Finite Group Databasebeta

← #34110233

Subgroups — #34110233

Order 20096.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757827C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic
3#23757610C211yesmaximal, characteristic, complemented
4#23757607C211yesmaximal, characteristic, complemented
5#23757552C211yesmaximal, characteristic, complemented
6#23757551C211yesmaximal, characteristic, complemented
7#23757477C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic
10#23757422C211yesmaximal, characteristic
11#23757375C211yesmaximal, characteristic, complemented
12#23757373C211yesmaximal, characteristic, complemented
13#23757371C211yesmaximal, characteristic, complemented
14#23757328C211yesmaximal, characteristic, complemented
15#440×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C22×C4×D157—21no—
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21#19097770C2211yescharacteristic, complemented
22#19097769C2211yescharacteristic, complemented
23#19097761—21no—
24#19097751C2211yescharacteristic
25#19097749—21no—
26#19097747C2211yescharacteristic
27C4×Dic314—21no—
28C4×Dic314C2211yescharacteristic, complemented
29C4×D628C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097728—21no—
35C4⋊C4×D157C2211yescharacteristic
36C22⋊C4×D157—21no—
37C42×D157C2211yescharacteristic
38#19097705C2211yescharacteristic, complemented
39#19097704C2211yescharacteristic, complemented
40#19097704C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097694C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097682C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic, complemented
54C4×D4×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×Dic314—21no—
61C2×Dic314C2311yescharacteristic
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64D4×D157—22no—
65D4×D157—22no—
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#14578943—22no—
75#14578942—22no—
76#14578941—22no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88C22⋊Dic157—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93C4×Dic157—21no—
94C4×Dic157D411yescharacteristic
95C4×Dic157D411yescharacteristic
96C4×Dic157C2311yescharacteristic
97C2×Q8×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C22×C4×C157—21no—
100SD16×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157—21no—
104C4⋊C4×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157—21no—
108C42×C157C2311yescharacteristic
109C22×D157—24no—
110C22×D157—21no—
111C22×D157—42no—
112C22×D157—42no—
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117D628—21no—
118D628#4112yes—
119C22⋊D157—24no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—42no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—22no—
128C4×D157—21no—
129C4×D157—21no—
130C4×D157—22no—
131C4×D157—42no—
132C4×D157—42no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157—21no—
137Q8×C157—21no—
138Q8×C157—21no—
139D4×C157—21no—
140D4×C157#4112yes—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—42no—
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—21no—
149C8×C157—21no—
150C22×C157—22no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157C22×D411yescharacteristic
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—22no—
157C2×D157—48no—
158C2×D157—41no—
159C2×D157—41no—
160C2×D157#13712yes—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157—41no—
166C4×C157C2×#4111yescharacteristic
167C4×C157C2×#4111yescharacteristic
168Dic157—21no—
169Dic157—21no—
170Dic157—41no—
171Dic157—41no—
172Dic157—41no—
173Dic157AGL(1,ℤ/8)12yes—
174C2×C157—42no—
175C2×C157#52511yescharacteristic
176C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C157C2×#13711yescharacteristic
178D157—22no—
179D157—42no—
180C157#311011yescharacteristic, complemented, Sylow
181#3110—1571nomaximal, Sylow
182#533—1571no—
183C2×C4×D4—1571no—
184#484—1571no—
185#484—1571no—
186#474—1571no—
187#474—1571no—
188#461—1571no—
189#461—1571no—
190#450—1571no—
191#450—1571no—
192#440—1571no—
193#440—1571no—
194#440—1571no—
195#440D15711yescharacteristic, complemented
196#432—1571no—
197C22×D4—1571no—
