Finite Group Databasebeta

← #28348065

Subgroups — #28348065

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×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3#20293268C211yesmaximal, characteristic, complemented
4#20293265C211yesmaximal, characteristic, complemented
5#20293012C211yesmaximal, characteristic, complemented
6#20292876C211yesmaximal, characteristic, complemented
7#20292875C211yesmaximal, characteristic, complemented
8#20292845C211yesmaximal, characteristic, complemented
9#20292844C211yesmaximal, characteristic, complemented
10#20292788C211yesmaximal, characteristic, complemented
11#20292763C211yesmaximal, characteristic, complemented
12#20292762C211yesmaximal, characteristic, complemented
13#20292707C211yesmaximal, characteristic, complemented
14#20292702C211yesmaximal, characteristic, complemented
15#452×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C22×D428C2211yescharacteristic, complemented
18C2×D4×D107C2211yescharacteristic, complemented
19#16812136—24no—
20#16812134—24no—
21C2×#13133805C2211yescharacteristic, complemented
22C2×#13133805C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic, complemented
24C2×D856C2211yescharacteristic
25C2×D856C2211yescharacteristic
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133802C2211yescharacteristic, complemented
29C2×#13133801C2211yescharacteristic, complemented
30#16812098C2211yescharacteristic, complemented
31#16812098C2211yescharacteristic, complemented
32#16812094—21no—
33#16812092—21no—
34#16812085C2211yescharacteristic, complemented
35#16812084—21no—
36#16812084C2211yescharacteristic
37#16812083C2211yescharacteristic
38#16812081—21no—
39#16812079—21no—
40#16812077—21no—
41D4×Dic107C2211yescharacteristic, complemented
42#16812071C2211yescharacteristic, complemented
43C4×C22⋊D107C2211yescharacteristic
44C2×(C2×D107)⋊C4—21no—
45C22⋊C4×D107—21no—
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812033C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, complemented
54C2×#41×C107C2211yescharacteristic
55C2×Q16×C107C2211yescharacteristic, complemented
56C2×SD16×C107C2211yescharacteristic, complemented
57C22⋊Q8×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133819—22no—
64#13133819—41no—
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—41no—
70C2×D428C2311yescharacteristic
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107D411yescharacteristic
76D4×D107—22no—
77D4×D107—41no—
78C22×Dic107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107C2311yescharacteristic
82#13133805—22no—
83#13133805—22no—
84#13133805—22no—
85D856—22no—
86D856—41no—
87#13133803—22no—
88#13133803D414yescomplemented
89#13133802—22no—
90#13133801—22no—
91C2×C107⋊C8C2311yescharacteristic
92C2×C107⋊C8C2311yescharacteristic
93C4⋊Dic107C2311yescharacteristic
94Dic107⋊C4—21no—
95Dic107⋊C4D411yescharacteristic
96C22⋊Dic107—21no—
97C22⋊Dic107D411yescharacteristic
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4D411yescharacteristic
107C4×Dic107C2311yescharacteristic
108#41×C107—22no—
109#41×C107—41no—
110C2×Q8×C107C2311yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107C2311yescharacteristic
116Q16×C107—41no—
117SD16×C107—22no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107—21no—
121C4⋊C4×C107C2311yescharacteristic
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—
137Q8×C107—21no—
138Q8×C107—21no—
139D4×C107—21no—
140D4×C107—42no—
141D4×C107C2×D412yes—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—42no—
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2411yescharacteristic
150C22⋊D107—41no—
151C22⋊D107—42no—
152C22⋊D107—81no—
153C2×Dic107—21no—
154C2×Dic107—81no—
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157C2×Dic107C2×D411yescharacteristic
158D428—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—21no—
163D428—42no—
164D428—42no—
165D428—81no—
166D428C2×D412yes—
167C4×D107—21no—
168C4×D107—21no—
169C4×D107—42no—
170C4×D107—81no—
171C8×C107—21no—
172C8×C107—21no—
173C107⋊C8—21no—
174C107⋊C8C2×D412yes—
175C22×C107—22no—
176C22×C107—21no—
177C22×C107—21no—
178C22×C107—21no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—22no—
184C2×D107—21no—
185C2×D107—21no—
186C2×D107—41no—
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—42no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—42no—
