Finite Group Databasebeta

← #34112713

Subgroups — #34112713

Order 20096.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757994C211yesmaximal, characteristic, complemented
2#23757924C211yesmaximal, characteristic
3#23757677C211yesmaximal, characteristic, complemented
4#23757607C211yesmaximal, characteristic, complemented
5#23757600C211yesmaximal, characteristic, complemented
6#23757598C211yesmaximal, characteristic, complemented
7#23757556C211yesmaximal, characteristic, complemented
8#23757552C211yesmaximal, characteristic, complemented
9#23757522C211yesmaximal, characteristic, complemented
10#23757521C211yesmaximal, characteristic, complemented
11#23757358C211yesmaximal, characteristic, complemented
12#23757348C211yesmaximal, characteristic, complemented
13#23757314C211yesmaximal, characteristic, complemented
14#23757313C211yesmaximal, characteristic, complemented
15#484×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×#14578941C2211yescharacteristic
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20#19097770C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097766—21no—
23#19097764—21no—
24#19097757—21no—
25#19097756—21no—
26#19097754—21no—
27#19097749—21no—
28#19097747C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097745C2211yescharacteristic
31Q8×Dic157C2211yescharacteristic
32#19097736—21no—
33#19097733C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097730C2211yescharacteristic
36#19097729C2211yescharacteristic
37C4⋊C4×D157C2211yescharacteristic
38#19097705C2211yescharacteristic
39#19097704C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097704C2211yescharacteristic
42#19097703C2211yescharacteristic
43#19097702C2211yescharacteristic
44#19097702C2211yescharacteristic
45#19097700C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22⋊Q8×C157C2211yescharacteristic
53C2×C4⋊C4×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157D411yescharacteristic
66C2×C4×D157—21no—
67C2×C4×D157—21no—
68C2×C4×D157C2311yescharacteristic
69#14578941—22no—
70#14578938—22no—
71#14578937—22no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C4⋊Dic157—21no—
75C4⋊Dic157C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933D411yescharacteristic
83C22⋊Dic157—21no—
84C22⋊Dic157D411yescharacteristic
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931D411yescharacteristic
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×Q8×C157C2311yescharacteristic
96C22×C4×C157—21no—
97C22×C4×C157C2311yescharacteristic
98C2×C8×C157C2311yescharacteristic
99C2×C8×C157C2311yescharacteristic
100C4⋊C4×C157—21no—
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—42no—
110D628—21no—
111D628—21no—
112D628—42no—
113C22⋊D157—22no—
114C22⋊D157—42no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157C2×D411yescharacteristic
119C2×Dic157C2×D411yescharacteristic
120C2×Dic157C2×D411yescharacteristic
121C4×D157—21no—
122C4×D157—21no—
123C4×D157—42no—
124C4×D157—42no—
125#11929434—21no—
126#11929434#4112yescomplemented
127C23×C157C2×D411yescharacteristic
128Q8×C157—21no—
129Q8×C157—21no—
130C2×C4×C157—21no—
131C2×C4×C157—21no—
132C2×C4×C157—21no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—42no—
137C2×C4×C157C2×D411yescharacteristic
138C2×C4×C157C2411yescharacteristic
139C8×C157—21no—
140C8×C157—21no—
141C22×C157—21no—
142C22×C157—21no—
143C22×C157—21no—
144C22×C157C22×D411yescharacteristic
145C2×D157—21no—
146C2×D157—21no—
147C2×D157—21no—
148C2×D157—41no—
149C2×D157—41no—
150C2×D157—41no—
151C4×C157—21no—
152C4×C157—41no—
153C4×C157—41no—
154C4×C157—41no—
155C4×C157—41no—
156C4×C157C2×#4111yescharacteristic
157C4×C1572−1+411yescharacteristic
158Dic157—21no—
159Dic157—21no—
160Dic157—41no—
161Dic157—41no—
162Dic157—41no—
163Dic157AGL(1,ℤ/8)12yes—
164C2×C157—41no—
165C2×C157#54711yescharacteristic
166C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C157#57811yescharacteristic
168D157—41no—
169D157—81no—
170C157#335311yescharacteristic, complemented, Sylow
