Finite Group Databasebeta

← #34112727

Subgroups — #34112727

Order 20096.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757986C211yesmaximal, characteristic, complemented
2#23757924C211yesmaximal, characteristic, complemented
3#23757670C211yesmaximal, characteristic, complemented
4#23757600C211yesmaximal, characteristic, complemented
5#23757597C211yesmaximal, characteristic, complemented
6#23757593C211yesmaximal, characteristic, complemented
7#23757591C211yesmaximal, characteristic, complemented
8#23757562C211yesmaximal, characteristic, complemented
9#23757552C211yesmaximal, characteristic, complemented
10#23757527C211yesmaximal, characteristic, complemented
11#23757526C211yesmaximal, characteristic, complemented
12#23757355C211yesmaximal, characteristic, complemented
13#23757349C211yesmaximal, characteristic, complemented
14#23757314C211yesmaximal, characteristic, complemented
15#428×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×D1256C2211yescharacteristic
18C2×#14578937C2211yescharacteristic
19#19097770C2211yescharacteristic
20#19097770C2211yescharacteristic
21#19097759—21no—
22#19097757—21no—
23#19097756—21no—
24#19097754—21no—
25#19097749—21no—
26#19097746C2211yescharacteristic
27#19097746C2211yescharacteristic
28#19097745C2211yescharacteristic
29#19097745C2211yescharacteristic
30#19097744—21no—
31#19097733C2211yescharacteristic
32#19097733C2211yescharacteristic
33C4×C22⋊D157C2211yescharacteristic
34#19097729C2211yescharacteristic
35#19097728—21no—
36C4⋊C4×D157C2211yescharacteristic
37#19097704C2211yescharacteristic
38#19097703C2211yescharacteristic
39#19097702C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097693C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097691C2211yescharacteristic
46#19097691C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×M16×C157C2211yescharacteristic
53#117×C157C2211yescharacteristic
54C2×C4⋊C4×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic
56#107×C157C2211yescharacteristic
57#106×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628—21no—
64C2×D628C2311yescharacteristic
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157D411yescharacteristic
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73D1256—22no—
74#14578937—22no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933D411yescharacteristic
83C22⋊Dic157D411yescharacteristic
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92C4×Dic157—21no—
93C4×Dic157C2311yescharacteristic
94C22×C4×C157—21no—
95C22×C4×C157C2311yescharacteristic
96M16×C157—22no—
97C2×C8×C157C2311yescharacteristic
98C2×C8×C157C2311yescharacteristic
99C4⋊C4×C157—21no—
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
105C42×C157—21no—
106C22×D157—21no—
107C22×D157—21no—
108C22×D157C2×D411yescharacteristic
109Dic314—42no—
110Dic314—81no—
111D628—21no—
112D628—21no—
113D628—21no—
114D628—21no—
115D628—42no—
116D628—81no—
117C22⋊D157—22no—
118C22⋊D157—42no—
119C22⋊D157—81no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C4×D157—21no—
126C4×D157—21no—
127C4×D157—42no—
128C4×D157—42no—
129C4×D157—42no—
130#11929434—21no—
131#11929434—21no—
132C23×C157C2×D411yescharacteristic
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—42no—
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2411yescharacteristic
142C8×C157—21no—
143C8×C157#4112yescomplemented
144C22×C157—21no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157C22×D411yescharacteristic
148C2×D157—21no—
149C2×D157—21no—
150C2×D157—21no—
151C2×D157—41no—
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—41no—
156C2×D157—41no—
157C4×C157—21no—
158C4×C157—41no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157C2×#4111yescharacteristic
163C4×C1572−1+411yescharacteristic
164Dic157—21no—
165Dic157—21no—
166Dic157—41no—
167Dic157—41no—
168Dic157AGL(1,ℤ/8)12yes—
169C2×C157—41no—
170C2×C157#54711yescharacteristic
171C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C157#57711yescharacteristic
173D157—41no—