198C23×C4—3141no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×SD16—1571no—
202C2×SD16C2×D15711yescharacteristic
203C4⋊Q8—1571no—
204C4⋊Q8—1571no—
205#125—3141no—
206C22⋊Q8—3141no—
207C22⋊Q8—1571no—
208C22⋊Q8—1571no—
209C4×Q8—3141no—
210C4×Q8—1571no—
211C4×Q8C2×D15711yescharacteristic, complemented
212C4×D4—3141no—
213C4×D4—3141no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4C2×D15711yescharacteristic
218#117—3141no—
219C2×C4⋊C4—1571no—
220C2×C22⋊C4—3141no—
221C2×C42—1571no—
222#107—1571no—
223#107—1571no—
224#107—1571no—
225#107C2×D15711yescharacteristic, complemented
226C4⋊C8—1571no—
227C4⋊C8—1571no—
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#103C2×D15711yescharacteristic, complemented
232#102—1571no—
233#102—1571no—
234#102—1571no—
235#102C2×D15711yescharacteristic
236C22⋊C8—1571no—
237C22⋊C8—1571no—
238#97—1571no—
239#97C2×D15711yescharacteristic, complemented
240C24—3141no—
241C2×Q8—3141no—
242C2×Q8—1571no—
243C2×Q8C22×D15711yescharacteristic
244C2×D4—3142no—
245C2×D4—3142no—
246C2×D4—3141no—
247C2×D4—1571no—
248C2×D4C22×D15711yescharacteristic
249C22×C4—21no—
250C22×C4—3141no—
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4—3141no—
254C22×C4—6284no—
255C22×C4—1571no—
256C22×C4—1571no—
257C22×C4—1571no—
258SD16—22no—
259SD16—3142no—
260SD16—3142no—
261SD16—3142no—
262C2×C8—1571no—
263C2×C8—1571no—
264C2×C8C22×D15711yescharacteristic
265C2×C8C22×D15711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4C22×D15711yescharacteristic
277C4⋊C4C22×D15711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C42—21no—
285C42—3141no—
286C42—1571no—
287C42—1571no—
288C42—1571no—
289C42C22×D15711yescharacteristic
290C23—21no—
291C23—3144no—
292C23—3141no—
293C23—6282no—
294C23—6282no—
295C23—1571no—
296Q8—21no—
297Q8—21no—
298Q8—3141no—
299Q8—3141no—
300Q8—6282no—
301D4—21no—
302D4—3144no—
303D4—3141no—
304D4—1572no—
305D4#1457895412yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3142no—
311C2×C4—3142no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—6282no—
318C2×C4—6282no—
319C2×C4—6282no—
320C2×C4—6282no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4—1571no—
324C2×C4D4×D15711yescharacteristic
325C2×C4D4×D15711yescharacteristic
326C2×C4C23×D15711yescharacteristic
327C8—21no—
328C8—21no—
329C8—3141no—
330C8—3141no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—3141no—
335C22—3141no—
336C22—3142no—
337C22—6288no—
338C22—6281no—
339C22—6281no—
340C22—1572no—
341C22C2×D4×D15711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3141no—
348C4—3141no—
349C4—6281no—
350C4—6281no—
351C4—6281no—
352C4—1572no—
353C4#41×D15711yescharacteristic
354C4C2×#1457895411yescharacteristic
355C2—42no—
356C2—3142no—
357C2—6282no—
358C2#2375778211yescharacteristic
359C2AGL(1,ℤ/8)×D15711yescharacteristic
360C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#440×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#440D157
hypocenterC157#3110
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23757827C2semidirect
3. #23757610C2semidirect
4. #23757607C2semidirect
5. #23757552C2semidirect
6. #23757551C2semidirect
7. #23757477C2semidirect
8. #23757476C2semidirect
11. #23757375C2semidirect
12. #23757373C2semidirect
13. #23757371C2semidirect
14. #23757328C2semidirect
15. #440×C157C2semidirect
21. #19097770C22semidirect
22. #19097769C22semidirect
28. C4×Dic314C22semidirect
38. #19097705C22semidirect
39. #19097704C22semidirect
40. #19097704C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097696C22semidirect
44. #19097695C22semidirect
45. #19097694C22semidirect
51. #19097682C22semidirect
53. C4×Q8×C157C22semidirect
55. #107×C157C22semidirect
56. #103×C157C22semidirect
58. #97×C157C22semidirect
180. C157#3110semidirect
195. #440D157semidirect
211. C4×Q8C2×D157semidirect
225. #107C2×D157semidirect
231. #103C2×D157semidirect
239. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578272
C4×D4×D1572
#237576102
#237576072
#237575522
#237575512
#237574772
#237574762
#237574252
#237574222
#237573752
#237573732
#237573712
#237573282
#440×C1572
#3110157