Finite Group Databasebeta

← #28346085

Subgroups — #28346085

Order 13696.

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

normal subgroups only

This group has 496 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
1C2×M16×D107C211yesmaximal, characteristic, complemented
2#20293134C211yesmaximal, characteristic
3C2×#16812085C211yesmaximal, characteristic, complemented
4#20292882C212yesmaximal, complemented
5#20292870C212yesmaximal, complemented
6#20292865C212yesmaximal, complemented
7#20292853C212yesmaximal, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292636C211yesmaximal, characteristic, complemented
10#20292628C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107C2211yescharacteristic
13C2×#13133802C2211yescharacteristic
14C2×#13133801C2212yes—
15M16×D107—24no—
16C2×C8×D107C2212yes—
17#16812090—21no—
18#16812085—21no—
19#16812085—21no—
20#16812085C2211yescharacteristic
21#16812085C2212yes—
22#16812085C2212yes—
23#16812085C2211yescharacteristic
24#16812080C2211yescharacteristic
25#16812080C2211yescharacteristic
26#16812077—21no—
27C4×C22⋊D107—21no—
28C4×C22⋊D107—21no—
29#16812068C2211yescharacteristic
30C2×C4⋊Dic107C2211yescharacteristic
31C2×Dic107⋊C4—21no—
32C2×(C2×D107)⋊C4—21no—
33#16812043C2212yescomplemented
34#16812041C2212yescomplemented
35#16812033C2212yescomplemented
36#16812033C2212yescomplemented
37#16812033C2212yescomplemented
38#16812030C2212yescomplemented
39C2×#41×C107C2211yescharacteristic
40C22×Q8×C107C2211yescharacteristic
41#137×C107—24no—
42C2×Q16×C107C2212yescomplemented
43C2×SD16×C107C2212yescomplemented
44C2×M16×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46C2×C22⋊D107—21no—
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C22×Dic107D411yescharacteristic
50C2×C4×D107—22no—
51C2×C4×D107—22no—
52C2×C4×D107D411yescharacteristic
53C2×C4×D107D411yescharacteristic
54C2×C4×D107C2311yescharacteristic
55C2×C4×D107C2311yescharacteristic
56#13133802—22no—
57#13133801—24no—
58C2×C107⋊C8C2312yes—
59C8×D107—24no—
60C4⋊Dic107—21no—
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65Dic107⋊C4—22no—
66Dic107⋊C4—21no—
67Dic107⋊C4—21no—
68C22⋊Dic107—21no—
69C22⋊Dic107—21no—
70C22⋊Dic107—21no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73(C2×D107)⋊C4—21no—
74(C2×D107)⋊C4—21no—
75C4×Dic107—21no—
76#41×C107—22no—
77#41×C107—22no—
78C2×Q8×C107—21no—
79C2×Q8×C107—22no—
80C2×Q8×C107—22no—
81C2×Q8×C107C2312yes—
82C2×Q8×C107C2311yescharacteristic
83C2×D4×C107—21no—
84C2×D4×C107C2311yescharacteristic
85C22×C4×C107—21no—
86C22×C4×C107—21no—
87C22×C4×C107C2311yescharacteristic
88Q16×C107—24no—
89SD16×C107—24no—
90M16×C107D414yescomplemented
91C2×C8×C107C2312yes—
92C23×C107—21no—
93C23×C107C2×D411yescharacteristic
94C22×D107—22no—
95C22×D107—22no—
96C22×D107—41no—
97C22×D107C2×D411yescharacteristic
98C22×D107C2×D411yescharacteristic
99Q8×C107—22no—
100Q8×C107—21no—
101Q8×C107—22no—
102Q8×C107—21no—
103Q8×C107—42no—
104D4×C107—21no—
105D4×C107—21no—
106D4×C107—42no—
107C2×C4×C107—21no—
108C2×C4×C107—22no—
109C2×C4×C107—42no—
110C2×C4×C107—42no—
111C2×C4×C107C2×D411yescharacteristic
112C2×C4×C107C2×D412yes—
113C2×C4×C107#4112yes—
114C2×C4×C107C2411yescharacteristic
115C22⋊D107—42no—
116C22⋊D107—42no—
117C2×Dic107—22no—
118C2×Dic107—21no—
119C2×Dic107—21no—
120C2×Dic107—22no—
121C2×Dic107—42no—
122C2×Dic107C2×D411yescharacteristic
123C2×Dic107C2×D411yescharacteristic
124C4×D107—22no—
125C4×D107—22no—
126C4×D107—21no—
127C4×D107—21no—
128C4×D107—21no—
129C4×D107—21no—
130C8×C107C2×D414yes—
131C107⋊C8—22no—
132C22×C107—21no—
133C22×C107—21no—
134C22×C107—41no—
135C22×C107—41no—
136C22×C107—41no—
137C22×C107#12112yes—
138C22×C107C22×D411yescharacteristic
139C2×D107—21no—
140C2×D107—21no—
141C2×D107—21no—
142C2×D107—21no—
143C2×D107—21no—
144C2×D107—44no—
145C2×D107#13712yescomplemented
146C4×C107—41no—
147C4×C107—42no—
148C4×C107#12112yes—
149C4×C107C22×D411yescharacteristic
150C4×C107C2×#4111yescharacteristic
151Dic107—21no—
152Dic107—21no—
153Dic107—41no—
154Dic107—42no—
155Dic107—41no—
156C2×C107—22no—
157C2×C107—81no—
