Finite Group Databasebeta

← #28346067

Subgroups — #28346067

Order 13696.

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

normal subgroups only

This group has 500 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
1C2×#16812129C211yesmaximal, characteristic, complemented
2C22×#13133802C211yesmaximal, characteristic, complemented
3#20293138C211yesmaximal, characteristic, complemented
4#20292883C212yesmaximal, complemented
5#20292878C212yesmaximal, complemented
6#20292875C212yesmaximal, complemented
7#20292871C212yesmaximal, complemented
8#20292663C211yesmaximal, characteristic, complemented
9#20292623C211yesmaximal, characteristic, complemented
10#20292621C211yesmaximal, characteristic, complemented
11#525×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812129—24no—
14C2×#13133808C2211yescharacteristic, complemented
15C2×#13133806C2211yescharacteristic, complemented
16C2×#13133805C2211yescharacteristic, complemented
17C2×#13133803C2211yescharacteristic, complemented
18C2×#13133802—24no—
19C2×#13133802C2212yes—
20C2×#13133802C2212yes—
21C22×C107⋊C8C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23#16812092—21no—
24#16812088C2212yes—
25#16812083C2212yes—
26#16812078—21no—
27C4×C22⋊D107—22no—
28#16812068C2211yescharacteristic
29#16812042C2212yescomplemented
30#16812040C2212yescomplemented
31#16812033C2212yescomplemented
32#16812032C2212yescomplemented
33#16812030C2212yescomplemented
34#16812029C2212yescomplemented
35C2×#41×C107C2211yescharacteristic
36C23×C4×C107C2211yescharacteristic
37#123×C107—21no—
38C22⋊Q8×C107C2212yes—
39#121×C107C2212yes—
40#120×C107—21no—
41C4×D4×C107—22no—
42#117×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—22no—
48C2×C22⋊D107—21no—
49C2×C4×D107—21no—
50#13133808—22no—
51#13133806—22no—
52#13133805—22no—
53#13133803—22no—
54#13133802—24no—
55#13133802—24no—
56C2×C107⋊C8—22no—
57C2×C107⋊C8D414yescomplemented
58C2×C107⋊C8C2312yes—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C4⋊Dic107C2312yes—
62Dic107⋊C4—22no—
63C22⋊Dic107—21no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—22no—
66C4×Dic107—21no—
67C24×C107D411yescharacteristic
68#41×C107—22no—
69#41×C107—22no—
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—22no—
72C2×D4×C107—21no—
73C2×D4×C107C2311yescharacteristic
74C22×C4×C107—22no—
75C22×C4×C107—22no—
76C22×C4×C107—21no—
77C22×C4×C107D412yes—
78C22×C4×C107D411yescharacteristic
79C22×C4×C107C2312yes—
80C22×C4×C107C2311yescharacteristic
81C4⋊C4×C107—22no—
82C4⋊C4×C107C2312yes—
83C22⋊C4×C107—21no—
84C22⋊C4×C107—22no—
85C22⋊C4×C107—22no—
86C42×C107—21no—
87C23×C107—22no—
88C23×C107—21no—
89C23×C107—42no—
90C23×C107C2×D412yes—
91C23×C107C2×D411yescharacteristic
92C22×D107—21no—
93Q8×C107—21no—
94Q8×C107—21no—
95D4×C107—21no—
96D4×C107—21no—
97D4×C107—44no—
98D4×C107—42no—
99C2×C4×C107—22no—
100C2×C4×C107—24no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—21no—
105C2×C4×C107—42no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107#4112yes—
110C2×C4×C107C2411yescharacteristic
111C22⋊D107—44no—
112C22⋊D107—42no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115D428—21no—
116D428—21no—
117C4×D107—42no—
118Dic214—21no—
119Dic214—21no—
120C107⋊C8—22no—
121C107⋊C8C2×D412yes—
122C107⋊C8C2×D412yes—
123C22×C107—22no—
124C22×C107—22no—
125C22×C107—22no—
126C22×C107—22no—
127C22×C107—44no—
128C22×C107—41no—
129C22×C107—42no—
130C22×C107#12112yes—
131C22×C107C22×D411yescharacteristic
132C2×D107—41no—
133C2×D107—42no—
134C4×C107—22no—
135C4×C107—41no—
136C4×C107—42no—
137C4×C107#12112yes—
138C4×C107C22×D411yescharacteristic
139C4×C107C2×#4111yescharacteristic
140Dic107—41no—
141Dic107—42no—
142C2×C107—22no—
143C2×C107—42no—
144C2×C107—81no—
