Finite Group Databasebeta

← #28348069

Subgroups — #28348069

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×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812136C211yesmaximal, characteristic, complemented
3#20293274C211yesmaximal, characteristic, complemented
4#20293268C211yesmaximal, characteristic, complemented
5#20293008C211yesmaximal, characteristic, complemented
6#20292874C211yesmaximal, characteristic, complemented
7#20292845C211yesmaximal, characteristic, complemented
8#20292841C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292787C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292709C211yesmaximal, characteristic, complemented
14#20292703C211yesmaximal, characteristic, complemented
15#468×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C22×D428C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic, complemented
20#16812137—24no—
21#16812136—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic
27C2×D856C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133803C2211yescharacteristic
30C2×#13133801C2211yescharacteristic, complemented
31#16812098C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic, complemented
33#16812094—21no—
34#16812091—21no—
35#16812089C2211yescharacteristic, complemented
36#16812084—21no—
37#16812084C2211yescharacteristic
38#16812080—21no—
39#16812079—21no—
40#16812077—21no—
41#16812071C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic, complemented
43C4×C22⋊D107C2211yescharacteristic
44C22⋊C4×D107—21no—
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic, complemented
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⋊D107D411yescharacteristic
79D4×D107—22no—
80D4×D107—41no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107—21no—
85C2×C4×D107C2311yescharacteristic
86#13133808—41no—
87#13133807—22no—
88#13133807—22no—
89#13133805—22no—
90D856—22no—
91D856D414yescomplemented
92#13133803—22no—
93#13133803—41no—
94#13133801—22no—
95C2×C107⋊C8C2311yescharacteristic
96C2×C107⋊C8C2311yescharacteristic
97C4⋊Dic107—21no—
98Dic107⋊C4D411yescharacteristic
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107D411yescharacteristic
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114SD16×C107—22no—
115M16×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—81no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135Q8×C107—21no—
136Q8×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—81no—
140C2×C4×C107—21no—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—41no—
148C22⋊D107—42no—
149C22⋊D107—42no—
150C2×Dic107—21no—
151C2×Dic107C2×D411yescharacteristic
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154D428—21no—
155D428—21no—
156D428—21no—
157D428—21no—
158D428—42no—
159D428—42no—
160D428—81no—
161D428C2×D412yes—
162D428C2×D412yes—
163C4×D107—21no—
164C4×D107—21no—
165C4×D107—42no—
166C4×D107—42no—
167C4×D107—81no—
168C4×D107—81no—
169Dic214—21no—
170Dic214—21no—
171C8×C107—21no—
172C8×C107C2×D412yes—
173C107⋊C8—21no—
174C107⋊C8—21no—
175C22×C107—21no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—22no—
184C2×D107—22no—
185C2×D107—21no—
186C2×D107—21no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—84no—
195C2×D107—84no—
196C4×C107—21no—
197C4×C107—41no—
198C4×C107—41no—
199C4×C107—41no—
200C4×C107C22×D411yescharacteristic
201C4×C107H(5,𝔽2)11yescharacteristic
202Dic107—21no—
203Dic107—21no—
204Dic107—21no—
205Dic107—41no—
206C2×C107—41no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—42no—
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#546—1071no—
221#495—1071no—
222#468—1071no—
223#468D10711yescharacteristic, complemented
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×#41—1071no—
234C22×D4—1071no—
235C22×D4—1071no—
236AGL(1,ℤ/8)—2144no—
237AGL(1,ℤ/8)—2144no—
238C2×Q16—1071no—
239C2×SD16—1071no—
240C2×SD16—1071no—
241C2×SD16—1071no—
242C2×SD16C2×D10711yescharacteristic, complemented
243C2×D8—1071no—
244C2×D8—1071no—
245C2×D8—1071no—
246C2×D8—1071no—
247C2×M16—1071no—
248C2×M16C2×D10711yescharacteristic, complemented
249#124—1071no—
250#124—1071no—
251#123—2141no—
252#123—2141no—
253C22⋊Q8—1071no—
254C22⋊Q8C2×D10711yescharacteristic, complemented
255#121—1071no—
256#121—2141no—
257#121—2141no—
