Finite Group Databasebeta

← #28345712

Subgroups — #28345712

Order 13696.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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×#13133819C211yesmaximal, characteristic, complemented
2C2×#16812136C212yesmaximal, complemented
3#121×D107C211yesmaximal, characteristic, complemented
4#20292775C212yesmaximal, complemented
5#20292771C212yesmaximal, complemented
6#20292766C211yesmaximal, characteristic, complemented
7#20292763C211yesmaximal, characteristic, complemented
8#20292760C211yesmaximal, characteristic, complemented
9#20292627C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#448×C107C211yesmaximal, characteristic, complemented
12C2×#13133819—28no—
13C2×#13133819—21no—
14C2×#13133819—21no—
15C2×#13133819C2212yes—
16C2×#13133819C2212yescomplemented
17C22×D428—21no—
18C22×D428C2211yescharacteristic, complemented
19C2×D4×D107—21no—
20C2×D4×D107C2211yescharacteristic, complemented
21C22×C4×D107—21no—
22C22×C4×D107C2211yescharacteristic
23#16812136—28no—
24C2×#13133805C2212yescomplemented
25C2×D856C2212yescomplemented
26C2×#13133803C2212yescomplemented
27C2×#13133801C2212yescomplemented
28#16812084C2211yescharacteristic, complemented
29#16812083C2211yescharacteristic
30#16812082—21no—
31#16812081—21no—
32#16812080C2211yescharacteristic, complemented
33C4⋊C4×D107C2211yescharacteristic, complemented
34C22⋊C4×D107—21no—
35#16812033C2212yescomplemented
36#16812031C2212yescomplemented
37#16812029C2212yescomplemented
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×D107—21no—
47C23×D107D411yescharacteristic
48#13133819—28no—
49#13133819—28no—
50#13133819—24no—
51#13133819—24no—
52#13133819—24no—
53#13133819D418yescomplemented
54C2×D428—28no—
55C2×D428—22no—
56C2×D428—22no—
57C2×D428—21no—
58C2×D428—24no—
59C2×D428—21no—
60C2×D428C2311yescharacteristic
61C2×D428C2312yes—
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65D4×D107—24no—
66D4×D107—44no—
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#13133805—24no—
78D856—24no—
79#13133803—24no—
80#13133801—24no—
81C2×C107⋊C8C2312yes—
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—24no—
101C22×D107—22no—
102C22×D107—22no—
103C22×D107—22no—
104C22×D107—21no—
105C22×D107—41no—
106C22×D107—42no—
107C22×D107—42no—
108C22×D107—42no—
109C22×D107—42no—
110C22×D107—42no—
111C22×D107C2×D411yescharacteristic
112C22×D107C2×D411yescharacteristic
113Q8×C107—24no—
114Q8×C107—22no—
115Q8×C107C2×D414yescomplemented
116D4×C107—22no—
117D4×C107—42no—
118C2×C4×C107—24no—
119C2×C4×C107—22no—
120C2×C4×C107—22no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2411yescharacteristic
124C22⋊D107—42no—
125C22⋊D107—42no—
126C22⋊D107—42no—
127C2×Dic107—22no—
128C2×Dic107—21no—
129C2×Dic107C2×D411yescharacteristic
130C2×Dic107C2×D411yescharacteristic
131D428—28no—
132D428—24no—
133D428—28no—
134D428—22no—
135D428—22no—
136D428—42no—
137D428C2×D414yescomplemented
138C4×D107—28no—
139C4×D107—22no—
140C4×D107—22no—
141C4×D107—28no—
142C4×D107—22no—
143C4×D107—42no—
144C4×D107C2×D414yes—
145C8×C107—22no—
146C107⋊C8—22no—
147C22×C107—21no—
148C22×C107—21no—
149C22×C107—41no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107AGL(1,ℤ/8)12yescomplemented
153C22×C107C22×D411yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—21no—
157C2×D107—24no—
158C2×D107—21no—
159C2×D107—21no—
160C2×D107—44no—
161C2×D107—48no—
162C2×D107—42no—
163C2×D107—42no—
164C2×D107—41no—
165C2×D107—41no—
166C2×D107—41no—
167C2×D107—84no—
168C2×D107#12012yes—
169C4×C107—24no—
170C4×C107—21no—
171C4×C107—41no—
172C4×C107C22×D412yes—
173Dic107—21no—
174Dic107—41no—
175Dic107#12012yes—
176C2×C107—22no—
177C2×C107—81no—
178C2×C107C2×#12011yescharacteristic
179C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C107#57511yescharacteristic
181D107—22no—
182D107—41no—
183D107—44no—
184D107—81no—
185#3006—1071nomaximal, Sylow
186C107#300611yescharacteristic, complemented, Sylow
187C22×#41—1071no—
188C2×AGL(1,ℤ/8)—1072no—
189C2×#121—1071no—
190#449—1072no—
191#449—1072no—
192#448—1071no—
193#448D10711yescharacteristic, complemented
194#447—1071no—
195#447—1071no—
196#417—1072no—
197#407—1071no—
198C2×#41—1072no—
199C2×#41—1072no—
200C2×#41—2141no—
201C2×#41—2141no—
202C2×#41—2148no—
203C22×Q8C2×D10711yescharacteristic, complemented
