Finite Group Databasebeta

← #28349950

Subgroups — #28349950

Order 13696.

1648 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
1C22×#13133819C211yesmaximal, characteristic, complemented
2C2×#41×D107C211yesmaximal, characteristic, complemented
3#20293296C211yesmaximal, characteristic, complemented
4Q8×C22⋊D107C211yesmaximal, characteristic, complemented
5#20293289C211yesmaximal, characteristic, complemented
6#20293279C213yesmaximal, complemented
7#20293273C213yesmaximal, complemented
8#20293270C213yesmaximal, complemented
9#20293261C213yesmaximal, complemented
10#20293176C213yesmaximal, complemented
11#20293140C213yesmaximal, complemented
12#20293137C213yesmaximal, complemented
13C4×#13133819C211yesmaximal, characteristic, complemented
14C4×D4×D107C213yesmaximal, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C2×#13133819—28no—
17C2×#13133819C2211yescharacteristic
18C2×#13133819C2213yescomplemented
19C2×#13133819C2213yescomplemented
20C2×#13133819C2211yescharacteristic, complemented
21C2×#13133818C2213yescomplemented
22C2×#13133817C2213yescomplemented
23#41×D107—28no—
24C2×Q8×D107C2211yescharacteristic, complemented
25C22×D428C2213yescomplemented
26C2×D4×D107C2213yes—
27C2×D4×D107C2213yescomplemented
28C22×C4×D107C2213yescomplemented
29C22×C4×D107C2213yescomplemented
30#16812107C2213yescomplemented
31#16812105C2213yescomplemented
32#16812097C2213yescomplemented
33#16812094C2216yescomplemented
34#16812091C2216yescomplemented
35#16812089C2213yescomplemented
36#16812088C2213yescomplemented
37#16812085C2213yescomplemented
38#16812084C2213yescomplemented
39#16812083C2213yescomplemented
40#16812082C2216yescomplemented
41#16812082C2216yescomplemented
42#16812080C2213yescomplemented
43#16812079C2216yescomplemented
44#16812077C2216yescomplemented
45Q8×Dic107C2211yescharacteristic, complemented
46C4×D428C2213yescomplemented
47C4×D428C2213yescomplemented
48D4×Dic107C2213yescomplemented
49#16812071C2213yescomplemented
50#16812071C2213yescomplemented
51C4×C22⋊D107C2213yes—
52C4×C22⋊D107C2213yescomplemented
53(C22⋊D107)⋊C4C2211yescharacteristic
54(C22⋊D107)⋊C4C2216yescomplemented
55(C22⋊D107)⋊C4C2213yescomplemented
56#16812067C2213yescomplemented
57C4⋊C4×D107C2213yescomplemented
58C22⋊C4×D107C2213yescomplemented
59C22⋊C4×D107C2213yescomplemented
60C42×D107C2213yescomplemented
61C2×#41×C107C2211yescharacteristic, complemented
62C22×Q8×C107C2211yescharacteristic, complemented
63#124×C107C2213yescomplemented
64C22⋊Q8×C107C2213yescomplemented
65#121×C107C2213yescomplemented
66C4×Q8×C107C2211yescharacteristic, complemented
67C4×D4×C107C2213yescomplemented
68C23×D107C2313yes—
69C23×D107C2313yes—
70#13133819—24no—
71#13133819—212no—
72#13133819—212no—
73#13133819D418yescomplemented
74#13133818—212no—
75#13133817—212no—
76C2×Dic214C2313yes—
77Q8×D107—24no—
78C2×D428—212no—
79C2×D428C2313yes—
80C2×D428C2316yes—
81C2×D428C2313yes—
82C2×D428C2313yes—
83C2×C22⋊D107C2313yes—
84C2×C22⋊D107C2316yes—
85C2×C22⋊D107C2316yes—
86C2×C22⋊D107C2311yescharacteristic
87D4×D107—212no—
88D4×D107—212no—
89C22×Dic107C2311yescharacteristic
90C22×Dic107C2313yes—
91C2×C4×D107—212no—
