Finite Group Databasebeta

← #34112091

Subgroups — #34112091

Order 20096.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757919C211yesmaximal, characteristic, complemented
2#23757635C211yesmaximal, characteristic, complemented
3#23757603C211yesmaximal, characteristic, complemented
4#23757591C211yesmaximal, characteristic, complemented
5#23757560C211yesmaximal, characteristic, complemented
6#23757524C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757416C211yesmaximal, characteristic, complemented
10#23757406C211yesmaximal, characteristic, complemented
11#23757383C211yesmaximal, characteristic, complemented
12#23757345C211yesmaximal, characteristic, complemented
13#23757272C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#547×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578942C2211yescharacteristic, complemented
19C2×#14578939C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C2×#14578938C2211yescharacteristic, complemented
22C22×#11929434C2211yescharacteristic
23#19097759C2211yescharacteristic
24#19097756—21no—
25#19097750C2211yescharacteristic
26#19097745C2211yescharacteristic
27#19097744—21no—
28#19097739—21no—
29C4×D628C2211yescharacteristic
30C22⋊C4×D157—21no—
31#19097702C2211yescharacteristic, complemented
32#19097702C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097694C2211yescharacteristic, complemented
36#19097694C2211yescharacteristic, complemented
37#19097694C2211yescharacteristic, complemented
38#19097692C2211yescharacteristic, complemented
39#19097691C2211yescharacteristic
40#19097691C2211yescharacteristic
41#19097691C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic, complemented
47#19097682C2211yescharacteristic, complemented
48C2×#41×C157—21no—
49C4⋊Q8×C157C2211yescharacteristic, complemented
50#123×C157—21no—
51#123×C157—21no—
52C22⋊Q8×C157—21no—
53#121×C157C2211yescharacteristic
54#121×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic, complemented
57C2×C4⋊C4×C157C2211yescharacteristic
58C2×C4⋊C4×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62C2×Dic314C2311yescharacteristic
63C2×D628—22no—
64C2×D628—21no—
65C2×D628—41no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72#14578942—41no—
73#14578939—22no—
74#14578939—41no—
75#14578938—41no—
76C2×#11929434—41no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#41×C157—44no—
89C2×Q8×C157—21no—
90C2×D4×C157—21no—
91C2×D4×C157—21no—
92C2×D4×C157D411yescharacteristic
93C2×D4×C157C2311yescharacteristic
94C22×C4×C157—21no—
95C22×C4×C157—21no—
96C22×C4×C157—21no—
97C22×C4×C157C2311yescharacteristic
98C22×C4×C157C2311yescharacteristic
99C4⋊C4×C157—21no—
100C4⋊C4×C157—21no—
101C4⋊C4×C157—21no—
102C4⋊C4×C157D411yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C4⋊C4×C157C2311yescharacteristic
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C22⋊C4×C157—21no—
108C22⋊C4×C157—21no—
109C22⋊C4×C157D411yescharacteristic
110C22⋊C4×C157D411yescharacteristic
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—81no—
117Dic314—21no—
118Dic314—21no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—42no—
123D628#4112yescomplemented
124C22⋊D157—42no—
125C22⋊D157—81no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C4×D157—42no—
129C4×D157—81no—
130#11929434—21no—
131#11929434—21no—
132#11929434—21no—
133#11929434—21no—
134C23×C157—21no—
135C23×C157C2×D411yescharacteristic
136C23×C157C2×D411yescharacteristic
137Q8×C157—42no—
138D4×C157—22no—
139D4×C157—21no—
140D4×C157—21no—
141D4×C157—42no—
142D4×C157—42no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157—42no—
152C2×C4×C157—42no—
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2×D411yescharacteristic
156C2×C4×C157C2×D411yescharacteristic
157C2×C4×C157C2411yescharacteristic
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—21no—
161C22×C157—21no—
162C22×C157—21no—
163C22×C157—41no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157AGL(1,ℤ/8)12yes—
167C22×C157C22×D411yescharacteristic
168C2×D157—22no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—84no—
175C4×C157—21no—
176C4×C157—21no—
177C4×C157—21no—
178C4×C157—21no—
179C4×C157—41no—
180C4×C157—41no—
181C4×C157—41no—
182C4×C157C2×#4111yescharacteristic
183C4×C157H(5,𝔽2)11yescharacteristic
184Dic157—41no—
185Dic157—41no—
186C2×C157—22no—
187C2×C157—41no—
188C2×C157—81no—
189C2×C157#54611yescharacteristic
190C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C157#57711yescharacteristic
192D157—42no—
193D157—81no—
194C157#330611yescharacteristic, complemented, Sylow
195#3306—1571nomaximal, Sylow
196#547D15711yescharacteristic, complemented
197#546—1571no—
