Finite Group Databasebeta

← #28348068

Subgroups — #28348068

Order 13696.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812135C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3#20293273C211yesmaximal, characteristic, complemented
4#20293009C211yesmaximal, characteristic, complemented
5#20292875C211yesmaximal, characteristic, complemented
6#20292871C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292839C211yesmaximal, characteristic, complemented
9#20292789C211yesmaximal, characteristic, complemented
10#20292787C211yesmaximal, characteristic, complemented
11#20292761C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14#20292705C211yesmaximal, characteristic, complemented
15#546×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic
17C22×D428C2211yescharacteristic, complemented
18#16812135—24no—
19#16812134—24no—
20C2×#13133808C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic, complemented
25C2×#13133803C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133802C2211yescharacteristic, complemented
30C2×#13133802C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic, complemented
32#16812094—21no—
33#16812088C2211yescharacteristic, complemented
34#16812083C2211yescharacteristic
35#16812082—21no—
36#16812077—21no—
37C4×D428C2211yescharacteristic, complemented
38C22⋊C4×D107—21no—
39#16812042C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812032C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic, complemented
44#16812030C2211yescharacteristic, complemented
45#16812029C2211yescharacteristic, complemented
46#16812029C2211yescharacteristic, complemented
47C22⋊(C107⋊C8)C2211yescharacteristic, complemented
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49C4×C107⋊C8C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic
51C22×D4×C107C2211yescharacteristic, complemented
52#124×C107C2211yescharacteristic, complemented
53#123×C107—21no—
54C22⋊Q8×C107C2211yescharacteristic, complemented
55#121×C107—21no—
56#121×C107C2211yescharacteristic
57#120×C107—21no—
58C4×D4×C107C2211yescharacteristic
59C4×D4×C107C2211yescharacteristic, complemented
60C2×C22⋊C4×C107—21no—
61C23×D107—21no—
62#13133818—22no—
63#13133818—41no—
64C2×Dic214C2311yescharacteristic
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74#13133808—41no—
75#13133806—22no—
76#13133806—22no—
77#13133805—22no—
78#13133805—22no—
79#13133803—22no—
80#13133803—22no—
81#13133803—41no—
82#13133803D414yescomplemented
83#13133802—22no—
84#13133802—22no—
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C2×C107⋊C8C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96C24×C107—21no—
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—22no—
101C2×D4×C107—21no—
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107—41no—
105C2×D4×C107D411yescharacteristic
106C2×D4×C107C2311yescharacteristic
107C2×D4×C107C2311yescharacteristic
108C22×C4×C107—21no—
109C22×C4×C107—21no—
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C4⋊C4×C107—21no—
113C4⋊C4×C107D411yescharacteristic
114C4⋊C4×C107C2311yescharacteristic
115C22⋊C4×C107—21no—
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C23×C107—42no—
126C23×C107—42no—
127C23×C107—81no—
128C23×C107C2×D411yescharacteristic
129C23×C107C2×D411yescharacteristic
130C22×D107—21no—
131C22×D107—21no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135Q8×C107—21no—
136Q8×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—41no—
141D4×C107—42no—
142D4×C107—42no—
143D4×C107—81no—
144D4×C107C2×D412yes—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—42no—
152C2×C4×C107—81no—
153C2×C4×C107C2×D411yescharacteristic
154C2×C4×C107C2×D411yescharacteristic
155C2×C4×C107C2×D411yescharacteristic
156C2×C4×C107C2×D411yescharacteristic
157C2×C4×C107C2411yescharacteristic
158C22⋊D107—42no—
159C22⋊D107—81no—
160C2×Dic107—21no—
161C2×Dic107—21no—
162D428—21no—
163D428—21no—
164D428—21no—
165D428—42no—
166D428C2×D412yes—
167C4×D107—42no—
168C4×D107—81no—
169Dic214—21no—
170Dic214—21no—
171C107⋊C8—21no—
172C107⋊C8—21no—
173C107⋊C8—21no—
174C107⋊C8C2×D412yes—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—22no—
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—21no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—84no—
188C22×C107C22×D411yescharacteristic
189C2×D107—22no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—84no—
196C4×C107—21no—
197C4×C107—21no—
198C4×C107—21no—
199C4×C107—21no—
200C4×C107—41no—
201C4×C107—41no—
202C4×C107C22×D411yescharacteristic
203C4×C107H(5,𝔽2)11yescharacteristic
204Dic107—41no—
205Dic107—41no—
206C2×C107—41no—
207C2×C107—41no—
208C2×C107—42no—
209C2×C107—81no—
210C2×C107D4211yescharacteristic
211C2×C107#57511yescharacteristic
212C2×C107#57611yescharacteristic
213D107—42no—
214D107—81no—
215#3273—1071nomaximal, Sylow
216C107#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—1071no—
218C2×AGL(1,ℤ/8)—1071no—
219#546—1071no—
