Finite Group Databasebeta

← #28348058

Subgroups — #28348058

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812136C211yesmaximal, characteristic, complemented
3#20293279C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20292998C211yesmaximal, characteristic, complemented
6#20292874C211yesmaximal, characteristic, complemented
7#20292831C211yesmaximal, characteristic, complemented
8#20292821C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic, complemented
10#20292777C211yesmaximal, characteristic, complemented
11#20292775C211yesmaximal, characteristic, complemented
12#20292770C211yesmaximal, characteristic, complemented
13#20292709C211yesmaximal, characteristic, complemented
14#20292703C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C2×#13133818C2211yescharacteristic, complemented
18C22×D428C2211yescharacteristic
19C2×D4×D107—21no—
20C2×D4×D107C2211yescharacteristic
21#16812137—24no—
22#16812136—24no—
23C2×#13133807C2211yescharacteristic
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133806C2211yescharacteristic, complemented
26C2×#13133805C2211yescharacteristic
27C2×#13133805C2211yescharacteristic, complemented
28C2×D856C2211yescharacteristic
29C2×D856C2211yescharacteristic, complemented
30C2×#13133803C2211yescharacteristic, complemented
31C2×#13133801C2211yescharacteristic, complemented
32#16812107C2211yescharacteristic, complemented
33#16812105C2211yescharacteristic
34#16812094—21no—
35#16812091—21no—
36#16812089C2211yescharacteristic, complemented
37#16812084C2211yescharacteristic
38#16812082—21no—
39#16812082—21no—
40#16812079—21no—
41#16812077—21no—
42#16812071C2211yescharacteristic
43#16812071C2211yescharacteristic, complemented
44C4×C22⋊D107C2211yescharacteristic
45C4⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#107×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133819—22no—
64#13133819—41no—
65#13133818—22no—
66#13133818—41no—
67C2×Dic214C2311yescharacteristic
68C2×D428—22no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—41no—
73C2×D428C2311yescharacteristic
74C2×D428C2311yescharacteristic
75C2×D428C2311yescharacteristic
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107D411yescharacteristic
81D4×D107—22no—
82D4×D107—41no—
83D4×D107—44no—
84C2×C4×D107—21no—
85C2×C4×D107—21no—
86C2×C4×D107—21no—
87C2×C4×D107—21no—
88C2×C4×D107C2311yescharacteristic
89#13133807—22no—
90#13133807—22no—
91#13133806—41no—
92#13133805—22no—
93#13133805—41no—
94D856—22no—
95D856D414yescomplemented
96#13133803—22no—
97#13133801—22no—
98C2×C107⋊C8C2311yescharacteristic
99C2×C107⋊C8C2311yescharacteristic
100C4⋊Dic107—21no—
101Dic107⋊C4—21no—
102Dic107⋊C4—21no—
103Dic107⋊C4D411yescharacteristic
104C22⋊Dic107D411yescharacteristic
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4D411yescharacteristic
110C4×Dic107C2311yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107C2311yescharacteristic
115SD16×C107—22no—
116M16×C107—22no—
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107—21no—
120C4⋊C4×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107C2×D411yescharacteristic
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—81no—
134C22×D107—81no—
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—42no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2411yescharacteristic
148C22⋊D107—22no—
149C22⋊D107—42no—
150C22⋊D107—42no—
151C22⋊D107—42no—
152C22⋊D107—81no—
153C2×Dic107—21no—
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157D428—21no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—42no—
162D428—42no—
163D428—42no—
164D428C2×D412yes—
165D428C2×D412yescomplemented
166C4×D107—21no—
167C4×D107—21no—
168C4×D107—42no—
169C4×D107—42no—
170C4×D107—42no—
171C4×D107—42no—
172Dic214—21no—
173Dic214—21no—
174C8×C107—21no—
175C8×C107C2×D412yescomplemented
176C107⋊C8—21no—
177C107⋊C8—21no—
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107C22×D411yescharacteristic
185C2×D107—21no—
186C2×D107—22no—
187C2×D107—22no—
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—84no—
198C2×D107—84no—
199C4×C107—21no—
200C4×C107—41no—
201C4×C107—41no—
202C4×C107—41no—
203C4×C107C22×D411yescharacteristic
204C4×C107H(5,𝔽2)11yescharacteristic
205Dic107—21no—
206Dic107—21no—
207Dic107—41no—
208Dic107AGL(1,ℤ/8)12yescomplemented
209C2×C107—41no—
210C2×C107—81no—
211C2×C107D4211yescharacteristic
212C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C107#57711yescharacteristic
214D107—41no—
215D107—42no—
216D107—42no—
217D107—81no—
218#3272—1071nomaximal, Sylow
219C107#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C2×AGL(1,ℤ/8)—1071no—
222#547—1071no—
223D42—1071no—
224#492—1071no—
225#468—1071no—
226#468D10711yescharacteristic, complemented
227#461—1071no—
228#459—1071no—
229#450—1071no—
230#450—1071no—
231#449—1071no—
232#449—1071no—
233C4×D8—1071no—