198#489—1571no—
199#483—1571no—
200#481—1571no—
201#476—1571no—
202#469—1571no—
203#465—1571no—
204#463—1571no—
205#449—1571no—
206#447—1571no—
207#442—1571no—
208#435—1571no—
209#417—1571no—
210C2×#102—1571no—
211C2×#41—21no—
212C2×#41—1571no—
213C22×D4—1571no—
214C2×SD16—1571no—
215C2×D8—1571no—
216C2×D8—1571no—
217C2×M16—1571no—
218C22×C8—1571no—
219C4⋊Q8C2×D15711yescharacteristic, complemented
220#124—1571no—
221#123—21no—
222#123—21no—
223#123—3141no—
224C22⋊Q8—21no—
225C22⋊Q8—1571no—
226#121—3141no—
227#121—1571no—
228#121C2×D15711yescharacteristic
229#121C2×D15711yescharacteristic, complemented
230#120—3141no—
231C4×D4—1571no—
232C4×D4C2×D15711yescharacteristic
233C4×D4C2×D15711yescharacteristic, complemented
234C2×C4⋊C4C2×D15711yescharacteristic
235C2×C4⋊C4C2×D15711yescharacteristic, complemented
236C2×C22⋊C4—3141no—
237#106—1571no—
238#106—1571no—
239#106—1571no—
240C4⋊C8—1571no—
241#103—1571no—
242#103—1571no—
243#103—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248#102—1571no—
249#102—1571no—
250#102—1571no—
251C22⋊C8—1571no—
252C22⋊C8—1571no—
253#97—1571no—
254C24—3141no—
255#41—44no—
256#41—3142no—
257#41—6281no—
258C2×Q8—21no—
259C2×Q8—1571no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3142no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—6281no—
267C2×D4—1571no—
268C2×D4—1571no—
269C2×D4C22⋊D15711yescharacteristic
270C2×D4C22×D15711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—3141no—
275C22×C4—3141no—
276C22×C4C22×D15711yescharacteristic
277C22×C4C22×D15711yescharacteristic
278SD16—6281no—
279D8—3142no—
280D8—6281no—
281M16—6281no—
282C2×C8—6281no—
283C2×C8—1571no—
284C2×C8—1571no—
285C2×C8—1571no—
286C2×C8—1571no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—3141no—
291C4⋊C4—1571no—
292C4⋊C4C22⋊D15711yescharacteristic
293C4⋊C4C22×D15711yescharacteristic
294C4⋊C4C22×D15711yescharacteristic
295C4⋊C4C22×D15711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—3141no—
304C22⋊C4C22⋊D15711yescharacteristic
305C22⋊C4C22⋊D15711yescharacteristic
306C42C22×D15711yescharacteristic
307C23—21no—
308C23—3141no—
309C23—3141no—
310C23—6282no—
311C23—6282no—
312C23—12561no—
313C23C2×C22⋊D15711yescharacteristic
314C23C2×C22⋊D15711yescharacteristic
315Q8—42no—
316Q8—3141no—
317Q8—3141no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—3141no—
324D4—3141no—
325D4—3141no—
326D4—6282no—
327D4—6282no—
328D4—1572no—
329D4—12561no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—3141no—
341C2×C4—3141no—
342C2×C4—6282no—
343C2×C4—12561no—
344C2×C4C2×C22⋊D15711yescharacteristic
345C2×C4C2×C22⋊D15711yescharacteristic
346C2×C4C2×C22⋊D15711yescharacteristic
347C2×C4C2×C22⋊D15711yescharacteristic
348C2×C4C23×D15711yescharacteristic
349C8—3141no—
350C8—3141no—
351C8—3141no—
352C8—3141no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3142no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—12564no—
368C22#1909779712yes—
369C22C22×C22⋊D15711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—6281no—
378C4—6281no—
379C4#41×D15711yescharacteristic
380C4#1909781711yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—6282no—
385C2—12561no—
386C2#2375791111yescharacteristic
387C2C2×#1909779711yescharacteristic
388C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#547×C157C2
socleC22×C157C22×D4
hypercenter#547D157
hypocenterC157#3306
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #23757919C2semidirect
2. #23757635C2semidirect
3. #23757603C2semidirect
4. #23757591C2semidirect
5. #23757560C2semidirect
6. #23757524C2semidirect
7. #23757500C2semidirect
8. #23757488C2semidirect
9. #23757416C2semidirect
10. #23757406C2semidirect
11. #23757383C2semidirect
12. #23757345C2semidirect
13. #23757272C2semidirect
14. C2×#19097691C2semidirect
15. #547×C157C2semidirect
18. C2×#14578942C22semidirect
21. C2×#14578938C22semidirect
31. #19097702C22semidirect
32. #19097702C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097694C22semidirect
36. #19097694C22semidirect
37. #19097694C22semidirect
38. #19097692C22semidirect
46. C22⋊#11929434C22semidirect
47. #19097682C22semidirect
49. C4⋊Q8×C157C22semidirect
54. #121×C157C22semidirect
56. C4×D4×C157C22semidirect
58. C2×C4⋊C4×C157C22semidirect
123. D628#41semidirect
194. C157#3306semidirect
196. #547D157semidirect
219. C4⋊Q8C2×D157semidirect
229. #121C2×D157semidirect
233. C4×D4C2×D157semidirect
235. C2×C4⋊C4C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579192
#237576352
#237576032
#237575912
#237575602
#237575242
#237575002
#237574882
#237574162
#237574062
#237573832
#237573452
#237572722
C2×#190976912
#547×C1572
#3306157