174D157—81no—
175D157—81no—
176C157#335411yescharacteristic, complemented, Sylow
177#3354—1571nomaximal, Sylow
178#555—1571no—
179#547—1571no—
180#498—1571no—
181#482—1571no—
182#482—1571no—
183#481—1571no—
184#481—1571no—
185#476—1571no—
186#474—1571no—
187#469—1571no—
188#469—1571no—
189C4×D8—1571no—
190#436—1571no—
191#428—1571no—
192#428D15711yescharacteristic, complemented
193C2×#41—3141no—
194C2×D8—1571no—
195C2×M16—1571no—
196C2×M16C2×D15711yescharacteristic
197C4⋊Q8—1571no—
198C4⋊Q8—1571no—
199#124—3141no—
200#124—3141no—
201#123—3141no—
202#123—3141no—
203C22⋊Q8—3141no—
204#121—3141no—
205#121—1571no—
206#121—1571no—
207#121—1571no—
208#121—1571no—
209C4×D4—1571no—
210C4×D4—1571no—
211C4×D4—1571no—
212#117—3141no—
213#117—1571no—
214#117C2×D15711yescharacteristic
215C2×C4⋊C4—1571no—
216C2×C4⋊C4C2×D15711yescharacteristic
217#107—1571no—
218#107C2×D15711yescharacteristic
219#107C2×D15711yescharacteristic
220#106—1571no—
221#106—1571no—
222#106—1571no—
223#106—1571no—
224#106C2×D15711yescharacteristic
225#106C2×D15711yescharacteristic
226C4⋊C8—1571no—
227#102—1571no—
228#102—1571no—
229#102—1571no—
230#102—1571no—
231#102—1571no—
232#102—1571no—
233C22⋊C8—1571no—
234C22⋊C8—1571no—
235C4×C8—1571no—
236#41—6284no—
237C2×Q8—3141no—
238C2×Q8—3141no—
239C2×D4—3141no—
240C2×D4—3141no—
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4—1571no—
245C2×D4—1571no—
246C22×C4—21no—
247C22×C4—3141no—
248C22×C4—3141no—
249C22×C4—3141no—
250C22×C4—1571no—
251C22×C4C22×D15711yescharacteristic
252D8—3142no—
253M16—22no—
254M16—3142no—
255C2×C8—1571no—
256C2×C8—1571no—
257C2×C8C22×D15711yescharacteristic
258C2×C8C22×D15711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—1571no—
264C4⋊C4—1571no—
265C4⋊C4—1571no—
266C4⋊C4C22×D15711yescharacteristic
267C4⋊C4C22×D15711yescharacteristic
268C4⋊C4C22×D15711yescharacteristic
269C4⋊C4C22×D15711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3141no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—1571no—
279C22⋊C4—1571no—
280C42—21no—
281C42—3141no—
282C42—1571no—
283C23—3141no—
284C23—3141no—
285C23—1571no—
286C23D4×D15711yescharacteristic
287Q8—6282no—
288Q8—12561no—
289D4—3142no—
290D4—3141no—
291D4—3141no—
292D4—3141no—
293D4—3141no—
294D4—6282no—
295D4—6282no—
296D4—12561no—
297D4—12561no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—6282no—
310C2×C4—6282no—
311C2×C4—6282no—
312C2×C4—1571no—
313C2×C4—1571no—
314C2×C4—1571no—
315C2×C4D4×D15711yescharacteristic
316C2×C4C23×D15711yescharacteristic
317C8—21no—
318C8—3141no—
319C8—3141no—
320C8#1457895412yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—3141no—
325C22—3141no—
326C22—3141no—
327C22—6281no—
328C22—6281no—
329C22—6281no—
330C22—6281no—
331C22—6281no—
332C22—6281no—
333C22C2×D4×D15711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—3141no—
340C4—3141no—
341C4—6281no—
342C4—6281no—
343C4—1572no—
344C4C2×#1457895411yescharacteristic
345C4#1909781911yescharacteristic
346C2—41no—
347C2—6281no—
348C2—12561no—
349C2—12561no—
350C2#2375792411yescharacteristic
351C2AGL(1,ℤ/8)×D15711yescharacteristic
352C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
hypercenter#428D157
hypocenterC157#3354
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #23757986C2semidirect
2. #23757924C2semidirect
3. #23757670C2semidirect
4. #23757600C2semidirect
5. #23757597C2semidirect
6. #23757593C2semidirect
7. #23757591C2semidirect
8. #23757562C2semidirect
9. #23757552C2semidirect
10. #23757527C2semidirect
11. #23757526C2semidirect
12. #23757355C2semidirect
13. #23757349C2semidirect
14. #23757314C2semidirect
15. #428×C157C2semidirect
143. C8×C157#41semidirect
176. C157#3354semidirect
192. #428D157semidirect
320. C8#14578954semidirect

Maximal subgroups

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

subgroupindex
#237579862
#237579242
#237576702
#237576002
#237575972
#237575932
#237575912
#237575622
#237575522
#237575272
#237575262
#237573552
#237573492
#237573142
#428×C1572
#3354157