Finite Group Databasebeta

← #34110317

Subgroups — #34110317

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#23757892C211yesmaximal, characteristic, complemented
2#23757745C211yesmaximal, characteristic, complemented
3#23757613C211yesmaximal, characteristic, complemented
4#23757595C211yesmaximal, characteristic, complemented
5#23757563C211yesmaximal, characteristic, complemented
6#23757554C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757422C211yesmaximal, characteristic, complemented
9#23757405C211yesmaximal, characteristic
10#23757382C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12#23757358C211yesmaximal, characteristic, complemented
13D8×Dic157C211yesmaximal, characteristic, complemented
14#23757333C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578941C2211yescharacteristic
18C2×#14578939C2211yescharacteristic
19#19097771C2211yescharacteristic, complemented
20#19097764—21no—
21#19097757—21no—
22#19097755—21no—
23#19097748—21no—
24#19097747C2211yescharacteristic
25#19097742C2211yescharacteristic
26Q8×Dic157C2211yescharacteristic, complemented
27C4×D628C2211yescharacteristic
28D4×Dic157C2211yescharacteristic
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#19097705C2211yescharacteristic, complemented
39#19097702C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41C4⋊#11929434C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097693C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic, complemented
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097681C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic
53C2×D8×C157C2211yescharacteristic
54#124×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×Dic314—21no—
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64D4×D157—22no—
65D4×D157—22no—
66C22×Dic157—21no—
67C22×Dic157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578941—22no—
72#14578939—22no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C4⋊Dic157—21no—
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933D411yescharacteristic
82#14578933D411yescharacteristic
83C22⋊Dic157—21no—
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931D411yescharacteristic
91#14578931D411yescharacteristic
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×Q8×C157C2311yescharacteristic
96C2×D4×C157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C22×C4×C157—21no—
99SD16×C157—22no—
100D8×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
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—81no—
114D628—21no—
115D628#4112yes—
116C22⋊D157—24no—
117C22⋊D157—81no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—81no—
122C2×Dic157—81no—
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C4×D157—21no—
127C4×D157—21no—
128C4×D157—81no—
129C4×D157—81no—
130#11929434—21no—
131#11929434—21no—
132C23×C157—21no—
133C23×C157—21no—
134Q8×C157—21no—
135Q8×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157#4112yes—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
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#463D15711yescharacteristic, complemented
191#450—1571no—
192#447—1571no—
193#442—1571no—
194#440—1571no—
195C4×SD16—1571no—
196C4×D8—1571no—
197#433—1571no—
198C22×D4—1571no—
199C2×SD16—1571no—
200C2×SD16C2×D15711yescharacteristic
201C2×D8—1571no—
202C2×D8C2×D15711yescharacteristic
203C4⋊Q8—1571no—
204#126—3141no—
205#124—3141no—
206#124C2×D15711yescharacteristic, complemented
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×D4—1571no—
218C4×D4C2×D15711yescharacteristic
219#117—1571no—
220#117—1571no—
221C2×C22⋊C4—3141no—
222C2×C22⋊C4—3141no—
223#107—1571no—
224#107—1571no—
225#106—1571no—
226#106—1571no—
227C4⋊C8—1571no—
228C4⋊C8C2×D15711yescharacteristic, complemented
229#103—1571no—
230#103C2×D15711yescharacteristic, complemented
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#102—1571no—
235#102—1571no—
236#102C2×D15711yescharacteristic, complemented
237C22⋊C8—1571no—
238C22⋊C8—1571no—
239#97—1571no—
240C4×C8—1571no—
241C24—3141no—
242C2×Q8—3141no—
243C2×Q8C22×D15711yescharacteristic
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—3141no—
253C22×C4—3141no—
254C22×C4—3141no—
255C22×C4—3141no—
256C22×C4—1571no—
257SD16—22no—
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⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
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—21no—
298Q8—21no—
299Q8—12561no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—3144no—
304D4—3141no—
305D4—1572no—
306D4—12561no—
307D4#1457895312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—1571no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—12561no—
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×#1457895311yescharacteristic
356C2—42no—
357C2—81no—
358C2—6281no—
359C2—6282no—
360C2#2375778311yescharacteristic
361C2#2375807511yescharacteristic
362C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
prime radical#23757405C2
hypercenter#463D157
hypocenterC157#3118
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #23757892C2semidirect
2. #23757745C2semidirect
3. #23757613C2semidirect
4. #23757595C2semidirect
5. #23757563C2semidirect
6. #23757554C2semidirect
7. #23757485C2semidirect
8. #23757422C2semidirect
10. #23757382C2semidirect
11. #23757374C2semidirect
12. #23757358C2semidirect
13. D8×Dic157C2semidirect
14. #23757333C2semidirect
15. #463×C157C2semidirect
19. #19097771C22semidirect
26. Q8×Dic157C22semidirect
29. D4×Dic157C22semidirect
37. #19097705C22semidirect
38. #19097705C22semidirect
39. #19097702C22semidirect
40. #19097702C22semidirect
41. C4⋊#11929434C22semidirect
42. #19097695C22semidirect
47. #19097692C22semidirect
50. #19097681C22semidirect
51. C8×Dic157C22semidirect
54. #124×C157C22semidirect
56. C4⋊C8×C157C22semidirect
57. #103×C157C22semidirect
58. #102×C157C22semidirect
181. C157#3118semidirect
190. #463D157semidirect
206. #124C2×D157semidirect
228. C4⋊C8C2×D157semidirect
230. #103C2×D157semidirect
236. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578922
#237577452
#237576132
#237575952
#237575632
#237575542
#237574852
#237574222
#237574052
#237573822
#237573742
#237573582
D8×Dic1572
#237573332
#463×C1572
#3118157