171#3353—1571nomaximal, Sylow
172#556—1571no—
173#547—1571no—
174#500—1571no—
175#484—1571no—
176#484D15711yescharacteristic, complemented
177#482—1571no—
178#482—1571no—
179#475—1571no—
180#474—1571no—
181#468—1571no—
182#468—1571no—
183C4×SD16—1571no—
184#436—1571no—
185#428—1571no—
186#428—1571no—
187C2×#41—3141no—
188C2×SD16—1571no—
189C2×M16—1571no—
190C2×M16—1571no—
191C4⋊Q8—1571no—
192C4⋊Q8—1571no—
193#126—3141no—
194#126—3141no—
195#124—3141no—
196#123—3141no—
197#123—3141no—
198C22⋊Q8—3141no—
199C22⋊Q8—1571no—
200C22⋊Q8C2×D15711yescharacteristic
201#121—1571no—
202#121—1571no—
203C4×Q8—3141no—
204C4×Q8—1571no—
205C4×D4—1571no—
206C4×D4—1571no—
207#117—1571no—
208#117—1571no—
209C2×C4⋊C4—1571no—
210C2×C4⋊C4C2×D15711yescharacteristic
211#107—1571no—
212#107—1571no—
213#107—1571no—
214#107—1571no—
215#107C2×D15711yescharacteristic
216#107C2×D15711yescharacteristic
217#106—1571no—
218#106—1571no—
219#106—1571no—
220C4⋊C8—1571no—
221#103—1571no—
222#103C2×D15711yescharacteristic
223#103C2×D15711yescharacteristic
224#102—1571no—
225#102—1571no—
226#102—1571no—
227C22⋊C8—1571no—
228C22⋊C8C2×D15711yescharacteristic
229C4×C8—1571no—
230#41—6284no—
231C2×Q8—3141no—
232C2×Q8—3141no—
233C2×Q8C22×D15711yescharacteristic
234C2×D4—3141no—
235C2×D4—3141no—
236C2×D4—1571no—
237C2×D4—1571no—
238C22×C4—21no—
239C22×C4—3141no—
240C22×C4—3141no—
241C22×C4—1571no—
242C22×C4C22×D15711yescharacteristic
243SD16—3142no—
244M16—3142no—
245M16—3142no—
246C2×C8—1571no—
247C2×C8—1571no—
248C2×C8C22×D15711yescharacteristic
249C2×C8C22×D15711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—3141no—
253C4⋊C4—3141no—
254C4⋊C4—3141no—
255C4⋊C4—3141no—
256C4⋊C4—3141no—
257C4⋊C4—1571no—
258C4⋊C4—1571no—
259C4⋊C4—1571no—
260C4⋊C4—1571no—
261C4⋊C4C22×D15711yescharacteristic
262C4⋊C4C22×D15711yescharacteristic
263C4⋊C4C22×D15711yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—3141no—
266C22⋊C4—3141no—
267C22⋊C4—3141no—
268C22⋊C4—3141no—
269C22⋊C4—3141no—
270C22⋊C4—3141no—
271C22⋊C4—1571no—
272C22⋊C4—1571no—
273C42—3141no—
274C42—3141no—
275C42—3141no—
276C42—1571no—
277C23—3141no—
278C23—1571no—
279C23D4×D15711yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—6282no—
283Q8—6282no—
284D4—3142no—
285D4—3141no—
286D4—3141no—
287D4—6282no—
288D4—6282no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—3141no—
297C2×C4—3141no—
298C2×C4—3141no—
299C2×C4—3141no—
300C2×C4—3141no—
301C2×C4—6282no—
302C2×C4—6282no—
303C2×C4—1571no—
304C2×C4—1571no—
305C2×C4—1571no—
306C2×C4D4×D15711yescharacteristic
307C2×C4C23×D15711yescharacteristic
308C8—21no—
309C8—21no—
310C8—3141no—
311C8—1572no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—3141no—
316C22—3141no—
317C22—3141no—
318C22—6281no—
319C22—6281no—
320C22—6281no—
321C22C2×D4×D15711yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3141no—
328C4—3141no—
329C4—6281no—
330C4—6281no—
331C4—6281no—
332C4—1572no—
333C4#41×D15711yescharacteristic
334C4#1909781811yescharacteristic
335C2—41no—
336C2—6281no—
337C2—12561no—
338C2#2375792111yescharacteristic
339C2AGL(1,ℤ/8)×D15711yescharacteristic
340C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
prime radical#23757924C2
hypercenter#484D157
hypocenterC157#3353
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #23757994C2semidirect
3. #23757677C2semidirect
4. #23757607C2semidirect
5. #23757600C2semidirect
6. #23757598C2semidirect
7. #23757556C2semidirect
8. #23757552C2semidirect
9. #23757522C2semidirect
10. #23757521C2semidirect
11. #23757358C2semidirect
12. #23757348C2semidirect
13. #23757314C2semidirect
14. #23757313C2semidirect
15. #484×C157C2semidirect
126. #11929434#41semidirect
170. C157#3353semidirect
176. #484D157semidirect

Maximal subgroups

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

subgroupindex
#237579942
#237579242
#237576772
#237576072
#237576002
#237575982
#237575562
#237575522
#237575222
#237575212
#237573582
#237573482
#237573142
#237573132
#484×C1572
#3353157