Finite Group Databasebeta

← #28345681

Subgroups — #28345681

Order 13696.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic, complemented
2C2×#16812131C211yesmaximal, characteristic, complemented
3C2×#16812129C211yesmaximal, characteristic, complemented
4#20293139C211yesmaximal, characteristic, complemented
5#20292792C211yesmaximal, characteristic, complemented
6#20292790C211yesmaximal, characteristic, complemented
7#20292784C211yesmaximal, characteristic, complemented
8#20292779C211yesmaximal, characteristic, complemented
9#20292775C211yesmaximal, characteristic, complemented
10#20292773C211yesmaximal, characteristic, complemented
11#20292767C211yesmaximal, characteristic, complemented
12#20292762C211yesmaximal, characteristic, complemented
13#20292621C211yesmaximal, characteristic, complemented
14C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
15#416×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818—21no—
18C2×#13133818C2211yescharacteristic
19C2×#13133818C2211yescharacteristic, complemented
20C2×#13133817—21no—
21C2×#13133817C2211yescharacteristic
22#41×D107—24no—
23#41×D107—24no—
24C2×Q8×D107C2211yescharacteristic
25C2×D4×D107—21no—
26C2×D4×D107C2211yescharacteristic
27C22×C4×D107—21no—
28C22×C4×D107C2211yescharacteristic
29#16812131—24no—
30#16812129—24no—
31C2×Dic428C2211yescharacteristic, complemented
32C2×#13133808C2211yescharacteristic, complemented
33C2×#13133807C2211yescharacteristic, complemented
34C2×#13133807C2211yescharacteristic, complemented
35C2×#13133806C2211yescharacteristic, complemented
36C2×#13133805C2211yescharacteristic, complemented
37C2×D856C2211yescharacteristic, complemented
38C2×#13133803C2211yescharacteristic, complemented
39C22×C107⋊C8C2211yescharacteristic, complemented
40#16812093—21no—
41#16812089C2211yescharacteristic
42#16812089C2211yescharacteristic, complemented
43#16812084C2211yescharacteristic
44#16812084C2211yescharacteristic, complemented
45#16812079—21no—
46C4×D428—21no—
47C4×D428—21no—
48#16812032C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic
53(C2×D107)⋊C8C2211yescharacteristic
54(C2×D107)⋊C8C2211yescharacteristic, complemented
55(C2×D107)⋊C8C2211yescharacteristic, complemented
56C2×#41×C107C2211yescharacteristic
57C22×C8×C107C2211yescharacteristic, complemented
58#117×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#103×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62#102×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64C23×D107D411yescharacteristic
65#13133819—22no—
66#13133819—22no—
67#13133818—28no—
68#13133818—22no—
69#13133818—22no—
70#13133818—22no—
71#13133818D414yescomplemented
72#13133817—28no—
73#13133817—22no—
74#13133817—22no—
75C2×Dic214—21no—
76C2×Dic214C2311yescharacteristic
77C2×Dic214C2311yescharacteristic
78Q8×D107—22no—
79Q8×D107—22no—
80C2×D428—21no—
81C2×D428—21no—
82C2×D428—21no—
83C2×D428C2311yescharacteristic
84C2×D428C2311yescharacteristic
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87C2×C22⋊D107—21no—
88C2×C22⋊D107—21no—
89D4×D107—28no—
90D4×D107—22no—
91D4×D107—22no—
92C22×Dic107—21no—
