Finite Group Databasebeta

← #34110083

Subgroups — #34110083

Order 20096.

410 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D157C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic
3#23757615C211yesmaximal, characteristic, complemented
4#23757614C211yesmaximal, characteristic, complemented
5#23757611C211yesmaximal, characteristic, complemented
6#23757610C211yesmaximal, characteristic, complemented
7#23757607C211yesmaximal, characteristic, complemented
8#23757605C211yesmaximal, characteristic, complemented
9#23757602C211yesmaximal, characteristic, complemented
10#23757314C211yesmaximal, characteristic, complemented
11#23757312C211yesmaximal, characteristic, complemented
12#23757280C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757235C211yesmaximal, characteristic
15#485×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20#19097751C2211yescharacteristic
21#19097750C2211yescharacteristic
22#19097749—21no—
23#19097748—21no—
24#19097747C2211yescharacteristic
25#19097730C2211yescharacteristic
26#19097729C2211yescharacteristic
27#19097729C2211yescharacteristic
28#19097728—21no—
29#19097727—21no—
30C4⋊C4×D157—21no—
31C4⋊C4×D157C2211yescharacteristic
32C4⋊C4×D157C2211yescharacteristic
33C4⋊C4×D157C2211yescharacteristic
34C22⋊C4×D157—21no—
35C22⋊C4×D157—21no—
36C42×D157—21no—
37#19097705C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097704C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097702C2211yescharacteristic, complemented
42#19097702C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097695C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic
52C22⋊Q8×C157C2211yescharacteristic
53#117×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic, complemented
55#106×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic
59C23×D157D411yescharacteristic
60C2×Dic314—21no—
61C2×Dic314C2311yescharacteristic
62Q8×D157—22no—
63Q8×D157—22no—
64C22×Dic157D411yescharacteristic
65C2×C4×D157—22no—
66C2×C4×D157—22no—
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157D411yescharacteristic
72C2×C4×D157D411yescharacteristic
73C2×C4×D157C2311yescharacteristic
74C2×C4×D157C2311yescharacteristic
75#14578937—22no—
76#14578937—22no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157—21no—
97C2×Q8×C157C2311yescharacteristic
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⋊C4×C157—21no—
107C42×C157—21no—
108C22×D157—22no—
109C22×D157—22no—
110C22×D157—41no—
111C22×D157C2×D411yescharacteristic
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116C2×Dic157—22no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C4×D157—22no—
124C4×D157—21no—
125C4×D157—21no—
126C4×D157—22no—
127C4×D157—42no—
128C4×D157—42no—
129C4×D157—42no—
130C4×D157—42no—
131C4×D157#4112yes—
132C4×D157#4112yes—
133#11929434—21no—
134#11929434—21no—
135C23×C157C2×D411yescharacteristic
136Q8×C157—21no—
137Q8×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157—21no—
148C22×C157—21no—
149C22×C157—21no—
150C22×C157—21no—
151C22×C157C22×D411yescharacteristic
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—44no—
157C2×D157#12312yes—
158C2×D157#13712yes—
159C4×C157—21no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157—41no—
163C4×C157—41no—
164C4×C157C2×#4111yescharacteristic
165C4×C157C2×#4111yescharacteristic
166Dic157—21no—
167Dic157—41no—
168Dic157—41no—
169Dic157—41no—
170Dic157—41no—
171Dic157#12312yes—
172C2×C157—41no—
173C2×C157C2×#12311yescharacteristic
174C2×C157C2×#13711yescharacteristic
175C2×C157#57511yescharacteristic
176D157—22no—
177D157—22no—
178C157#309511yescharacteristic, complemented, Sylow
179#3095—1571nomaximal, Sylow
180C2×C22⋊Q8—1571no—
181C2×#117—1571no—
182#485—1571no—
183#485—1571no—
184#485—1571no—
185#485D15711yescharacteristic, complemented
186#484—1571no—
187#484—1571no—
188#483—1571no—
189#483—1571no—
190#428—1571no—
191#428—1571no—
192#419—1571no—
193#419—1571no—
194#407—1571no—
195C22×Q8—1571no—
196C23×C4—1571no—
197C2×M16—1571no—
198C2×M16—1571no—
199C22⋊Q8—3141no—
