Finite Group Databasebeta

← #28348067

Subgroups — #28348067

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×#16812133C211yesmaximal, characteristic, complemented
3#20293272C211yesmaximal, characteristic, complemented
4#20293270C211yesmaximal, characteristic, complemented
5#20293009C211yesmaximal, characteristic, complemented
6#20292876C211yesmaximal, characteristic, complemented
7#20292872C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13C4×#13133803C211yesmaximal, characteristic, complemented
14#20292704C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic, complemented
20#16812136—24no—
21#16812133—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133801C2211yescharacteristic, complemented
30C2×#13133801C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic, complemented
32#16812093—21no—
33#16812091—21no—
34#16812089C2211yescharacteristic, complemented
35#16812085C2211yescharacteristic, complemented
36#16812084C2211yescharacteristic
37#16812083—21no—
38#16812082—21no—
39#16812080C2211yescharacteristic
40#16812079—21no—
41#16812079—21no—
42C4×D428C2211yescharacteristic
43C4×D428C2211yescharacteristic, complemented
44C2×(C2×D107)⋊C4—21no—
45C22⋊C4×D107—21no—
46#16812042C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C4×C107⋊C8C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133819—22no—
64#13133819—41no—
65#13133817—22no—
66#13133817—41no—
67C2×Dic214C2311yescharacteristic
68C2×D428—21no—
69C2×D428—21no—
70C2×D428D411yescharacteristic
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—41no—
81C22×Dic107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85C2×C4×D107C2311yescharacteristic
86#13133808—41no—
87#13133807—22no—
88#13133806—22no—
89#13133805—22no—
90D856—22no—
91#13133803—41no—
92#13133803D414yescomplemented
93#13133801—22no—
94#13133801—22no—
95C2×C107⋊C8C2311yescharacteristic
96C2×C107⋊C8C2311yescharacteristic
97C4⋊Dic107—21no—
98C4⋊Dic107D411yescharacteristic
99C4⋊Dic107C2311yescharacteristic
100Dic107⋊C4—21no—
101C22⋊Dic107—21no—
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
109(C2×D107)⋊C4D411yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114SD16×C107—22no—
115D8×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107C2311yescharacteristic
119C22⋊C4×C107—21no—
120C22⋊C4×C107—21no—
121C22⋊C4×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—81no—
132C22×D107—81no—
133C22×D107C2×D411yescharacteristic
134C22×D107C2×D411yescharacteristic
135Q8×C107—21no—
136Q8×C107—21no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107C2×D412yes—
141C2×C4×C107—21no—
142C2×C4×C107—21no—
143C2×C4×C107—81no—
144C2×C4×C107C2×D411yescharacteristic
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2411yescharacteristic
147C22⋊D107—42no—
148C22⋊D107—42no—
149C22⋊D107—42no—
150C22⋊D107—81no—
151C2×Dic107—21no—
152C2×Dic107—21no—
153C2×Dic107—42no—
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156D428—21no—
157D428—21no—
158D428—21no—
159D428—41no—
160D428—42no—
161D428—81no—
162D428C2×D412yes—
163C4×D107—21no—
164C4×D107—21no—
165C4×D107—21no—
166C4×D107—21no—
167C4×D107—42no—
168C4×D107—81no—
169Dic214—21no—
170Dic214—21no—
171C8×C107—21no—
172C8×C107—21no—
173C107⋊C8—21no—
174C107⋊C8C2×D412yes—
175C22×C107—22no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107C22×D411yescharacteristic
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—22no—
185C2×D107—21no—
186C2×D107—21no—
187C2×D107—21no—
188C2×D107—21no—
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—21no—
198C4×C107—41no—
199C4×C107—41no—
200C4×C107C22×D411yescharacteristic
201C4×C107H(5,𝔽2)11yescharacteristic
202Dic107—21no—
203Dic107—21no—
204Dic107—41no—
205Dic107—41no—
206C2×C107—42no—
207C2×C107—81no—
208C2×C107D4211yescharacteristic
209C2×C107#57511yescharacteristic
210C2×C107#57611yescharacteristic
211D107—41no—
212D107—41no—
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#468—1071no—
224#463—1071no—
225#463D10711yescharacteristic, complemented
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×D8C2×D10711yescharacteristic
247C2×M16—1071no—
248C2×M16—1071no—
249#124—1071no—
250#124C2×D10711yescharacteristic, complemented
251#123—2141no—
252#123—2141no—
253C22⋊Q8—1071no—
254C22⋊Q8—1071no—
