Finite Group Databasebeta

← #34112724

Subgroups — #34112724

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#23757987C211yesmaximal, characteristic, complemented
2#23757924C211yesmaximal, characteristic, complemented
3#23757670C211yesmaximal, characteristic, complemented
4#23757599C211yesmaximal, characteristic, complemented
5#23757598C211yesmaximal, characteristic, complemented
6#23757590C211yesmaximal, characteristic, complemented
7#23757562C211yesmaximal, characteristic, complemented
8#23757551C211yesmaximal, characteristic, complemented
9#23757525C211yesmaximal, characteristic, complemented
10#23757524C211yesmaximal, characteristic, complemented
11#23757353C211yesmaximal, characteristic, complemented
12#23757348C211yesmaximal, characteristic, complemented
13#23757313C211yesmaximal, characteristic, complemented
14#23757312C211yesmaximal, characteristic, complemented
15#481×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×#14578939C2211yescharacteristic
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20#19097770C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097758—21no—
23#19097757—21no—
24#19097756—21no—
25#19097754—21no—
26#19097749—21no—
27#19097746C2211yescharacteristic
28#19097745C2211yescharacteristic
29#19097743—21no—
30#19097742C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32#19097733C2211yescharacteristic
33C4×C22⋊D157C2211yescharacteristic
34#19097730C2211yescharacteristic
35#19097729C2211yescharacteristic
36#19097727—21no—
37C4⋊C4×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097703C2211yescharacteristic
42#19097702C2211yescharacteristic
43#19097702C2211yescharacteristic
44#19097702C2211yescharacteristic
45#19097700C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52#121×C157C2211yescharacteristic
53C2×C4⋊C4×C157C2211yescharacteristic
54#106×C157C2211yescharacteristic
55#106×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157D411yescharacteristic
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578939—22no—
72#14578938—22no—
73#14578937—22no—
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933D411yescharacteristic
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157D411yescharacteristic
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×D4×C157—21no—
96C2×D4×C157C2311yescharacteristic
97C22×C4×C157—21no—
98C22×C4×C157C2311yescharacteristic
99C2×C8×C157C2311yescharacteristic
100C2×C8×C157C2311yescharacteristic
101C4⋊C4×C157—21no—
102C4⋊C4×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C4⋊C4×C157C2311yescharacteristic
105C22⋊C4×C157—21no—
106C22×D157—21no—
107C22×D157C2×D411yescharacteristic
108Dic314—42no—
109Dic314—81no—
110D628—21no—
111D628—21no—
112D628—42no—
113C22⋊D157—22no—
114C22⋊D157—42no—
115C22⋊D157—81no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—42no—
120C2×Dic157C2×D411yescharacteristic
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C4×D157—21no—
124C4×D157—21no—
125C4×D157—42no—
126C4×D157—42no—
127#11929434—21no—
128#11929434#4112yescomplemented
129C23×C157—21no—
130C23×C157C2×D411yescharacteristic
131D4×C157—21no—
132D4×C157—21no—
133D4×C157—81no—
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—21no—
144C22×C157—21no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157—41no—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157C22×D411yescharacteristic
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—41no—
155C2×D157—41no—
156C2×D157—41no—
157C4×C157—21no—
158C4×C157—41no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157C2×#4111yescharacteristic
162C4×C1572−1+411yescharacteristic
163Dic157—21no—
164Dic157—21no—
165Dic157—41no—
166Dic157—41no—
167Dic157—41no—
168Dic157AGL(1,ℤ/8)12yes—
169C2×C157—41no—
170C2×C157—81no—
171C2×C157#54711yescharacteristic
172C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
173C2×C157#57711yescharacteristic
174D157—41no—
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#481D15711yescharacteristic, complemented
185#476—1571no—
186#474—1571no—
187#469—1571no—
188#469—1571no—
189C4×D8—1571no—
190#436—1571no—
191#428—1571no—
192#428—1571no—
193C2×#41—3141no—
194C2×D8—1571no—
195C2×M16—1571no—
196C2×M16—1571no—
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#121C2×D15711yescharacteristic
209C4×D4—1571no—
210C4×D4—1571no—
211C4×D4—1571no—
212#117—3141no—
213#117—1571no—
214#117—1571no—
215C2×C4⋊C4—1571no—
216C2×C4⋊C4C2×D15711yescharacteristic
217#107—1571no—
218#107—1571no—
219#107—1571no—
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#102C2×D15711yescharacteristic
232#102C2×D15711yescharacteristic
233C22⋊C8—1571no—
234C22⋊C8C2×D15711yescharacteristic
235C4×C8—1571no—
236#41—6284no—
237C2×Q8—3141no—
238C2×Q8—3141no—
239C2×D4—21no—
240C2×D4—3141no—
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4—1571no—
245C2×D4C22×D15711yescharacteristic
246C22×C4—21no—
247C22×C4—3141no—
248C22×C4—3141no—
249C22×C4—3141no—
250C22×C4—1571no—
251C22×C4C22×D15711yescharacteristic
252D8—3142no—
253M16—3142no—
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⋊C4—1571no—
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—3141no—
281C42—3141no—
282C42—1571no—
283C23—21no—
284C23—3141no—
285C23—1571no—
286C23D4×D15711yescharacteristic
287Q8—6282no—
288Q8—12561no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—3142no—
293D4—3141no—
294D4—3141no—
295D4—6282no—
296D4—6282no—
297D4—12561no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—3141no—
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—21no—
319C8—3141no—
320C8—1572no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—3141no—
328C22—3141no—
329C22—3141no—
330C22—6281no—
331C22—6281no—
332C22—6281no—
333C22C2×D4×D15711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—3141no—
339C4—3141no—
340C4—6281no—
341C4—6281no—
342C4—6281no—
343C4—1572no—
344C4#41×D15711yescharacteristic
345C4#1909781811yescharacteristic
346C2—41no—
347C2—81no—
348C2—6281no—
349C2—12561no—
350C2#2375792111yescharacteristic
351C2AGL(1,ℤ/8)×D15711yescharacteristic
352C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#481×C157C2
socleC22×C157C22×D4
hypercenter#481D157
hypocenterC157#3354
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #23757987C2semidirect
2. #23757924C2semidirect
3. #23757670C2semidirect
4. #23757599C2semidirect
5. #23757598C2semidirect
6. #23757590C2semidirect
7. #23757562C2semidirect
8. #23757551C2semidirect
9. #23757525C2semidirect
10. #23757524C2semidirect
11. #23757353C2semidirect
12. #23757348C2semidirect
13. #23757313C2semidirect
14. #23757312C2semidirect
15. #481×C157C2semidirect
128. #11929434#41semidirect
176. C157#3354semidirect
184. #481D157semidirect

Maximal subgroups

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

subgroupindex
#237579872
#237579242
#237576702
#237575992
#237575982
#237575902
#237575622
#237575512
#237575252
#237575242
#237573532
#237573482
#237573132
#237573122
#481×C1572
#3354157