Finite Group Databasebeta

← #34109931

Subgroups — #34109931

Order 20096.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757829C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic
3#23757574C211yesmaximal, characteristic, complemented
4#23757573C211yesmaximal, characteristic, complemented
5#23757570C211yesmaximal, characteristic, complemented
6#23757562C211yesmaximal, characteristic, complemented
7#23757561C211yesmaximal, characteristic, complemented
8#23757552C211yesmaximal, characteristic, complemented
9#23757551C211yesmaximal, characteristic, complemented
10#23757314C211yesmaximal, characteristic, complemented
11#23757312C211yesmaximal, characteristic, complemented
12#23757295C211yesmaximal, characteristic, complemented
13#23757276C211yesmaximal, characteristic
14#23757276C211yesmaximal, characteristic
15#478×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C22×C4×D157—21no—
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20#19097770C2211yescharacteristic, complemented
21#19097770C2211yescharacteristic, complemented
22#19097763C2211yescharacteristic, complemented
23#19097749—21no—
24#19097749—21no—
25#19097736—21no—
26#19097736—21no—
27C4×D628C2211yescharacteristic
28D4×Dic157C2211yescharacteristic
29#19097733C2211yescharacteristic
30C4×C22⋊D157—21no—
31#19097731—21no—
32#19097729C2211yescharacteristic
33#19097729C2211yescharacteristic
34C4⋊C4×D157C2211yescharacteristic
35C22⋊C4×D157—21no—
36C42×D157C2211yescharacteristic
37#19097705C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097704C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097702C2211yescharacteristic, complemented
42#19097702C2211yescharacteristic, complemented
43C4⋊#11929434C2211yescharacteristic, complemented
44#19097700C2211yescharacteristic, complemented
45#19097700C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097691C2211yescharacteristic
51#19097691C2211yescharacteristic
52#125×C157C2211yescharacteristic, complemented
53C4×D4×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic, complemented
55#106×C157C2211yescharacteristic, complemented
56C4⋊C8×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59C23×D157—21no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64D4×D157—22no—
65D4×D157Q814yescomplemented
66C22×Dic157—21no—
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—44no—
71C2×C4×D157D411yescharacteristic
72C2×C4×D157D411yescharacteristic
73C2×C4×D157C2311yescharacteristic
74#14578937—22no—
75#14578937—22no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86C22⋊Dic157—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157D411yescharacteristic
94C4×Dic157D411yescharacteristic
95C4×Dic157C2311yescharacteristic
96C2×D4×C157C2311yescharacteristic
97C22×C4×C157—21no—
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—
106C42×C157C2311yescharacteristic
107C22×D157—21no—
108C22×D157—24no—
109C22×D157—42no—
110C22×D157—42no—
111C22×D157C2×D411yescharacteristic
112Dic314—42no—
113Dic314—42no—
114D628—21no—
115D628C2×Q812yes—
116C22⋊D157—24no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—42no—
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C4×D157—22no—
125C4×D157—41no—
126C4×D157—42no—
127C4×D157—42no—
128C4×D157—42no—
129C4×D157C2×Q812yes—
130C4×D157#4112yes—
131#11929434—21no—
132#11929434—21no—
133C23×C157—21no—
134D4×C157—21no—
135D4×C157C2×Q812yes—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—42no—
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2411yescharacteristic
143C8×C157—21no—
144C8×C157—21no—
145C22×C157—22no—
146C22×C157—41no—
147C22×C157—41no—
148C22×C157C22×D411yescharacteristic
149C2×D157—21no—
150C2×D157—21no—
151C2×D157—22no—
152C2×D157—48no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157C22⋊Q812yes—
156C4×C157—21no—
157C4×C157—21no—
158C4×C157—41no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157C22×Q811yescharacteristic
162C4×C157C2×#4111yescharacteristic
163Dic157—21no—
164Dic157—41no—
165Dic157—41no—
166Dic157—41no—
167Dic157C22⋊Q812yes—
168Dic157AGL(1,ℤ/8)12yes—
169C2×C157—42no—
170C2×C157C2×C22⋊Q811yescharacteristic
171C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C157#57511yescharacteristic
173D157—22no—
174D157—42no—
175C157#307711yescharacteristic, complemented, Sylow
176#3077—1571nomaximal, Sylow
177#533—1571no—
178C2×C4×D4—1571no—
179#478—1571no—
180#478—1571no—
