Finite Group Databasebeta

← #28345315

Subgroups — #28345315

Order 13696.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D107C211yesmaximal, characteristic
2#20293103C211yesmaximal, characteristic, complemented
3#20293099C211yesmaximal, characteristic
4#20292730C211yesmaximal, characteristic, complemented
5#20292727C211yesmaximal, characteristic, complemented
6#20292725C211yesmaximal, characteristic, complemented
7#20292724C211yesmaximal, characteristic, complemented
8C4×#13133807C211yesmaximal, characteristic, complemented
9SD16×Dic107C211yesmaximal, characteristic, complemented
10#20292714C211yesmaximal, characteristic, complemented
11#20292662C211yesmaximal, characteristic, complemented
12#20292634C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic
14#20292581C211yesmaximal, characteristic, complemented
15C4×SD16×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×#13133807C2211yescharacteristic
19C2×#13133806C2211yescharacteristic
20C2×#13133805C2211yescharacteristic
21SD16×D107C418yescomplemented
22C2×#13133801C2211yescharacteristic
23C2×C8×D107C2211yescharacteristic
24Q8×Dic107C2211yescharacteristic, complemented
25C4×Dic214—21no—
26C4×Dic214C2211yescharacteristic, complemented
27#16812074—21no—
28#16812074C2211yescharacteristic, complemented
29C4×D428C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31#16812071C2211yescharacteristic
32C4×C22⋊D107—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067—21no—
35#16812067C2211yescharacteristic
36#16812067C2211yescharacteristic
37#16812066—21no—
38#16812066C2211yescharacteristic
39C2×(C2×D107)⋊C4—21no—
40C22⋊C4×D107—21no—
41#16812042C2211yescharacteristic, complemented
42#16812042C2211yescharacteristic, complemented
43#16812041C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812032C2211yescharacteristic, complemented
47#16812031C2211yescharacteristic
48#16812030C2211yescharacteristic
49#16812029C2211yescharacteristic
50#16812020C2211yescharacteristic, complemented
51#16812019C2211yescharacteristic, complemented
52C8×Dic107C2211yescharacteristic, complemented
53C2×SD16×C107C2211yescharacteristic
54C4×Q8×C107C2211yescharacteristic, complemented
55C4×D4×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic
59C4×C8×C107C2211yescharacteristic, complemented
60C23×D107—21no—
61C2×Dic214—21no—
62C2×Dic214C2311yescharacteristic
63Q8×D107—22no—
64Q8×D107C2×C414yescomplemented
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67D4×D107—22no—
68D4×D107C2×C414yes—
69C22×Dic107—21no—
70C2×C4×D107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133807C2×C414yescomplemented
75#13133806C2×C414yescomplemented
76#13133805C2×C414yescomplemented
77#13133801—22no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107C2×C414yescomplemented
81C4⋊Dic107—21no—
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87Dic107⋊C4D411yescharacteristic
88C22⋊Dic107—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4D411yescharacteristic
94(C2×D107)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107—21no—
100C4×Dic107C2311yescharacteristic
101C2×Q8×C107C2311yescharacteristic
102C2×D4×C107C2311yescharacteristic
103C22×C4×C107—21no—
104SD16×C107C2×C414yescomplemented
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C42×C107—21no—
112C42×C107C2311yescharacteristic
113C23×C107—21no—
114C22×D107—21no—
115C22×D107—24no—
116C22×D107—81no—
117C22×D107—81no—
118C22×D107C2×D411yescharacteristic
119Q8×C107—21no—
120Q8×C107C22×C412yes—
121D4×C107—21no—
122D4×C107C22×C412yes—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—81no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—24no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—21no—
134C2×Dic107—81no—
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—21no—
139D428C22×C412yes—
140C4×D107—24no—
141C4×D107—81no—
142C4×D107—81no—
143C4×D107C22×C412yes—
144C4×D107#4112yes—
145Dic214—21no—
146Dic214—24no—
147Dic214C22×C412yes—
148C8×C107—21no—
149C8×C107C22×C412yes—
150C107⋊C8—21no—
151C107⋊C8C22×C412yes—
152C22×C107—22no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—22no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—84no—
162C2×D107C4×D412yes—
163C4×C107—22no—
164C4×C107—21no—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107C23×C411yescharacteristic
169C4×C107C2×#4111yescharacteristic
170Dic107—22no—
171Dic107—21no—
172Dic107—21no—
173Dic107—41no—
174Dic107—41no—
175Dic107C4×D412yes—
176C2×C107—42no—
177C2×C107C2×C4×D411yescharacteristic
178C2×C107#57711yescharacteristic
179C2×C107#57711yescharacteristic
180D107—22no—
181D107—42no—
182#2949—1071nomaximal, Sylow
183C107#294911yescharacteristic, complemented, Sylow
184C22×SD16—1071no—
185#519—1071no—
186#518—1071no—
187#440—1071no—
188#440—1071no—
189#440—1071no—
190#440—1071no—
191C4×SD16—1071no—