92C2×C4×D107—212no—
93C2×C4×D107C2313yes—
94C2×C4×D107C2313yes—
95C2×C4×D107C2313yes—
96C2×C4×D107C2311yescharacteristic
97C2×C4×D107C2313yes—
98C2×C4×D107C2313yes—
99C2×C4×D107C2313yes—
100C2×C4×D107C2313yes—
101C2×C4×D107C2313yes—
102C4⋊Dic107C2313yes—
103C4⋊Dic107C2313yes—
104Dic107⋊C4C2311yescharacteristic
105Dic107⋊C4C2313yes—
106Dic107⋊C4C2316yes—
107C22⋊Dic107C2313yes—
108C22⋊Dic107C2313yes—
109(C2×D107)⋊C4C2311yescharacteristic
110(C2×D107)⋊C4C2313yes—
111(C2×D107)⋊C4C2313yes—
112(C2×D107)⋊C4C2313yes—
113(C2×D107)⋊C4C2316yes—
114(C2×D107)⋊C4C2316yes—
115C4×Dic107C2311yescharacteristic
116C4×Dic107C2313yes—
117C4×Dic107C2313yes—
118#41×C107—24no—
119C2×Q8×C107—24no—
120C2×Q8×C107C2313yes—
121C2×Q8×C107C2311yescharacteristic
122C2×D4×C107C2313yes—
123C2×D4×C107C2313yes—
124C22×C4×C107C2313yes—
125C22×C4×C107C2313yes—
126C4⋊C4×C107C2313yes—
127C4⋊C4×C107C2313yes—
128C22⋊C4×C107C2311yescharacteristic
129C22⋊C4×C107C2313yes—
130C22⋊C4×C107C2316yes—
131C42×C107C2313yes—
132C23×C107C2413yes—
133C23×C107C2411yescharacteristic
134C22×D107—26no—
135C22×D107—26no—
136C22×D107—26no—
137C22×D107—26no—
138C22×D107—43no—
139C22×D107—43no—
140C22×D107C2413yes—
141C22×D107C2413yes—
142C22×D107C2411yescharacteristic
143Q8×C107—26no—
144Q8×C107C2×D414yescomplemented
145D4×C107—26no—
146D4×C107—26no—
147C2×C4×C107—26no—
148C2×C4×C107—26no—
149C2×C4×C107C2411yescharacteristic
150C2×C4×C107C2413yes—
151C2×C4×C107C2413yes—
152C2×C4×C107C2411yescharacteristic
153C2×C4×C107C2413yes—
154C22⋊D107—212no—
155C22⋊D107—26no—
156C22⋊D107—46no—
157C22⋊D107#4114yescomplemented
158C2×Dic107—22no—
159C2×Dic107—26no—
160C2×Dic107C2413yes—
161C2×Dic107C2413yes—
162C2×Dic107C2411yescharacteristic
163C2×Dic107C2411yescharacteristic
164C2×Dic107C2411yescharacteristic
165D428—212no—
166D428—26no—
167D428—26no—
168D428C2×D4112yescomplemented
169C4×D107—26no—
170C4×D107—26no—
171C4×D107—26no—
172C4×D107—26no—
173C4×D107—22no—
174C4×D107—26no—
175C4×D107—26no—
176C4×D107—43no—
177C4×D107C2×D4112yescomplemented
178Dic214—26no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—23no—
182C22×C107—23no—
183C22×C107—23no—
184C22×C107C2×#4112yescomplemented
185C22×C107C2511yescharacteristic
186C2×D107—21no—
187C2×D107—23no—
188C2×D107—23no—
189C2×D107—21no—
190C2×D107—23no—
191C2×D107—23no—
192C2×D107—23no—
193C2×D107—412no—
194C2×D107—412no—
195C2×D107C22×D416yes—
196C2×D107C2×#4112yescomplemented
197C4×C107—21no—
198C4×C107—23no—
199C4×C107—23no—
200C4×C107—21no—
201C4×C107C22×D416yes—
202Dic107—23no—
203Dic107—21no—
204Dic107—23no—
205Dic107C22×D412yes—
206Dic107C2×#4112yescomplemented
207C2×C107—22no—
208C2×C107—43no—
209C2×C107C23×D411yescharacteristic
210C2×C107C22×#4111yescharacteristic
211C2×C107#58511yescharacteristic
212D107—22no—
213D107—26no—
214D107—43no—
215D4×#41—1071nomaximal, Sylow
216C107D4×#4111yescharacteristic, complemented, Sylow
217C22×#41—1071no—
218C22×#41—1071no—
219#550—1071no—
220D4×Q8—1071no—
221#548—1071no—
222#548D10711yescharacteristic, complemented
223#547—1073no—
224#546—1073no—
225#546—1073no—
226D42—1073no—
227#532—1073no—
228#525—1073no—
229#525—1073no—
230C4×#41—1071no—
