Finite Group Databasebeta

← #34113625

Subgroups — #34113625

Order 20096.

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×#14578955C211yesmaximal, characteristic, complemented
2C2×#41×D157C211yesmaximal, characteristic, complemented
3#23757942C211yesmaximal, characteristic, complemented
4Q8×C22⋊D157C211yesmaximal, characteristic, complemented
5#23757935C211yesmaximal, characteristic, complemented
6#23757925C213yesmaximal, complemented
7#23757919C213yesmaximal, complemented
8#23757916C213yesmaximal, complemented
9#23757907C213yesmaximal, complemented
10#23757822C213yesmaximal, complemented
11#23757786C213yesmaximal, complemented
12#23757783C213yesmaximal, complemented
13C4×#14578955C211yesmaximal, characteristic, complemented
14C4×D4×D157C213yesmaximal, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578955—28no—
17C2×#14578955C2211yescharacteristic
18C2×#14578955C2213yescomplemented
19C2×#14578955C2211yescharacteristic, complemented
20C2×#14578955C2213yescomplemented
21C2×#14578954C2213yescomplemented
22C2×#14578953C2213yescomplemented
23#41×D157—28no—
24C2×Q8×D157C2211yescharacteristic, complemented
25C22×D628C2213yescomplemented
26C2×D4×D157C2213yes—
27C2×D4×D157C2213yescomplemented
28C22×C4×D157C2213yescomplemented
29C22×C4×D157C2213yescomplemented
30#19097769C2213yescomplemented
31#19097767C2213yescomplemented
32#19097759C2213yescomplemented
33#19097756C2216yescomplemented
34#19097753C2216yescomplemented
35#19097751C2213yescomplemented
36#19097750C2213yescomplemented
37#19097747C2213yescomplemented
38#19097746C2213yescomplemented
39#19097745C2213yescomplemented
40#19097744C2216yescomplemented
41#19097744C2216yescomplemented
42#19097742C2213yescomplemented
43#19097741C2216yescomplemented
44#19097739C2216yescomplemented
45Q8×Dic157C2211yescharacteristic, complemented
46C4×D628C2213yescomplemented
47C4×D628C2213yescomplemented
48D4×Dic157C2213yescomplemented
49#19097733C2213yescomplemented
50#19097733C2213yescomplemented
51C4×C22⋊D157C2213yes—
52C4×C22⋊D157C2213yescomplemented
53#19097731C2211yescharacteristic
54#19097731C2213yescomplemented
55#19097731C2216yescomplemented
56#19097729C2213yescomplemented
57C4⋊C4×D157C2213yescomplemented
58C22⋊C4×D157C2213yescomplemented
59C22⋊C4×D157C2213yescomplemented
60C42×D157C2213yescomplemented
61C2×#41×C157C2211yescharacteristic, complemented
62C22×Q8×C157C2211yescharacteristic, complemented
63#124×C157C2213yescomplemented
64C22⋊Q8×C157C2213yescomplemented
65#121×C157C2213yescomplemented
66C4×Q8×C157C2211yescharacteristic, complemented
67C4×D4×C157C2213yescomplemented
68C23×D157C2313yes—
69C23×D157C2313yes—
70#14578955—212no—
71#14578955—212no—
72#14578955—24no—
73#14578955D418yescomplemented
74#14578954—212no—
75#14578953—212no—
76C2×Dic314C2313yes—
77Q8×D157—24no—
78C2×D628—212no—
79C2×D628C2313yes—
80C2×D628C2316yes—
81C2×D628C2313yes—
82C2×D628C2313yes—
83C2×C22⋊D157C2313yes—
84C2×C22⋊D157C2316yes—
85C2×C22⋊D157C2316yes—
86C2×C22⋊D157C2311yescharacteristic
87D4×D157—212no—
88D4×D157—212no—
89C22×Dic157C2311yescharacteristic
90C22×Dic157C2313yes—
91C2×C4×D157—212no—
92C2×C4×D157—212no—