192C4×SD16—1071no—
193C4×SD16—1071no—
194C4×SD16D10711yescharacteristic, complemented
195#427—1071no—
196#419—1071no—
197#418—1071no—
198#405—1071no—
199C22×Q8—1071no—
200C22×D4—1071no—
201C2×SD16—1071no—
202C2×SD16—1071no—
203C2×SD16—1078no—
204C2×SD16—1071no—
205C2×SD16C2×D10711yescharacteristic
206C2×M16—1071no—
207C22×C8—1071no—
208C4×Q8—1071no—
209C4×Q8—1071no—
210C4×Q8—1071no—
211C4×Q8—2141no—
212C4×Q8—2141no—
213C4×Q8C2×D10711yescharacteristic, complemented
214C4×D4—1071no—
215C4×D4—1071no—
216C4×D4—1071no—
217C4×D4—2141no—
218C4×D4—2141no—
219C4×D4C2×D10711yescharacteristic
220#117—1071no—
221#117—1071no—
222#117—1071no—
223#117—2141no—
224#117—2141no—
225C2×C22⋊C4—2141no—
226C2×C22⋊C4—2141no—
227#107—1071no—
228#107—1071no—
229#106—1071no—
230#106C2×D10711yescharacteristic, complemented
231#103—1071no—
232#103—1071no—
233#103—1071no—
234#103C2×D10711yescharacteristic, complemented
235#102—1071no—
236#102—1071no—
237#102—1071no—
238#102C2×D10711yescharacteristic
239#97—1071no—
240#97—1071no—
241C4×C8—1071no—
242C4×C8C2×D10711yescharacteristic, complemented
243C24—2141no—
244C2×Q8—1074no—
245C2×Q8—1071no—
246C2×Q8—2141no—
247C2×Q8—2142no—
248C2×Q8C22×D10711yescharacteristic
249C2×D4—1074no—
250C2×D4—1071no—
251C2×D4—2142no—
252C2×D4—2141no—
253C2×D4C22×D10711yescharacteristic
254C22×C4—21no—
255C22×C4—1071no—
256C22×C4—2141no—
257C22×C4—2141no—
258C22×C4—2141no—
259C22×C4—2141no—
260SD16—1074no—
261SD16—1074no—
262SD16—1074no—
263SD16C4×D10714yescomplemented
264M16—2142no—
265C2×C8—1071no—
266C2×C8—1074no—
267C2×C8—1071no—
268C2×C8C22×D10711yescharacteristic
269C2×C8C22×D10711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—1071no—
272C4⋊C4—1071no—
273C4⋊C4—1071no—
274C4⋊C4—1071no—
275C4⋊C4—2141no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4C22×D10711yescharacteristic
279C4⋊C4C22×D10711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—1071no—
282C22⋊C4—1071no—
283C22⋊C4—2141no—
284C22⋊C4—2141no—
285C22⋊C4—2141no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C42—21no—
289C42—1071no—
290C42—2141no—
291C42—2141no—
292C42—2141no—
293C42—2141no—
294C42—2141no—
295C42C22×D10711yescharacteristic
296C23—21no—
297C23—8561no—
298C23—8561no—
299C23—1071no—
300C23—2144no—
301C23—2141no—
302Q8—21no—
303Q8—1072no—
304Q8—2144no—
305Q8—2141no—
306Q8C2×C4×D10712yes—
307D4—21no—
308D4—1072no—
309D4—2144no—
310D4—2141no—
311D4C2×C4×D10712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—8561no—
317C2×C4—8561no—
318C2×C4—8561no—
319C2×C4—1071no—
320C2×C4—1072no—
321C2×C4—1072no—
322C2×C4—1071no—
323C2×C4—1071no—
324C2×C4—2144no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4D4×D10711yescharacteristic
329C2×C4D4×D10711yescharacteristic
330C2×C4C23×D10711yescharacteristic
331C8—21no—
332C8—1072no—
333C8—2141no—
334C8C2×C4×D10712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—8564no—
339C22—1072no—
340C22—4281no—
341C22—4281no—
342C22—2142no—
343C22—2141no—
344C22—2141no—
345C22C2×D4×D10711yescharacteristic
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—1072no—
352C4—4281no—
353C4—4281no—
354C4—2141no—
355C4—2141no—
356C4—2142no—
357C4C22×C4×D10711yescharacteristic
358C4#41×D10711yescharacteristic
359C2—42no—
360C2—4282no—
361C2—2142no—
362C2C4×D4×D10711yescharacteristic
363C2#2029343311yescharacteristic
364C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
cosocleC2×C4×C107C24
hypercenterC4×SD16D107
hypocenterC107#2949
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20293103C2semidirect
4. #20292730C2semidirect
5. #20292727C2semidirect
6. #20292725C2semidirect
7. #20292724C2semidirect
8. C4×#13133807C2semidirect
9. SD16×Dic107C2semidirect
10. #20292714C2semidirect
11. #20292662C2semidirect
12. #20292634C2semidirect
14. #20292581C2semidirect
15. C4×SD16×C107C2semidirect
21. SD16×D107C4semidirect
24. Q8×Dic107C22semidirect
26. C4×Dic214C22semidirect
28. #16812074C22semidirect
41. #16812042C22semidirect
42. #16812042C22semidirect
43. #16812041C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812032C22semidirect
50. #16812020C22semidirect
51. #16812019C22semidirect
52. C8×Dic107C22semidirect
54. C4×Q8×C107C22semidirect
56. #106×C107C22semidirect
57. #103×C107C22semidirect
59. C4×C8×C107C22semidirect
64. Q8×D107C2×C4semidirect
74. #13133807C2×C4semidirect
75. #13133806C2×C4semidirect
76. #13133805C2×C4semidirect
80. C8×D107C2×C4semidirect
104. SD16×C107C2×C4semidirect
183. C107#2949semidirect
194. C4×SD16D107semidirect
213. C4×Q8C2×D107semidirect
230. #106C2×D107semidirect
234. #103C2×D107semidirect
242. C4×C8C2×D107semidirect
263. SD16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1072
#202931032
#202930992
#202927302
#202927272
#202927252
#202927242
C4×#131338072
SD16×Dic1072
#202927142
#202926622
#202926342
#202926302
#202925812
C4×SD16×C1072
#2949107