Finite Group Databasebeta

← #28345716

Subgroups — #28345716

Order 13696.

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

normal subgroups only

This group has 754 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
1C22×#13133817C211yesmaximal, characteristic
2C2×#16812138C212yesmaximal, complemented
3C22⋊Q8×D107C211yesmaximal, characteristic, complemented
4#20292790C212yesmaximal, complemented
5#20292786C212yesmaximal, complemented
6#20292785C211yesmaximal, characteristic, complemented
7#20292782C211yesmaximal, characteristic, complemented
8#20292778C211yesmaximal, characteristic, complemented
9#20292625C212yesmaximal, complemented
10#20292589C211yesmaximal, characteristic, complemented
11#450×C107C211yesmaximal, characteristic, complemented
12C2×#13133817—28no—
13C2×#13133817—21no—
14C2×#13133817—21no—
15C2×#13133817C2212yes—
16C2×#13133817C2212yes—
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic, complemented
19C22×C22⋊D107—21no—
20C23×Dic107—21no—
21C22×C4×D107C2211yescharacteristic
22#16812138—28no—
23C2×Dic428C2212yescomplemented
24C2×#13133808C2212yescomplemented
25C2×#13133806C2212yescomplemented
26C2×#13133801C2212yescomplemented
27#16812089C2211yescharacteristic, complemented
28#16812088C2211yescharacteristic
29#16812087—21no—
30#16812086—21no—
31#16812085C2211yescharacteristic, complemented
32C4⋊C4×D107—21no—
33C4⋊C4×D107C2211yescharacteristic, complemented
34C22⋊C4×D107—21no—
35#16812034C2212yescomplemented
36#16812032C2212yescomplemented
37#16812030C2212yescomplemented
38C22⋊(C107⋊C8)C2211yescharacteristic
39(C2×D107)⋊C8C2212yescomplemented
40C22×D4×C107C2211yescharacteristic
41C2×SD16×C107C2212yescomplemented
42C22⋊Q8×C107C2211yescharacteristic
43#102×C107C2212yescomplemented
44C22⋊C8×C107C2211yescharacteristic
45C23×D107D411yescharacteristic
46#13133817—28no—
47#13133817—28no—
48#13133817—24no—
49#13133817—24no—
50#13133817—24no—
51#13133817D418yes—
52C2×Dic214—21no—
53C2×Dic214—24no—
54C2×Dic214—21no—
55C2×Dic214C2311yescharacteristic
56C2×Dic214C2312yes—
57Q8×D107—24no—
58C2×C22⋊D107—28no—
59C2×C22⋊D107—22no—
60C2×C22⋊D107—22no—
61C22×Dic107—28no—
62C22×Dic107—21no—
63C22×Dic107—22no—
64C22×Dic107—22no—
65C22×Dic107D411yescharacteristic
66C2×C4×D107—24no—
67C2×C4×D107—21no—
68C2×C4×D107—21no—
69C2×C4×D107D411yescharacteristic
70C2×C4×D107D411yescharacteristic
71C2×C4×D107C2311yescharacteristic
72C2×C4×D107C2311yescharacteristic
73Dic428—24no—
74#13133808—24no—
75#13133806—24no—
76#13133801—24no—
77C2×C107⋊C8C2312yes—
78C4⋊Dic107—21no—
79C4⋊Dic107C2311yescharacteristic
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83C22⋊Dic107—21no—
84(C2×D107)⋊C4—21no—
85(C2×D107)⋊C4—21no—
86C24×C107—21no—
87C2×Q8×C107C2311yescharacteristic
88C2×D4×C107—24no—
89C2×D4×C107—21no—
90C2×D4×C107C2312yes—
91C22×C4×C107C2311yescharacteristic
92SD16×C107—24no—
93C2×C8×C107C2312yes—
94C4⋊C4×C107—21no—
95C4⋊C4×C107C2311yescharacteristic
96C22⋊C4×C107—21no—