93C22×Dic107D411yescharacteristic
94C2×C4×D107—28no—
95C2×C4×D107—21no—
96C2×C4×D107—21no—
97C2×C4×D107—22no—
98C2×C4×D107—22no—
99C2×C4×D107—21no—
100C2×C4×D107—21no—
101C2×C4×D107D411yescharacteristic
102C2×C4×D107D411yescharacteristic
103C2×C4×D107C2311yescharacteristic
104C2×C4×D107C2311yescharacteristic
105Dic428—22no—
106#13133808—22no—
107#13133807—22no—
108#13133807—22no—
109#13133806—22no—
110#13133805—22no—
111D856—22no—
112#13133803—22no—
113C2×C107⋊C8—22no—
114C2×C107⋊C8C2311yescharacteristic
115C2×C107⋊C8C2311yescharacteristic
116C4⋊Dic107—21no—
117C4⋊Dic107—21no—
118(C2×D107)⋊C4—21no—
119(C2×D107)⋊C4—21no—
120(C2×D107)⋊C4—21no—
121(C2×D107)⋊C4—21no—
122#41×C107—22no—
123#41×C107D414yescomplemented
124C2×Q8×C107C2311yescharacteristic
125C2×D4×C107—21no—
126C2×D4×C107C2311yescharacteristic
127C22×C4×C107—21no—
128C22×C4×C107C2311yescharacteristic
129C2×C8×C107—22no—
130C2×C8×C107C2311yescharacteristic
131C2×C8×C107C2311yescharacteristic
132C4⋊C4×C107C2311yescharacteristic
133C4⋊C4×C107C2311yescharacteristic
134C22⋊C4×C107—21no—
135C42×C107—21no—
136C23×C107—21no—
137C23×C107C2×D411yescharacteristic
138C22×D107—24no—
139C22×D107—22no—
140C22×D107—21no—
141C22×D107—22no—
142C22×D107—21no—
143C22×D107—41no—
144C22×D107—42no—
145C22×D107—42no—
146C22×D107C2×D411yescharacteristic
147C22×D107C2×D411yescharacteristic
148Q8×C107—21no—
149Q8×C107C2×D412yes—
150D4×C107—24no—
151D4×C107—21no—
152D4×C107C2×D412yes—
153C2×C4×C107—21no—
154C2×C4×C107—24no—
155C2×C4×C107—21no—
156C2×C4×C107—21no—
157C2×C4×C107C2×D412yes—
158C2×C4×C107C2×D411yescharacteristic
159C2×C4×C107C2×D412yes—
160C2×C4×C107C2411yescharacteristic
161C22⋊D107—24no—
162C22⋊D107—24no—
163C22⋊D107—24no—
164C22⋊D107—42no—
165C2×Dic107—24no—
166C2×Dic107—22no—
167C2×Dic107—21no—
168C2×Dic107—21no—
169C2×Dic107C2×D411yescharacteristic
170C2×Dic107C2×D411yescharacteristic
171D428—24no—
172D428—21no—
173D428—21no—
174D428—21no—
175D428—42no—
176D428—42no—
177D428C2×D412yes—
178C4×D107—24no—
179C4×D107—24no—
180C4×D107—22no—
181C4×D107—24no—
182C4×D107—22no—
183C4×D107—21no—
184C4×D107—21no—
185C4×D107—21no—
186C4×D107—21no—
187C4×D107—42no—
188Dic214—24no—
189Dic214—21no—
190Dic214—21no—
191Dic214—21no—
192Dic214C2×D412yes—
193C8×C107—21no—
194C8×C107—21no—
195C107⋊C8—21no—
196C107⋊C8—21no—
197C22×C107—21no—
198C22×C107—21no—
199C22×C107—22no—
200C22×C107—41no—
201C22×C107—41no—
202C22×C107#12012yes—
203C22×C107C22×D411yescharacteristic
204C2×D107—21no—
205C2×D107—21no—
206C2×D107—21no—
207C2×D107—22no—
208C2×D107—21no—
209C2×D107—21no—
210C2×D107—44no—
211C2×D107—48no—
212C2×D107—41no—
213C2×D107—41no—
214C2×D107—41no—
215C2×D107—41no—
216C2×D107—41no—
217C2×D107#13512yescomplemented
218C4×C107—22no—
219C4×C107—41no—
220C4×C107—41no—
221C4×C107#12012yes—
222C4×C107C22×D411yescharacteristic
223C4×C107C22×D411yescharacteristic
224Dic107—21no—
225Dic107—22no—
226Dic107—21no—
227Dic107—41no—
228C2×C107—22no—
229C2×C107—42no—
230C2×C107C2×#12011yescharacteristic
231C2×C107C2×#13511yescharacteristic
232C2×C107#57511yescharacteristic
233D107—22no—
234D107—41no—
235D107—42no—
236D107—81no—