93C2×C4×D157C2313yes—
94C2×C4×D157C2313yes—
95C2×C4×D157C2313yes—
96C2×C4×D157C2311yescharacteristic
97C2×C4×D157C2313yes—
98C2×C4×D157C2313yes—
99C2×C4×D157C2313yes—
100C2×C4×D157C2313yes—
101C2×C4×D157C2313yes—
102C4⋊Dic157C2313yes—
103C4⋊Dic157C2313yes—
104#14578933C2311yescharacteristic
105#14578933C2313yes—
106#14578933C2316yes—
107C22⋊Dic157C2313yes—
108C22⋊Dic157C2313yes—
109#14578931C2311yescharacteristic
110#14578931C2313yes—
111#14578931C2313yes—
112#14578931C2313yes—
113#14578931C2316yes—
114#14578931C2316yes—
115C4×Dic157C2311yescharacteristic
116C4×Dic157C2313yes—
117C4×Dic157C2313yes—
118#41×C157—24no—
119C2×Q8×C157—24no—
120C2×Q8×C157C2313yes—
121C2×Q8×C157C2311yescharacteristic
122C2×D4×C157C2313yes—
123C2×D4×C157C2313yes—
124C22×C4×C157C2313yes—
125C22×C4×C157C2313yes—
126C4⋊C4×C157C2313yes—
127C4⋊C4×C157C2313yes—
128C22⋊C4×C157C2311yescharacteristic
129C22⋊C4×C157C2313yes—
130C22⋊C4×C157C2316yes—
131C42×C157C2313yes—
132C22×D157—26no—
133C22×D157—26no—
134C22×D157—26no—
135C22×D157—26no—
136C22×D157—43no—
137C22×D157—43no—
138C22×D157C2413yes—
139C22×D157C2413yes—
140C22×D157C2411yescharacteristic
141Dic314—26no—
142D628—212no—
143D628—26no—
144D628—26no—
145D628C2×D4112yescomplemented
146C22⋊D157—212no—
147C22⋊D157—26no—
148C22⋊D157—46no—
149C22⋊D157#4114yescomplemented
150C2×Dic157—22no—
151C2×Dic157—26no—
152C2×Dic157C2411yescharacteristic
153C2×Dic157C2413yes—
154C2×Dic157C2413yes—
155C2×Dic157C2411yescharacteristic
156C2×Dic157C2411yescharacteristic
157C4×D157—26no—
158C4×D157—26no—
159C4×D157—26no—
160C4×D157—26no—
161C4×D157—22no—
162C4×D157—26no—
163C4×D157—26no—
164C4×D157—43no—
165C4×D157C2×D4112yescomplemented
166C23×C157C2413yes—
167C23×C157C2411yescharacteristic
168Q8×C157—26no—
169Q8×C157C2×D414yescomplemented
170D4×C157—26no—
171D4×C157—26no—
172C2×C4×C157—26no—
173C2×C4×C157—26no—
174C2×C4×C157C2411yescharacteristic
175C2×C4×C157C2411yescharacteristic
176C2×C4×C157C2413yes—
177C2×C4×C157C2413yes—
178C2×C4×C157C2413yes—
179C22×C157—21no—
180C22×C157—21no—
181C22×C157—23no—
182C22×C157—23no—
183C22×C157—23no—
184C22×C157C2×#4112yescomplemented
185C22×C157C2511yescharacteristic
186C2×D157—21no—
187C2×D157—23no—
188C2×D157—23no—
189C2×D157—21no—
190C2×D157—23no—
191C2×D157—23no—
192C2×D157—23no—
193C2×D157—412no—
194C2×D157—412no—
195C2×D157C22×D416yes—
196C2×D157C2×#4112yescomplemented
197C4×C157—21no—
198C4×C157—23no—
199C4×C157—23no—
200C4×C157—21no—
201C4×C157C22×D416yes—
202Dic157—21no—
203Dic157—23no—
204Dic157—23no—
205Dic157C22×D412yes—
206Dic157C2×#4112yescomplemented
207C2×C157—22no—
208C2×C157—43no—
209C2×C157C23×D411yescharacteristic
210C2×C157C22×#4111yescharacteristic
211C2×C157#58511yescharacteristic
212D157—22no—
213D157—26no—
214D157—43no—
215C157D4×#4111yescharacteristic, complemented, Sylow
216D4×#41—1571nomaximal, Sylow
217C22×#41—1571no—
218C22×#41—1571no—
219#550—1571no—
220D4×Q8—1571no—
221#548—1571no—
222#548D15711yescharacteristic, complemented
223#547—1573no—
224#546—1573no—
225#546—1573no—
226D42—1573no—
227#532—1573no—
228#525—1573no—
229#525—1573no—
230C4×#41—1571no—