220#546D10711yescharacteristic, complemented
221#495—1071no—
222#468—1071no—
223#468—1071no—
224#463—1071no—
225#463—1071no—
226#452—1071no—
227#452—1071no—
228#447—1071no—
229#447—1071no—
230C4×D8—1071no—
231#436—1071no—
232C2×#41—1071no—
233C2×#41C2×D10711yescharacteristic
234C22×D4—1071no—
235C22×D4C2×D10711yescharacteristic, complemented
236AGL(1,ℤ/8)—2144no—
237AGL(1,ℤ/8)—2144no—
238C2×Q16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8—1071no—
246C2×D8—1071no—
247C2×M16—1071no—
248C2×M16—1071no—
249#124—1071no—
250#124C2×D10711yescharacteristic, complemented
251#123—21no—
252#123—2141no—
253C22⋊Q8—1071no—
254C22⋊Q8C2×D10711yescharacteristic, complemented
255#121—21no—
256#121—1071no—
257#121—2141no—
258#121C2×D10711yescharacteristic
259#120—21no—
260#120—2141no—
261C4×D4—1071no—
262C4×D4C2×D10711yescharacteristic
263C4×D4C2×D10711yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—2141no—
266#107—1071no—
267C4⋊C8—1071no—
268#103—1071no—
269#103—1071no—
270#102—1071no—
271#102—1071no—
272#102—1071no—
273#102—1071no—
274C22⋊C8—1071no—
275C22⋊C8—1071no—
276C4×C8—1071no—
277C24—21no—
278C24—2141no—
279#41—22no—
280#41—41no—
281#41—4281no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8C22×D10711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—1071no—
291C2×D4—1071no—
292C2×D4—4281no—
293C2×D4—2142no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4C22⋊D10711yescharacteristic
298C2×D4C22×D10711yescharacteristic
299C2×D4C22×D10711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4C22×D10711yescharacteristic
305C22×C4C22×D10711yescharacteristic
306Q16—4281no—
307SD16—2142no—
308SD16—2142no—
309SD16—2142no—
310SD16—2142no—
311D8—1074no—
312D8—4281no—
313D8—2142no—
314D8—2142no—
315M16—2142no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8—1071no—
320C2×C8—1071no—
321C4⋊C4—21no—
322C4⋊C4—1071no—
323C4⋊C4—2141no—
324C4⋊C4C22⋊D10711yescharacteristic
325C4⋊C4C22×D10711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4C22⋊D10711yescharacteristic
337C22⋊C4C22⋊D10711yescharacteristic
338C42C22×D10711yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—8561no—
345C23—4282no—
346C23—4282no—
347C23—2141no—
348C23—2141no—
349C23C2×C22⋊D10711yescharacteristic
350C23C2×C22⋊D10711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2141no—
354Q8—2141no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—8561no—
363D4—1072no—
364D4—4282no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369D4D4×D10712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—8561no—
379C2×C4—4282no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4C2×C22⋊D10711yescharacteristic
383C2×C4C2×C22⋊D10711yescharacteristic
384C2×C4C2×C22⋊D10711yescharacteristic
385C2×C4C2×C22⋊D10711yescharacteristic
386C2×C4C23×D10711yescharacteristic
387C8—1072no—
388C8—2141no—
389C8—2141no—
390C8—2141no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—8564no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—2142no—
411C22C22×C22⋊D10711yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—4281no—
419C4—4281no—
420C4C2×D4×D10711yescharacteristic
421C4#1681215411yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—8561no—
427C2—4282no—
428C2D4×C22⋊D10711yescharacteristic
429C2#2029342511yescharacteristic
430C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#546×C107C2
socleC22×C107C22×D4
hypercenter#546D107
hypocenterC107#3273
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C2×#16812134C2semidirect
3. #20293273C2semidirect
4. #20293009C2semidirect
5. #20292875C2semidirect
6. #20292871C2semidirect
7. #20292842C2semidirect
8. #20292839C2semidirect
9. #20292789C2semidirect
10. #20292787C2semidirect
11. #20292761C2semidirect
12. #20292760C2semidirect
13. C4×#13133803C2semidirect
14. #20292705C2semidirect
15. #546×C107C2semidirect
17. C22×D428C22semidirect
20. C2×#13133808C22semidirect
21. C2×#13133806C22semidirect
22. C2×#13133806C22semidirect
23. C2×#13133805C22semidirect
24. C2×#13133805C22semidirect
29. C2×#13133802C22semidirect
30. C2×#13133802C22semidirect
31. #16812097C22semidirect
33. #16812088C22semidirect
37. C4×D428C22semidirect
39. #16812042C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812032C22semidirect
43. #16812030C22semidirect
44. #16812030C22semidirect
45. #16812029C22semidirect
46. #16812029C22semidirect
47. C22⋊(C107⋊C8)C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. C4×C107⋊C8C22semidirect
51. C22×D4×C107C22semidirect
52. #124×C107C22semidirect
54. C22⋊Q8×C107C22semidirect
59. C4×D4×C107C22semidirect
82. #13133803D4semidirect
216. C107#3273semidirect
220. #546D107semidirect
235. C22×D4C2×D107semidirect
250. #124C2×D107semidirect
254. C22⋊Q8C2×D107semidirect
263. C4×D4C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×#168121342
#202932732
#202930092
#202928752
#202928712
#202928422
#202928392
#202927892
#202927872
#202927612
#202927602
C4×#131338032
#202927052
#546×C1072
#3273107