Finite Group Databasebeta

← #28348131

Subgroups — #28348131

Order 13696.

620 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133803C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293261C211yesmaximal, characteristic, complemented
4D4×C22⋊D107C211yesmaximal, characteristic, complemented
5#20293002C211yesmaximal, characteristic, complemented
6#20292862C211yesmaximal, characteristic, complemented
7#20292861C211yesmaximal, characteristic, complemented
8#20292821C211yesmaximal, characteristic, complemented
9#20292820C211yesmaximal, characteristic, complemented
10#20292763C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
15#447×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic, complemented
17C22×C22⋊D107—21no—
18C2×D4×D107—21no—
19C2×D4×D107C2211yescharacteristic, complemented
20C2×D4×D107C2211yescharacteristic, complemented
21C2×D856C2211yescharacteristic, complemented
22C2×D856C2211yescharacteristic, complemented
23C2×#13133803—24no—
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29D8×D107—24no—
30C22×C107⋊C8C2211yescharacteristic, complemented
31C2×C8×D107C2211yescharacteristic, complemented
32#16812105C2211yescharacteristic, complemented
33#16812105C2211yescharacteristic, complemented
34#16812084C2211yescharacteristic, complemented
35#16812083C2211yescharacteristic, complemented
36#16812082—21no—
37#16812081—21no—
38#16812080C2211yescharacteristic, complemented
39#16812079—21no—
40#16812078—21no—
41#16812077—21no—
42#16812077—21no—
43D4×Dic107C2211yescharacteristic, complemented
44#16812071C2211yescharacteristic, complemented
45C4×C22⋊D107C2211yescharacteristic
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—44no—
80C2×C22⋊D107D411yescharacteristic
81D4×D107—22no—
82D4×D107—22no—
83D4×D107—41no—
84D4×D107—41no—
85D4×D107—44no—
86C22×Dic107—21no—
87C2×C4×D107—21no—
88C2×C4×D107C2311yescharacteristic
89D856—22no—
90D856—41no—
91#13133803—22no—
92#13133803—22no—
93#13133803—22no—
94#13133803—22no—
95#13133803D414yescomplemented
96C2×C107⋊C8—22no—
97C2×C107⋊C8C2311yescharacteristic
98C2×C107⋊C8C2311yescharacteristic
99C8×D107—22no—
100C4⋊Dic107C2311yescharacteristic
101Dic107⋊C4D411yescharacteristic
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104C22⋊Dic107D411yescharacteristic
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107C2311yescharacteristic
110C24×C107—21no—
111C2×D4×C107—22no—
112C2×D4×C107—21no—
113C2×D4×C107—21no—
114C2×D4×C107—41no—
115C2×D4×C107C2311yescharacteristic
116C2×D4×C107C2311yescharacteristic
117C2×D4×C107C2311yescharacteristic
118C22×C4×C107C2311yescharacteristic
119D8×C107—22no—
120D8×C107—41no—
121C2×C8×C107C2311yescharacteristic
122C2×C8×C107C2311yescharacteristic
123C4⋊C4×C107C2311yescharacteristic
124C22⋊C4×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—21no—
128C23×C107—42no—
129C23×C107—42no—
130C23×C107—81no—
131C23×C107C2×D411yescharacteristic
132C22×D107—21no—
133C22×D107—21no—
134C22×D107—21no—
135C22×D107—42no—
136C22×D107—42no—
137C22×D107—42no—
138C22×D107—42no—
139C22×D107—42no—
140C22×D107—42no—
141C22×D107—81no—
142C22×D107—81no—
143C22×D107—81no—
144C22×D107C2×D411yescharacteristic
145D4×C107—21no—
146D4×C107—21no—
147D4×C107—21no—
148D4×C107—21no—
149D4×C107—21no—
150D4×C107—42no—
151D4×C107—42no—
152D4×C107C2×D412yescomplemented
153C2×C4×C107—21no—
154C2×C4×C107—21no—
155C2×C4×C107—21no—
156C2×C4×C107C2×D411yescharacteristic
157C2×C4×C107C2411yescharacteristic
158C22⋊D107—42no—
159C22⋊D107—42no—
160C22⋊D107—42no—
161C22⋊D107—42no—
162C22⋊D107—42no—
163C22⋊D107—42no—
164C22⋊D107—81no—
165C22⋊D107—81no—
166C22⋊D107D814yescomplemented
167C2×Dic107—21no—
168C2×Dic107—42no—
169C2×Dic107C2×D411yescharacteristic
170C2×Dic107C2×D411yescharacteristic