204C22×D4—1071no—
205C22×D4—1071no—
206C22×D4—2141no—
207C22×D4—2141no—
208C23×C4—1071no—
209C23×C4—2141no—
210AGL(1,ℤ/8)—2148no—
211C2×SD16—1072no—
212C2×SD16C2×D10712yescomplemented
213C2×D8—1072no—
214C2×D8—1072no—
215C2×M16—1072no—
216#121—1071no—
217#121—1071no—
218#121—1071no—
219#121—2141no—
220#121—2141no—
221#121C2×D10711yescharacteristic
222C2×C4⋊C4—1071no—
223C2×C22⋊C4—2141no—
224#103—1072no—
225#103C2×D10712yescomplemented
226#102—1072no—
227#102—1072no—
228C22⋊C8—1071no—
229C22⋊C8—1072no—
230C22⋊C8C2×D10711yescharacteristic, complemented
231C24—1071no—
232C24—2141no—
233C24—2141no—
234#41—1078no—
235#41—2144no—
236#41—2144no—
237#41—2148no—
238#41—2148no—
239#41—2144no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D10712yes—
243C2×D4—21no—
244C2×D4—1072no—
245C2×D4—1071no—
246C2×D4—4284no—
247C2×D4—2144no—
248C2×D4—2141no—
249C2×D4—2142no—
250C2×D4—2142no—
251C2×D4—2148no—
252C2×D4—2144no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—1071no—
262C22×C4—1071no—
263C22×C4—1071no—
264C22×C4—2144no—
265C22×C4—2142no—
266C22×C4—2142no—
267C22×C4—2141no—
268C22×C4—2148no—
269C22×C4C22×D10711yescharacteristic
270SD16—24no—
271SD16—2144no—
272D8—2144no—
273D8—2144no—
274M16—2144no—
275C2×C8—1072no—
276C2×C8C22×D10712yes—
277C4⋊C4—1071no—
278C4⋊C4—2141no—
279C4⋊C4C22×D10711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—2141no—
282C22⋊C4—2141no—
283C22⋊C4—2141no—
284C23—21no—
285C23—1071no—
286C23—1071no—
287C23—4282no—
288C23—4281no—
289C23—4282no—
290C23—4282no—
291C23—4282no—
292C23—4282no—
293C23—2142no—
294C23—2144no—
295C23—2141no—
296C23—2142no—
297C23—2142no—
298C23D4×D10711yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D10714yescomplemented
302D4—22no—
303D4—42no—
304D4—1074no—
305D4—4282no—
306D4—4282no—
307D4—4282no—
308D4—4282no—
309D4—2148no—
310D4—2148no—
311D4—2144no—
312D4—2142no—
313D4—2142no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—1071no—
319C2×C4—1074no—
320C2×C4—1071no—
321C2×C4—4282no—
322C2×C4—2142no—
323C2×C4—2148no—
324C2×C4—2142no—
325C2×C4—2148no—
326C2×C4—2142no—
327C2×C4—2141no—
328C2×C4—2142no—
329C2×C4D4×D10711yescharacteristic
330C2×C4C23×D10711yescharacteristic
331C8—22no—
332C8—2142no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—8564no—
339C22—1072no—
340C22—4288no—
341C22—4284no—
342C22—4281no—
343C22—4281no—
344C22—4281no—
345C22—4282no—
346C22—4282no—
347C22—2141no—
348C22—2141no—
349C22—2144no—
350C22—2141no—
351C22—2141no—
352C22—2141no—
353C22#1681213612yescomplemented
354C22C2×D4×D10711yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—1072no—
359C4—4281no—
360C4—2141no—
361C4C2×D4×D10712yes—
362C2—22no—
363C2—81no—
364C2—8561no—
365C2—4284no—
366C2—4281no—
367C2—2142no—
368C2#120×D10711yescharacteristic
369C2C2×#1681213611yescharacteristic
370C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#448×C107C2
socleC22×C107C22×D4
hypercenter#448D107
hypocenterC107#3006
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×#13133819C2semidirect
2. C2×#16812136C2semidirect
3. #121×D107C2semidirect
4. #20292775C2semidirect
5. #20292771C2semidirect
6. #20292766C2semidirect
7. #20292763C2semidirect
8. #20292760C2semidirect
9. #20292627C2semidirect
10. #20292589C2semidirect
11. #448×C107C2semidirect
16. C2×#13133819C22semidirect
18. C22×D428C22semidirect
20. C2×D4×D107C22semidirect
24. C2×#13133805C22semidirect
25. C2×D856C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133801C22semidirect
28. #16812084C22semidirect
32. #16812080C22semidirect
33. C4⋊C4×D107C22semidirect
35. #16812033C22semidirect
36. #16812031C22semidirect
37. #16812029C22semidirect
38. C22⋊(C107⋊C8)C22semidirect
39. (C2×D107)⋊C8C22semidirect
40. C22×Q8×C107C22semidirect
41. C2×SD16×C107C22semidirect
43. #103×C107C22semidirect
44. C22⋊C8×C107C22semidirect
53. #13133819D4semidirect
115. Q8×C107C2×D4semidirect
137. D428C2×D4semidirect
152. C22×C107AGL(1,ℤ/8)semidirect
186. C107#3006semidirect
193. #448D107semidirect
203. C22×Q8C2×D107semidirect
212. C2×SD16C2×D107semidirect
225. #103C2×D107semidirect
230. C22⋊C8C2×D107semidirect
301. Q8D4×D107semidirect
353. C22#16812136semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#168121362
#121×D1072
#202927752
#202927712
#202927662
#202927632
#202927602
#202926272
#202925892
#448×C1072
#3006107