237#3002—1071nomaximal, Sylow
238C107#300211yescharacteristic, complemented, Sylow
239C22×#41—1071no—
240C2×#135—1071no—
241C2×#135—1071no—
242#525—1071no—
243#452—1071no—
244#452—1071no—
245#451—1071no—
246#450—1071no—
247#449—1071no—
248#449—1071no—
249#448—1071no—
250#447—1071no—
251#416—1071no—
252#416D10711yescharacteristic, complemented
253C2×C22⋊C8—1071no—
254C2×#41—1071no—
255C2×#41—1071no—
256C2×#41—1071no—
257C2×#41—1071no—
258C2×#41—2141no—
259C2×#41—2141no—
260C2×#41—2144no—
261C2×#41—2144no—
262C2×#41C2×D10711yescharacteristic
263C22×Q8—1071no—
264C22×D4—1071no—
265C22×D4—2141no—
266C23×C4—1071no—
267C23×C4—2141no—
268#135—2144no—
269#135—2144no—
270C2×Q16—1071no—
271C2×Q16—1071no—
272C2×SD16—1071no—
273C2×SD16—1071no—
274C2×SD16—1071no—
275C2×SD16—1071no—
276C2×D8—1071no—
277C2×D8—1071no—
278C22×C8—1071no—
279C22×C8C2×D10711yescharacteristic, complemented
280#123—2141no—
281C22⋊Q8—1071no—
282C22⋊Q8—1071no—
283#121—1071no—
284#121—1071no—
285#120—2141no—
286C4×D4—2141no—
287C4×D4—2141no—
288#117C2×D10711yescharacteristic, complemented
289#103—1071no—
290#103—1071no—
291#103C2×D10711yescharacteristic, complemented
292#103C2×D10711yescharacteristic, complemented
293#102—1071no—
294#102—1071no—
295#102C2×D10711yescharacteristic, complemented
296#102C2×D10711yescharacteristic, complemented
297C22⋊C8—1071no—
298C22⋊C8—1071no—
299C22⋊C8—1071no—
300C22⋊C8—1071no—
301C24—1071no—
302C24—2141no—
303#41—22no—
304#41—1074no—
305#41—2148no—
306#41—2142no—
307#41—2142no—
308#41—2142no—
309#41—2148no—
310#41—2142no—
311#41—2142no—
312#41—2142no—
313#41—2142no—
314#41D42814yescomplemented
315C2×Q8—1071no—
316C2×Q8—1071no—
317C2×Q8—2141no—
318C2×Q8—2142no—
319C2×Q8—2142no—
320C2×Q8C22×D10711yescharacteristic
321C2×D4—21no—
322C2×D4—1071no—
323C2×D4—1071no—
324C2×D4—2141no—
325C2×D4—2141no—
326C2×D4—2141no—
327C2×D4—2141no—
328C2×D4—2142no—
329C2×D4—2148no—
330C2×D4—2141no—
331C2×D4—2141no—
332C2×D4—2141no—
333C2×D4—2142no—
334C2×D4C22×D10711yescharacteristic
335C22×C4—21no—
336C22×C4—1071no—
337C22×C4—1071no—
338C22×C4—1071no—
339C22×C4—1071no—
340C22×C4—1071no—
341C22×C4—2141no—
342C22×C4—2141no—
343C22×C4—2148no—
344C22×C4—2141no—
345C22×C4—2141no—
346C22×C4—2142no—
347C22×C4—2141no—
348C22×C4—2142no—
349C22×C4C22×D10711yescharacteristic
350Q16—2142no—
351Q16—2142no—
352SD16—2142no—
353SD16—2142no—
354SD16—2142no—
355SD16—2142no—
356D8—2142no—
357D8—2142no—
358C2×C8—22no—
359C2×C8—1071no—
360C2×C8—1071no—
361C2×C8—2142no—
362C2×C8C22×D10711yescharacteristic
363C2×C8C22×D10711yescharacteristic
364C4⋊C4—2141no—
365C4⋊C4—2141no—
366C4⋊C4C22×D10711yescharacteristic
367C4⋊C4C22×D10711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—2141no—
370C22⋊C4—2141no—
371C22⋊C4—2141no—
372C22⋊C4—2141no—
373C42—21no—
374C23—21no—
375C23—1071no—
376C23—1071no—
377C23—4281no—
378C23—4282no—
379C23—4282no—
380C23—2142no—
381C23—2144no—
382C23—2141no—
383C23—2142no—
384C23—2141no—
385C23D4×D10711yescharacteristic
386Q8—21no—
387Q8—1072no—
388Q8—2144no—
389Q8—2141no—
390Q8—2141no—
391Q8—2141no—
392Q8C2×D42812yes—
393D4—24no—
394D4—21no—
395D4—1072no—