258#121C2×D10711yescharacteristic
259#120—2141no—
260#120—2141no—
261C4×D4—1071no—
262C4×D4—1071no—
263C4×D4—1071no—
264C2×C22⋊C4—2141no—
265C2×C22⋊C4—2141no—
266#107C2×D10711yescharacteristic, complemented
267C4⋊C8—1071no—
268#103—1071no—
269#103C2×D10711yescharacteristic, complemented
270#102—1071no—
271#102—1071no—
272#102—1071no—
273#102C2×D10711yescharacteristic, complemented
274C22⋊C8—1071no—
275C22⋊C8—1071no—
276C4×C8—1071no—
277C24—2141no—
278C24—2141no—
279#41—4281no—
280#41—4281no—
281#41—2142no—
282#41—2142no—
283C2×Q8—1071no—
284C2×Q8C22×D10711yescharacteristic
285C2×D4—21no—
286C2×D4—1071no—
287C2×D4—1071no—
288C2×D4—1071no—
289C2×D4—1071no—
290C2×D4—4281no—
291C2×D4—4281no—
292C2×D4—2141no—
293C2×D4—2142no—
294C2×D4—2141no—
295C2×D4—2141no—
296C2×D4—2142no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4C22×D10711yescharacteristic
300C22×C4—1071no—
301C22×C4—2141no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4C22×D10711yescharacteristic
306Q16—4281no—
307SD16—22no—
308SD16—2142no—
309SD16—2142no—
310SD16—2142no—
311D8—1074no—
312D8—4281no—
313D8—2142no—
314D8—2142no—
315M16—22no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8C22×D10711yescharacteristic
320C2×C8C22×D10711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1071no—
323C4⋊C4—2141no—
324C4⋊C4C22×D10711yescharacteristic
325C4⋊C4C22×D10711yescharacteristic
326C22⋊C4—21no—
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—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C42—1071no—
339C23—21no—
340C23—8561no—
341C23—8561no—
342C23—1071no—
343C23—4282no—
344C23—4282no—
345C23—4282no—
346C23—4282no—
347C23—2141no—
348C23—2141no—
349C23—2141no—
350C23D4×D10711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—2141no—
354Q8—2141no—
355D4—21no—
356D4—21no—
357D4—81no—
358D4—8561no—
359D4—1072no—
360D4—1072no—
361D4—4282no—
362D4—4282no—
363D4—4282no—
364D4—4281no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369D4—2141no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—8561no—
376C2×C4—8561no—
377C2×C4—1071no—
378C2×C4—1071no—
379C2×C4—1071no—
380C2×C4—4282no—
381C2×C4—4282no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4D4×D10711yescharacteristic
386C2×C4C23×D10711yescharacteristic
387C8—21no—
388C8—2141no—
389C8—2141no—
390C8D4×D10712yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—8564no—
398C22—8564no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—4281no—
406C22—2141no—
407C22—2141no—
408C22—2142no—
409C22—2142no—
410C22—2141no—
411C22C2×D4×D10711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4281no—
417C4—2141no—
418C4—2141no—
419C4—2141no—
420C4C2×D4×D10711yescharacteristic
421C4#1681215511yescharacteristic
422C2—41no—
423C2—81no—
424C2—8561no—
425C2—4281no—
426C2—4282no—
427C2—4282no—
428C2#2029326111yescharacteristic
429C2#2029342611yescharacteristic
430C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#468×C107C2
socleC22×C107C22×D4
hypercenter#468D107
hypocenterC107#3273
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. #20293274C2semidirect
4. #20293268C2semidirect
5. #20293008C2semidirect
6. #20292874C2semidirect
7. #20292845C2semidirect
8. #20292841C2semidirect
9. #20292792C2semidirect
10. #20292787C2semidirect
11. #20292762C2semidirect
12. #20292760C2semidirect
13. #20292709C2semidirect
14. #20292703C2semidirect
15. #468×C107C2semidirect
18. C22×D428C22semidirect
19. C2×D4×D107C22semidirect
22. C2×#13133808C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133805C22semidirect
30. C2×#13133801C22semidirect
31. #16812098C22semidirect
32. #16812096C22semidirect
35. #16812089C22semidirect
41. #16812071C22semidirect
42. #16812071C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812029C22semidirect
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
91. D856D4semidirect
216. C107#3273semidirect
223. #468D107semidirect
242. C2×SD16C2×D107semidirect
248. C2×M16C2×D107semidirect
254. C22⋊Q8C2×D107semidirect
266. #107C2×D107semidirect
269. #103C2×D107semidirect
273. #102C2×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
#202932742
#202932682
#202930082
#202928742
#202928452
#202928412
#202927922
#202927872
#202927622
#202927602
#202927092
#202927032
#468×C1072
#3273107