145C2×C107C2×#12111yescharacteristic
146C2×C107#57512yes—
147D107—81no—
148#3054—1071nomaximal, Sylow
149C107#305411yescharacteristic, complemented, Sylow
150C2×#135—1071no—
151C22×M16—1071no—
152#525—1071no—
153#525D10711yescharacteristic, complemented
154#470—1072no—
155#469—1072no—
156#468—1072no—
157#468—1072no—
158#427—1071no—
159#416—1071no—
160#416—1071no—
161C2×#41—1071no—
162C2×#41C2×D10711yescharacteristic
163C23×C4C2×D10711yescharacteristic
164#135—2144no—
165C2×Q16—1071no—
166C2×SD16—1071no—
167C2×SD16—1071no—
168C2×D8—1071no—
169C2×M16—1072no—
170C2×M16—1072no—
171C2×M16—2144no—
172C22×C8—1071no—
173C22×C8—1071no—
174#123—21no—
175#123—2141no—
176C22⋊Q8—1072no—
177C22⋊Q8C2×D10712yes—
178#121—1072no—
179#121C2×D10712yes—
180#120—21no—
181#120—2141no—
182C4×D4—22no—
183C4×D4—2142no—
184#117—1071no—
185#117C2×D10711yescharacteristic
186#107—1072no—
187#106—1072no—
188#103—1072no—
189#103—1072no—
190#102—1072no—
191#102—1072no—
192C24C22⋊D10711yescharacteristic
193#41—22no—
194#41—22no—
195#41—2142no—
196#41—2142no—
197C2×Q8—1071no—
198C2×Q8C22×D10711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—1071no—
202C2×D4—2142no—
203C2×D4—2141no—
204C2×D4C22×D10711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2141no—
209C22×C4C22⋊D10711yescharacteristic
210C22×C4C22⋊D10712yes—
211C22×C4C22×D10712yes—
212C22×C4C22×D10711yescharacteristic
213Q16—2142no—
214SD16—2142no—
215SD16—2142no—
216D8—2142no—
217M16—2144no—
218M16—2144no—
219C2×C8—1074no—
220C2×C8—1071no—
221C2×C8—1071no—
222C2×C8—1072no—
223C2×C8—2142no—
224C4⋊C4—22no—
225C4⋊C4—1072no—
226C4⋊C4—2142no—
227C4⋊C4C22×D10712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2141no—
232C22⋊C4—2142no—
233C22⋊C4—2142no—
234C42—21no—
235C42—2141no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—2141no—
240C23C2×C22⋊D10712yes—
241C23C2×C22⋊D10711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2141no—
245Q8—2141no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—4284no—
251D4—4282no—
252D4—2141no—
253D4—2141no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—4282no—
262C2×C4—2142no—
263C2×C4—2141no—
264C2×C4D4×D10712yes—
265C2×C4C2×C22⋊D10712yes—
266C2×C4C2×C22⋊D10711yescharacteristic
267C2×C4#1313381712yes—
268C2×C4C23×D10711yescharacteristic
269C8—1072no—
270C8—1072no—
271C8—2142no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—42no—
278C22—41no—
279C22—4282no—
280C22—4281no—
281C22#1681208112yes—
282C22C22×C22⋊D10711yescharacteristic
283C4—22no—
284C4—42no—
285C4—41no—
286C4—4282no—
287C4—4281no—
288C4#1681208112yes—
289C4C2×D4×D10711yescharacteristic
290C4C2×#1313381711yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—8561no—
295C2C2×#1681208111yescharacteristic
296C2#2029342512yes—
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#525×C107C2
socleC22×C107C22×D4
hypercenter#525D107
hypocenterC107#3054
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. C22×#13133802C2semidirect
3. #20293138C2semidirect
4. #20292883C2semidirect
5. #20292878C2semidirect
6. #20292875C2semidirect
7. #20292871C2semidirect
8. #20292663C2semidirect
9. #20292623C2semidirect
10. #20292621C2semidirect
11. #525×C107C2semidirect
14. C2×#13133808C22semidirect
15. C2×#13133806C22semidirect
16. C2×#13133805C22semidirect
17. C2×#13133803C22semidirect
29. #16812042C22semidirect
30. #16812040C22semidirect
31. #16812033C22semidirect
32. #16812032C22semidirect
33. #16812030C22semidirect
34. #16812029C22semidirect
57. C2×C107⋊C8D4semidirect
149. C107#3054semidirect
153. #525D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
C22×#131338022
#202931382
#202928832
#202928782
#202928752
#202928712
#202926632
#202926232
#202926212
#525×C1072
#3054107