Finite Group Databasebeta

← #34111710

Subgroups — #34111710

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C22×#14578944C211yesmaximal, characteristic, complemented
3#23757936C211yesmaximal, characteristic, complemented
4#23757934C211yesmaximal, characteristic
5#23757666C211yesmaximal, characteristic, complemented
6#23757530C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757498C211yesmaximal, characteristic, complemented
9#23757481C211yesmaximal, characteristic, complemented
10#23757438C211yesmaximal, characteristic, complemented
11#23757427C211yesmaximal, characteristic, complemented
12#23757414C211yesmaximal, characteristic, complemented
13#23757377C211yesmaximal, characteristic, complemented
14#23757341C211yesmaximal, characteristic, complemented
15#451×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097803—24no—
20C2×#14578944—24no—
21C2×#14578944C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578943C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28C2×#14578937C2211yescharacteristic
29C22×#11929434C2211yescharacteristic
30#19097770—21no—
31#19097769C2211yescharacteristic
32#19097760—21no—
33#19097760C2211yescharacteristic
34#19097751C2211yescharacteristic
35#19097749—21no—
36#19097748—21no—
37#19097747—21no—
38#19097747C2211yescharacteristic
39#19097745—21no—
40#19097745C2211yescharacteristic
41Q8×Dic157C2211yescharacteristic
42#19097736C2211yescharacteristic
43C4×C22⋊D157—21no—
44C4×C22⋊D157C2211yescharacteristic
45#19097731—21no—
46#19097702C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097695C2211yescharacteristic
51#19097691C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097681C2211yescharacteristic
54C22×Q8×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic
56C2×Q16×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60C22⋊C8×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63C2×Dic314—21no—
64C2×Dic314—22no—
65C2×Dic314—21no—
66C2×Dic314—41no—
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—41no—
71C2×D628—21no—
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157D411yescharacteristic
75C22×Dic157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157C2311yescharacteristic
79#14578944—22no—
80#14578944—22no—
81#14578944—22no—
82#14578944D414yescomplemented
83#14578943—22no—
84#14578943—41no—
85#14578941—22no—
86#14578937—22no—
87C2×#11929434—22no—
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C4⋊Dic157—21no—
91C4⋊Dic157—21no—
92C4⋊Dic157C2311yescharacteristic
93#14578933—21no—
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933D411yescharacteristic
98C22⋊Dic157D411yescharacteristic
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931D411yescharacteristic
105C4×Dic157—21no—
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108C2×Q8×C157—21no—
109C2×Q8×C157—22no—
110C2×Q8×C157—41no—
111C2×Q8×C157C2311yescharacteristic
112C2×Q8×C157C2311yescharacteristic
113C2×Q8×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116Q16×C157—22no—
117Q16×C157—41no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157—21no—
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—42no—
129Dic314—42no—
130Dic314C2×D412yes—
131D628—21no—
132D628—21no—
133D628—42no—
134C22⋊D157—22no—
135C22⋊D157—42no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—42no—
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C4×D157—21no—
144C4×D157—21no—
145C4×D157—42no—
146C4×D157—42no—
147#11929434—21no—
148#11929434C2×D412yes—
149C23×C157C2×D411yescharacteristic
150Q8×C157—21no—
151Q8×C157—21no—
152Q8×C157—21no—
153Q8×C157—21no—
154Q8×C157—21no—
155Q8×C157—42no—
156Q8×C157C2×D412yes—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—42no—
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2411yescharacteristic
166C8×C157—21no—
167C8×C157—21no—
168C22×C157—21no—
169C22×C157—21no—
170C22×C157#13712yescomplemented
171C22×C157C22×D411yescharacteristic
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—41no—
178C4×C157—21no—
179C4×C157—22no—
180C4×C157—41no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157C22×D411yescharacteristic
184C4×C157H(5,𝔽2)11yescharacteristic
185Dic157—21no—
186Dic157—22no—
187Dic157—21no—
188Dic157—21no—
189Dic157—41no—
190Dic157—41no—
191C2×C157—22no—
192C2×C157D4211yescharacteristic
193C2×C157C2×#13711yescharacteristic
194C2×C157#57811yescharacteristic
195D157—41no—
196D157—81no—