97C23×C107—24no—
98C23×C107—22no—
99C23×C107—42no—
100C23×C107—42no—
101C23×C107C2×D411yescharacteristic
102C22×D107—22no—
103C22×D107—22no—
104C22×D107—41no—
105C22×D107C2×D411yescharacteristic
106C22×D107C2×D411yescharacteristic
107Q8×C107—22no—
108D4×C107—24no—
109D4×C107—22no—
110D4×C107C2×D414yes—
111C2×C4×C107—22no—
112C2×C4×C107—21no—
113C2×C4×C107—21no—
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2411yescharacteristic
116C22⋊D107—28no—
117C22⋊D107—28no—
118C2×Dic107—24no—
119C2×Dic107—28no—
120C2×Dic107—22no—
121C2×Dic107—22no—
122C2×Dic107—28no—
123C2×Dic107—21no—
124C2×Dic107—21no—
125C2×Dic107C2×D411yescharacteristic
126C2×Dic107C2×D411yescharacteristic
127C4×D107—22no—
128C4×D107—22no—
129C4×D107—22no—
130C4×D107—42no—
131C4×D107—42no—
132C4×D107C2×D414yes—
133Dic214—24no—
134Dic214—22no—
135Dic214—22no—
136Dic214—42no—
137Dic214C2×D414yes—
138C8×C107—22no—
139C107⋊C8—22no—
140C22×C107—21no—
141C22×C107—21no—
142C22×C107—24no—
143C22×C107—48no—
144C22×C107—42no—
145C22×C107—42no—
146C22×C107#13712yescomplemented
147C22×C107C22×D411yescharacteristic
148C2×D107—21no—
149C2×D107—21no—
150C2×D107—21no—
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—44no—
154C2×D107#12012yes—
155C4×C107—21no—
156C4×C107—41no—
157C4×C107—41no—
158C4×C107C22×D412yes—
159Dic107—24no—
160Dic107—21no—
161Dic107—41no—
162Dic107—41no—
163Dic107#12012yes—
164C2×C107—22no—
165C2×C107—44no—
166C2×C107C2×#12011yescharacteristic
167C2×C107C2×#13711yescharacteristic
168C2×C107#57511yescharacteristic
169D107—22no—
170D107—41no—
171#3007—1071nomaximal, Sylow
172C107#300711yescharacteristic, complemented, Sylow
173C22×#41—1071no—
174C2×#137—1072no—
175C2×C22⋊Q8—1071no—
176#452—1072no—
177#452—1072no—
178#451—1071no—
179#451—1071no—
180#450—1071no—
181#450D10711yescharacteristic, complemented
182#417—1072no—
183#407—1071no—
184C2×#41—1072no—
185C2×#41—1072no—
186C2×#41—2141no—
187C2×#41—2141no—
188C2×#41—2148no—
189C22×Q8—1071no—
190C22×Q8—1071no—
191C22×D4—2141no—
192C22×D4C2×D10711yescharacteristic
193C23×C4—1071no—
194C23×C4—2141no—
195#137—2148no—
196C2×Q16—1072no—
197C2×Q16—1072no—
198C2×SD16—1072no—
199C2×SD16C2×D10712yescomplemented
200C2×M16—1072no—
201C22⋊Q8—1071no—
202C22⋊Q8—1071no—
203C22⋊Q8—1071no—
204C22⋊Q8—2141no—
205C22⋊Q8—2141no—
206C22⋊Q8C2×D10711yescharacteristic
207C2×C4⋊C4—1071no—
208C2×C4⋊C4—2141no—
209C2×C22⋊C4—2141no—
210#103—1072no—
211#103—1072no—
212#102—1072no—
213#102C2×D10712yescomplemented
214C22⋊C8—1071no—
215C22⋊C8—1072no—
216C22⋊C8C2×D10711yescharacteristic
217C24—21no—
218C24—1071no—
219#41—1078no—
220#41—2144no—
221#41—2144no—
222#41—2148no—
223#41—2148no—
224#41—2144no—
225C2×Q8—1072no—
226C2×Q8—1071no—
227C2×Q8—2141no—
228C2×Q8—2144no—
229C2×Q8—2144no—
230C2×Q8—2141no—
231C2×Q8C22×D10711yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—2142no—
235C2×D4—2142no—
236C2×D4—2148no—
237C2×D4C22×D10712yes—
238C22×C4—1071no—
239C22×C4—1071no—
240C22×C4—1071no—