200C22⋊Q8—3141no—
201C22⋊Q8—1571no—
202C22⋊Q8—1571no—
203C22⋊Q8—1571no—
204C22⋊Q8C2×D15711yescharacteristic
205#117—3141no—
206#117—3141no—
207#117—1571no—
208#117—1571no—
209#117—1571no—
210#117C2×D15711yescharacteristic
211C2×C4⋊C4—3141no—
212C2×C4⋊C4—1571no—
213C2×C4⋊C4—1571no—
214C2×C4⋊C4—1571no—
215C2×C22⋊C4—3141no—
216C2×C22⋊C4—3141no—
217C2×C42—3141no—
218#107—1571no—
219#107—1571no—
220#107—1571no—
221#107C2×D15711yescharacteristic, complemented
222#106—1571no—
223#106—1571no—
224#106—1571no—
225#106C2×D15711yescharacteristic, complemented
226#103—1571no—
227#103—1571no—
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#103—1571no—
232#103C2×D15711yescharacteristic, complemented
233#103C2×D15711yescharacteristic, complemented
234C22⋊C8—1571no—
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237C22⋊C8C2×D15711yescharacteristic
238C24—1571no—
239C2×Q8—3141no—
240C2×Q8—3142no—
241C2×Q8—3142no—
242C2×Q8—1571no—
243C2×Q8C22×D15711yescharacteristic
244C22×C4—3142no—
245C22×C4—3142no—
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4—3141no—
249C22×C4—3141no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4—1571no—
253C22×C4—1571no—
254C22×C4—1571no—
255C22×C4C22×D15711yescharacteristic
256M16—3142no—
257M16—3142no—
258C2×C8—1571no—
259C2×C8—1571no—
260C2×C8C22×D15711yescharacteristic
261C2×C8C22×D15711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3141no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—1571no—
270C4⋊C4—1571no—
271C4⋊C4—1571no—
272C4⋊C4C22×D15711yescharacteristic
273C4⋊C4C22×D15711yescharacteristic
274C4⋊C4C22×D15711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C42—21no—
284C42—3141no—
285C42—3141no—
286C42—3141no—
287C23—3142no—
288C23—3142no—
289C23—6281no—
290C23—1571no—
291C23—1571no—
292C23D4×D15711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3141no—
296Q8—3141no—
297Q8—6282no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—3142no—
305C2×C4—3142no—
306C2×C4—3142no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—6282no—
314C2×C4—6282no—
315C2×C4—6282no—
316C2×C4—6282no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—1572no—
320C2×C4—1572no—
321C2×C4D4×D15711yescharacteristic
322C2×C4C23×D15711yescharacteristic
323C8—21no—
324C8—21no—
325C8—3141no—
326C8—3141no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—3141no—
331C22—3141no—
332C22—3141no—
333C22—3141no—
334C22—6284no—
335C22—1572no—
336C22—1572no—
337C22C2×D4×D15711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3141no—
344C4—6281no—
345C4—6281no—
346C4—6281no—
347C4—6281no—
348C4—1572no—
349C4#41×D15711yescharacteristic
350C4#41×D15711yescharacteristic
351C2—41no—
352C2—3142no—
353C2—3142no—
354C2#123×D15711yescharacteristic
355C2#137×D15711yescharacteristic
356C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#485×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#485D157
hypocenterC157#3095
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #23757615C2semidirect
4. #23757614C2semidirect
5. #23757611C2semidirect
6. #23757610C2semidirect
7. #23757607C2semidirect
8. #23757605C2semidirect
9. #23757602C2semidirect
10. #23757314C2semidirect
11. #23757312C2semidirect
12. #23757280C2semidirect
13. #23757280C2semidirect
15. #485×C157C2semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097704C22semidirect
40. #19097703C22semidirect
41. #19097702C22semidirect
42. #19097702C22semidirect
43. #19097696C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097695C22semidirect
47. #19097694C22semidirect
48. #19097694C22semidirect
54. #107×C157C22semidirect
55. #106×C157C22semidirect
56. #103×C157C22semidirect
57. #103×C157C22semidirect
178. C157#3095semidirect
185. #485D157semidirect
221. #107C2×D157semidirect
225. #106C2×D157semidirect
232. #103C2×D157semidirect
233. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1572
#117×D1572
#237576152
#237576142
#237576112
#237576102
#237576072
#237576052
#237576022
#237573142
#237573122
#237572802
#237572352
#485×C1572
#3095157