231C2×C4×D4—1573no—
232C2×#41—3148no—
233C2×#41—3148no—
234C2×#41—1573no—
235C2×#41—1573no—
236C2×#41—1571no—
237C2×#41—1571no—
238C2×#41—1573no—
239C2×#41—1573no—
240C2×#41C2×D15711yescharacteristic, complemented
241C22×Q8—1571no—
242C22×Q8C2×D15711yescharacteristic, complemented
243C22×D4—1573no—
244C22×D4—1573no—
245C22×D4—1573no—
246C23×C4—1573no—
247C23×C4—1573no—
248C4⋊Q8—1573no—
249#127—1573no—
250#124—1573no—
251#124C2×D15713yescomplemented
252#123—1576no—
253#123—1576no—
254C22⋊Q8—1573no—
255C22⋊Q8—1573no—
256C22⋊Q8—1573no—
257C22⋊Q8C2×D15713yescomplemented
258#121—1573no—
259#121—1573no—
260#121—1573no—
261#121—1576no—
262#121—1576no—
263#121C2×D15713yescomplemented
264#120—1576no—
265#120—1576no—
266C4×Q8—1571no—
267C4×Q8C2×D15711yescharacteristic, complemented
268C4×D4—1573no—
269C4×D4—1573no—
270C4×D4—1573no—
271C4×D4—1573no—
272C4×D4—1573no—
273C4×D4—1571no—
274C4×D4—1573no—
275C4×D4—1576no—
276C4×D4—1573no—
277C4×D4—1573no—
278C4×D4C2×D15713yescomplemented
279#117—1573no—
280C2×C4⋊C4—1573no—
281C2×C22⋊C4—1573no—
282C2×C22⋊C4—1573no—
283C2×C42—1573no—
284C24—1573no—
285C24—1573no—
286#41—24no—
287#41—31412no—
288#41—3144no—
289#41—31412no—
290#41—31412no—
291#41—31412no—
292#41—1578no—
293C2×Q8—24no—
294C2×Q8—3144no—
295C2×Q8—1573no—
296C2×Q8C22×D15713yes—
297C2×Q8C22×D15711yescharacteristic
298C2×D4—31412no—
299C2×D4—31412no—
300C2×D4—31412no—
301C2×D4—1573no—
302C2×D4—1576no—
303C2×D4—1576no—
304C2×D4—1571no—
305C2×D4—1573no—
306C2×D4—1576no—
307C2×D4—1573no—
308C2×D4—1573no—
309C2×D4C22×D15713yes—
310C2×D4C22×D15713yes—
311C22×C4—31412no—
312C22×C4—31412no—
313C22×C4—1571no—
314C22×C4—1573no—
315C22×C4—1573no—
316C22×C4—1573no—
317C22×C4—1573no—
318C22×C4—1571no—
319C22×C4—1573no—
320C22×C4—1573no—
321C22×C4—1573no—
322C22×C4—1573no—
323C22×C4—1573no—
324C22×C4C22×D15713yes—
325C22×C4C22×D15713yes—
326C4⋊C4—1571no—
327C4⋊C4—1573no—
328C4⋊C4—1573no—
329C4⋊C4—1573no—
330C4⋊C4—1576no—
331C4⋊C4C22×D15713yes—
332C4⋊C4C22×D15713yes—
333C22⋊C4—1573no—
334C22⋊C4—1573no—
335C22⋊C4—1571no—
336C22⋊C4—1573no—
337C22⋊C4—1573no—
338C22⋊C4—1573no—
339C22⋊C4—1576no—
340C22⋊C4—1576no—
341C22⋊C4C22×D15711yescharacteristic
342C22⋊C4C22×D15713yes—
343C22⋊C4C22×D15716yes—
344C42—1571no—
345C42—1573no—
346C42—1573no—
347C42C22×D15713yes—
348C23—3146no—
349C23—3146no—
350C23—3146no—
351C23—3146no—
352C23—6283no—
353C23—6283no—
354C23—1573no—
355C23—1571no—
356C23—1573no—
357C23C23×D15713yes—
358C23C23×D15711yescharacteristic
359Q8—26no—
360Q8—3146no—
361Q8D4×D15714yescomplemented
362D4—26no—
363D4—26no—
364D4—31412no—
365D4—3146no—
366D4—31412no—
367D4—3146no—
368D4—3146no—
369D4—6286no—
370D4—1574no—
371D4—15712no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—3146no—
375C2×C4—3146no—
376C2×C4—3142no—
377C2×C4—3146no—
378C2×C4—3146no—
379C2×C4—3142no—
380C2×C4—3146no—
381C2×C4—3146no—
382C2×C4—3146no—
383C2×C4—6283no—
384C2×C4—1573no—
385C2×C4—1571no—
386C2×C4—1573no—
387C2×C4—1571no—
388C2×C4—1571no—
389C2×C4—15712no—
390C2×C4C23×D15711yescharacteristic
391C2×C4C23×D15713yes—
392C2×C4C23×D15711yescharacteristic
393C2×C4C23×D15713yes—
