Finite Group Databasebeta

← #34112070

Subgroups — #34112070

Order 20096.

396 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757994C211yesmaximal, characteristic, complemented
2#23757941C211yesmaximal, characteristic
3#23757618C211yesmaximal, characteristic, complemented
4#23757599C211yesmaximal, characteristic, complemented
5#23757594C211yesmaximal, characteristic, complemented
6#23757557C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757471C211yesmaximal, characteristic, complemented
9#23757433C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757365C211yesmaximal, characteristic, complemented
12C8×C22⋊D157C211yesmaximal, characteristic, complemented
13#23757270C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578944C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097767—21no—
23#19097767C2211yescharacteristic
24#19097766—21no—
25#19097764—21no—
26#19097757—21no—
27#19097747C2211yescharacteristic
28#19097746C2211yescharacteristic
29#19097744—21no—
30#19097743—21no—
31Q8×Dic157C2211yescharacteristic
32#19097736—21no—
33#19097736C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097731—21no—
36#19097730C2211yescharacteristic
37#19097729C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097702C2211yescharacteristic
40#19097700C2211yescharacteristic
41#19097696C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097694C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×SD16×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C22⋊Q8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#106×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578954—22no—
60#14578954—41no—
61#14578953—22no—
62#14578953—41no—
63C2×Dic314—21no—
64C2×Dic314C2311yescharacteristic
65C2×D628—21no—
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157D411yescharacteristic
72C22×Dic157—21no—
73C2×C4×D157—21no—
74C2×C4×D157C2311yescharacteristic
75#14578944—22no—
76#14578941—41no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C8×D157—41no—
80C4⋊Dic157—21no—
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157—21no—
97C4×Dic157C2311yescharacteristic
98C2×Q8×C157C2311yescharacteristic
99C2×D4×C157—21no—
100C2×D4×C157C2311yescharacteristic
101C22×C4×C157C2311yescharacteristic
102SD16×C157—41no—
103C2×C8×C157—41no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157—21no—
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—42no—
115Dic314#4112yes—
116D628—21no—
117D628—21no—
118D628—81no—
119C22⋊D157—22no—
120C22⋊D157—42no—
121C22⋊D157—42no—
122C22⋊D157—42no—
123C22⋊D157—81no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—42no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157—21no—
137C23×C157C2×D411yescharacteristic
138Q8×C157—21no—
139Q8×C157—21no—
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—81no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—22no—
173Dic157—21no—
174Dic157—41no—
175Dic157—41no—
176Dic157#13512yes—
177C2×C157—41no—
178C2×C157—81no—
179C2×C157#54611yescharacteristic
180C2×C157C2×#13511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—41no—
183D157—81no—
184C157#330411yescharacteristic, complemented, Sylow
185#3304—1571nomaximal, Sylow
186#556—1571no—
187#550—1571no—
188#486—1571no—
189#482—1571no—
190#482—1571no—
191#475—1571no—
192#465—1571no—
193#465D15711yescharacteristic, complemented
194#460—1571no—
195#452—1571no—
196#452—1571no—
197C4×Q16—1571no—
198C8×D4—1571no—
199#416—1571no—
200#416—1571no—
201C2×#41—1571no—
202C2×#41—1571no—
203C2×Q16—1571no—
204C2×SD16—1571no—
205C2×SD16C2×D15711yescharacteristic
206C22×C8—1571no—
207C22×C8C2×D15711yescharacteristic
208C4⋊Q8—1571no—
209#127—3141no—
210#127—1571no—
211#126—3141no—
212#126—3141no—
213#124—3141no—
214C22⋊Q8—1571no—
215C22⋊Q8C2×D15711yescharacteristic
216#121—3141no—
217#121—3141no—
218#121—1571no—
219#121C2×D15711yescharacteristic
220C4×Q8—3141no—
221C4×Q8—1571no—
222C4×Q8—1571no—
223C4×D4—3141no—
224C4×D4—1571no—
225#117—1571no—
226#117—1571no—
227#107—1571no—
228#106—1571no—
229#106C2×D15711yescharacteristic
230C4⋊C8—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103C2×D15711yescharacteristic
235#102—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102C2×D15711yescharacteristic
241C22⋊C8—1571no—
242C22⋊C8—1571no—
243C4×C8—1571no—
244#41—3142no—
245#41—3142no—
246#41—6281no—
247#41—6281no—
248C2×Q8—3141no—
249C2×Q8—1571no—
250C2×Q8C22×D15711yescharacteristic
251C2×D4—21no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—3141no—
256C2×D4—3141no—
257C2×D4—1571no—
258C2×D4—1571no—
259C2×D4C22×D15711yescharacteristic
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264Q16—3142no—
265SD16—41no—
266SD16—6281no—
267C2×C8—41no—
268C2×C8—6281no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15711yescharacteristic
272C2×C8C22×D15711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—1571no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4C22×D15711yescharacteristic
282C4⋊C4C22×D15711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—1571no—
290C22⋊C4—1571no—
291C42—3141no—
292C42—3141no—
293C42—3141no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—3141no—
298C23—1571no—
299C23D4×D15711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3141no—
303Q8—6282no—
304Q8—1572no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—3142no—
309D4—3141no—
310D4—3141no—
311D4—6282no—
312D4—6282no—
313D4—6282no—
314D4—12561no—
315D4—12561no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—1571no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4D4×D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—3141no—
336C8—3141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—3141no—
344C22—3141no—
345C22—3141no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3141no—
355C4—3142no—
356C4—3141no—
357C4—6281no—
358C4—6281no—
359C4—1572no—
360C4#41×D15711yescharacteristic
361C4#1909781611yescharacteristic
362C2—41no—
363C2—81no—
364C2—6281no—
365C2—12561no—
366C2#2375791211yescharacteristic
367C2#135×D15711yescharacteristic
368C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757941C2
hypercenter#465D157
hypocenterC157#3304
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23757994C2semidirect
3. #23757618C2semidirect
4. #23757599C2semidirect
5. #23757594C2semidirect
6. #23757557C2semidirect
7. #23757505C2semidirect
8. #23757471C2semidirect
9. #23757433C2semidirect
10. #23757432C2semidirect
11. #23757365C2semidirect
12. C8×C22⋊D157C2semidirect
13. #23757270C2semidirect
14. #23757266C2semidirect
15. #465×C157C2semidirect
184. C157#3304semidirect
193. #465D157semidirect

Maximal subgroups

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

subgroupindex
#237579942
#237579412
#237576182
#237575992
#237575942
#237575572
#237575052
#237574712
#237574332
#237574322
#237573652
C8×C22⋊D1572
#237572702
#237572662
#465×C1572
#3304157