Finite Group Databasebeta

← #28346087

Subgroups — #28346087

Order 13696.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812140C211yesmaximal, characteristic, complemented
2C22×#13133802C211yesmaximal, characteristic, complemented
3#20293138C211yesmaximal, characteristic
4#20292883C212yesmaximal, complemented
5#20292875C212yesmaximal, complemented
6#20292866C212yesmaximal, complemented
7#20292858C212yesmaximal, complemented
8#20292667C211yesmaximal, characteristic, complemented
9#20292636C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812140—24no—
14C2×#13133808C2212yescomplemented
15C2×#13133805C2212yescomplemented
16C2×#13133802—24no—
17C2×#13133802C2212yes—
18C2×#13133802C2211yescharacteristic
19C2×#13133802C2211yescharacteristic
20C22×C107⋊C8C2212yes—
21#16812092—21no—
22#16812088C2211yescharacteristic
23#16812088C2211yescharacteristic
24#16812083C2211yescharacteristic
25#16812083C2211yescharacteristic
26#16812078—21no—
27C4×C22⋊D107—21no—
28C4×C22⋊D107—21no—
29#16812068C2211yescharacteristic
30#16812042C2212yescomplemented
31#16812040C2212yescomplemented
32#16812033C2212yescomplemented
33#16812033C2212yescomplemented
34#16812032C2212yescomplemented
35#16812029C2212yescomplemented
36C22×Q8×C107C2211yescharacteristic
37C23×C4×C107C2211yescharacteristic
38C22⋊Q8×C107—21no—
39C22⋊Q8×C107—21no—
40C22⋊Q8×C107C2212yes—
41C22⋊Q8×C107C2212yes—
42C2×C4⋊C4×C107—21no—
43C2×C4⋊C4×C107C2211yescharacteristic
44C2×C22⋊C4×C107—21no—
45#13133818—22no—
46#13133818—22no—
47C2×Dic214C2311yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50C2×C22⋊D107—21no—
51C2×C22⋊D107—21no—
52C2×C4×D107—21no—
53#13133808—24no—
54#13133805—24no—
55#13133802—22no—
56#13133802—24no—
57#13133802D414yescomplemented
58C2×C107⋊C8—24no—
59C2×C107⋊C8C2312yes—
60C2×C107⋊C8C2312yes—
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—21no—
64Dic107⋊C4—21no—
65C22⋊Dic107—21no—
66C22⋊Dic107—21no—
67C22⋊Dic107—21no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4—21no—
70C4×Dic107—21no—
71C24×C107D411yescharacteristic
72C2×Q8×C107—21no—
73C2×Q8×C107—22no—
74C2×Q8×C107—22no—
75C2×Q8×C107C2312yes—
76C22×C4×C107—22no—
77C22×C4×C107—22no—
78C22×C4×C107—21no—
79C22×C4×C107—21no—
80C22×C4×C107D411yescharacteristic
81C22×C4×C107D411yescharacteristic
82C22×C4×C107D411yescharacteristic
83C22×C4×C107C2311yescharacteristic
84C22×C4×C107C2311yescharacteristic
85C22×C4×C107C2311yescharacteristic
86C4⋊C4×C107—22no—
87C4⋊C4×C107—22no—
88C4⋊C4×C107—21no—
89C4⋊C4×C107C2312yes—
90C22⋊C4×C107—22no—
91C22⋊C4×C107—22no—
92C23×C107—22no—
93C23×C107—22no—
94C23×C107—41no—
95C23×C107C2×D411yescharacteristic
96C23×C107C2×D411yescharacteristic
97C23×C107C2×D411yescharacteristic
98C22×D107—21no—
99Q8×C107—22no—
100Q8×C107—22no—
101Q8×C107—42no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—22no—
105C2×C4×C107—21no—
106C2×C4×C107—21no—
107C2×C4×C107—21no—
108C2×C4×C107—21no—
109C2×C4×C107—22no—
110C2×C4×C107—22no—
111C2×C4×C107—42no—
112C2×C4×C107—42no—
113C2×C4×C107C2×D412yes—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107#4112yes—
118C2×C4×C107C2411yescharacteristic
119C22⋊D107—42no—
120C22⋊D107—42no—
121C22⋊D107—42no—
122C2×Dic107—21no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125D428—21no—
126D428—21no—
127C4×D107—42no—
128Dic214—21no—
129Dic214—21no—
130C107⋊C8—22no—
131C107⋊C8C2×D414yes—
132C22×C107—21no—
133C22×C107—21no—
134C22×C107—21no—
135C22×C107—21no—
136C22×C107—21no—
137C22×C107—21no—
138C22×C107—21no—
139C22×C107—44no—
140C22×C107#12112yes—
141C22×C107#13712yescomplemented
142C22×C107C22×D411yescharacteristic
143C2×D107—41no—
144C2×D107—41no—
145C2×D107—41no—
146C4×C107—21no—
147C4×C107—21no—
148C4×C107—42no—
149C4×C107—42no—
150C4×C107#12112yes—
151C4×C107C22×D411yescharacteristic
152C4×C107C2×#4111yescharacteristic
153Dic107—41no—
154Dic107—41no—
155Dic107—41no—
156C2×C107—22no—
157C2×C107—22no—
158C2×C107—41no—
159C2×C107C2×#12111yescharacteristic
160C2×C107C2×#13711yescharacteristic
161C2×C107#57511yescharacteristic
162D107—81no—
