Finite Group Databasebeta

← #34112068

Subgroups — #34112068

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#23757942C211yesmaximal, characteristic
3#23757618C211yesmaximal, characteristic, complemented
4#23757598C211yesmaximal, characteristic, complemented
5#23757595C211yesmaximal, characteristic, complemented
6#23757556C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757471C211yesmaximal, characteristic, complemented
10#23757434C211yesmaximal, characteristic, complemented
11Q16×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757269C211yesmaximal, characteristic, complemented
14#23757268C211yesmaximal, characteristic, complemented
15#452×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578941C2211yescharacteristic
18C22×#11929434C2211yescharacteristic
19C2×C8×D157C2211yescharacteristic
20#19097770C2211yescharacteristic
21#19097767—21no—
22#19097767C2211yescharacteristic
23#19097766—21no—
24#19097764—21no—
25#19097757—21no—
26#19097747C2211yescharacteristic
27#19097745—21no—
28#19097745C2211yescharacteristic
29#19097742—21no—
30#19097742C2211yescharacteristic
31Q8×Dic157C2211yescharacteristic
32Q8×Dic157C2211yescharacteristic
33#19097736—21no—
34C4×C22⋊D157—21no—
35C4×C22⋊D157C2211yescharacteristic
36#19097730C2211yescharacteristic
37#19097729C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097695C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×Q16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C22⋊Q8×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578955—22no—
60#14578955—41no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157D411yescharacteristic
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578941—41no—
72C2×#11929434—41no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C8×D157—41no—
76C4⋊Dic157—21no—
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157D411yescharacteristic
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931D411yescharacteristic
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96#41×C157—22no—
97#41×C157—41no—
98C2×Q8×C157C2311yescharacteristic
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104Q16×C157—22no—
105SD16×C157—41no—
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C4⋊C4×C157—21no—
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—42no—
114D628—21no—
115D628—21no—
116D628—42no—
117C22⋊D157—22no—
118C22⋊D157—42no—
119C22⋊D157—81no—
120C22⋊D157—81no—
121C22⋊D157—81no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C4×D157—21no—
129C4×D157—21no—
130C4×D157—42no—
131#11929434—21no—
132#11929434—21no—
133C23×C157—21no—
134C23×C157C2×D411yescharacteristic
135Q8×C157—21no—
136Q8×C157—21no—
137Q8×C157—21no—
138Q8×C157#4112yes—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—42no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—42no—
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—22no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—41no—
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#465—1571no—
194#460—1571no—
195#452—1571no—
196#452D15711yescharacteristic, complemented
197C4×Q16—1571no—
198C8×D4—1571no—
199#416—1571no—
200#416—1571no—
201C2×#41—1571no—
202C2×#41C2×D15711yescharacteristic
203C2×Q16C2×D15711yescharacteristic
204C2×SD16—1571no—
205C2×SD16C2×D15711yescharacteristic
206C22×C8—1571no—
207C22×C8—1571no—
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#121—1571no—
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#106—1571no—
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⋊C8C2×D15711yescharacteristic
243C4×C8—1571no—
244#41—22no—
245#41—41no—
246#41—3142no—
247#41—6281no—
248C2×Q8—3141no—
249C2×Q8C22×D15711yescharacteristic
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—21no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264Q16—22no—
265SD16—41no—
266SD16—6281no—
267C2×C8—6281no—
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⋊C4—1571no—
282C4⋊C4C22×D15711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3141no—
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—21no—
303Q8—6282no—
304Q8#1457895312yes—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—3142no—
309D4—3141no—
310D4—3141no—
311D4—6282no—
312D4—6282no—
313D4—12561no—
314D4—12561no—
315D4—12561no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
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—22no—
352C4—41no—
353C4—41no—
354C4—3141no—
355C4—3141no—
356C4—6281no—
357C4—6281no—
358C4—6281no—
359C4—1572no—
360C4C2×#1457895311yescharacteristic
361C4#1909781611yescharacteristic
362C2—41no—
363C2—81no—
364C2—6281no—
365C2—12561no—
366C2#2375791011yescharacteristic
367C2#135×D15711yescharacteristic
368C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#452×C157C2
socleC22×C157C22×D4
prime radical#23757942C2
hypercenter#452D157
hypocenterC157#3304
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #23757994C2semidirect
3. #23757618C2semidirect
4. #23757598C2semidirect
5. #23757595C2semidirect
6. #23757556C2semidirect
7. #23757504C2semidirect
8. #23757499C2semidirect
9. #23757471C2semidirect
10. #23757434C2semidirect
11. Q16×Dic157C2semidirect
12. #23757336C2semidirect
13. #23757269C2semidirect
14. #23757268C2semidirect
15. #452×C157C2semidirect
184. C157#3304semidirect
196. #452D157semidirect

Maximal subgroups

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

subgroupindex
#237579942
#237579422
#237576182
#237575982
#237575952
#237575562
#237575042
#237574992
#237574712
#237574342
Q16×Dic1572
#237573362
#237572692
#237572682
#452×C1572
#3304157