Finite Group Databasebeta

← #28346814

Subgroups — #28346814

Order 13696.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293257C211yesmaximal, characteristic, complemented
2#20293195C211yesmaximal, characteristic
3#20292993C211yesmaximal, characteristic, complemented
4#20292990C211yesmaximal, characteristic, complemented
5#20292988C211yesmaximal, characteristic, complemented
6#20292986C211yesmaximal, characteristic, complemented
7#20292983C211yesmaximal, characteristic, complemented
8#20292974C211yesmaximal, characteristic, complemented
9#20292972C211yesmaximal, characteristic, complemented
10#20292630C211yesmaximal, characteristic, complemented
11#20292627C211yesmaximal, characteristic, complemented
12#20292622C211yesmaximal, characteristic, complemented
13#102×D107C211yesmaximal, characteristic, complemented
14#20292581C211yesmaximal, characteristic, complemented
15#488×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×#13133801C2211yescharacteristic
19C2×C8×D107C2211yescharacteristic
20#16812109C2211yescharacteristic, complemented
21#16812108—21no—
22#16812108C2211yescharacteristic, complemented
23#16812106C2211yescharacteristic
24#16812105C2211yescharacteristic
25#16812100C2211yescharacteristic, complemented
26#16812099—21no—
27#16812098C2211yescharacteristic
28#16812089—21no—
29#16812087—21no—
30#16812082—21no—
31#16812082—21no—
32#16812079—21no—
33#16812079—21no—
34#16812067C2211yescharacteristic
35#16812066C2211yescharacteristic
36C4⋊C4×D107C2211yescharacteristic
37#16812033C2211yescharacteristic, complemented
38#16812033C2211yescharacteristic, complemented
39#16812031C2211yescharacteristic
40#16812031C2211yescharacteristic
41#16812031C2211yescharacteristic, complemented
42#16812031C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic
44#16812030C2211yescharacteristic
45#16812029C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52#125×C107C2211yescharacteristic, complemented
53#124×C107C2211yescharacteristic
54#103×C107C2211yescharacteristic, complemented
55#103×C107C2211yescharacteristic, complemented
56#102×C107C2211yescharacteristic
57#102×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—22no—
62C2×Dic214—21no—
63C2×Dic214—21no—
64C2×D428—21no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72D4×D107—22no—
73D4×D107—22no—
74C2×C4×D107—21no—
75C2×C4×D107—21no—
76C2×C4×D107C2311yescharacteristic
77#13133801—22no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—22no—
81C4⋊Dic107—21no—
82C4⋊Dic107—21no—
83C4⋊Dic107C2311yescharacteristic
84C4⋊Dic107C2311yescharacteristic
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4—21no—
88Dic107⋊C4D411yescharacteristic
89Dic107⋊C4D411yescharacteristic
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×Dic107C2311yescharacteristic
97C2×Q8×C107C2311yescharacteristic
98C2×D4×C107C2311yescharacteristic
99C2×C8×C107C2311yescharacteristic
100C2×C8×C107C2311yescharacteristic
101C4⋊C4×C107—21no—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C22⋊C4×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107—21no—
110C22×D107—21no—
111C22×D107—42no—
112C22×D107—81no—
113C22×D107—81no—
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107—21no—
117D4×C107—21no—
118D4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—42no—
126C22⋊D107—81no—
127C22⋊D107—81no—
128C2×Dic107—21no—
129C2×Dic107—21no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133D428—21no—
134D428—21no—
135D428—21no—
136D428—21no—
137D428—42no—
138D428—81no—
139D428—81no—
140C4×D107—42no—
141C4×D107—81no—
142C4×D107#4112yes—
143C4×D107#4112yes—
144Dic214—81no—
145Dic214—81no—
146C8×C107—21no—
147C8×C107—21no—
148C107⋊C8—21no—
149C107⋊C8—21no—
150C22×C107—41no—
151C22×C107—41no—
152C22×C107—41no—
153C22×C107C22×D411yescharacteristic
154C2×D107—21no—
155C2×D107—21no—
156C2×D107—41no—
157C2×D107—41no—
158C2×D107—41no—
159C2×D107—41no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—84no—
163C2×D107#12412yes—
164C4×C107—21no—
165C4×C107—21no—
166C4×C107—41no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107C2×#4111yescharacteristic
170C4×C107C2×#4111yescharacteristic
171Dic107—21no—
172Dic107—21no—
173Dic107—41no—
174Dic107—41no—
175Dic107#12412yes—
176C2×C107—81no—
177C2×C107C2×#12411yescharacteristic
178C2×C107#57611yescharacteristic
179C2×C107#57711yescharacteristic
180D107—22no—
181D107—81no—
182D107—81no—
183#3136—1071nomaximal, Sylow
184C107#313611yescharacteristic, complemented, Sylow
185#544—1071no—
186#535—1071no—
187#490—1071no—
188#490—1071no—
189#489—1071no—