163#3056—1071nomaximal, Sylow
164C107#305611yescharacteristic, complemented, Sylow
165C2×#137—1071no—
166C22×M16—1071no—
167#525—1071no—
168C2×C22⋊Q8D10711yescharacteristic, complemented
169#470—1072no—
170#468—1072no—
171#467—1072no—
172#465—1072no—
173#428—1071no—
174#419—1071no—
175#417—1071no—
176C2×#41—1071no—
177C22×Q8C2×D10711yescharacteristic
178C23×C4C2×D10711yescharacteristic
179#137—2144no—
180C2×Q16—1072no—
181C2×SD16—1072no—
182C2×M16—1071no—
183C2×M16—1071no—
184C2×M16—1072no—
185C2×M16—2144no—
186C22×C8—1072no—
187#123—2141no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—1071no—
191C22⋊Q8—1071no—
192C22⋊Q8C2×D10712yes—
193C22⋊Q8C2×D10712yes—
194#121—1071no—
195#121—1071no—
196#120—2141no—
197C4×D4—2141no—
198C4×D4—2141no—
199#117—1071no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×D10711yescharacteristic
202C2×C22⋊C4—21no—
203#107—1072no—
204#106—1072no—
205#103—1072no—
206#103—1072no—
207#103—1072no—
208#102—1072no—
209C24C22⋊D10711yescharacteristic
210#41—2142no—
211#41—2142no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—1071no—
216C2×Q8C22×D10712yes—
217C2×D4—1071no—
218C2×D4—2141no—
219C2×D4—2141no—
220C2×D4—2141no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—2141no—
226C22×C4C22⋊D10711yescharacteristic
227C22×C4C22⋊D10711yescharacteristic
228C22×C4C22⋊D10711yescharacteristic
229C22×C4C22×D10711yescharacteristic
230C22×C4C22×D10711yescharacteristic
231C22×C4C22×D10711yescharacteristic
232Q16—2144no—
233SD16—2144no—
234M16—1074no—
235M16—2144no—
236M16—2142no—
237C2×C8—1072no—
238C2×C8—1072no—
239C2×C8—2144no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—1071no—
244C4⋊C4—1071no—
245C4⋊C4—2141no—
246C4⋊C4—2141no—
247C4⋊C4C22×D10712yes—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2141no—
251C22⋊C4—2141no—
252C22⋊C4—2141no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C42—2141no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—2141no—
260C23C2×C22⋊D10711yescharacteristic
261C23C2×C22⋊D10711yescharacteristic
262C23C2×C22⋊D10711yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—42no—
266Q8—2141no—
267Q8—2141no—
268D4—4282no—
269D4—4282no—
270D4—4282no—
271D4—2141no—
272D4—2141no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—4282no—
285C2×C4—2141no—
286C2×C4—2141no—
287C2×C4—2141no—
288C2×C4D4×D10712yes—
289C2×C4C2×C22⋊D10711yescharacteristic
290C2×C4C2×C22⋊D10711yescharacteristic
291C2×C4C2×C22⋊D10711yescharacteristic
292C2×C4#1313381712yes—
293C2×C4C23×D10711yescharacteristic
294C8—1074no—
295C8—2142no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—44no—
304C22—4281no—
305C22—4281no—
306C22—4281no—
307C22#1681208112yes—
308C22#1681214012yescomplemented
309C22C22×C22⋊D10711yescharacteristic
310C4—21no—
311C4—21no—
312C4—42no—
313C4—42no—
314C4—4281no—
315C4—4281no—
316C4—4281no—
317C4#1681208112yes—
318C4C2×D4×D10711yescharacteristic
319C4C2×#1313381711yescharacteristic
320C2—22no—
321C2—22no—
322C2—41no—
323C2—8561no—
324C2C2×#1681208111yescharacteristic
325C2C2×#1681214011yescharacteristic
326C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C22⋊Q8×C107C2
socleC22×C107C22×D4
prime radical#20293138C2
hypercenterC2×C22⋊Q8D107
hypocenterC107#3056
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C22×#13133802C2semidirect
4. #20292883C2semidirect
5. #20292875C2semidirect
6. #20292866C2semidirect
7. #20292858C2semidirect
8. #20292667C2semidirect
9. #20292636C2semidirect
10. #20292626C2semidirect
11. C2×C22⋊Q8×C107C2semidirect
14. C2×#13133808C22semidirect
15. C2×#13133805C22semidirect
30. #16812042C22semidirect
31. #16812040C22semidirect
32. #16812033C22semidirect
33. #16812033C22semidirect
34. #16812032C22semidirect
35. #16812029C22semidirect
57. #13133802D4semidirect
141. C22×C107#137semidirect
164. C107#3056semidirect
168. C2×C22⋊Q8D107semidirect
308. C22#16812140semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C22×#131338022
#202931382
#202928832
#202928752
#202928662
#202928582
#202926672
#202926362
#202926262
C2×C22⋊Q8×C1072
#3056107