158C2×C107C2×#12111yescharacteristic
159C2×C107C2×#13711yescharacteristic
160C2×C107#57511yescharacteristic
161D107—22no—
162D107—41no—
163#3056—1071nomaximal, Sylow
164C107#305611yescharacteristic, complemented, Sylow
165C2×#137D10711yescharacteristic, complemented
166C22×M16—1071no—
167#525—1071no—
168C2×C22⋊Q8—1071no—
169#470—1072no—
170#468—1072no—
171#467—1072no—
172#465—1072no—
173#428—1071no—
174#419—1071no—
175#417—1071no—
176C2×#41C2×D10711yescharacteristic
177C22×Q8C2×D10711yescharacteristic
178C23×C4—1071no—
179#137—24no—
180C2×Q16C2×D10712yescomplemented
181C2×SD16C2×D10712yescomplemented
182C2×M16—1071no—
183C2×M16—1072no—
184C2×M16—2144no—
185C2×M16C2×D10711yescharacteristic
186C22×C8—1072no—
187#123—2141no—
188C22⋊Q8—1072no—
189C22⋊Q8—1072no—
190C22⋊Q8—1071no—
191C22⋊Q8—1071no—
192C22⋊Q8—2141no—
193C22⋊Q8—2141no—
194#121—1071no—
195#121—1071no—
196#120—2141no—
197C4×D4—2141no—
198C4×D4—2141no—
199#117—1071no—
200C2×C4⋊C4—1071no—
201C2×C4⋊C4—2141no—
202C2×C22⋊C4—2141no—
203#107—1072no—
204#106—1072no—
205#103—1072no—
206#103—1072no—
207#103—1072no—
208#102—1072no—
209C24—1071no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D10712yes—
216C2×Q8C22×D10711yescharacteristic
217C2×D4—21no—
218C2×D4—2141no—
219C2×D4—2141no—
220C2×D4C22×D10711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—1071no—
224C22×C4—1071no—
225C22×C4—1071no—
226C22×C4—1071no—
227C22×C4—1071no—
228C22×C4—2141no—
229C22×C4—2142no—
230C22×C4—2142no—
231C22×C4C22×D10711yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—2142no—
235M16—2144no—
236M16C22⋊D10714yescomplemented
237C2×C8—1072no—
238C2×C8—2144no—
239C2×C8C22×D10712yes—
240C4⋊C4—1072no—
241C4⋊C4—1071no—
242C4⋊C4—1071no—
243C4⋊C4—2142no—
244C4⋊C4—2142no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4—2141no—
248C22⋊C4—2142no—
249C22⋊C4—2142no—
250C22⋊C4—2141no—
251C22⋊C4—2141no—
252C22⋊C4—2141no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C42—2141no—
256C23—21no—
257C23—1071no—
258C23—1071no—
259C23—4281no—
260C23—2142no—
261C23—2142no—
262C23D4×D10711yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—4282no—
272D4—4282no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—1071no—
278C2×C4—1071no—
279C2×C4—4282no—
280C2×C4—2141no—
281C2×C4—2141no—
282C2×C4—2142no—
283C2×C4—2142no—
284C2×C4—2142no—
285C2×C4—2142no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4—2141no—
289C2×C4—2141no—
290C2×C4D4×D10711yescharacteristic
291C2×C4C2×C22⋊D10712yes—
292C2×C4#1313381712yes—
293C2×C4C23×D10711yescharacteristic
294C8—2142no—
295C8C2×C22⋊D10714yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—1072no—
302C22—4284no—
303C22—2141no—
304C22—2141no—
305C22—2141no—
306C22—2141no—
307C22—2141no—
308C22#1681208012yes—
309C22C2×D4×D10711yescharacteristic
310C4—41no—
311C4—42no—
312C4—4281no—
313C4—4282no—
314C4—4281no—
315C4—2141no—
316C4—2141no—
317C4#1681208012yes—
318C4C22×C22⋊D10711yescharacteristic
319C4C2×#1313381711yescharacteristic
320C2—22no—
321C2—81no—
322C2—4281no—
323C2—2142no—
324C2C2×#1681208011yescharacteristic
325C2#137×D10711yescharacteristic
326C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293134C2
hypercenterC2×#137D107
hypocenterC107#3056
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D107C2semidirect
3. C2×#16812085C2semidirect
4. #20292882C2semidirect
5. #20292870C2semidirect
6. #20292865C2semidirect
7. #20292853C2semidirect
8. #20292669C2semidirect
9. #20292636C2semidirect
10. #20292628C2semidirect
11. C2×#137×C107C2semidirect
33. #16812043C22semidirect
34. #16812041C22semidirect
35. #16812033C22semidirect
36. #16812033C22semidirect
37. #16812033C22semidirect
38. #16812030C22semidirect
42. C2×Q16×C107C22semidirect
43. C2×SD16×C107C22semidirect
90. M16×C107D4semidirect
145. C2×D107#137semidirect
164. C107#3056semidirect
165. C2×#137D107semidirect
180. C2×Q16C2×D107semidirect
181. C2×SD16C2×D107semidirect
236. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1072
#202931342
C2×#168120852
#202928822
#202928702
#202928652
#202928532
#202926692
#202926362
#202926282
C2×#137×C1072
#3056107