234#436—1071no—
235C2×#41—1071no—
236C2×#41—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241AGL(1,ℤ/8)—2144no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—1071no—
247C2×SD16C2×D10711yescharacteristic, complemented
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8—1071no—
251C2×M16—1071no—
252C2×M16C2×D10711yescharacteristic, complemented
253C4⋊Q8—1071no—
254#127—1071no—
255#123—2141no—
256#123—2141no—
257C22⋊Q8—1071no—
258C22⋊Q8C2×D10711yescharacteristic, complemented
259#121—1071no—
260#121—2141no—
261#121—2141no—
262#121C2×D10711yescharacteristic
263#120—2141no—
264#120—2141no—
265C4×D4—1071no—
266C4×D4—1071no—
267C4×D4—1071no—
268C2×C4⋊C4—2141no—
269#107C2×D10711yescharacteristic, complemented
270C4⋊C8—1071no—
271#103—1071no—
272#103C2×D10711yescharacteristic, complemented
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102C2×D10711yescharacteristic, complemented
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279C4×C8—1071no—
280C24—2141no—
281C24—2141no—
282#41—4281no—
283#41—4281no—
284#41—2142no—
285#41—2142no—
286C2×Q8—1071no—
287C2×Q8C22×D10711yescharacteristic
288C2×D4—21no—
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—4281no—
294C2×D4—4284no—
295C2×D4—4281no—
296C2×D4—2142no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2142no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4C22×D10711yescharacteristic
306C22×C4—1071no—
307C22×C4—2141no—
308C22×C4—2141no—
309C22×C4—2141no—
310C22×C4—2141no—
311C22×C4C22×D10711yescharacteristic
312SD16—22no—
313SD16—4281no—
314SD16—4281no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—1074no—
319D8—2142no—
320D8—2142no—
321M16—22no—
322M16—2142no—
323C2×C8—1071no—
324C2×C8—1071no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—1071no—
329C4⋊C4—2141no—
330C4⋊C4—2141no—
331C4⋊C4—2141no—
332C4⋊C4C22×D10711yescharacteristic
333C4⋊C4C22×D10711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—1071no—
337C22⋊C4—1071no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C22⋊C4—2141no—
342C42—1071no—
343C23—21no—
344C23—8561no—
345C23—8561no—
346C23—1071no—
347C23—4282no—
348C23—4282no—
349C23—4282no—
350C23—4282no—
351C23—2141no—
352C23—2141no—
353C23—2141no—
354C23D4×D10711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2141no—
358Q8—2141no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—8561no—
363D4—1072no—
364D4—1072no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—2142no—
372D4—2141no—
373D4—2141no—
374D4—2141no—
375D4—2141no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—1071no—
382C2×C4—1071no—
383C2×C4—1071no—
384C2×C4—4282no—
385C2×C4—4282no—
386C2×C4—4282no—
387C2×C4—4282no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4—2141no—
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—2141no—
395C8—2141no—
396C8D4×D10712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—8564no—
404C22—8564no—
405C22—4281no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—2141no—
413C22—2141no—
414C22—2142no—
415C22—2141no—
416C22—2142no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—1072no—
423C4—4281no—
424C4—2141no—
425C4—2141no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215511yescharacteristic
428C2—41no—
429C2—81no—
430C2—8561no—
431C2—4282no—
432C2—4281no—
433C2—4282no—
434C2#2029326111yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812136C2semidirect
3. #20293279C2semidirect
4. #20293261C2semidirect
5. #20292998C2semidirect
6. #20292874C2semidirect
7. #20292831C2semidirect
8. #20292821C2semidirect
9. #20292779C2semidirect
10. #20292777C2semidirect
11. #20292775C2semidirect
12. #20292770C2semidirect
13. #20292709C2semidirect
14. #20292703C2semidirect
15. #468×C107C2semidirect
16. C2×#13133819C22semidirect
17. C2×#13133818C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133806C22semidirect
27. C2×#13133805C22semidirect
29. C2×D856C22semidirect
30. C2×#13133803C22semidirect
31. C2×#13133801C22semidirect
32. #16812107C22semidirect
36. #16812089C22semidirect
43. #16812071C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
49. #16812029C22semidirect
50. #16812029C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×M16×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #107×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
95. D856D4semidirect
165. D428C2×D4semidirect
175. C8×C107C2×D4semidirect
208. Dic107AGL(1,ℤ/8)semidirect
219. C107#3272semidirect
226. #468D107semidirect
247. C2×SD16C2×D107semidirect
252. C2×M16C2×D107semidirect
258. C22⋊Q8C2×D107semidirect
269. #107C2×D107semidirect
272. #103C2×D107semidirect
276. #102C2×D107semidirect
396. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121362
#202932792
#202932612
#202929982
#202928742
#202928312
#202928212
#202927792
#202927772
#202927752
#202927702
#202927092
#202927032
#468×C1072
#3272107