394C2×C4C23×D15713yes—
395C22—21no—
396C22—21no—
397C22—23no—
398C22—23no—
399C22—23no—
400C22—3141no—
401C22—3143no—
402C22—3143no—
403C22—3141no—
404C22—3143no—
405C22—3143no—
406C22—3143no—
407C22—62812no—
408C22—62812no—
409C22—1576no—
410C22—1572no—
411C22#41×D15712yescomplemented
412C22C24×D15711yescharacteristic
413C4—21no—
414C4—23no—
415C4—23no—
416C4—21no—
417C4—3141no—
418C4—3143no—
419C4—3143no—
420C4—1572no—
421C4—1572no—
422C4C2×D4×D15716yes—
423C2—22no—
424C2—43no—
425C2—3142no—
426C2—3146no—
427C2—6283no—
428C2C22×D4×D15711yescharacteristic
429C2C2×#41×D15711yescharacteristic
430C2#2375811011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157D4×#41
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. C2×#41×D157C2semidirect
3. #23757942C2semidirect
4. Q8×C22⋊D157C2semidirect
5. #23757935C2semidirect
6. #23757925C2semidirect
7. #23757919C2semidirect
8. #23757916C2semidirect
9. #23757907C2semidirect
10. #23757822C2semidirect
11. #23757786C2semidirect
12. #23757783C2semidirect
13. C4×#14578955C2semidirect
14. C4×D4×D157C2semidirect
15. #548×C157C2semidirect
18. C2×#14578955C22semidirect
19. C2×#14578955C22semidirect
20. C2×#14578955C22semidirect
21. C2×#14578954C22semidirect
22. C2×#14578953C22semidirect
24. C2×Q8×D157C22semidirect
25. C22×D628C22semidirect
27. C2×D4×D157C22semidirect
28. C22×C4×D157C22semidirect
29. C22×C4×D157C22semidirect
30. #19097769C22semidirect
31. #19097767C22semidirect
32. #19097759C22semidirect
33. #19097756C22semidirect
34. #19097753C22semidirect
35. #19097751C22semidirect
36. #19097750C22semidirect
37. #19097747C22semidirect
38. #19097746C22semidirect
39. #19097745C22semidirect
40. #19097744C22semidirect
41. #19097744C22semidirect
42. #19097742C22semidirect
43. #19097741C22semidirect
44. #19097739C22semidirect
45. Q8×Dic157C22semidirect
46. C4×D628C22semidirect
47. C4×D628C22semidirect
48. D4×Dic157C22semidirect
49. #19097733C22semidirect
50. #19097733C22semidirect
52. C4×C22⋊D157C22semidirect
54. #19097731C22semidirect
55. #19097731C22semidirect
56. #19097729C22semidirect
57. C4⋊C4×D157C22semidirect
58. C22⋊C4×D157C22semidirect
59. C22⋊C4×D157C22semidirect
60. C42×D157C22semidirect
61. C2×#41×C157C22semidirect
62. C22×Q8×C157C22semidirect
63. #124×C157C22semidirect
64. C22⋊Q8×C157C22semidirect
65. #121×C157C22semidirect
66. C4×Q8×C157C22semidirect
67. C4×D4×C157C22semidirect
73. #14578955D4semidirect
145. D628C2×D4semidirect
149. C22⋊D157#41semidirect
165. C4×D157C2×D4semidirect
169. Q8×C157C2×D4semidirect
184. C22×C157C2×#41semidirect
196. C2×D157C2×#41semidirect
206. Dic157C2×#41semidirect
215. C157D4×#41semidirect
222. #548D157semidirect
240. C2×#41C2×D157semidirect
242. C22×Q8C2×D157semidirect
251. #124C2×D157semidirect
257. C22⋊Q8C2×D157semidirect
263. #121C2×D157semidirect
267. C4×Q8C2×D157semidirect
278. C4×D4C2×D157semidirect
361. Q8D4×D157semidirect
411. C22#41×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#41×D1572
#237579422
Q8×C22⋊D1572
#237579352
#237579252
#237579192
#237579162
#237579072
#237578222
#237577862
#237577832
C4×#145789552
C4×D4×D1572
#548×C1572
D4×#41157