Finite Group Databasebeta

← #28345715

Subgroups — #28345715

Order 13696.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133818C211yesmaximal, characteristic, complemented
2C2×#16812137C212yesmaximal, complemented
3C2×#16812083C211yesmaximal, characteristic, complemented
4#20292774C214yesmaximal, complemented
5#20292768C212yesmaximal, complemented
6#20292763C212yesmaximal, complemented
7#20292629C212yesmaximal, complemented
8#407×C107C211yesmaximal, characteristic, complemented
9C2×#13133818—28no—
10C2×#13133818—21no—
11C2×#13133818—21no—
12C2×#13133818C2212yes—
13C2×#13133818C2212yescomplemented
14C22×Dic214C2211yescharacteristic, complemented
15C22×D428C2211yescharacteristic, complemented
16C22×D428C2211yescharacteristic, complemented
17C22×C22⋊D107—21no—
18C22×C22⋊D107—21no—
19C22×C4×D107—21no—
20#16812137—28no—
21C2×#13133807C2214yescomplemented
22C2×D856C2214yescomplemented
23#16812083—21no—
24#16812083—21no—
25#16812083C2212yes—
26#16812083C2212yescomplemented
27C2×C4⋊Dic107C2211yescharacteristic, complemented
28C2×(C2×D107)⋊C4—21no—
29#16812034C2214yescomplemented
30#16812031C2214yescomplemented
31C23×C4×C107C2211yescharacteristic
32C2×M16×C107C2212yescomplemented
33C22⋊C8×C107C2212yescomplemented
34C22⋊C8×C107C2212yescomplemented
35C23×D107—21no—
36C23×D107—21no—
37#13133818—28no—
38#13133818—28no—
39#13133818—24no—
40#13133818—24no—
41#13133818—24no—
42#13133818D418yescomplemented
43C2×Dic214—21no—
44C2×Dic214—24no—
45C2×Dic214C2312yes—
46C2×D428—21no—
47C2×D428—24no—
48C2×D428—24no—
49C2×D428—21no—
50C2×D428C2312yes—
51C2×D428C2312yes—
52C2×C22⋊D107—28no—
53C2×C22⋊D107—22no—
54C2×C22⋊D107—22no—
55C2×C22⋊D107—22no—
56C2×C22⋊D107—22no—
57C2×C22⋊D107—44no—
58C22×Dic107—21no—
59C22×Dic107—21no—
60C2×C4×D107—28no—
61C2×C4×D107—22no—
62C2×C4×D107—22no—
63#13133807—28no—
64D856—28no—
65C4⋊Dic107—21no—
66C4⋊Dic107C2312yes—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69C24×C107D411yescharacteristic
70C22×C4×C107—24no—
71C22×C4×C107D411yescharacteristic
72C22×C4×C107D411yescharacteristic
73C22×C4×C107D411yescharacteristic
74C22×C4×C107C2311yescharacteristic
75C22×C4×C107C2311yescharacteristic
76C22×C4×C107C2311yescharacteristic
77M16×C107—24no—
78C2×C8×C107C2314yes—
79C23×C107—22no—
80C23×C107—22no—
81C23×C107—41no—
82C23×C107C2×D411yescharacteristic
83C23×C107C2×D411yescharacteristic
84C23×C107C2×D411yescharacteristic
85C22×D107—24no—
86C22×D107—22no—
87C22×D107—22no—
88C22×D107—42no—
89C22×D107—42no—
90C22×D107—42no—
91C22×D107—42no—
92C22×D107—42no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—22no—
98C2×C4×C107C2×D414yes—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—28no—
104C22⋊D107—28no—
105C22⋊D107—44no—
106C22⋊D107—44no—
107C2×Dic107—24no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—42no—
111D428—24no—
112D428—22no—
113D428—24no—
114D428—42no—
115D428C2×D414yescomplemented
116C4×D107—28no—
117C4×D107—28no—
118Dic214—24no—
119Dic214—22no—
120Dic214C2×D414yescomplemented
121C8×C107—24no—
122C22×C107—21no—
123C22×C107—21no—
124C22×C107—21no—
125C22×C107—21no—
126C22×C107—21no—
127C22×C107—21no—
128C22×C107—21no—
129C22×C107—44no—
130C22×C107#12012yes—
131C22×C107AGL(1,ℤ/8)12yescomplemented
132C22×C107C22×D411yescharacteristic
133C2×D107—24no—
134C2×D107—48no—
135C2×D107—42no—
136C2×D107—42no—
137C2×D107—42no—
138C2×D107—42no—
139C2×D107—42no—
140C2×D107—84no—
141C4×C107—21no—
142C4×C107—21no—
143C4×C107#12012yes—
144C4×C107C22×D412yes—
145Dic107—24no—
146Dic107—42no—
147C2×C107—22no—
148C2×C107—22no—
149C2×C107—41no—
150C2×C107C2×#12011yescharacteristic
151C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C107#57511yescharacteristic
153D107—44no—
154D107—82no—
155#3006—1071nomaximal, Sylow
156C107#300611yescharacteristic, complemented, Sylow
157C22×#41—1071no—
158C2×AGL(1,ℤ/8)—1072no—
159C2×#121—1071no—
160#449—1074no—
161#448—1072no—
162#447—1072no—
163#417—1072no—
