Finite Group Databasebeta

← #34111973

Subgroups — #34111973

Order 20096.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757963C211yesmaximal, characteristic, complemented
2#23757919C211yesmaximal, characteristic
3#23757628C211yesmaximal, characteristic, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757598C211yesmaximal, characteristic, complemented
6#23757574C211yesmaximal, characteristic, complemented
7#23757514C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757496C211yesmaximal, characteristic, complemented
10#23757437C211yesmaximal, characteristic, complemented
11#23757426C211yesmaximal, characteristic, complemented
12C4×#14578943C211yesmaximal, characteristic, complemented
13#23757270C211yesmaximal, characteristic, complemented
14C2×#19097693C211yesmaximal, characteristic, complemented
15C8×D4×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×Dic628C2211yescharacteristic, complemented
19C2×#14578943C2211yescharacteristic
20C2×#14578943C2211yescharacteristic
21#19097764—21no—
22#19097763C2211yescharacteristic, complemented
23#19097761—21no—
24#19097759C2211yescharacteristic
25#19097756—21no—
26#19097752—21no—
27#19097750C2211yescharacteristic
28#19097750C2211yescharacteristic
29#19097750C2211yescharacteristic, complemented
30#19097745C2211yescharacteristic
31#19097744—21no—
32#19097739—21no—
33C4×Dic314C2211yescharacteristic, complemented
34C4×D628C2211yescharacteristic
35#19097730C2211yescharacteristic, complemented
36#19097727—21no—
37C2×C4⋊Dic157C2211yescharacteristic
38C22⋊C4×D157—21no—
39#19097705C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097703C2211yescharacteristic, complemented
42#19097696C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097696C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097693C2211yescharacteristic
52C22×C8×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic, complemented
54C4×D4×C157C2211yescharacteristic
55C4⋊C8×C157C2211yescharacteristic, complemented
56C22⋊C8×C157C2211yescharacteristic
57C22⋊C8×C157C2211yescharacteristic, complemented
58C4×C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62C2×Dic314C2311yescharacteristic
63C2×Dic314C2311yescharacteristic
64C2×D628—22no—
65C2×D628—21no—
66C2×D628—41no—
67C2×D628C2311yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74Dic628—41no—
75#14578943—22no—
76#14578943—41no—
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
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—
89C22⋊Dic157—21no—
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96C4×Dic157—21no—
97C4×Dic157—21no—
98C2×D4×C157D411yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C2×C8×C157—41no—
102C2×C8×C157—41no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116Dic314—21no—
117Dic314—21no—
118Dic314—21no—
119Dic314—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—42no—
124D628#4112yes—
125C22⋊D157—42no—
126C22⋊D157—81no—
127C2×Dic157—21no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—42no—
134C4×D157—42no—
135C4×D157—81no—
136C23×C157C2×D411yescharacteristic
137C23×C157C2×D411yescharacteristic
138D4×C157—22no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—21no—
149C8×C157—21no—
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157#13512yes—
158C22×C157C22×D411yescharacteristic
159C2×D157—22no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—84no—
166C4×C157—21no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157C2×#4111yescharacteristic
171C4×C157H(5,𝔽2)11yescharacteristic
172Dic157—41no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177Dic157—41no—
178C2×C157—22no—
179C2×C157—41no—
180C2×C157#54611yescharacteristic
181C2×C157C2×#13511yescharacteristic
182C2×C157#57711yescharacteristic
183D157—42no—
184D157—81no—
185C157#329611yescharacteristic, complemented, Sylow
186#3296—1571nomaximal, Sylow
187#552—1571no—
188#546—1571no—
189#488—1571no—
190#483—1571no—
191#482—1571no—
192#478—1571no—
193#467—1571no—
194#465—1571no—
195#464—1571no—
196#452—1571no—
197#450—1571no—
198C4×SD16—1571no—
199C8×D4D15711yescharacteristic, complemented
