Finite Group Databasebeta

← #34110316

Subgroups — #34110316

Order 20096.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757888C211yesmaximal, characteristic, complemented
2#23757745C211yesmaximal, characteristic, complemented
3#23757616C211yesmaximal, characteristic, complemented
4#23757596C211yesmaximal, characteristic, complemented
5#23757561C211yesmaximal, characteristic, complemented
6#23757552C211yesmaximal, characteristic, complemented
7#23757487C211yesmaximal, characteristic, complemented
8#23757486C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic, complemented
10#23757405C211yesmaximal, characteristic
11#23757383C211yesmaximal, characteristic, complemented
12#23757373C211yesmaximal, characteristic, complemented
13C4×#14578943C211yesmaximal, characteristic, complemented
14#23757332C211yesmaximal, characteristic, complemented
15C4×D8×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578943C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20#19097770C2211yescharacteristic, complemented
21#19097764—21no—
22#19097758C2211yescharacteristic, complemented
23#19097757—21no—
24#19097754—21no—
25#19097751C2211yescharacteristic
26#19097750—21no—
27#19097742C2211yescharacteristic
28C4×Dic314C2211yescharacteristic, complemented
29C4×D628C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic
35#19097728C2211yescharacteristic
36C2×#14578931—21no—
37C22⋊C4×D157—21no—
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097694C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic, complemented
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097682C2211yescharacteristic, complemented
52C2×D8×C157C2211yescharacteristic
53C4×D4×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic, complemented
55#107×C157C2211yescharacteristic, complemented
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic, complemented
58C4×C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×Dic314—21no—
61C2×Dic314C2311yescharacteristic
62C2×D628C2311yescharacteristic
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65D4×D157—22no—
66D4×D157—22no—
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578943—22no—
72#14578942—22no—
73#14578939—22no—
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80#14578933—21no—
81#14578933D411yescharacteristic
82#14578933D411yescharacteristic
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96C2×D4×C157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C22×C4×C157—21no—
99C22×C4×C157—21no—
100D8×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C4⋊C4×C157C2311yescharacteristic
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C42×C157C2311yescharacteristic
108C22×D157—21no—
109C22×D157—24no—
110C22×D157—81no—
111C22×D157—81no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—81no—
116D628—21no—
117D628#4112yes—
118C22⋊D157—24no—
119C22⋊D157—81no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—81no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—21no—
128C4×D157—21no—
129C4×D157—81no—
130C4×D157—81no—
131#11929434—21no—
132#11929434—21no—
133C23×C157—21no—
134C23×C157—21no—
135D4×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157#4112yes—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—81no—
142C2×C4×C157—81no—
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157—21no—
148C22×C157—22no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157C22×D411yescharacteristic
155C2×D157—21no—
156C2×D157—22no—
157C2×D157—21no—
158C2×D157—21no—
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—84no—
162C4×C157—21no—
163C4×C157—21no—
164C4×C157—41no—
165C4×C157—41no—
166C4×C157C2×#4111yescharacteristic
167C4×C157C2×#4111yescharacteristic
168Dic157—21no—
169Dic157—21no—
170Dic157—21no—
171Dic157—41no—
172Dic157—41no—
173Dic157—41no—
174C2×C157—42no—
175C2×C157—81no—
176C2×C157#52511yescharacteristic
177C2×C157#57611yescharacteristic
178C2×C157#57711yescharacteristic
179D157—41no—
180D157—42no—
181C157#311811yescharacteristic, complemented, Sylow
182#3118—1571nomaximal, Sylow
183#542—1571no—
184#518—1571no—
185#485—1571no—
186#482—1571no—
187#476—1571no—