164#407D10711yescharacteristic, complemented
165C2×#41—1072no—
166C2×#41—1072no—
167C2×#41—2148no—
168C2×#41—2141no—
169C2×#41—2141no—
170C22×Q8—1071no—
171C22×D4—1071no—
172C22×D4—1071no—
173C22×D4—2141no—
174C22×D4—2141no—
175C23×C4—2141no—
176C23×C4C2×D10711yescharacteristic
177AGL(1,ℤ/8)—2148no—
178C2×SD16—1074no—
179C2×D8—1074no—
180C2×M16C2×D10712yescomplemented
181#121—1072no—
182#121—1072no—
183#121—2141no—
184#121—2141no—
185C2×C4⋊C4—1071no—
186C2×C22⋊C4—2141no—
187#103—1074no—
188#102—1074no—
189C22⋊C8C2×D10712yescomplemented
190C22⋊C8C2×D10712yescomplemented
191C24—2141no—
192C24—2141no—
193C24D42811yescharacteristic
194#41—1078no—
195#41—2144no—
196#41—2144no—
197#41—2148no—
198#41—2148no—
199#41—2144no—
200C2×Q8—1072no—
201C2×Q8—2144no—
202C2×Q8—2141no—
203C2×D4—1072no—
204C2×D4—1072no—
205C2×D4—4284no—
206C2×D4—2144no—
207C2×D4—2142no—
208C2×D4—2142no—
209C2×D4—2148no—
210C2×D4—2142no—
211C2×D4—2142no—
212C2×D4—2144no—
213C2×D4—2141no—
214C2×D4—2141no—
215C22×C4—24no—
216C22×C4—2148no—
217C22×C4—2141no—
218C22×C4—2141no—
219C22×C4—2142no—
220C22×C4—2142no—
221C22×C4D42811yescharacteristic
222C22×C4D42811yescharacteristic
223C22×C4D42811yescharacteristic
224C22×C4C22×D10711yescharacteristic
225C22×C4C22×D10711yescharacteristic
226C22×C4C22×D10711yescharacteristic
227SD16—2148no—
228D8—2148no—
229M16—24no—
230C2×C8C22×D10714yes—
231C4⋊C4—1072no—
232C4⋊C4—2141no—
233C22⋊C4—2142no—
234C22⋊C4—2142no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—4282no—
239C23—4282no—
240C23—4282no—
241C23—4282no—
242C23—4282no—
243C23—2144no—
244C23—2142no—
245C23—2142no—
246C23C2×D42811yescharacteristic
247C23C2×D42811yescharacteristic
248C23C2×D42811yescharacteristic
249Q8—1074no—
250Q8—2144no—
251Q8—2142no—
252D4—1074no—
253D4—4284no—
254D4—4284no—
255D4—4282no—
256D4—2144no—
257D4—2148no—
258D4—2148no—
259D4—2144no—
260D4—2142no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—4282no—
267C2×C4—2148no—
268C2×C4—2142no—
269C2×C4—2144no—
270C2×C4—2142no—
271C2×C4—2148no—
272C2×C4D4×D10714yes—
273C2×C4C2×D42811yescharacteristic
274C2×C4C2×D42811yescharacteristic
275C2×C4C2×D42811yescharacteristic
276C2×C4C23×D10711yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—8564no—
287C22—4288no—
288C22—4282no—
289C22—4282no—
290C22—4282no—
291C22—4282no—
292C22—4282no—
293C22—2144no—
294C22#1681207912yes—
295C22#1681213712yescomplemented
296C22C22×D42811yescharacteristic
297C4—21no—
298C4—21no—
299C4—4282no—
300C4—2144no—
301C4#1681207912yes—
302C4C2×D4×D10712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—8562no—
307C2—4284no—
308C2C2×#1681207911yescharacteristic
309C2C2×#1681213711yescharacteristic
310C2#2029342711yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#407×C107C2
socleC22×C107C22×D4
hypercenter#407D107
hypocenterC107#3006
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13133818C2semidirect
2. C2×#16812137C2semidirect
3. C2×#16812083C2semidirect
4. #20292774C2semidirect
5. #20292768C2semidirect
6. #20292763C2semidirect
7. #20292629C2semidirect
8. #407×C107C2semidirect
13. C2×#13133818C22semidirect
14. C22×Dic214C22semidirect
15. C22×D428C22semidirect
16. C22×D428C22semidirect
21. C2×#13133807C22semidirect
22. C2×D856C22semidirect
26. #16812083C22semidirect
27. C2×C4⋊Dic107C22semidirect
29. #16812034C22semidirect
30. #16812031C22semidirect
32. C2×M16×C107C22semidirect
33. C22⋊C8×C107C22semidirect
34. C22⋊C8×C107C22semidirect
42. #13133818D4semidirect
115. D428C2×D4semidirect
120. Dic214C2×D4semidirect
131. C22×C107AGL(1,ℤ/8)semidirect
156. C107#3006semidirect
164. #407D107semidirect
180. C2×M16C2×D107semidirect
189. C22⋊C8C2×D107semidirect
190. C22⋊C8C2×D107semidirect
295. C22#16812137semidirect

Maximal subgroups

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

subgroupindex
C22×#131338182
C2×#168121372
C2×#168120832
#202927742
#202927682
#202927632
#202926292
#407×C1072
#3006107