200#416—1571no—
201C2×#102—1571no—
202C2×#41—1571no—
203C22×D4—1571no—
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C22×C8C2×D15711yescharacteristic
208C22×C8C2×D15711yescharacteristic, complemented
209#126—3141no—
210#125—3141no—
211#125—1571no—
212#124—1571no—
213#123—3141no—
214#123—3141no—
215C22⋊Q8—1571no—
216C22⋊Q8—1571no—
217C22⋊Q8—1571no—
218#121—3141no—
219#121—1571no—
220#120—3141no—
221C4×Q8—1571no—
222C4×D4—1571no—
223C4×D4C2×D15711yescharacteristic
224#117—3141no—
225#117—1571no—
226C2×C4⋊C4—1571no—
227C2×C22⋊C4—3141no—
228#107—1571no—
229#106—1571no—
230#106—1571no—
231C4⋊C8C2×D15711yescharacteristic, complemented
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241#102—1571no—
242C22⋊C8C2×D15711yescharacteristic
243C22⋊C8C2×D15711yescharacteristic, complemented
244C4×C8C2×D15711yescharacteristic, complemented
245C24—3141no—
246#41—3142no—
247#41—6281no—
248C2×Q8—1571no—
249C2×Q8—1571no—
250C2×D4—3142no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—6281no—
255C2×D4—1571no—
256C2×D4—1571no—
257C2×D4D62811yescharacteristic
258C22×C4—3141no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4C22×D15711yescharacteristic
262C22×C4C22×D15711yescharacteristic
263Q16—6281no—
264SD16—3142no—
265SD16—6281no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D15711yescharacteristic
269C2×C8C22×D15711yescharacteristic
270C2×C8C22×D15711yescharacteristic
271C2×C8C22×D15711yescharacteristic
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4—1571no—
282C4⋊C4—1571no—
283C4⋊C4D62811yescharacteristic
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4D62811yescharacteristic
293C22⋊C4D62811yescharacteristic
294C42—3141no—
295C42—3141no—
296C42C22×D15711yescharacteristic
297C23—3141no—
298C23—3141no—
299C23—6282no—
300C23—6282no—
301C23—12561no—
302C23C2×D62811yescharacteristic
303C23C2×D62811yescharacteristic
304Q8—3141no—
305Q8—3141no—
306Q8—3141no—
307Q8—3141no—
308D4—22no—
309D4—3141no—
310D4—3141no—
311D4—3141no—
312D4—6282no—
313D4—6282no—
314D4—1572no—
315D4—12561no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—12561no—
329C2×C4C2×D62811yescharacteristic
330C2×C4C2×D62811yescharacteristic
331C2×C4C2×D62811yescharacteristic
332C2×C4C2×D62811yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3142no—
344C22—6281no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22—12564no—
350C22#1909779312yes—
351C22C22×D62811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—6281no—
357C4—6281no—
358C4—6281no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4#41×D15711yescharacteristic
363C4#1909781611yescharacteristic
364C2—22no—
365C2—41no—
366C2—6282no—
367C2—12561no—
368C2#2375791811yescharacteristic
369C2C2×#1909779311yescharacteristic
370C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
prime radical#23757919C2
hypercenterC8×D4D157
hypocenterC157#3296
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23757963C2semidirect
3. #23757628C2semidirect
4. #23757604C2semidirect
5. #23757598C2semidirect
6. #23757574C2semidirect
7. #23757514C2semidirect
8. #23757502C2semidirect
9. #23757496C2semidirect
10. #23757437C2semidirect
11. #23757426C2semidirect
12. C4×#14578943C2semidirect
13. #23757270C2semidirect
14. C2×#19097693C2semidirect
15. C8×D4×C157C2semidirect
18. C2×Dic628C22semidirect
22. #19097763C22semidirect
29. #19097750C22semidirect
33. C4×Dic314C22semidirect
35. #19097730C22semidirect
39. #19097705C22semidirect
40. #19097703C22semidirect
41. #19097703C22semidirect
42. #19097696C22semidirect
43. #19097696C22semidirect
44. #19097696C22semidirect
45. #19097696C22semidirect
53. C22×C8×C157C22semidirect
55. C4⋊C8×C157C22semidirect
57. C22⋊C8×C157C22semidirect
58. C4×C8×C157C22semidirect
185. C157#3296semidirect
199. C8×D4D157semidirect
208. C22×C8C2×D157semidirect
231. C4⋊C8C2×D157semidirect
243. C22⋊C8C2×D157semidirect
244. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579632
#237579192
#237576282
#237576042
#237575982
#237575742
#237575142
#237575022
#237574962
#237574372
#237574262
C4×#145789432
#237572702
C2×#190976932
C8×D4×C1572
#3296157