241C22×C4—1071no—
242C22×C4—1071no—
243C22×C4—2144no—
244C22×C4—2141no—
245C22×C4—2141no—
246C22×C4—2142no—
247C22×C4—2142no—
248C22×C4—2148no—
249C22×C4—2141no—
250C22×C4C22×D10711yescharacteristic
251Q16—2144no—
252Q16—2144no—
253SD16—24no—
254SD16—2144no—
255M16—2144no—
256C2×C8—1072no—
257C2×C8C22×D10712yes—
258C4⋊C4—21no—
259C4⋊C4—1071no—
260C4⋊C4—2141no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4C22×D10711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2141no—
267C22⋊C4—2141no—
268C22⋊C4—2141no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—1071no—
274C23—1071no—
275C23—4281no—
276C23—2142no—
277C23—2142no—
278C23D4×D10711yescharacteristic
279Q8—22no—
280Q8—1074no—
281Q8—4282no—
282Q8—2144no—
283Q8—2142no—
284Q8—2142no—
285D4—24no—
286D4—22no—
287D4—2148no—
288D4—2148no—
289D4D4×D10714yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—1071no—
294C2×C4—1074no—
295C2×C4—1071no—
296C2×C4—4282no—
297C2×C4—4282no—
298C2×C4—2142no—
299C2×C4—2142no—
300C2×C4—2142no—
301C2×C4—2148no—
302C2×C4—2141no—
303C2×C4—2141no—
304C2×C4—2144no—
305C2×C4—2148no—
306C2×C4—2142no—
307C2×C4—2142no—
308C2×C4D4×D10711yescharacteristic
309C2×C4C23×D10711yescharacteristic
310C8—22no—
311C8—2142no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—1072no—
319C22—4284no—
320C22—2141no—
321C22—2141no—
322C22—2141no—
323C22—2141no—
324C22—2141no—
325C22#1681213812yescomplemented
326C22C2×D4×D10711yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—1072no—
331C4—4281no—
332C4—4281no—
333C4—2141no—
334C4—2144no—
335C4C2×D4×D10712yes—
336C2—22no—
337C2—44no—
338C2—4281no—
339C2—2142no—
340C2#120×D10711yescharacteristic
341C2C2×#1681213811yescharacteristic
342C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#450×C107C2
socleC22×C107C22×D4
prime radicalC22×#13133817C2
hypercenter#450D107
hypocenterC107#3007
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#16812138C2semidirect
3. C22⋊Q8×D107C2semidirect
4. #20292790C2semidirect
5. #20292786C2semidirect
6. #20292785C2semidirect
7. #20292782C2semidirect
8. #20292778C2semidirect
9. #20292625C2semidirect
10. #20292589C2semidirect
11. #450×C107C2semidirect
18. C2×Q8×D107C22semidirect
23. C2×Dic428C22semidirect
24. C2×#13133808C22semidirect
25. C2×#13133806C22semidirect
26. C2×#13133801C22semidirect
27. #16812089C22semidirect
31. #16812085C22semidirect
33. C4⋊C4×D107C22semidirect
35. #16812034C22semidirect
36. #16812032C22semidirect
37. #16812030C22semidirect
39. (C2×D107)⋊C8C22semidirect
41. C2×SD16×C107C22semidirect
43. #102×C107C22semidirect
146. C22×C107#137semidirect
172. C107#3007semidirect
181. #450D107semidirect
199. C2×SD16C2×D107semidirect
213. #102C2×D107semidirect
325. C22#16812138semidirect

Maximal subgroups

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

subgroupindex
C22×#131338172
C2×#168121382
C22⋊Q8×D1072
#202927902
#202927862
#202927852
#202927822
#202927782
#202926252
#202925892
#450×C1072
#3007107