Finite Group Databasebeta

← #28348007

Subgroups — #28348007

Order 13696.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×#16812128C211yesmaximal, characteristic, complemented
3#20293297C211yesmaximal, characteristic, complemented
4#20293272C211yesmaximal, characteristic, complemented
5#20293014C211yesmaximal, characteristic, complemented
6#20292872C211yesmaximal, characteristic, complemented
7#20292862C211yesmaximal, characteristic, complemented
8#20292850C211yesmaximal, characteristic, complemented
9#20292790C211yesmaximal, characteristic, complemented
10#20292773C211yesmaximal, characteristic, complemented
11#20292769C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292697C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×#13133817C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic
20#16812133—24no—
21#16812128—24no—
22C2×Dic428C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic
24C2×#13133806C2211yescharacteristic
25C2×#13133806C2211yescharacteristic, complemented
26C2×#13133806C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133801C2211yescharacteristic
30C2×C8×D107C2211yescharacteristic, complemented
31#16812105—21no—
32#16812097C2211yescharacteristic
33#16812093—21no—
34#16812089C2211yescharacteristic
35#16812084C2211yescharacteristic
36#16812083—21no—
37#16812081—21no—
38#16812081—21no—
39#16812080C2211yescharacteristic
40#16812080C2211yescharacteristic, complemented
41#16812079—21no—
42C4×Dic214C2211yescharacteristic, complemented
43C4×D428C2211yescharacteristic
44D4×Dic107—21no—
45C2×(C2×D107)⋊C4—21no—
46#16812042C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic
50#16812030C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812020C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic
56#127×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—22no—
64#13133817—22no—
65#13133817—41no—
66#13133817—41no—
67#13133817—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×D428—21no—
71C2×D428D411yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79D4×D107—22no—
80D4×D107—41no—
81C22×Dic107—21no—
82C22×Dic107—21no—
83C22×Dic107—21no—
84C2×C4×D107C2311yescharacteristic
85C2×C4×D107C2311yescharacteristic
86Dic428—22no—
87#13133807—22no—
88#13133806—22no—
89#13133806—41no—
90#13133806D414yescomplemented
91#13133803—22no—
92#13133803—41no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C8×D107—22no—
97C4⋊Dic107—21no—
98C4⋊Dic107D411yescharacteristic
99C4⋊Dic107C2311yescharacteristic
100Dic107⋊C4—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107—21no—
110C2×D4×C107—21no—
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116D8×C107—22no—
117D8×C107—22no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130C22×D107C2×D411yescharacteristic
131C22×D107C2×D411yescharacteristic
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—42no—
138D4×C107—81no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
141C2×C4×C107—81no—
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2411yescharacteristic
145C22⋊D107—42no—
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—81no—
150C22⋊D107—81no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—42no—
155C2×Dic107—42no—
156C2×Dic107—42no—
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159D428—21no—
160D428—21no—
161D428—41no—
162D428—81no—
163C4×D107—21no—
164C4×D107—21no—
165C4×D107—21no—
166C4×D107—21no—
167Dic214—21no—
168Dic214—21no—
169Dic214—21no—
170Dic214C2×D412yes—
171C8×C107—21no—
172C8×C107—21no—
173C107⋊C8—21no—
174C107⋊C8C2×D412yes—
175C22×C107—22no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107C22×D411yescharacteristic
185C2×D107—21no—
186C2×D107—21no—
187C2×D107—21no—
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C4×C107—21no—
196C4×C107—21no—
197C4×C107—41no—
198C4×C107C22×D411yescharacteristic
199C4×C107H(5,𝔽2)11yescharacteristic
200Dic107—22no—
201Dic107—21no—
202Dic107—21no—
203Dic107—41no—
204Dic107—41no—
205C2×C107—42no—
206C2×C107—81no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—41no—
213D107—81no—
214#3268—1071nomaximal, Sylow
215C107#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—1071no—
217C2×#135—1071no—
218#550—1071no—
219#546—1071no—
220#496—1071no—
221#468—1071no—
222#466—1071no—
223#464—1071no—
224#459D10711yescharacteristic, complemented
225#452—1071no—
226#449—1071no—
227#449—1071no—
228#447—1071no—
229#440—1071no—
230#435—1071no—
231C2×#41—1071no—
232C2×#41—1071no—
233C2×#41—1071no—
234C22×D4—1071no—
235AGL(1,ℤ/8)—2144no—