188#474—1571no—
189#463—1571no—
190#463—1571no—
191#450—1571no—
192#447—1571no—
193#442—1571no—
194#440—1571no—
195C4×SD16—1571no—
196C4×D8D15711yescharacteristic, complemented
197#433—1571no—
198C22×D4—1571no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×D8—1571no—
202C2×D8C2×D15711yescharacteristic
203C4⋊Q8—1571no—
204#126—3141no—
205#124—3141no—
206#124—1571no—
207#123—3141no—
208C22⋊Q8—3141no—
209C22⋊Q8—1571no—
210#121—1571no—
211C4×Q8—1571no—
212C4×D4—3141no—
213C4×D4—3141no—
214C4×D4—1571no—
215C4×D4—1571no—
216C4×D4—1571no—
217C4×D4C2×D15711yescharacteristic
218C4×D4C2×D15711yescharacteristic, complemented
219#117—1571no—
220#117—1571no—
221C2×C22⋊C4—3141no—
222C2×C22⋊C4—3141no—
223#107—1571no—
224#107C2×D15711yescharacteristic, complemented
225#106—1571no—
226#106—1571no—
227C4⋊C8—1571no—
228C4⋊C8—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102—1571no—
235#102C2×D15711yescharacteristic
236#102C2×D15711yescharacteristic, complemented
237C22⋊C8—1571no—
238C22⋊C8—1571no—
239#97—1571no—
240C4×C8C2×D15711yescharacteristic, complemented
241C24—3141no—
242C2×Q8—3141no—
243C2×Q8—1571no—
244C2×D4—3142no—
245C2×D4—3142no—
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4C22×D15711yescharacteristic
250C2×D4C22×D15711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—1571no—
257SD16—3142no—
258SD16—3142no—
259D8—22no—
260D8—3142no—
261C2×C8—1571no—
262C2×C8—1571no—
263C2×C8C22×D15711yescharacteristic
264C2×C8C22×D15711yescharacteristic
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—1571no—
269C4⋊C4—1571no—
270C4⋊C4—1571no—
271C4⋊C4—1571no—
272C4⋊C4C22×D15711yescharacteristic
273C4⋊C4C22×D15711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—1571no—
285C22⋊C4—1571no—
286C42—3141no—
287C42—3141no—
288C42—1571no—
289C42C22×D15711yescharacteristic
290C23—21no—
291C23—21no—
292C23—3144no—
293C23—3141no—
294C23—1571no—
295C23—12561no—
296C23—12561no—
297Q8—3141no—
298Q8—3141no—
299Q8—12561no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—3144no—
304D4—3141no—
305D4—1572no—
306D4—12561no—
307D4#1457895412yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—1571no—
320C2×C4—12561no—
321C2×C4—12561no—
322C2×C4—12561no—
323C2×C4D4×D15711yescharacteristic
324C2×C4D4×D15711yescharacteristic
325C2×C4C23×D15711yescharacteristic
326C8—21no—
327C8—21no—
328C8—3141no—
329C8—3141no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—3141no—
337C22—3141no—
338C22—3142no—
339C22—3141no—
340C22—6281no—
341C22—6281no—
342C22—12564no—
343C22C2×D4×D15711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—3141no—
349C4—3141no—
350C4—3141no—
351C4—6281no—
352C4—6281no—
353C4—6281no—
354C4#41×D15711yescharacteristic
355C4C2×#1457895411yescharacteristic
356C2—42no—
357C2—81no—
358C2—6281no—
359C2—6282no—
360C2#2375778211yescharacteristic
361C2#2375807411yescharacteristic
362C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC4×D8×C157C2
socleC22×C157C22×D4
prime radical#23757405C2
hypercenterC4×D8D157
hypocenterC157#3118
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #23757888C2semidirect
2. #23757745C2semidirect
3. #23757616C2semidirect
4. #23757596C2semidirect
5. #23757561C2semidirect
6. #23757552C2semidirect
7. #23757487C2semidirect
8. #23757486C2semidirect
9. #23757425C2semidirect
11. #23757383C2semidirect
12. #23757373C2semidirect
13. C4×#14578943C2semidirect
14. #23757332C2semidirect
15. C4×D8×C157C2semidirect
20. #19097770C22semidirect
22. #19097758C22semidirect
28. C4×Dic314C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097696C22semidirect
44. #19097694C22semidirect
47. #19097692C22semidirect
51. #19097682C22semidirect
54. C4×D4×C157C22semidirect
55. #107×C157C22semidirect
57. #102×C157C22semidirect
58. C4×C8×C157C22semidirect
181. C157#3118semidirect
196. C4×D8D157semidirect
218. C4×D4C2×D157semidirect
224. #107C2×D157semidirect
236. #102C2×D157semidirect
240. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578882
#237577452
#237576162
#237575962
#237575612
#237575522
#237574872
#237574862
#237574252
#237574052
#237573832
#237573732
C4×#145789432
#237573322
C4×D8×C1572
#3118157