171C2×Dic107C2×D411yescharacteristic
172D428—21no—
173D428—21no—
174D428—21no—
175D428—21no—
176D428—21no—
177D428—42no—
178D428—42no—
179D428C2×D412yescomplemented
180C4×D107—21no—
181C4×D107—21no—
182C4×D107—42no—
183C8×C107—21no—
184C8×C107—21no—
185C107⋊C8—21no—
186C107⋊C8C2×D412yescomplemented
187C22×C107—21no—
188C22×C107—21no—
189C22×C107—22no—
190C22×C107—41no—
191C22×C107—41no—
192C22×C107—41no—
193C22×C107—41no—
194C22×C107—41no—
195C22×C107—41no—
196C22×C107—41no—
197C22×C107—41no—
198C22×C107—84no—
199C22×C107C2×D812yescomplemented
200C22×C107C22×D411yescharacteristic
201C2×D107—21no—
202C2×D107—21no—
203C2×D107—22no—
204C2×D107—41no—
205C2×D107—41no—
206C2×D107—41no—
207C2×D107—41no—
208C2×D107—41no—
209C2×D107—41no—
210C2×D107—41no—
211C2×D107—41no—
212C2×D107—84no—
213C2×D107—84no—
214C2×D107—84no—
215C2×D107C2×D812yescomplemented
216C4×C107—21no—
217C4×C107—41no—
218C4×C107C22×D411yescharacteristic
219C4×C107H(5,𝔽2)11yescharacteristic
220Dic107—21no—
221Dic107—21no—
222Dic107—41no—
223Dic107C2×D812yescomplemented
224C2×C107—22no—
225C2×C107—42no—
226C2×C107—81no—
227C2×C107—81no—
228C2×C107D4211yescharacteristic
229C2×C107C22×D811yescharacteristic
230C2×C107#57611yescharacteristic
231D107—22no—
232D107—42no—
233D107—81no—
234D107—81no—
235D4×D8—1071nomaximal, Sylow
236C107D4×D811yescharacteristic, complemented, Sylow
237C22×D8—1071no—
238C22×D8—1071no—
239D42—1071no—
240D42—1071no—
241#493—1071no—
242#466—1071no—
243#466—1071no—
244#459—1071no—
245#459—1071no—
246#447—1071no—
247#447—1071no—
248#447—1071no—
249#447D10711yescharacteristic, complemented
250C4×D8—1071no—
251C8×D4—1071no—
252C22×D4—1071no—
253C22×D4—1071no—
254C22×D4—1071no—
255C22×D4—2141no—
256C22×D4—2141no—
257C22×D4C2×D10711yescharacteristic, complemented
258C2×D8—1071no—
259C2×D8—1071no—
260C2×D8—1071no—
261C2×D8—1071no—
262C2×D8—1071no—
263C2×D8—1071no—
264C2×D8—1071no—
265C2×D8—2144no—
266C2×D8—2144no—
267C2×D8C2×D10711yescharacteristic, complemented
268C2×D8C2×D10711yescharacteristic, complemented
269C22×C8—1071no—
270C22×C8—1071no—
271#127—1071no—
272#127—1071no—
273#121—1071no—
274#121—1071no—
275#121—1071no—
276#121—2141no—
277#121—2141no—
278#121C2×D10711yescharacteristic, complemented
279#120—2141no—
280#120—2141no—
281#120—2141no—
282#120—2141no—
283C4×D4—1071no—
284C4×D4—1071no—
285C4×D4—1071no—
286#107—1071no—
287C4⋊C8—1071no—
288#102—1071no—
289#102—1071no—
290#102—1071no—
291#102—1071no—
292#102C2×D10711yescharacteristic, complemented
293#102C2×D10711yescharacteristic, complemented
294C22⋊C8—1071no—
295C22⋊C8C2×D10711yescharacteristic, complemented
296C4×C8—1071no—
297C24—21no—
298C24—2141no—
299C24—2141no—
300C24—2141no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1071no—
306C2×D4—1071no—
307C2×D4—1071no—
308C2×D4—1071no—
309C2×D4—4281no—
310C2×D4—4284no—
311C2×D4—4284no—
312C2×D4—4281no—
313C2×D4—4281no—
314C2×D4—2142no—
315C2×D4—2141no—
316C2×D4—2141no—
317C2×D4—2141no—
318C2×D4—2141no—
319C2×D4—2141no—
320C2×D4—2141no—
321C2×D4—2141no—
322C2×D4—2142no—
323C2×D4—2141no—
324C2×D4—2141no—
325C2×D4—2142no—
326C2×D4—2141no—
327C2×D4C22×D10711yescharacteristic
328C2×D4C22×D10711yescharacteristic
329C2×D4C22×D10711yescharacteristic
330C22×C4—1071no—
331C22×C4—2141no—
332C22×C4—2141no—
333C22×C4C22×D10711yescharacteristic
334D8—22no—
335D8—41no—
336D8—1074no—
337D8—4281no—
338D8—2142no—
339D8—2142no—
340D8—2142no—
341D8—2142no—
342D8—2142no—
343C2×C8—1071no—
344C2×C8—1071no—
345C2×C8—2142no—
346C2×C8—2142no—
347C2×C8C22×D10711yescharacteristic
348C2×C8C22×D10711yescharacteristic
349C4⋊C4—1071no—
350C4⋊C4—1071no—
351C4⋊C4C22×D10711yescharacteristic
352C22⋊C4—21no—
353C22⋊C4—1071no—
354C22⋊C4—1071no—
355C22⋊C4—2141no—