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×D4C2×D10711yescharacteristic, complemented
264C2×C22⋊C4—2141no—
265C2×C22⋊C4—2141no—
266#107—1071no—
267C4⋊C8C2×D10711yescharacteristic, complemented
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—1071no—
286C2×D4—1071no—
287C2×D4—1071no—
288C2×D4—4281no—
289C2×D4—4281no—
290C2×D4—2141no—
291C2×D4—2141no—
292C2×D4—2141no—
293C2×D4—2141no—
294C2×D4—2142no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2142no—
298C2×D4C22×D10711yescharacteristic
299C2×D4C22×D10711yescharacteristic
300C22×C4—21no—
301C22×C4—1071no—
302C22×C4—1071no—
303C22×C4—2141no—
304C22×C4—2141no—
305C22×C4—2141no—
306Q16—4281no—
307SD16—22no—
308SD16—2142no—
309SD16—2142no—
310SD16—2142no—
311D8—22no—
312D8—1074no—
313D8—4281no—
314D8—2142no—
315M16—2142no—
316M16—2142no—
317C2×C8—1071no—
318C2×C8—1071no—
319C2×C8C22×D10711yescharacteristic
320C2×C8C22×D10711yescharacteristic
321C4⋊C4—1071no—
322C4⋊C4—1071no—
323C4⋊C4—2141no—
324C4⋊C4—2141no—
325C4⋊C4C22×D10711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1071no—
330C22⋊C4—1071no—
331C22⋊C4—2141no—
332C22⋊C4—2141no—
333C22⋊C4—2141no—
334C22⋊C4—2141no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C42C22×D10711yescharacteristic
339C23—21no—
340C23—21no—
341C23—8561no—
342C23—8561no—
343C23—1071no—
344C23—1071no—
345C23—4282no—
346C23—4282no—
347C23—4282no—
348C23—4282no—
349C23—2141no—
350C23—2141no—
351Q8—21no—
352Q8—21no—
353Q8—2141no—
354Q8—2141no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—8561no—
359D4—8561no—
360D4—1072no—
361D4—4281no—
362D4—4282no—
363D4—4282no—
364D4—4282no—
365D4—4282no—
366D4—2141no—
367D4—2141no—
368D4—2141no—
369D4D4×D10712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—8561no—
374C2×C4—1071no—
375C2×C4—1071no—
376C2×C4—4282no—
377C2×C4—4282no—
378C2×C4—2141no—
379C2×C4—2141no—
380C2×C4—2141no—
381C2×C4—2141no—
382C2×C4—2141no—
383C2×C4—2141no—
384C2×C4D4×D10711yescharacteristic
385C2×C4D4×D10711yescharacteristic
386C2×C4C23×D10711yescharacteristic
387C8—21no—
388C8—21no—
389C8—1072no—
390C8—2141no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—8564no—
398C22—8564no—
399C22—4281no—
400C22—4281no—
401C22—4281no—
402C22—4281no—
403C22—4281no—
404C22—2141no—
405C22—2141no—
406C22—2141no—
407C22—2141no—
408C22—2142no—
409C22—2141no—
410C22—2141no—
411C22C2×D4×D10711yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—4281no—
417C4—4281no—
418C4—2141no—
419C4—2141no—
420C4C2×D4×D10711yescharacteristic
421C4#1681215511yescharacteristic
422C2—42no—
423C2—81no—
424C2—8561no—
425C2—4281no—
426C2—4281no—
427C2—4282no—
428C2#2029326111yescharacteristic
429C2#2029342411yescharacteristic
430C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
hypercenter#463D107
hypocenterC107#3273
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812133C2semidirect
3. #20293272C2semidirect
4. #20293270C2semidirect
5. #20293009C2semidirect
6. #20292876C2semidirect
7. #20292872C2semidirect
8. #20292843C2semidirect
9. #20292792C2semidirect
10. #20292786C2semidirect
11. #20292762C2semidirect
12. #20292759C2semidirect
13. C4×#13133803C2semidirect
14. #20292704C2semidirect
15. #463×C107C2semidirect
18. C2×D4×D107C22semidirect
19. C2×D4×D107C22semidirect
22. C2×#13133808C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133806C22semidirect
25. C2×#13133805C22semidirect
29. C2×#13133801C22semidirect
30. C2×#13133801C22semidirect
31. #16812097C22semidirect
34. #16812089C22semidirect
35. #16812085C22semidirect
43. C4×D428C22semidirect
46. #16812042C22semidirect
47. #16812034C22semidirect
48. #16812031C22semidirect
49. #16812030C22semidirect
50. #16812029C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C4×C107⋊C8C22semidirect
54. C2×SD16×C107C22semidirect
56. #124×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
92. #13133803D4semidirect
216. C107#3273semidirect
225. #463D107semidirect
242. C2×SD16C2×D107semidirect
250. #124C2×D107semidirect
263. C4×D4C2×D107semidirect
267. C4⋊C8C2×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×#168121362
C2×#168121332
#202932722
#202932702
#202930092
#202928762
#202928722
#202928432
#202927922
#202927862
#202927622
#202927592
C4×#131338032
#202927042
#463×C1072
#3273107