231C2×C4×D4—1073no—
232C2×#41—1071no—
233C2×#41—1073no—
234C2×#41—1073no—
235C2×#41—1071no—
236C2×#41—1073no—
237C2×#41—1073no—
238C2×#41—2148no—
239C2×#41—2148no—
240C2×#41C2×D10711yescharacteristic, complemented
241C22×Q8—1071no—
242C22×Q8C2×D10711yescharacteristic, complemented
243C22×D4—1073no—
244C22×D4—1073no—
245C22×D4—1073no—
246C23×C4—1073no—
247C23×C4—1073no—
248C4⋊Q8—1073no—
249#127—1073no—
250#124—1073no—
251#124C2×D10713yescomplemented
252#123—1076no—
253#123—1076no—
254C22⋊Q8—1073no—
255C22⋊Q8—1073no—
256C22⋊Q8—1073no—
257C22⋊Q8C2×D10713yescomplemented
258#121—1073no—
259#121—1073no—
260#121—1073no—
261#121—1076no—
262#121—1076no—
263#121C2×D10713yescomplemented
264#120—1076no—
265#120—1076no—
266C4×Q8—1071no—
267C4×Q8C2×D10711yescharacteristic, complemented
268C4×D4—1073no—
269C4×D4—1073no—
270C4×D4—1073no—
271C4×D4—1073no—
272C4×D4—1073no—
273C4×D4—1071no—
274C4×D4—1073no—
275C4×D4—1076no—
276C4×D4—1073no—
277C4×D4—1073no—
278C4×D4C2×D10713yescomplemented
279#117—1073no—
280C2×C4⋊C4—1073no—
281C2×C22⋊C4—1073no—
282C2×C22⋊C4—1073no—
283C2×C42—1073no—
284C24—1073no—
285C24—1073no—
286#41—24no—
287#41—1078no—
288#41—21412no—
289#41—21412no—
290#41—21412no—
291#41—21412no—
292#41—2144no—
293C2×Q8—24no—
294C2×Q8—1073no—
295C2×Q8—2144no—
296C2×Q8C22×D10713yes—
297C2×Q8C22×D10711yescharacteristic
298C2×D4—1071no—
299C2×D4—1076no—
300C2×D4—1073no—
301C2×D4—1076no—
302C2×D4—1073no—
303C2×D4—1073no—
304C2×D4—1076no—
305C2×D4—1073no—
306C2×D4—21412no—
307C2×D4—21412no—
308C2×D4—21412no—
309C2×D4C22×D10713yes—
310C2×D4C22×D10713yes—
311C22×C4—1073no—
312C22×C4—1071no—
313C22×C4—1073no—
314C22×C4—1073no—
315C22×C4—1073no—
316C22×C4—1073no—
317C22×C4—1073no—
318C22×C4—1071no—
319C22×C4—1073no—
320C22×C4—1073no—
321C22×C4—1073no—
322C22×C4—21412no—
323C22×C4—21412no—
324C22×C4C22×D10713yes—
325C22×C4C22×D10713yes—
326C4⋊C4—1071no—
327C4⋊C4—1073no—
328C4⋊C4—1073no—
329C4⋊C4—1073no—
330C4⋊C4—1076no—
331C4⋊C4C22×D10713yes—
332C4⋊C4C22×D10713yes—
333C22⋊C4—1073no—
334C22⋊C4—1073no—
335C22⋊C4—1071no—
336C22⋊C4—1073no—
337C22⋊C4—1073no—
338C22⋊C4—1073no—
339C22⋊C4—1076no—
340C22⋊C4—1076no—
341C22⋊C4C22×D10711yescharacteristic
342C22⋊C4C22×D10713yes—
343C22⋊C4C22×D10716yes—
344C42—1071no—
345C42—1073no—
346C42—1073no—
347C42C22×D10713yes—
348C23—1071no—
349C23—1073no—
350C23—1073no—
351C23—4283no—
352C23—4283no—
353C23—2146no—
354C23—2146no—
355C23—2146no—
356C23—2146no—
357C23C23×D10713yes—
358C23C23×D10711yescharacteristic
359Q8—26no—
360Q8—2146no—
361Q8D4×D10714yescomplemented
362D4—26no—
363D4—26no—
364D4—1074no—
365D4—10712no—
366D4—4286no—
367D4—21412no—
368D4—2146no—
369D4—2146no—
370D4—21412no—
371D4—2146no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—1071no—
375C2×C4—1071no—
376C2×C4—10712no—
377C2×C4—1073no—
378C2×C4—1071no—
379C2×C4—1073no—
380C2×C4—4283no—
381C2×C4—2142no—
382C2×C4—2146no—
383C2×C4—2146no—
384C2×C4—2146no—
385C2×C4—2146no—
386C2×C4—2142no—
387C2×C4—2146no—
388C2×C4—2146no—
389C2×C4—2146no—
390C2×C4C23×D10711yescharacteristic
391C2×C4C23×D10713yes—
392C2×C4C23×D10711yescharacteristic
393C2×C4C23×D10713yes—