356C22⋊C4—2141no—
357C22⋊C4—2141no—
358C22⋊C4—2141no—
359C22⋊C4—2141no—
360C42—1071no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—8561no—
368C23—8561no—
369C23—8561no—
370C23—1071no—
371C23—4282no—
372C23—4282no—
373C23—4282no—
374C23—4282no—
375C23—4282no—
376C23—4282no—
377C23—2141no—
378C23—2141no—
379C23—2141no—
380C23D4×D10711yescharacteristic
381D4—21no—
382D4—21no—
383D4—21no—
384D4—21no—
385D4—21no—
386D4—42no—
387D4—42no—
388D4—8561no—
389D4—8561no—
390D4—1074no—
391D4—1072no—
392D4—4282no—
393D4—4282no—
394D4—4282no—
395D4—4282no—
396D4—4282no—
397D4—4282no—
398D4—4282no—
399D4—4282no—
400D4—2141no—
401D4—2141no—
402D4—2141no—
403D4—2141no—
404D4—2141no—
405D4D4×D10712yescomplemented
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—1071no—
410C2×C4—1071no—
411C2×C4—1071no—
412C2×C4—4282no—
413C2×C4—4282no—
414C2×C4—2141no—
415C2×C4—2141no—
416C2×C4—2141no—
417C2×C4D4×D10711yescharacteristic
418C2×C4C23×D10711yescharacteristic
419C8—21no—
420C8—21no—
421C8—1072no—
422C8—2141no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—84no—
435C22—8564no—
436C22—8564no—
437C22—8564no—
438C22—1072no—
439C22—4281no—
440C22—4281no—
441C22—4281no—
442C22—4281no—
443C22—4281no—
444C22—4281no—
445C22—4281no—
446C22—4281no—
447C22—2142no—
448C22—2141no—
449C22—2141no—
450C22D8×D10712yescomplemented
451C22C2×D4×D10711yescharacteristic
452C4—21no—
453C4—41no—
454C4—1072no—
455C4—4281no—
456C4—2141no—
457C4—2141no—
458C4C2×D4×D10711yescharacteristic
459C4#1681215411yescharacteristic
460C2—22no—
461C2—42no—
462C2—81no—
463C2—81no—
464C2—8561no—
465C2—8561no—
466C2—4282no—
467C2—2142no—
468C2#2029325911yescharacteristic
469C2C2×D8×D10711yescharacteristic
470C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#447×C107C2
socleC22×C107C22×D4
hypercenter#447D107
hypocenterC107D4×D8
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#13133803C2semidirect
2. C2×D8×D107C2semidirect
3. #20293261C2semidirect
4. D4×C22⋊D107C2semidirect
5. #20293002C2semidirect
6. #20292862C2semidirect
7. #20292861C2semidirect
8. #20292821C2semidirect
9. #20292820C2semidirect
10. #20292763C2semidirect
11. #20292762C2semidirect
12. #20292759C2semidirect
13. #20292707C2semidirect
14. (C22⋊D107)⋊C8C2semidirect
15. #447×C107C2semidirect
16. C22×D428C22semidirect
19. C2×D4×D107C22semidirect
20. C2×D4×D107C22semidirect
21. C2×D856C22semidirect
22. C2×D856C22semidirect
25. C2×#13133803C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
30. C22×C107⋊C8C22semidirect
31. C2×C8×D107C22semidirect
32. #16812105C22semidirect
33. #16812105C22semidirect
34. #16812084C22semidirect
35. #16812083C22semidirect
38. #16812080C22semidirect
43. D4×Dic107C22semidirect
44. #16812071C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812031C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C22×D4×C107C22semidirect
55. C2×D8×C107C22semidirect
56. C2×D8×C107C22semidirect
57. #121×C107C22semidirect
58. #102×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
95. #13133803D4semidirect
152. D4×C107C2×D4semidirect
166. C22⋊D107D8semidirect
179. D428C2×D4semidirect
186. C107⋊C8C2×D4semidirect
199. C22×C107C2×D8semidirect
215. C2×D107C2×D8semidirect
223. Dic107C2×D8semidirect
236. C107D4×D8semidirect
249. #447D107semidirect
257. C22×D4C2×D107semidirect
267. C2×D8C2×D107semidirect
268. C2×D8C2×D107semidirect
278. #121C2×D107semidirect
292. #102C2×D107semidirect
293. #102C2×D107semidirect
295. C22⋊C8C2×D107semidirect
405. D4D4×D107semidirect
450. C22D8×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338032
C2×D8×D1072
#202932612
D4×C22⋊D1072
#202930022
#202928622
#202928612
#202928212
#202928202
#202927632
#202927622
#202927592
#202927072
(C22⋊D107)⋊C82
#447×C1072
D4×D8107