236#135—2144no—
237C2×Q16—1071no—
238C2×SD16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16—1071no—
242C2×D8—1071no—
243C2×D8—1071no—
244C2×D8C2×D10711yescharacteristic
245C2×D8C2×D10711yescharacteristic
246C2×M16—1071no—
247C22×C8—1071no—
248#127—2141no—
249#127C2×D10711yescharacteristic, complemented
250#124—1071no—
251#123—2141no—
252C22⋊Q8—1071no—
253#121—1071no—
254#121—1071no—
255#121—1071no—
256#121—2141no—
257#121—2141no—
258#121—2141no—
259#120—2141no—
260C4×Q8—1071no—
261C4×D4—1071no—
262C4×D4—2141no—
263C4×D4C2×D10711yescharacteristic
264C2×C22⋊C4—2141no—
265#107—1071no—
266C4⋊C8C2×D10711yescharacteristic, complemented
267#103—1071no—
268#103—1071no—
269#102—1071no—
270#102—1071no—
271#102C2×D10711yescharacteristic, complemented
272#102C2×D10711yescharacteristic, complemented
273C22⋊C8—1071no—
274C22⋊C8—1071no—
275#97—1071no—
276C24—2141no—
277#41—4281no—
278#41—4281no—
279#41—4281no—
280#41—2142no—
281#41—2142no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8—1071no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—4281no—
290C2×D4—2141no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2142no—
298C2×D4C22×D10711yescharacteristic
299C2×D4C22×D10711yescharacteristic
300C2×D4C22×D10711yescharacteristic
301C22×C4—21no—
302C22×C4—1071no—
303C22×C4—1071no—
304C22×C4—2141no—
305C22×C4—2141no—
306C22×C4—2141no—
307Q16—2142no—
308SD16—1074no—
309SD16—4281no—
310SD16—2142no—
311SD16—2142no—
312D8—22no—
313D8—22no—
314D8—4281no—
315D8—2142no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—2142no—
320C2×C8C22×D10711yescharacteristic
321C2×C8C22×D10711yescharacteristic
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4—2141no—
326C4⋊C4C22×D10711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—1071no—
329C22⋊C4—1071no—
330C22⋊C4—2141no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C42—2141no—
337C42C22×D10711yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—8561no—
342C23—1071no—
343C23—1071no—
344C23—4282no—
345C23—4282no—
346C23—2141no—
347Q8—1072no—
348Q8—2141no—
349Q8—2141no—
350Q8—2141no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—8561no—
359D4—8561no—
360D4—8561no—
361D4—4281no—
362D4—4282no—
363D4—4282no—
364D4—4282no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4D4×D10712yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—1071no—
372C2×C4—1071no—
373C2×C4—4282no—
374C2×C4—4282no—
375C2×C4—4282no—
376C2×C4—2141no—
377C2×C4—2141no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4D4×D10711yescharacteristic
384C2×C4D4×D10711yescharacteristic
385C2×C4C23×D10711yescharacteristic
386C8—21no—
387C8—21no—
388C8—1072no—
389C8—2141no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—8564no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—2141no—
404C22—2141no—
405C22—2141no—
406C22—2141no—
407C22—2141no—
408C22—2141no—
409C22C2×D4×D10711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—4281no—
414C4—4281no—
415C4—2141no—
416C4—2141no—
417C4—2142no—
418C4C2×D4×D10711yescharacteristic
419C4#1681215511yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—8561no—
424C2—4281no—
425C2—4281no—
426C2#2029326111yescharacteristic
427C2#2029342311yescharacteristic
428C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3268
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×#16812128C2semidirect
3. #20293297C2semidirect
4. #20293272C2semidirect
5. #20293014C2semidirect
6. #20292872C2semidirect
7. #20292862C2semidirect
8. #20292850C2semidirect
9. #20292790C2semidirect
10. #20292773C2semidirect
11. #20292769C2semidirect
12. #20292759C2semidirect
13. #20292727C2semidirect
14. #20292697C2semidirect
15. #459×C107C2semidirect
18. C2×#13133817C22semidirect
22. C2×Dic428C22semidirect
25. C2×#13133806C22semidirect
26. C2×#13133806C22semidirect
30. C2×C8×D107C22semidirect
40. #16812080C22semidirect
42. C4×Dic214C22semidirect
46. #16812042C22semidirect
47. #16812034C22semidirect
48. #16812032C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812020C22semidirect
56. #127×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #102×C107C22semidirect
60. #102×C107C22semidirect
90. #13133806D4semidirect
215. C107#3268semidirect
224. #459D107semidirect
249. #127C2×D107semidirect
266. C4⋊C8C2×D107semidirect
271. #102C2×D107semidirect
272. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×#168121282
#202932972
#202932722
#202930142
#202928722
#202928622
#202928502
#202927902
#202927732
#202927692
#202927592
#202927272
#202926972
#459×C1072
#3268107