Finite Group Databasebeta

← #34111848

Subgroups — #34111848

Order 20096.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097793C211yesmaximal, characteristic, complemented
2C22×#14578943C211yesmaximal, characteristic, complemented
3#23757932C211yesmaximal, characteristic, complemented
4#23757919C211yesmaximal, characteristic, complemented
5#23757657C211yesmaximal, characteristic, complemented
6#23757502C211yesmaximal, characteristic, complemented
7#23757502C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757489C211yesmaximal, characteristic, complemented
10#23757437C211yesmaximal, characteristic, complemented
11#23757431C211yesmaximal, characteristic, complemented
12#23757420C211yesmaximal, characteristic, complemented
13#23757409C211yesmaximal, characteristic, complemented
14C4×#14578943C211yesmaximal, characteristic, complemented
15C8×D4×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic, complemented
18C22×Dic314C2211yescharacteristic
19C22×D628C2211yescharacteristic
20#19097793—24no—
21C2×Dic628C2211yescharacteristic, complemented
22C2×Dic628C2211yescharacteristic, complemented
23C2×#14578943—24no—
24C2×#14578943C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578943C2211yescharacteristic
27C2×#14578943C2211yescharacteristic
28C2×#14578943C2211yescharacteristic, complemented
29C2×D1256C2211yescharacteristic
30C2×D1256C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097759C2211yescharacteristic, complemented
33#19097757—21no—
34#19097756—21no—
35#19097750C2211yescharacteristic
36#19097750C2211yescharacteristic, complemented
37#19097749—21no—
38#19097745C2211yescharacteristic
39#19097745C2211yescharacteristic
40#19097744—21no—
41#19097743—21no—
42#19097739—21no—
43C4×Dic314C2211yescharacteristic, complemented
44C4×D628C2211yescharacteristic
45#19097731—21no—
46C22⋊C4×D157—21no—
47#19097703C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097696C2211yescharacteristic, complemented
50#19097696C2211yescharacteristic, complemented
51#19097693C2211yescharacteristic
52#19097693C2211yescharacteristic
53#19097693C2211yescharacteristic, complemented
54C22×C8×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic, complemented
56C4×D4×C157C2211yescharacteristic
57C4⋊C8×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic
59C22⋊C8×C157C2211yescharacteristic, complemented
60C4×C8×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578954—22no—
63#14578954—22no—
64#14578954—41no—
65#14578954—41no—
66C2×Dic314—22no—
67C2×Dic314—21no—
68C2×Dic314—41no—
69C2×Dic314C2311yescharacteristic
70C2×Dic314C2311yescharacteristic
71C2×Dic314C2311yescharacteristic
72C2×D628—22no—
73C2×D628—21no—
74C2×D628—41no—
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×D628C2311yescharacteristic
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81C2×C22⋊D157—21no—
82C22×Dic157—21no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157—21no—
86Dic628—22no—
87Dic628—41no—
88#14578943—22no—
89#14578943—22no—
90#14578943—22no—
91#14578943—22no—
92#14578943D414yescomplemented
93D1256—22no—
94D1256—41no—
95C4⋊Dic157C2311yescharacteristic
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
98#14578933—21no—
99#14578933—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109C4×Dic157—21no—
110C2×D4×C157D411yescharacteristic
111C22×C4×C157C2311yescharacteristic
112C22×C4×C157C2311yescharacteristic
113C2×C8×C157—22no—
114C2×C8×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
119C4⋊C4×C157D411yescharacteristic
120C22⋊C4×C157D411yescharacteristic
121C22⋊C4×C157D411yescharacteristic
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—81no—
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—21no—
133Dic314—21no—
134Dic314—42no—
135Dic314C2×D412yes—
136D628—21no—
137D628—21no—
138D628—21no—
139D628—21no—
140D628—21no—
141D628—42no—
142D628C2×D412yes—
143C22⋊D157—42no—
144C22⋊D157—42no—
145C22⋊D157—42no—
146C22⋊D157—81no—
147C2×Dic157—21no—
148C2×Dic157—21no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—21no—
152C2×Dic157—42no—
153C4×D157—42no—
154C4×D157—42no—
155C4×D157—81no—
156C23×C157C2×D411yescharacteristic
157C23×C157C2×D411yescharacteristic
158D4×C157—22no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2411yescharacteristic
168C8×C157—21no—
169C8×C157—21no—
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157#13512yescomplemented
178C22×C157C22×D411yescharacteristic
179C2×D157—22no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—21no—
192C4×C157—21no—
193C4×C157C22×D411yescharacteristic
194C4×C157H(5,𝔽2)11yescharacteristic
195Dic157—22no—
196Dic157—41no—
197Dic157—41no—
198Dic157—41no—
199Dic157—41no—
200C2×C157—22no—
201C2×C157—41no—
202C2×C157D4211yescharacteristic
203C2×C157C2×#13511yescharacteristic
204C2×C157#57711yescharacteristic
205D157—42no—
206D157—81no—
207D157—81no—
208C157#328211yescharacteristic, complemented, Sylow
209#3282—1571nomaximal, Sylow
210C2×#135—1571no—
211C22×SD16—1571no—
