Finite Group Databasebeta

← #34112708

Subgroups — #34112708

Order 20096.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757964C211yesmaximal, characteristic
2#23757961C211yesmaximal, characteristic
3#23757672C211yesmaximal, characteristic, complemented
4#23757617C211yesmaximal, characteristic, complemented
5#23757612C211yesmaximal, characteristic, complemented
6#23757610C211yesmaximal, characteristic, complemented
7#23757607C211yesmaximal, characteristic, complemented
8#23757581C211yesmaximal, characteristic, complemented
9#23757530C211yesmaximal, characteristic, complemented
10#23757530C211yesmaximal, characteristic, complemented
11C4×#14578944C211yesmaximal, characteristic, complemented
12#23757350C211yesmaximal, characteristic
13#23757314C211yesmaximal, characteristic, complemented
14#23757312C211yesmaximal, characteristic, complemented
15#479×C157C211yesmaximal, characteristic, complemented
16C2×#14578944C2211yescharacteristic
17C2×#14578937C2211yescharacteristic
18C2×#14578937C2211yescharacteristic
19#19097766—21no—
20#19097765—21no—
21#19097763C2211yescharacteristic
22#19097762—21no—
23#19097761—21no—
24#19097756—21no—
25#19097756—21no—
26#19097751C2211yescharacteristic
27#19097751C2211yescharacteristic
28#19097747C2211yescharacteristic
29#19097747C2211yescharacteristic
30C4×Dic314C2211yescharacteristic
31C4×D628C2211yescharacteristic
32#19097729—21no—
33#19097729C2211yescharacteristic
34#19097729C2211yescharacteristic
35#19097728—21no—
36C4⋊C4×D157C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097704C2211yescharacteristic
42#19097703C2211yescharacteristic
43#19097702C2211yescharacteristic
44#19097702C2211yescharacteristic
45#19097696C2211yescharacteristic
46#19097696C2211yescharacteristic
47#19097695C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51C4×#11929434C2211yescharacteristic
52#125×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59C2×Dic314C2311yescharacteristic
60C2×D628D411yescharacteristic
61C2×C4×D157—21no—
62C2×C4×D157—21no—
63C2×C4×D157C2311yescharacteristic
64C2×C4×D157C2311yescharacteristic
65#14578944—22no—
66#14578937—22no—
67#14578937—22no—
68C2×#11929434C2311yescharacteristic
69C2×#11929434C2311yescharacteristic
70C4⋊Dic157—21no—
71C4⋊Dic157—21no—
72C4⋊Dic157D411yescharacteristic
73C4⋊Dic157C2311yescharacteristic
74C4⋊Dic157C2311yescharacteristic
75C4⋊Dic157C2311yescharacteristic
76#14578933—21no—
77#14578933—21no—
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578931—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931D411yescharacteristic
90#14578931D411yescharacteristic
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C2×Q8×C157C2311yescharacteristic
95C2×C8×C157C2311yescharacteristic
96C2×C8×C157C2311yescharacteristic
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157—21no—
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C42×C157—21no—
104C42×C157C2311yescharacteristic
105C22×D157C2×D411yescharacteristic
106C22×D157C2×D411yescharacteristic
107Dic314—21no—
108Dic314—21no—
109D628—41no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157C2×D411yescharacteristic
115C2×Dic157C2×D411yescharacteristic
116C4×D157—21no—
117C4×D157—21no—
118C4×D157—21no—
119C4×D157—21no—
120C4×D157—42no—
121C4×D157—42no—
122#11929434—21no—
123#11929434#4112yes—
124Q8×C157—21no—
125Q8×C157—21no—
126C2×C4×C157—21no—
127C2×C4×C157—21no—
128C2×C4×C157—21no—
129C2×C4×C157—21no—
130C2×C4×C157C2×D411yescharacteristic
131C2×C4×C157C2×D411yescharacteristic
132C2×C4×C157C2411yescharacteristic
133C8×C157—21no—
134C8×C157—21no—
135C22×C157C22×D411yescharacteristic
136C2×D157—21no—
137C2×D157—21no—
138C2×D157—21no—
139C2×D157—21no—
140C2×D157—21no—
141C2×D157—21no—
142C4×C157—21no—
143C4×C157—21no—
144C4×C157—41no—
145C4×C157—41no—
146C4×C157—41no—
147C4×C157—41no—
148C4×C157C2×#4111yescharacteristic
149C4×C1572−1+411yescharacteristic
150Dic157—21no—
151Dic157—21no—
152Dic157—41no—
153Dic157—41no—
154Dic157—41no—
155Dic157—41no—