197C157#327011yescharacteristic, complemented, Sylow
198#3270—1571nomaximal, Sylow
199C2×#137—1571no—
200C22×Q16—1571no—
201D4×Q8—1571no—
202#548—1571no—
203#497—1571no—
204#470—1571no—
205#465—1571no—
206#464—1571no—
207#462—1571no—
208#452—1571no—
209#451—1571no—
210#451D15711yescharacteristic, complemented
211#448—1571no—
212#441—1571no—
213#435—1571no—
214C2×#41—1571no—
215C22×Q8—1571no—
216C22×Q8—1571no—
217C22×Q8C2×D15711yescharacteristic
218#137—3144no—
219C2×Q16—3144no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×Q16—1571no—
223C2×Q16—1571no—
224C2×Q16C2×D15711yescharacteristic
225C2×Q16C2×D15711yescharacteristic
226C2×SD16—1571no—
227C2×SD16—1571no—
228C2×SD16—1571no—
229C2×M16—1571no—
230C22×C8—1571no—
231C4⋊Q8—3141no—
232C4⋊Q8—1571no—
233#124—3141no—
234#124—1571no—
235C22⋊Q8—3141no—
236C22⋊Q8—3141no—
237C22⋊Q8—3141no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240C22⋊Q8C2×D15711yescharacteristic
241#121—3141no—
242#121—1571no—
243C4×Q8—1571no—
244C4×Q8—1571no—
245C4×D4—3141no—
246C4×D4—3141no—
247C4×D4—1571no—
248#106—1571no—
249C4⋊C8—1571no—
250#103—1571no—
251#103—1571no—
252#103—1571no—
253#103C2×D15711yescharacteristic
254#103C2×D15711yescharacteristic
255#102—1571no—
256C22⋊C8—1571no—
257C22⋊C8C2×D15711yescharacteristic
258#97—1571no—
259#41—3142no—
260#41—6281no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—3141no—
265C2×Q8—3142no—
266C2×Q8—3141no—
267C2×Q8—3142no—
268C2×Q8—6281no—
269C2×Q8—6281no—
270C2×Q8—1571no—
271C2×Q8—1571no—
272C2×Q8C22×D15711yescharacteristic
273C2×Q8C22×D15711yescharacteristic
274C2×Q8C22×D15711yescharacteristic
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4—1571no—
279C22×C4—21no—
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4—1571no—
284C22×C4C22×D15711yescharacteristic
285Q16—22no—
286Q16—41no—
287Q16—3142no—
288Q16—3142no—
289Q16—3142no—
290Q16—1574no—
291SD16—3142no—
292SD16—3142no—
293SD16—6281no—
294M16—3142no—
295C2×C8—3142no—
296C2×C8—1571no—
297C2×C8—1571no—
298C2×C8C22×D15711yescharacteristic
299C2×C8C22×D15711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—3141no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—1571no—
308C4⋊C4—1571no—
309C4⋊C4C22×D15711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—3141no—
312C22⋊C4—3141no—
313C22⋊C4—3141no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4—1571no—
317C22⋊C4—1571no—
318C42—3141no—
319C42—3141no—
320C42—1571no—
321C23—3141no—
322C23—1571no—
323C23D4×D15711yescharacteristic
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—42no—
330Q8—3141no—
331Q8—3141no—
332Q8—3141no—
333Q8—6282no—
334Q8—6282no—
335Q8—1572no—
336Q8D4×D15712yes—
337D4—3142no—
338D4—3141no—
339D4—3141no—
340D4—6282no—
341D4—6282no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—3141no—
354C2×C4—6282no—
355C2×C4—6282no—
356C2×C4—6282no—
357C2×C4—1571no—
358C2×C4—1571no—
359C2×C4—1571no—
360C2×C4D4×D15711yescharacteristic
361C2×C4C23×D15711yescharacteristic
362C8—21no—
363C8—21no—
364C8—3141no—
365C8—1572no—
366C22—21no—
367C22—21no—
368C22—3141no—
369C22—3141no—
370C22—3141no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22#1909780312yescomplemented
375C22C2×D4×D15711yescharacteristic
376C4—21no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3141no—
382C4—3141no—
383C4—3142no—
384C4—3141no—
385C4—6281no—
386C4—6281no—
387C4C2×D4×D15711yescharacteristic
388C4#1909781611yescharacteristic
389C2—22no—
390C2—6281no—
391C2—12561no—
392C2#2375790511yescharacteristic
393C2C2×#1909780311yescharacteristic
394C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
prime radical#23757934C2
hypercenter#451D157
hypocenterC157#3270
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C22×#14578944C2semidirect
3. #23757936C2semidirect
5. #23757666C2semidirect
6. #23757530C2semidirect
7. #23757504C2semidirect
8. #23757498C2semidirect
9. #23757481C2semidirect
10. #23757438C2semidirect
11. #23757427C2semidirect
12. #23757414C2semidirect
13. #23757377C2semidirect
14. #23757341C2semidirect
15. #451×C157C2semidirect
82. #14578944D4semidirect
170. C22×C157#137semidirect
197. C157#3270semidirect
210. #451D157semidirect
374. C22#19097803semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C22×#145789442
#237579362
#237579342
#237576662
#237575302
#237575042
#237574982
#237574812
#237574382
#237574272
#237574142
#237573772
#237573412
#451×C1572
#3270157