212#548—1571no—
213#546—1571no—
214#495—1571no—
215#465—1571no—
216#465—1571no—
217#464—1571no—
218#463—1571no—
219#452—1571no—
220#451—1571no—
221#449—1571no—
222#447—1571no—
223C4×SD16—1571no—
224C8×D4D15711yescharacteristic, complemented
225C2×#41—1571no—
226C2×#41—1571no—
227C22×Q8—1571no—
228C22×D4—1571no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16—1571no—
232C2×SD16—3144no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16—1571no—
238C2×D8—1571no—
239C2×D8—1571no—
240C22×C8C2×D15711yescharacteristic
241C22×C8C2×D15711yescharacteristic, complemented
242#124—3141no—
243#124—1571no—
244#124—1571no—
245#123—3141no—
246C22⋊Q8—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8—1571no—
249#121—3141no—
250#121—3141no—
251#121—1571no—
252#121—1571no—
253#120—3141no—
254C4×Q8—1571no—
255C4×D4—3141no—
256C4×D4—1571no—
257C4×D4C2×D15711yescharacteristic
258C2×C22⋊C4—3141no—
259#106—1571no—
260C4⋊C8C2×D15711yescharacteristic, complemented
261#103—1571no—
262#103—1571no—
263#103—1571no—
264#102—1571no—
265#102—1571no—
266#102—1571no—
267C22⋊C8C2×D15711yescharacteristic
268C22⋊C8C2×D15711yescharacteristic, complemented
269C4×C8C2×D15711yescharacteristic, complemented
270C24—3141no—
271#41—3142no—
272#41—3142no—
273#41—6281no—
274#41—6281no—
275C2×Q8—3142no—
276C2×Q8—3141no—
277C2×Q8—6281no—
278C2×Q8—1571no—
279C2×Q8—1571no—
280C2×Q8—1571no—
281C2×D4—3142no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—6281no—
288C2×D4—1571no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4D62811yescharacteristic
292C22×C4—3141no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4C22×D15711yescharacteristic
297C22×C4C22×D15711yescharacteristic
298Q16—3142no—
299Q16—6281no—
300SD16—3142no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—1574no—
305D8—3142no—
306D8—6281no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8C22×D15711yescharacteristic
310C2×C8C22×D15711yescharacteristic
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—3141no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—1571no—
317C4⋊C4—1571no—
318C4⋊C4D62811yescharacteristic
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4D62811yescharacteristic
329C22⋊C4D62811yescharacteristic
330C42—3141no—
331C42C22×D15711yescharacteristic
332C23—3141no—
333C23—3141no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—12561no—
338C23C2×D62811yescharacteristic
339C23C2×D62811yescharacteristic
340Q8—3141no—
341Q8—3141no—
342Q8—3141no—
343Q8—3141no—
344Q8—3141no—
345Q8—6282no—
346Q8—1572no—
347D4—22no—
348D4—3141no—
349D4—3141no—
350D4—3141no—
351D4—3141no—
352D4—3141no—
353D4—6282no—
354D4—6282no—
355D4—6282no—
356D4—6282no—
357D4—1572no—
358D4—12561no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—12561no—
372C2×C4C2×D62811yescharacteristic
373C2×C4C2×D62811yescharacteristic
374C2×C4C2×D62811yescharacteristic
375C2×C4C2×D62811yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D15712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—3142no—
387C22—6281no—
388C22—6281no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22—12564no—
396C22#1909779312yescomplemented
397C22C22×D62811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—3142no—
403C4—6281no—
404C4—6281no—
405C4—6281no—
406C4—6281no—
407C4C2×D4×D15711yescharacteristic
408C4#1909781711yescharacteristic
409C2—22no—
410C2—41no—
411C2—6282no—
412C2—12561no—
413C2—12561no—
414C2D4×D62811yescharacteristic
415C2C2×#1909779311yescharacteristic
416C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
hypercenterC8×D4D157
hypocenterC157#3282
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19097793C2semidirect
2. C22×#14578943C2semidirect
3. #23757932C2semidirect
4. #23757919C2semidirect
5. #23757657C2semidirect
6. #23757502C2semidirect
7. #23757502C2semidirect
8. #23757496C2semidirect
9. #23757489C2semidirect
10. #23757437C2semidirect
11. #23757431C2semidirect
12. #23757420C2semidirect
13. #23757409C2semidirect
14. C4×#14578943C2semidirect
15. C8×D4×C157C2semidirect
17. C2×#14578954C22semidirect
21. C2×Dic628C22semidirect
22. C2×Dic628C22semidirect
28. C2×#14578943C22semidirect
32. #19097759C22semidirect
36. #19097750C22semidirect
43. C4×Dic314C22semidirect
47. #19097703C22semidirect
48. #19097696C22semidirect
49. #19097696C22semidirect
50. #19097696C22semidirect
53. #19097693C22semidirect
55. C22×C8×C157C22semidirect
57. C4⋊C8×C157C22semidirect
59. C22⋊C8×C157C22semidirect
60. C4×C8×C157C22semidirect
92. #14578943D4semidirect
177. C22×C157#135semidirect
208. C157#3282semidirect
224. C8×D4D157semidirect
241. C22×C8C2×D157semidirect
260. C4⋊C8C2×D157semidirect
268. C22⋊C8C2×D157semidirect
269. C4×C8C2×D157semidirect
396. C22#19097793semidirect

Maximal subgroups

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

subgroupindex
C2×#190977932
C22×#145789432
#237579322
#237579192
#237576572
#237575022
#237574962
#237574892
#237574372
#237574312
#237574202
#237574092
C4×#145789432
C8×D4×C1572
#3282157