Finite Group Databasebeta

← #34111663

Subgroups — #34111663

Order 20096.

500 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×#19097795C211yesmaximal, characteristic, complemented
2C2×#19097790C211yesmaximal, characteristic, complemented
3#23757927C211yesmaximal, characteristic, complemented
4#23757918C211yesmaximal, characteristic, complemented
5#23757667C211yesmaximal, characteristic, complemented
6#23757532C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757478C211yesmaximal, characteristic, complemented
10#23757436C211yesmaximal, characteristic, complemented
11#23757436C211yesmaximal, characteristic, complemented
12#23757425C211yesmaximal, characteristic, complemented
13#23757419C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#442×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578953C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic
20#19097795—24no—
21#19097790—24no—
22C2×Dic628C2211yescharacteristic, complemented
23C2×Dic628C2211yescharacteristic, complemented
24C2×#14578943C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578942C2211yescharacteristic
28C2×#14578942C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic
30C2×#14578937C2211yescharacteristic
31C2×C8×D157C2211yescharacteristic, complemented
32#19097771C2211yescharacteristic, complemented
33#19097759C2211yescharacteristic
34#19097755—21no—
35#19097755—21no—
36#19097755—21no—
37#19097751C2211yescharacteristic
38#19097751C2211yescharacteristic
39#19097751C2211yescharacteristic, complemented
40#19097746C2211yescharacteristic
41#19097745—21no—
42#19097745—21no—
43#19097741—21no—
44C4×D628C2211yescharacteristic
45C2×C4⋊Dic157—21no—
46C2×#14578931—21no—
47C4⋊#11929434C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097694C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097683C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic
55C4×D4×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic, complemented
57#106×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic, complemented
60#97×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578953—22no—
63#14578953—22no—
64#14578953—22no—
65#14578953—41no—
66#14578953—41no—
67#14578953—41no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70C2×Dic314C2311yescharacteristic
71C2×D628—21no—
72C2×D628D411yescharacteristic
73C2×D628C2311yescharacteristic
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78D4×D157—22no—
79D4×D157—41no—
80C22×Dic157—21no—
81C22×Dic157—21no—
82C22×Dic157—21no—
83C2×C4×D157C2311yescharacteristic
84C2×C4×D157C2311yescharacteristic
85Dic628—22no—
86Dic628—41no—
87#14578943—22no—
88#14578943—41no—
89#14578942—22no—
90#14578942—22no—
91#14578942—22no—
92#14578939—22no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C8×D157—22no—
97C4⋊Dic157—21no—
98C4⋊Dic157—21no—
99C4⋊Dic157—21no—
100C4⋊Dic157—21no—
101C4⋊Dic157—21no—
102C4⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931D411yescharacteristic
110#14578931D411yescharacteristic
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157—21no—
114C22×C4×C157—21no—
115D8×C157D414yescomplemented
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157C2311yescharacteristic
119C4⋊C4×C157C2311yescharacteristic
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157—81no—
127C22×D157C2×D411yescharacteristic
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—21no—
133Dic314—21no—
134Dic314—21no—
135D628—21no—
136D628—21no—
137D628—41no—
138D628—81no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C22⋊D157—42no—
142C22⋊D157—42no—
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—42no—
147C2×Dic157—42no—
148C2×Dic157—42no—
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—21no—
154C4×D157—21no—
155#11929434—21no—
156#11929434—21no—
157C23×C157—21no—
158C23×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157C2×D412yes—
162D4×C157C2×D412yes—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—42no—
166C2×C4×C157—81no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—22no—
173C22×C157—22no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157C22×D411yescharacteristic
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—21no—
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—41no—
192C4×C157—41no—
193C4×C157C22×D411yescharacteristic
194C4×C157H(5,𝔽2)11yescharacteristic
195Dic157—21no—
196Dic157—21no—
197Dic157—41no—
198Dic157—41no—
199Dic157—41no—
200C2×C157—42no—
201C2×C157—42no—
202C2×C157D4211yescharacteristic
203C2×C157#57511yescharacteristic
204C2×C157#57711yescharacteristic
205D157—41no—
206D157—41no—
207D157—81no—
208C157#326711yescharacteristic, complemented, Sylow
209#3267—1571nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1571no—