213D107—81no—
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#468—1071no—
224#463—1071no—
225#463—1071no—
226#452—1071no—
227#452D10711yescharacteristic, complemented
228#447—1071no—
229#447—1071no—
230C4×D8—1071no—
231#436—1071no—
232C2×#41—1071no—
233C2×#41C2×D10711yescharacteristic
234C22×D4—1071no—
235C22×D4—1071no—
236AGL(1,ℤ/8)—2144no—
237AGL(1,ℤ/8)—2144no—
238C2×Q16C2×D10711yescharacteristic, complemented
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×M16—1071no—
249#124—1071no—
250#124—1071no—
251#123—2141no—
252#123—2141no—
253C22⋊Q8—1071no—
254C22⋊Q8C2×D10711yescharacteristic, complemented
255#121—1071no—
256#121—1071no—
257#121—2141no—
258#121—2141no—
259#120—2141no—
260#120—2141no—
261C4×D4—1071no—
262C4×D4—1071no—
263C4×D4—1071no—
264C2×C22⋊C4—2141no—
265C2×C22⋊C4—2141no—
266#107—1071no—
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⋊C8C2×D10711yescharacteristic, complemented
276C4×C8—1071no—
277C24—2141no—
278C24—2141no—
279#41—22no—
280#41—41no—
281#41—4281no—
282#41—2142no—
283C2×Q8C22×D10711yescharacteristic
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—2141no—
297C2×D4—2142no—
298C2×D4—2141no—
299C2×D4C22×D10711yescharacteristic
300C22×C4—21no—
301C22×C4—1071no—
302C22×C4—2141no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4C22×D10711yescharacteristic
306Q16—41no—
307SD16—22no—
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×C8C22×D10711yescharacteristic
320C2×C8C22×D10711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—1071no—
323C4⋊C4—1071no—
324C4⋊C4—2141no—
325C4⋊C4C22×D10711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—1071no—
328C22⋊C4—1071no—
329C22⋊C4—2141no—
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—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—8561no—
358D4—8561no—
359D4—1072no—
360D4—4282no—
361D4—4282no—
362D4—4282no—
363D4—4281no—
364D4—2141no—
365D4—2141no—
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—42no—
376C2×C4—8561no—
377C2×C4—8561no—
378C2×C4—1071no—
379C2×C4—1071no—
380C2×C4—1071no—
381C2×C4—4282no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4—2141no—
385C2×C4D4×D10711yescharacteristic
386C2×C4C23×D10711yescharacteristic
387C8—21no—
388C8—21no—
389C8—1072no—
390C8—2141no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—8564no—
398C22—8564no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—4281no—
405C22—4281no—
406C22—4281no—
407C22—2141no—
408C22—2141no—
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#1681215411yescharacteristic
422C2—41no—
423C2—42no—
424C2—8561no—
425C2—8561no—
426C2—4281no—
427C2—4282no—
428C2#2029325911yescharacteristic
429C2#2029342611yescharacteristic
430C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#452×C107C2
socleC22×C107C22×D4
hypercenter#452D107
hypocenterC107#3273
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812134C2semidirect
3. #20293268C2semidirect
4. #20293265C2semidirect
5. #20293012C2semidirect
6. #20292876C2semidirect
7. #20292875C2semidirect
8. #20292845C2semidirect
9. #20292844C2semidirect
10. #20292788C2semidirect
11. #20292763C2semidirect
12. #20292762C2semidirect
13. #20292707C2semidirect
14. #20292702C2semidirect
15. #452×C107C2semidirect
17. C22×D428C22semidirect
18. C2×D4×D107C22semidirect
21. C2×#13133805C22semidirect
22. C2×#13133805C22semidirect
23. C2×#13133805C22semidirect
28. C2×#13133802C22semidirect
29. C2×#13133801C22semidirect
30. #16812098C22semidirect
31. #16812098C22semidirect
34. #16812085C22semidirect
41. D4×Dic107C22semidirect
42. #16812071C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812033C22semidirect
49. #16812031C22semidirect
50. #16812031C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
55. C2×Q16×C107C22semidirect
56. C2×SD16×C107C22semidirect
57. C22⋊Q8×C107C22semidirect
58. #103×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
88. #13133803D4semidirect
216. C107#3273semidirect
227. #452D107semidirect
238. C2×Q16C2×D107semidirect
242. C2×SD16C2×D107semidirect
254. C22⋊Q8C2×D107semidirect
269. #103C2×D107semidirect
273. #102C2×D107semidirect
275. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121342
#202932682
#202932652
#202930122
#202928762
#202928752
#202928452
#202928442
#202927882
#202927632
#202927622
#202927072
#202927022
#452×C1072
#3273107