181#478—1571no—
182#478D15711yescharacteristic, complemented
183#476—1571no—
184#476—1571no—
185#474—1571no—
186#474—1571no—
187#428—1571no—
188#428—1571no—
189#423—1571no—
190#418—1571no—
191#418—1571no—
192C22×D4—1571no—
193C23×C4—3141no—
194C2×M16—1571no—
195C2×M16—1571no—
196C4⋊Q8—1571no—
197C4⋊Q8—1571no—
198#125—1571no—
199#125C2×D15711yescharacteristic, complemented
200C22⋊Q8—3141no—
201C22⋊Q8—3141no—
202C4×Q8—3141no—
203C4×Q8—3141no—
204C4×D4—3141no—
205C4×D4—3141no—
206C4×D4—1571no—
207C4×D4—1571no—
208C4×D4—1571no—
209C4×D4C2×D15711yescharacteristic
210#117—1571no—
211#117—1571no—
212C2×C4⋊C4—1571no—
213C2×C22⋊C4—3141no—
214C2×C42—1571no—
215#107—1571no—
216#107—1571no—
217#107—1571no—
218#107C2×D15711yescharacteristic, complemented
219#106—1571no—
220#106—1571no—
221#106—1571no—
222#106C2×D15711yescharacteristic, complemented
223C4⋊C8—1571no—
224C4⋊C8—1571no—
225C4⋊C8—1571no—
226C4⋊C8C2×D15711yescharacteristic, complemented
227#102—1571no—
228#102—1571no—
229#102—1571no—
230#102—1571no—
231#102—1571no—
232#102—1571no—
233#102C2×D15711yescharacteristic
234#102C2×D15711yescharacteristic
235C24—3141no—
236C2×Q8—3141no—
237C2×Q8—3141no—
238C2×D4—3142no—
239C2×D4—3141no—
240C2×D4—1574no—
241C2×D4—1571no—
242C2×D4C22×D15711yescharacteristic
243C22×C4—21no—
244C22×C4—3141no—
245C22×C4—3141no—
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4—6284no—
249C22×C4—1571no—
250C22×C4—1571no—
251C22×C4—1571no—
252M16—3142no—
253M16—3142no—
254C2×C8—1571no—
255C2×C8—1571no—
256C2×C8C22×D15711yescharacteristic
257C2×C8C22×D15711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—3141no—
264C4⋊C4—3141no—
265C4⋊C4—1571no—
266C4⋊C4—1571no—
267C4⋊C4—1571no—
268C4⋊C4C22×D15711yescharacteristic
269C4⋊C4C22×D15711yescharacteristic
270C4⋊C4C22×D15711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C42—3141no—
278C42—3141no—
279C42—1571no—
280C42—1571no—
281C42—1571no—
282C42C22×D15711yescharacteristic
283C23—21no—
284C23—3144no—
285C23—3141no—
286C23—6282no—
287C23—6282no—
288C23—1571no—
289Q8—6282no—
290Q8—6282no—
291D4—21no—
292D4—3144no—
293D4—3141no—
294D4—1572no—
295D4Q8×D15712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—3142no—
301C2×C4—3141no—
302C2×C4—3141no—
303C2×C4—3141no—
304C2×C4—6281no—
305C2×C4—6282no—
306C2×C4—6282no—
307C2×C4—6282no—
308C2×C4—6282no—
309C2×C4—1571no—
310C2×C4—1571no—
311C2×C4—1571no—
312C2×C4—1572no—
313C2×C4—1572no—
314C2×C4D4×D15711yescharacteristic
315C2×C4D4×D15711yescharacteristic
316C2×C4C23×D15711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3141no—
320C8—3141no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3141no—
325C22—3141no—
326C22—3142no—
327C22—6288no—
328C22—6281no—
329C22—6281no—
330C22—1572no—
331C22C2×D4×D15711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3141no—
338C4—6281no—
339C4—6281no—
340C4—6281no—
341C4—1572no—
342C4—1572no—
343C4C2×Q8×D15711yescharacteristic
344C4#41×D15711yescharacteristic
345C2—42no—
346C2—3142no—
347C2—6282no—
348C2C22⋊Q8×D15711yescharacteristic
349C2AGL(1,ℤ/8)×D15711yescharacteristic
350C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#478×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#478D157
hypocenterC157#3077
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23757829C2semidirect
3. #23757574C2semidirect
4. #23757573C2semidirect
5. #23757570C2semidirect
6. #23757562C2semidirect
7. #23757561C2semidirect
8. #23757552C2semidirect
9. #23757551C2semidirect
10. #23757314C2semidirect
11. #23757312C2semidirect
12. #23757295C2semidirect
15. #478×C157C2semidirect
20. #19097770C22semidirect
21. #19097770C22semidirect
22. #19097763C22semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097704C22semidirect
40. #19097703C22semidirect
41. #19097702C22semidirect
42. #19097702C22semidirect
43. C4⋊#11929434C22semidirect
44. #19097700C22semidirect
45. #19097700C22semidirect
52. #125×C157C22semidirect
54. #107×C157C22semidirect
55. #106×C157C22semidirect
56. C4⋊C8×C157C22semidirect
65. D4×D157Q8semidirect
175. C157#3077semidirect
182. #478D157semidirect
199. #125C2×D157semidirect
218. #107C2×D157semidirect
222. #106C2×D157semidirect
226. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578292
C4×D4×D1572
#237575742
#237575732
#237575702
#237575622
#237575612
#237575522
#237575512
#237573142
#237573122
#237572952
#237572762
#478×C1572
#3077157