396D4—4282no—
397D4—4282no—
398D4—4282no—
399D4—2144no—
400D4—2144no—
401D4—2144no—
402D4—2144no—
403D4—2141no—
404D4—2141no—
405D4—2141no—
406D4C2×D42812yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—1071no—
412C2×C4—1071no—
413C2×C4—4282no—
414C2×C4—2144no—
415C2×C4—2144no—
416C2×C4—2141no—
417C2×C4—2144no—
418C2×C4—2142no—
419C2×C4—2141no—
420C2×C4—2142no—
421C2×C4—2142no—
422C2×C4—2144no—
423C2×C4—2141no—
424C2×C4—2141no—
425C2×C4—2141no—
426C2×C4—2141no—
427C2×C4D4×D10712yes—
428C2×C4D4×D10711yescharacteristic
429C2×C4C2×D42812yes—
430C2×C4C23×D10711yescharacteristic
431C8—21no—
432C8—21no—
433C8—2141no—
434C8—2141no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—1072no—
441C22—4284no—
442C22—4288no—
443C22—4281no—
444C22—4281no—
445C22—4281no—
446C22—4281no—
447C22—4281no—
448C22—2141no—
449C22—2141no—
450C22—2142no—
451C22—2141no—
452C22—2141no—
453C22—2141no—
454C22#1681207912yes—
455C22C2×D4×D10711yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—4281no—
460C4—2141no—
461C4—2141no—
462C4—2142no—
463C4#1681207912yes—
464C4C2×D4×D10711yescharacteristic
465C4C22×D42811yescharacteristic
466C2—22no—
467C2—42no—
468C2—8561no—
469C2—4282no—
470C2—4281no—
471C2—2142no—
472C2C2×#1681207911yescharacteristic
473C2#135×D10711yescharacteristic
474C2#2029342411yescharacteristic
subgroupG/H
centerC2×C4D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
hypercenter#416D107
hypocenterC107#3002
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D107C2semidirect
2. C2×#16812131C2semidirect
3. C2×#16812129C2semidirect
4. #20293139C2semidirect
5. #20292792C2semidirect
6. #20292790C2semidirect
7. #20292784C2semidirect
8. #20292779C2semidirect
9. #20292775C2semidirect
10. #20292773C2semidirect
11. #20292767C2semidirect
12. #20292762C2semidirect
13. #20292621C2semidirect
14. C2×(C2×D107)⋊C8C2semidirect
15. #416×C107C2semidirect
19. C2×#13133818C22semidirect
31. C2×Dic428C22semidirect
32. C2×#13133808C22semidirect
33. C2×#13133807C22semidirect
34. C2×#13133807C22semidirect
35. C2×#13133806C22semidirect
36. C2×#13133805C22semidirect
37. C2×D856C22semidirect
38. C2×#13133803C22semidirect
39. C22×C107⋊C8C22semidirect
42. #16812089C22semidirect
44. #16812084C22semidirect
48. #16812032C22semidirect
49. #16812032C22semidirect
50. #16812029C22semidirect
51. #16812029C22semidirect
54. (C2×D107)⋊C8C22semidirect
55. (C2×D107)⋊C8C22semidirect
57. C22×C8×C107C22semidirect
58. #117×C107C22semidirect
59. #103×C107C22semidirect
60. #103×C107C22semidirect
61. #102×C107C22semidirect
62. #102×C107C22semidirect
71. #13133818D4semidirect
123. #41×C107D4semidirect
217. C2×D107#135semidirect
238. C107#3002semidirect
252. #416D107semidirect
279. C22×C8C2×D107semidirect
288. #117C2×D107semidirect
291. #103C2×D107semidirect
292. #103C2×D107semidirect
295. #102C2×D107semidirect
296. #102C2×D107semidirect
314. #41D428semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1072
C2×#168121312
C2×#168121292
#202931392
#202927922
#202927902
#202927842
#202927792
#202927752
#202927732
#202927672
#202927622
#202926212
C2×(C2×D107)⋊C82
#416×C1072
#3002107