C23semidirect
95. C2×C8×C107C23semidirect
97. C4⋊C4×C107C23semidirect
156. C22×C107AGL(1,ℤ/8)semidirect
173. C2×D107AGL(1,ℤ/8)semidirect
191. C107#3003semidirect
197. #450D107semidirect
216. C2×SD16C2×D107semidirect
222. C22⋊Q8C2×D107semidirect
232. #102C2×D107semidirect
235. C22⋊C8C2×D107semidirect
245. C2×Q8C22×D107semidirect
279. C2×C8C22×D107semidirect
283. C4⋊C4C22×D107semidirect
363. C22#16812136semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1072
C2×#168121362
#202931402
#202927772
#202927762
#202927632
#202927622
#202927612
#202926302
#202925892
#450×C1072
#3003107