211C2×#135—1571no—
212#547—1571no—
213#546—1571no—
214#497—1571no—
215#470—1571no—
216#465—1571no—
217#463—1571no—
218#461—1571no—
219#452—1571no—
220#452—1571no—
221#450—1571no—
222#449—1571no—
223#442D15711yescharacteristic, complemented
224#435—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C2×#41—1571no—
228C22×D4—1571no—
229AGL(1,ℤ/8)—3144no—
230#135—3144no—
231C2×Q16—1571no—
232C2×Q16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16—1571no—
238C2×D8—1571no—
239C2×D8C2×D15711yescharacteristic
240C2×M16—1571no—
241C22×C8—1571no—
242C4⋊Q8—1571no—
243#124—1571no—
244#123—3141no—
245#123—3141no—
246#123—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8—1571no—
249C22⋊Q8—1571no—
250#121—3141no—
251#121—3141no—
252#121—1571no—
253#120—3141no—
254C4×D4—1571no—
255C4×D4C2×D15711yescharacteristic
256C4×D4C2×D15711yescharacteristic, complemented
257C2×C4⋊C4—3141no—
258C2×C22⋊C4—3141no—
259#106C2×D15711yescharacteristic, complemented
260C4⋊C8—1571no—
261#103—1571no—
262#103—1571no—
263#103—1571no—
264#102—1571no—
265#102C2×D15711yescharacteristic
266#102C2×D15711yescharacteristic, complemented
267C22⋊C8—1571no—
268C22⋊C8—1571no—
269#97C2×D15711yescharacteristic, complemented
270C24—3141no—
271#41—3142no—
272#41—3142no—
273#41—3142no—
274#41—6281no—
275#41—6281no—
276#41—6281no—
277C2×Q8—1571no—
278C2×Q8—1571no—
279C2×Q8—1571no—
280C2×D4—3141no—
281C2×D4—3142no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—6281no—
287C2×D4—1571no—
288C2×D4—1571no—
289C2×D4C22×D15711yescharacteristic
290C2×D4C22×D15711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4—1571no—
298Q16—3142no—
299Q16—6281no—
300SD16—3142no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—6281no—
305D8—3142no—
306D8D62814yescomplemented
307M16—3142no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—3141no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—3141no—
318C4⋊C4—1571no—
319C4⋊C4C22×D15711yescharacteristic
320C4⋊C4C22×D15711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—1571no—
330C22⋊C4—1571no—
331C42C22×D15711yescharacteristic
332C23—21no—
333C23—21no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—1571no—
338C23—1571no—
339C23—12561no—
340Q8—3141no—
341Q8—3141no—
342Q8—3141no—
343Q8—3141no—
344Q8—3141no—
345Q8—3141no—
346D4—21no—
347D4—21no—
348D4—3141no—
349D4—3141no—
350D4—6281no—
351D4—6282no—
352D4—6282no—
353D4—6282no—
354D4—6282no—
355D4—12561no—
356D4C2×D62812yes—
357D4C2×D62812yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—3141no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—6282no—
372C2×C4—1571no—
373C2×C4—1571no—
374C2×C4D4×D15711yescharacteristic
375C2×C4D4×D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—3141no—
379C8—3141no—
380C8C2×D62812yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3141no—
388C22—3141no—
389C22—3141no—
390C22—3141no—
391C22—3141no—
392C22—3141no—
393C22—6281no—
394C22—6281no—
395C22—6281no—
396C22—12564no—
397C22C2×D4×D15711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3141no—
403C4—3141no—
404C4—6281no—
405C4—6281no—
406C4—6281no—
407C4C22×D62811yescharacteristic
408C4#1909781711yescharacteristic
409C2—42no—
410C2—42no—
411C2—6281no—
412C2—6281no—
413C2—12561no—
414C2D4×D62811yescharacteristic
415C2#2375806911yescharacteristic
416C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#442×C157C2
socleC22×C157C22×D4
hypercenter#442D157
hypocenterC157#3267
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×#19097790C2semidirect
3. #23757927C2semidirect
4. #23757918C2semidirect
5. #23757667C2semidirect
6. #23757532C2semidirect
7. #23757503C2semidirect
8. #23757488C2semidirect
9. #23757478C2semidirect
10. #23757436C2semidirect
11. #23757436C2semidirect
12. #23757425C2semidirect
13. #23757419C2semidirect
14. #23757346C2semidirect
15. #442×C157C2semidirect
18. C2×#14578953C22semidirect
22. C2×Dic628C22semidirect
23. C2×Dic628C22semidirect
28. C2×#14578942C22semidirect
31. C2×C8×D157C22semidirect
32. #19097771C22semidirect
39. #19097751C22semidirect
47. C4⋊#11929434C22semidirect
48. #19097694C22semidirect
49. #19097694C22semidirect
50. #19097694C22semidirect
53. #19097683C22semidirect
56. C4×D4×C157C22semidirect
57. #106×C157C22semidirect
59. #102×C157C22semidirect
60. #97×C157C22semidirect
115. D8×C157D4semidirect
208. C157#3267semidirect
223. #442D157semidirect
256. C4×D4C2×D157semidirect
259. #106C2×D157semidirect
266. #102C2×D157semidirect
269. #97C2×D157semidirect
306. D8D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×#190977902
#237579272
#237579182
#237576672
#237575322
#237575032
#237574882
#237574782
#237574362
#237574252
#237574192
#237573462
#442×C1572
#3267157