Finite Group Databasebeta

← #34110377

Subgroups — #34110377

Order 20096.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757877C211yesmaximal, characteristic, complemented
2#23757745C211yesmaximal, characteristic, complemented
3#23757596C211yesmaximal, characteristic, complemented
4#23757595C211yesmaximal, characteristic, complemented
5#23757574C211yesmaximal, characteristic, complemented
6#23757573C211yesmaximal, characteristic, complemented
7#23757465C211yesmaximal, characteristic, complemented
8#23757405C211yesmaximal, characteristic, complemented
9#23757405C211yesmaximal, characteristic, complemented
10#23757385C211yesmaximal, characteristic, complemented
11#23757382C211yesmaximal, characteristic, complemented
12D8×Dic157C211yesmaximal, characteristic, complemented
13#23757353C211yesmaximal, characteristic, complemented
14#23757333C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18C2×#14578939C2211yescharacteristic
19#19097763C2211yescharacteristic, complemented
20#19097752—21no—
21#19097752—21no—
22#19097743—21no—
23#19097743—21no—
24#19097742C2211yescharacteristic
25#19097742C2211yescharacteristic
26C4×D628C2211yescharacteristic
27D4×Dic157C2211yescharacteristic
28D4×Dic157C2211yescharacteristic, complemented
29D4×Dic157C2211yescharacteristic, complemented
30#19097733C2211yescharacteristic
31C4×C22⋊D157—21no—
32#19097731—21no—
33#19097729C2211yescharacteristic
34#19097728C2211yescharacteristic
35C2×#14578931—21no—
36C22⋊C4×D157—21no—
37#19097705C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41C4⋊#11929434C2211yescharacteristic, complemented
42#19097693C2211yescharacteristic
43#19097693C2211yescharacteristic
44#19097692C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic, complemented
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097681C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C2×D8×C157C2211yescharacteristic
53C2×D8×C157C2211yescharacteristic
54#127×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64D4×D157—22no—
65D4×D157—22no—
66C22×Dic157—21no—
67C22×Dic157—21no—
68C22×Dic157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157C2311yescharacteristic
72#14578939—22no—
73#14578939—22no—
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
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—
87C22⋊Dic157—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×D4×C157—21no—
96C2×D4×C157—21no—
97C2×D4×C157C2311yescharacteristic
98C2×D4×C157C2311yescharacteristic
99C2×D4×C157C2311yescharacteristic
100C22×C4×C157—21no—
101D8×C157—22no—
102D8×C157—22no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157C2311yescharacteristic
108C22×D157—21no—
109C22×D157—24no—
110C22×D157—81no—
111C22×D157—81no—
112C22×D157C2×D411yescharacteristic
113D628—21no—
114D628#4112yes—
115C22⋊D157—24no—
116C22⋊D157—81no—
117C22⋊D157—81no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—81no—
122C2×Dic157—81no—
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—
135C23×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—81no—
142D4×C157—81no—
143D4×C157#4112yes—
144C2×C4×C157—21no—
145C2×C4×C157—81no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C8×C157—21no—
150C8×C157—21no—
151C22×C157—22no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157C22×D411yescharacteristic
161C2×D157—21no—
162C2×D157—22no—
163C2×D157—21no—
164C2×D157—21no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—84no—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157—41no—
171C4×C157C2×#4111yescharacteristic
172C4×C157C2×#4111yescharacteristic
173Dic157—21no—
174Dic157—21no—
175Dic157—21no—
176Dic157—41no—
177Dic157—41no—
178Dic157—41no—
179C2×C157—42no—
180C2×C157—81no—
181C2×C157—81no—
182C2×C157#52511yescharacteristic
183C2×C157#57611yescharacteristic
184C2×C157#57611yescharacteristic
185D157—41no—
186D157—42no—
187C157#312211yescharacteristic, complemented, Sylow
188#3122—1571nomaximal, Sylow
189#540—1571no—
190#518—1571no—
191#482—1571no—
192#482—1571no—
193#478—1571no—
194#478—1571no—
195#459—1571no—
196#459D15711yescharacteristic, complemented
197#447—1571no—
198#447—1571no—