156C2×C157#54711yescharacteristic
157C2×C157#57511yescharacteristic
158C2×C157#57811yescharacteristic
159D157—41no—
160D157—41no—
161C157#335211yescharacteristic, complemented, Sylow
162#3352—1571nomaximal, Sylow
163#552—1571no—
164#552—1571no—
165#499—1571no—
166#485—1571no—
167#485—1571no—
168#484—1571no—
169#484—1571no—
170#479—1571no—
171#479D15711yescharacteristic, complemented
172#470—1571no—
173#470—1571no—
174C4×Q16—1571no—
175#436—1571no—
176#428—1571no—
177#428—1571no—
178C2×Q16—1571no—
179C2×M16—1571no—
180C2×M16—1571no—
181#126—3141no—
182#126—3141no—
183#125—3141no—
184#125—3141no—
185#125—1571no—
186#125C2×D15711yescharacteristic
187#123—3141no—
188#123—3141no—
189C22⋊Q8—1571no—
190C22⋊Q8—1571no—
191C22⋊Q8—1571no—
192C22⋊Q8—1571no—
193C4×Q8—1571no—
194C4×Q8C2×D15711yescharacteristic
195C4×D4—1571no—
196#117—3141no—
197#117—3141no—
198#117—1571no—
199#117—1571no—
200C2×C4⋊C4—1571no—
201C2×C4⋊C4—1571no—
202#107—1571no—
203#107—1571no—
204#107—1571no—
205#107—1571no—
206#107C2×D15711yescharacteristic
207#106—1571no—
208#106—1571no—
209#106—1571no—
210#106C2×D15711yescharacteristic
211C4⋊C8C2×D15711yescharacteristic
212#103—1571no—
213#103—1571no—
214#103—1571no—
215#103—1571no—
216#103C2×D15711yescharacteristic
217#103C2×D15711yescharacteristic
218C22⋊C8—1571no—
219C22⋊C8—1571no—
220C4×C8—1571no—
221C2×Q8—1571no—
222C2×Q8C22×D15711yescharacteristic
223C2×D4—1571no—
224C22×C4—3141no—
225C22×C4—3141no—
226C22×C4—1571no—
227C22×C4—1571no—
228Q16—3142no—
229M16—3142no—
230M16—3142no—
231C2×C8—1571no—
232C2×C8—1571no—
233C2×C8C22×D15711yescharacteristic
234C2×C8C22×D15711yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—3141no—
239C4⋊C4—3141no—
240C4⋊C4—3141no—
241C4⋊C4—3141no—
242C4⋊C4—3141no—
243C4⋊C4—3141no—
244C4⋊C4—3141no—
245C4⋊C4—3141no—
246C4⋊C4—3141no—
247C4⋊C4—1571no—
248C4⋊C4—1571no—
249C4⋊C4—1571no—
250C4⋊C4—1571no—
251C4⋊C4C22×D15711yescharacteristic
252C4⋊C4C22×D15711yescharacteristic
253C4⋊C4C22×D15711yescharacteristic
254C22⋊C4—3141no—
255C22⋊C4—3141no—
256C22⋊C4—3141no—
257C22⋊C4—3141no—
258C22⋊C4—3141no—
259C22⋊C4—3141no—
260C22⋊C4—1571no—
261C22⋊C4—1571no—
262C42—21no—
263C42—3141no—
264C42—3141no—
265C42—3141no—
266C42C22×D15711yescharacteristic
267C23—1571no—
268C23—1571no—
269Q8—21no—
270Q8—21no—
271Q8—3141no—
272Q8—3141no—
273D4—6281no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—3141no—
279C2×C4—3141no—
280C2×C4—3141no—
281C2×C4—3141no—
282C2×C4—3141no—
283C2×C4—3141no—
284C2×C4—3141no—
285C2×C4—3141no—
286C2×C4—6282no—
287C2×C4—6282no—
288C2×C4—1571no—
289C2×C4—1571no—
290C2×C4D4×D15711yescharacteristic
291C2×C4D4×D15711yescharacteristic
292C2×C4C23×D15711yescharacteristic
293C8—21no—
294C8—21no—
295C8—3141no—
296C8—1572no—
297C22—3141no—
298C22—3141no—
299C22—3141no—
300C22—3141no—
301C22—3141no—
302C22—3141no—
303C22C2×D4×D15711yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—3141no—
311C4—3141no—
312C4—6281no—
313C4—6281no—
314C4—6281no—
315C4—6281no—
316C4#41×D15711yescharacteristic
317C4#1909781911yescharacteristic
318C2—6281no—
319C2—6281no—
320C2#2375792511yescharacteristic
321C2#2375807011yescharacteristic
322C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#479×C157C2
socleC22×C157C22×D4
prime radicalC2×D628D4
hypercenter#479D157
hypocenterC157#3352
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
3. #23757672C2semidirect
4. #23757617C2semidirect
5. #23757612C2semidirect
6. #23757610C2semidirect
7. #23757607C2semidirect
8. #23757581C2semidirect
9. #23757530C2semidirect
10. #23757530C2semidirect
11. C4×#14578944C2semidirect
13. #23757314C2semidirect
14. #23757312C2semidirect
15. #479×C157C2semidirect
161. C157#3352semidirect
171. #479D157semidirect

Maximal subgroups

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

subgroupindex
#237579642
#237579612
#237576722
#237576172
#237576122
#237576102
#237576072
#237575812
#237575302
C4×#145789442
#237573502
#237573142
#237573122
#479×C1572
#3352157