Finite Group Databasebeta

← #28345703

Subgroups — #28345703

Order 13696.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic, complemented
3#20293138C211yesmaximal, characteristic, complemented
4#20292789C214yesmaximal, complemented
5#20292772C214yesmaximal, complemented
6#20292636C211yesmaximal, characteristic, complemented
7#20292632C211yesmaximal, characteristic, complemented
8#20292591C211yesmaximal, characteristic, complemented
9C22×#41×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812140—24no—
12#16812135—24no—
13C2×#13133808C2212yescomplemented
14C2×#13133806C2212yescomplemented
15C2×#13133805C2212yescomplemented
16C2×#13133803C2212yescomplemented
17C2×#13133802C2211yescharacteristic
18C2×#13133802C2211yescharacteristic
19#16812092—21no—
20#16812088C2212yescomplemented
21#16812083C2212yescomplemented
22#16812078—21no—
23C4×C22⋊D107—22no—
24#16812068C2211yescharacteristic
25#16812033C2214yescomplemented
26#16812030C2214yescomplemented
27C22⋊(C107⋊C8)C2214yescomplemented
28C2×#41×C107—24no—
29C2×#41×C107—24no—
30C2×#41×C107—22no—
31C2×#41×C107C2214yes—
32C22×Q8×C107C2211yescharacteristic
33C22×D4×C107—21no—
34C22×D4×C107C2211yescharacteristic
35C23×C4×C107—21no—
36C23×C4×C107C2211yescharacteristic
37#13133818—22no—
38#13133818—22no—
39C2×Dic214C2311yescharacteristic
40C2×D428C2311yescharacteristic
41C2×C22⋊D107—22no—
42C2×C22⋊D107—21no—
43C2×C4×D107—21no—
44#13133808—24no—
45#13133806—24no—
46#13133805—24no—
47#13133803—24no—
48#13133802—22no—
49#13133802—22no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52C4⋊Dic107C2312yes—
53Dic107⋊C4—22no—
54C22⋊Dic107—21no—
55C22⋊Dic107—22no—
56(C2×D107)⋊C4—22no—
57C4×Dic107—21no—
58C24×C107—21no—
59C24×C107D411yescharacteristic
60#41×C107—216no—
61#41×C107—28no—
62#41×C107—28no—
63C2×Q8×C107—22no—
64C2×Q8×C107—21no—
65C2×Q8×C107D414yescomplemented
66C2×Q8×C107C2312yes—
67C2×D4×C107—28no—
68C2×D4×C107—24no—
69C2×D4×C107—21no—
70C2×D4×C107—22no—
71C2×D4×C107D414yescomplemented
72C2×D4×C107C2312yes—
73C22×C4×C107—28no—
74C22×C4×C107—21no—
75C22×C4×C107—24no—
76C22×C4×C107—22no—
77C22×C4×C107—22no—
78C22×C4×C107D412yes—
79C22×C4×C107D411yescharacteristic
80C22×C4×C107C2312yes—
81C22×C4×C107C2311yescharacteristic
82C23×C107—22no—
83C23×C107—24no—
84C23×C107—22no—
85C23×C107—42no—
86C23×C107—44no—
87C23×C107C2×D412yes—
88C23×C107C2×D411yescharacteristic
89C22×D107—21no—
90Q8×C107—24no—
91Q8×C107—22no—
92Q8×C107C2×D414yes—
93D4×C107—216no—
94D4×C107—24no—
95D4×C107—22no—
96D4×C107C2×D414yes—
97C2×C4×C107—216no—
98C2×C4×C107—22no—
99C2×C4×C107—24no—
100C2×C4×C107—22no—
101C2×C4×C107—24no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107C2×D412yes—
105C2×C4×C107C2×D411yescharacteristic
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2411yescharacteristic
109C22⋊D107—44no—
110C22⋊D107—42no—
111C2×Dic107—22no—
112C2×Dic107—21no—
113D428—21no—
114D428—21no—
115C4×D107—42no—
116Dic214—21no—
117Dic214—21no—
118C107⋊C8—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—24no—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—44no—
126C22×C107—48no—
127C22×C107—44no—
128C22×C107#12012yes—
129C22×C107C22×D411yescharacteristic
130C2×D107—41no—
131C2×D107—42no—
132C4×C107—22no—
133C4×C107—24no—
134C4×C107#12012yes—
135C4×C107C22×D411yescharacteristic
136C4×C107C22×D411yescharacteristic
137Dic107—41no—
138Dic107—42no—
139C2×C107—22no—
140C2×C107—42no—
141C2×C107—44no—
142C2×C107C2×#12011yescharacteristic
143C2×C107#57512yes—
144D107—81no—