199#442—1571no—
200#442—1571no—
201C4×D8—1571no—
202C4×D8—1571no—
203#433—1571no—
204C22×D4—1571no—
205C2×D8—1571no—
206C2×D8—1571no—
207C2×D8C2×D15711yescharacteristic
208C2×D8C2×D15711yescharacteristic
209#127C2×D15711yescharacteristic, complemented
210#125—1571no—
211#123—3141no—
212#123—3141no—
213#121—3141no—
214#121—3141no—
215#121—1571no—
216#121—1571no—
217C4×D4—3141no—
218C4×D4—3141no—
219C4×D4—1571no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4—1571no—
223C4×D4—1571no—
224C4×D4C2×D15711yescharacteristic
225#117—1571no—
226#117—1571no—
227C2×C22⋊C4—3141no—
228C2×C22⋊C4—3141no—
229#107—1571no—
230#107—1571no—
231#106—1571no—
232#106—1571no—
233C4⋊C8—1571no—
234C4⋊C8C2×D15711yescharacteristic, complemented
235#102—1571no—
236#102—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241#102C2×D15711yescharacteristic, complemented
242#102C2×D15711yescharacteristic, complemented
243C22⋊C8—1571no—
244C22⋊C8—1571no—
245#97—1571no—
246C4×C8—1571no—
247C24—3141no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—3142no—
251C2×D4—3142no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4C22×D15711yescharacteristic
257C2×D4C22×D15711yescharacteristic
258C2×D4C22×D15711yescharacteristic
259C22×C4—21no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—3141no—
263C22×C4—3141no—
264C22×C4—3141no—
265C22×C4—1571no—
266D8—22no—
267D8—22no—
268D8—3142no—
269D8—3142no—
270C2×C8—1571no—
271C2×C8—1571no—
272C2×C8C22×D15711yescharacteristic
273C2×C8C22×D15711yescharacteristic
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—1571no—
277C4⋊C4—1571no—
278C4⋊C4—1571no—
279C4⋊C4—1571no—
280C4⋊C4—1571no—
281C4⋊C4C22×D15711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—1571no—
292C22⋊C4—1571no—
293C42—3141no—
294C42—1571no—
295C42C22×D15711yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—3144no—
300C23—3141no—
301C23—1571no—
302C23—12561no—
303C23—12561no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—3144no—
312D4—3141no—
313D4—1572no—
314D4—12561no—
315D4—12561no—
316D4#1457895312yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—1571no—
325C2×C4—1571no—
326C2×C4—1571no—
327C2×C4—12561no—
328C2×C4—12561no—
329C2×C4—12561no—
330C2×C4—12561no—
331C2×C4—12561no—
332C2×C4D4×D15711yescharacteristic
333C2×C4D4×D15711yescharacteristic
334C2×C4C23×D15711yescharacteristic
335C8—21no—
336C8—21no—
337C8—3141no—
338C8—3141no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—3141no—
349C22—3142no—
350C22—3141no—
351C22—3141no—
352C22—6281no—
353C22—6281no—
354C22—12564no—
355C22C2×D4×D15711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—3141no—
360C4—3141no—
361C4—3141no—
362C4—6281no—
363C4—6281no—
364C4—6281no—
365C4#41×D15711yescharacteristic
366C4C2×#1457895311yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—6281no—
371C2—6282no—
372C2#2375778311yescharacteristic
373C2#2375807411yescharacteristic
374C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3122
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23757877C2semidirect
2. #23757745C2semidirect
3. #23757596C2semidirect
4. #23757595C2semidirect
5. #23757574C2semidirect
6. #23757573C2semidirect
7. #23757465C2semidirect
8. #23757405C2semidirect
9. #23757405C2semidirect
10. #23757385C2semidirect
11. #23757382C2semidirect
12. D8×Dic157C2semidirect
13. #23757353C2semidirect
14. #23757333C2semidirect
15. #459×C157C2semidirect
19. #19097763C22semidirect
28. D4×Dic157C22semidirect
29. D4×Dic157C22semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. C4⋊#11929434C22semidirect
46. #19097692C22semidirect
47. #19097692C22semidirect
50. #19097681C22semidirect
51. C8×Dic157C22semidirect
54. #127×C157C22semidirect
56. C4⋊C8×C157C22semidirect
57. #102×C157C22semidirect
58. #102×C157C22semidirect
187. C157#3122semidirect
196. #459D157semidirect
209. #127C2×D157semidirect
234. C4⋊C8C2×D157semidirect
241. #102C2×D157semidirect
242. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578772
#237577452
#237575962
#237575952
#237575742
#237575732
#237574652
#237574052
#237573852
#237573822
D8×Dic1572
#237573532
#237573332
#459×C1572
#3122157