190#489—1071no—
191#488—1071no—
192#488D10711yescharacteristic, complemented
193#486—1071no—
194#486—1071no—
195#418—1071no—
196#417—1071no—
197#416—1071no—
198C2×#102—1071no—
199#405—1071no—
200C2×#41—1071no—
201C22×D4—1071no—
202C2×M16—1071no—
203C22×C8—1071no—
204C4⋊Q8—1071no—
205C4⋊Q8—1071no—
206C4⋊Q8—2141no—
207#127—1071no—
208#127—1071no—
209#125—1071no—
210#125—2141no—
211#125C2×D10711yescharacteristic, complemented
212#124—1071no—
213#124C2×D10711yescharacteristic
214C22⋊Q8—2141no—
215C22⋊Q8—2141no—
216#121—2141no—
217#121—2141no—
218#120—2141no—
219#120—2141no—
220#117—1071no—
221#117—1071no—
222C2×C4⋊C4—1071no—
223#103—1071no—
224#103—1071no—
225#103C2×D10711yescharacteristic, complemented
226#103C2×D10711yescharacteristic, complemented
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102—1071no—
231#102—1071no—
232#102—1071no—
233#102—1071no—
234#102—1071no—
235#102—1071no—
236#102—1071no—
237#102C2×D10711yescharacteristic
238#102C2×D10711yescharacteristic
239#97—1071no—
240#97—1071no—
241C4×C8—1071no—
242C4×C8C2×D10711yescharacteristic, complemented
243C24—2141no—
244#41—2142no—
245#41—2142no—
246C2×Q8—2141no—
247C2×Q8—2141no—
248C2×Q8C22×D10711yescharacteristic
249C2×D4—1071no—
250C2×D4—1071no—
251C2×D4—2141no—
252C2×D4—2142no—
253C2×D4—2142no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2141no—
257C2×D4—2141no—
258C2×D4—2141no—
259C2×D4C22×D10711yescharacteristic
260C22×C4—1071no—
261C22×C4—2141no—
262C22×C4—2141no—
263M16—2142no—
264C2×C8—1071no—
265C2×C8—1071no—
266C2×C8—2142no—
267C2×C8C22×D10711yescharacteristic
268C2×C8C22×D10711yescharacteristic
269C4⋊C4—21no—
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⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4C22×D10711yescharacteristic
281C4⋊C4C22×D10711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—1071no—
285C22⋊C4—1071no—
286C22⋊C4—2141no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C42—1071no—
290C42—2141no—
291C42C22×D10711yescharacteristic
292C23—21no—
293C23—8561no—
294C23—8561no—
295C23—1071no—
296C23—4282no—
297C23—2141no—
298C23—2141no—
299Q8—21no—
300Q8—21no—
301Q8—8561no—
302Q8—8561no—
303D4—21no—
304D4—21no—
305D4—8561no—
306D4—8561no—
307D4—8561no—
308D4—8561no—
309D4—4282no—
310D4—4282no—
311D4—2141no—
312D4—2141no—
313D4—2141no—
314D4—2141no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—8561no—
319C2×C4—1071no—
320C2×C4—1072no—
321C2×C4—1072no—
322C2×C4—1071no—
323C2×C4—1071no—
324C2×C4—4282no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4D4×D10711yescharacteristic
328C2×C4D4×D10711yescharacteristic
329C2×C4C23×D10711yescharacteristic
330C8—21no—
331C8—21no—
332C8—2141no—
333C8—2141no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—8564no—
338C22—1072no—
339C22—4281no—
340C22—4281no—
341C22—4281no—
342C22—4281no—
343C22—4281no—
344C22—4281no—
345C22—2141no—
346C22—2141no—
347C22C2×D4×D10711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—1072no—
354C4—4281no—
355C4—4281no—
356C4—2141no—
357C4—2141no—
358C4#41×D10711yescharacteristic
359C4#41×D10711yescharacteristic
360C2—81no—
361C2—8561no—
362C2—8561no—
363C2—2142no—
364C2#124×D10711yescharacteristic
365C2#2029342911yescharacteristic
366C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#488×C107C2
socleC22×C107C22×D4
prime radical#20293195C2
hypercenter#488D107
hypocenterC107#3136
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #20293257C2semidirect
3. #20292993C2semidirect
4. #20292990C2semidirect
5. #20292988C2semidirect
6. #20292986C2semidirect
7. #20292983C2semidirect
8. #20292974C2semidirect
9. #20292972C2semidirect
10. #20292630C2semidirect
11. #20292627C2semidirect
12. #20292622C2semidirect
13. #102×D107C2semidirect
14. #20292581C2semidirect
15. #488×C107C2semidirect
20. #16812109C22semidirect
22. #16812108C22semidirect
25. #16812100C22semidirect
37. #16812033C22semidirect
38. #16812033C22semidirect
41. #16812031C22semidirect
42. #16812031C22semidirect
47. #16812029C22semidirect
48. #16812029C22semidirect
49. #16812020C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
52. #125×C107C22semidirect
54. #103×C107C22semidirect
55. #103×C107C22semidirect
58. C4×C8×C107C22semidirect
184. C107#3136semidirect
192. #488D107semidirect
211. #125C2×D107semidirect
225. #103C2×D107semidirect
226. #103C2×D107semidirect
242. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932572
#202931952
#202929932
#202929902
#202929882
#202929862
#202929832
#202929742
#202929722
#202926302
#202926272
#202926222
#102×D1072
#202925812
#488×C1072
#3136107