394C2×C4C23×D10713yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—1076no—
401C22—1072no—
402C22—42812no—
403C22—42812no—
404C22—2143no—
405C22—2143no—
406C22—2143no—
407C22—2141no—
408C22—2143no—
409C22—2143no—
410C22—2141no—
411C22#41×D10712yescomplemented
412C22C24×D10711yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—1072no—
418C4—1072no—
419C4—2141no—
420C4—2143no—
421C4—2143no—
422C4C2×D4×D10716yes—
423C2—22no—
424C2—43no—
425C2—4283no—
426C2—2142no—
427C2—2146no—
428C2C22×D4×D10711yescharacteristic
429C2C2×#41×D10711yescharacteristic
430C2#2029346411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#548×C107C2
hypercenter#548D107
hypocenterC107D4×#41
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#13133819C2semidirect
2. C2×#41×D107C2semidirect
3. #20293296C2semidirect
4. Q8×C22⋊D107C2semidirect
5. #20293289C2semidirect
6. #20293279C2semidirect
7. #20293273C2semidirect
8. #20293270C2semidirect
9. #20293261C2semidirect
10. #20293176C2semidirect
11. #20293140C2semidirect
12. #20293137C2semidirect
13. C4×#13133819C2semidirect
14. C4×D4×D107C2semidirect
15. #548×C107C2semidirect
18. C2×#13133819C22semidirect
19. C2×#13133819C22semidirect
20. C2×#13133819C22semidirect
21. C2×#13133818C22semidirect
22. C2×#13133817C22semidirect
24. C2×Q8×D107C22semidirect
25. C22×D428C22semidirect
27. C2×D4×D107C22semidirect
28. C22×C4×D107C22semidirect
29. C22×C4×D107C22semidirect
30. #16812107C22semidirect
31. #16812105C22semidirect
32. #16812097C22semidirect
33. #16812094C22semidirect
34. #16812091C22semidirect
35. #16812089C22semidirect
36. #16812088C22semidirect
37. #16812085C22semidirect
38. #16812084C22semidirect
39. #16812083C22semidirect
40. #16812082C22semidirect
41. #16812082C22semidirect
42. #16812080C22semidirect
43. #16812079C22semidirect
44. #16812077C22semidirect
45. Q8×Dic107C22semidirect
46. C4×D428C22semidirect
47. C4×D428C22semidirect
48. D4×Dic107C22semidirect
49. #16812071C22semidirect
50. #16812071C22semidirect
52. C4×C22⋊D107C22semidirect
54. (C22⋊D107)⋊C4C22semidirect
55. (C22⋊D107)⋊C4C22semidirect
56. #16812067C22semidirect
57. C4⋊C4×D107C22semidirect
58. C22⋊C4×D107C22semidirect
59. C22⋊C4×D107C22semidirect
60. C42×D107C22semidirect
61. C2×#41×C107C22semidirect
62. C22×Q8×C107C22semidirect
63. #124×C107C22semidirect
64. C22⋊Q8×C107C22semidirect
65. #121×C107C22semidirect
66. C4×Q8×C107C22semidirect
67. C4×D4×C107C22semidirect
73. #13133819D4semidirect
144. Q8×C107C2×D4semidirect
157. C22⋊D107#41semidirect
168. D428C2×D4semidirect
177. C4×D107C2×D4semidirect
184. C22×C107C2×#41semidirect
196. C2×D107C2×#41semidirect
206. Dic107C2×#41semidirect
216. C107D4×#41semidirect
222. #548D107semidirect
240. C2×#41C2×D107semidirect
242. C22×Q8C2×D107semidirect
251. #124C2×D107semidirect
257. C22⋊Q8C2×D107semidirect
263. #121C2×D107semidirect
267. C4×Q8C2×D107semidirect
278. C4×D4C2×D107semidirect
361. Q8D4×D107semidirect
411. C22#41×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338192
C2×#41×D1072
#202932962
Q8×C22⋊D1072
#202932892
#202932792
#202932732
#202932702
#202932612
#202931762
#202931402
#202931372
C4×#131338192
C4×D4×D1072
#548×C1072
D4×#41107