145#3005—1071nomaximal, Sylow
146C107#300511yescharacteristic, complemented, Sylow
147C22×#41D10711yescharacteristic, complemented
148C2×#137—1071no—
149C2×AGL(1,ℤ/8)—1071no—
150#525—1071no—
151#452—1074no—
152#449—1074no—
153#419—1071no—
154#418—1071no—
155#407—1071no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1071no—
160C2×#41C2×D10714yes—
161C22×Q8C2×D10711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D10711yescharacteristic
164C23×C4—21no—
165C23×C4C2×D10711yescharacteristic
166#137—2144no—
167AGL(1,ℤ/8)—2144no—
168C2×Q16—1072no—
169C2×SD16—1072no—
170C2×SD16—1072no—
171C2×D8—1072no—
172C2×M16—1071no—
173C2×M16—1071no—
174#123—2141no—
175C22⋊Q8—1072no—
176#121—1072no—
177#120—2141no—
178C4×D4—2142no—
179#117—1071no—
180#103—1074no—
181#102—1074no—
182C22⋊C8—1074no—
183C24—21no—
184C24C22⋊D10711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—2142no—
189#41—2142no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1071no—
193C2×Q8C22⋊D10714yescomplemented
194C2×Q8C22×D10712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—1071no—
200C2×D4—2142no—
201C2×D4—2141no—
202C2×D4C22⋊D10714yescomplemented
203C2×D4C22×D10712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2141no—
210C22×C4C22⋊D10712yes—
211C22×C4C22⋊D10711yescharacteristic
212C22×C4C22×D10712yes—
213C22×C4C22×D10711yescharacteristic
214Q16—2144no—
215SD16—2144no—
216SD16—2144no—
217D8—2144no—
218M16—2142no—
219M16—2142no—
220C2×C8—1072no—
221C2×C8—1072no—
222C4⋊C4—1072no—
223C4⋊C4—2142no—
224C22⋊C4—2141no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C42—2141no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—2141no—
234C23C2×C22⋊D10712yes—
235C23C2×C22⋊D10711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—2141no—
239Q8—2141no—
240Q8C2×C22⋊D10714yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—4284no—
245D4—4282no—
246D4—2141no—
247D4—2141no—
248D4C2×C22⋊D10714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—4282no—
257C2×C4—2142no—
258C2×C4—2141no—
259C2×C4C2×C22⋊D10712yes—
260C2×C4C2×C22⋊D10711yescharacteristic
261C2×C4C2×C22⋊D10712yes—
262C2×C4C2×C22⋊D10712yes—
263C2×C4C23×D10711yescharacteristic
264C8—2142no—
265C8—2142no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—4282no—
275C22—4281no—
276C22#1681207812yes—
277C22C22×C22⋊D10711yescharacteristic
278C4—22no—
279C4—24no—
280C4—4282no—
281C4—4281no—
282C4#1681207812yes—
283C4C22×C22⋊D10711yescharacteristic
284C4C22×C22⋊D10711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—8561no—
289C2C2×#1681207811yescharacteristic
290C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×#41×C107C2
socleC22×C107C22×D4
hypercenterC22×#41D107
hypocenterC107#3005
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×#16812135C2semidirect
3. #20293138C2semidirect
4. #20292789C2semidirect
5. #20292772C2semidirect
6. #20292636C2semidirect
7. #20292632C2semidirect
8. #20292591C2semidirect
9. C22×#41×C107C2semidirect
13. C2×#13133808C22semidirect
14. C2×#13133806C22semidirect
15. C2×#13133805C22semidirect
16. C2×#13133803C22semidirect
20. #16812088C22semidirect
21. #16812083C22semidirect
25. #16812033C22semidirect
26. #16812030C22semidirect
27. C22⋊(C107⋊C8)C22semidirect
65. C2×Q8×C107D4semidirect
71. C2×D4×C107D4semidirect
146. C107#3005semidirect
147. C22×#41D107semidirect
193. C2×Q8C22⋊D107semidirect
202. C2×D4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121352
#202931382
#202927892
#202927722
#202926362
#202926322
#202925912
C22×#41×C1072
#3005107