Finite Group Databasebeta

← #34112101

Subgroups — #34112101

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#23757923C211yesmaximal, characteristic, complemented
2#23757917C211yesmaximal, characteristic, complemented
3#23757631C211yesmaximal, characteristic, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757590C211yesmaximal, characteristic, complemented
6#23757561C211yesmaximal, characteristic, complemented
7#23757523C211yesmaximal, characteristic, complemented
8#23757500C211yesmaximal, characteristic, complemented
9#23757486C211yesmaximal, characteristic, complemented
10#23757415C211yesmaximal, characteristic, complemented
11#23757385C211yesmaximal, characteristic, complemented
12#23757341C211yesmaximal, characteristic, complemented
13#23757271C211yesmaximal, characteristic, complemented
14C2×#19097692C211yesmaximal, characteristic, complemented
15#447×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×#14578953C2211yescharacteristic
18C2×#14578942C2211yescharacteristic, complemented
19C2×#14578937C2211yescharacteristic, complemented
20C22×#11929434C2211yescharacteristic
21#19097770C2211yescharacteristic, complemented
22#19097758C2211yescharacteristic
23#19097753—21no—
24#19097752—21no—
25#19097752—21no—
26#19097750C2211yescharacteristic
27#19097748—21no—
28#19097743—21no—
29#19097742C2211yescharacteristic
30#19097742C2211yescharacteristic, complemented
31#19097740—21no—
32D4×Dic157C2211yescharacteristic
33D4×Dic157C2211yescharacteristic, complemented
34C4×C22⋊D157C2211yescharacteristic
35C2×C4⋊Dic157C2211yescharacteristic
36C4⋊C4×D157C2211yescharacteristic, complemented
37C2×C22⋊Dic157—21no—
38#19097703C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097696C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097694C2211yescharacteristic, complemented
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52C22×D4×C157C2211yescharacteristic
53C2×D8×C157C2211yescharacteristic
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61#14578953—44no—
62C2×Dic314—21no—
63C2×Dic314C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157D411yescharacteristic
68C22×Dic157—21no—
69C22×Dic157—21no—
70C22×Dic157—21no—
71C22×Dic157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74#14578942—41no—
75#14578937—41no—
76C2×#11929434—41no—
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92C22⋊Dic157—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157D411yescharacteristic
95#14578931D411yescharacteristic
96C4×Dic157C2311yescharacteristic
97C24×C157—21no—
98C2×D4×C157—22no—
99C2×D4×C157—21no—
100C2×D4×C157—21no—
101C2×D4×C157—41no—
102C2×D4×C157C2311yescharacteristic
103C2×D4×C157C2311yescharacteristic
104C2×D4×C157C2311yescharacteristic
105C22×C4×C157C2311yescharacteristic
106D8×C157—22no—
107D8×C157—41no—
108C2×C8×C157C2311yescharacteristic
109C2×C8×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116C22⋊D157—22no—
117C22⋊D157—42no—
118C22⋊D157—42no—
119C22⋊D157—81no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—42no—
125C2×Dic157—42no—
126C2×Dic157—42no—
127C2×Dic157—81no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—21no—
132C4×D157—21no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157—21no—
137C23×C157—21no—
138C23×C157—21no—
139C23×C157—42no—
140C23×C157—42no—
141C23×C157—81no—
142C23×C157C2×D411yescharacteristic
143D4×C157—21no—
144D4×C157—21no—
145D4×C157—21no—
146D4×C157—21no—
147D4×C157—21no—
148D4×C157—42no—
149D4×C157—42no—
150D4×C157#4112yescomplemented
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—21no—
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2411yescharacteristic
156C8×C157—21no—
157C8×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—22no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157—41no—
168C22×C157—41no—
169C22×C157—84no—
170C22×C157AGL(1,ℤ/8)12yes—
171C22×C157C22×D411yescharacteristic
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—21no—
175C4×C157—21no—
176C4×C157—41no—
177C4×C157C2×#4111yescharacteristic
178C4×C157H(5,𝔽2)11yescharacteristic
179Dic157—21no—
180Dic157—21no—
181Dic157—21no—
182Dic157—41no—
183Dic157—41no—
184Dic157—41no—
185Dic157—41no—
186C2×C157—22no—
187C2×C157—42no—
188C2×C157—81no—
189C2×C157—81no—
190C2×C157#54611yescharacteristic
191C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C157#57711yescharacteristic
193D157—41no—
194C157#330611yescharacteristic, complemented, Sylow
195#3306—1571nomaximal, Sylow
196#547—1571no—
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#447D15711yescharacteristic, complemented
207#442—1571no—
208#435—1571no—
209#417—1571no—
210C2×#102—1571no—
211C2×#41—3141no—
212C2×#41—1571no—
213C22×D4C2×D15711yescharacteristic
214C2×SD16—1571no—
215C2×D8C2×D15711yescharacteristic
216C2×D8C2×D15711yescharacteristic
217C2×M16—1571no—
218C22×C8—1571no—
219C4⋊Q8—1571no—
220#124—1571no—
221#123—3141no—
222#123—3141no—
223#123—3141no—
224C22⋊Q8—3141no—
225C22⋊Q8—1571no—
226#121—3141no—
227#121—1571no—
228#121—1571no—
229#121C2×D15711yescharacteristic
230#120—3141no—
231C4×D4—1571no—
232C4×D4—1571no—
233C4×D4—1571no—
234C2×C4⋊C4—1571no—
235C2×C4⋊C4—1571no—
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#102C2×D15711yescharacteristic
250#102C2×D15711yescharacteristic
251C22⋊C8—1571no—
252C22⋊C8C2×D15711yescharacteristic
253#97—1571no—
254C24—21no—
255#41—3142no—
256#41—6281no—
257#41—6284no—
258C2×Q8—3141no—
259C2×Q8—1571no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—3141no—
267C2×D4—1571no—
268C2×D4C22×D15711yescharacteristic
269C2×D4C22×D15711yescharacteristic
270C2×D4C22×D15711yescharacteristic
271C22×C4—3141no—
272C22×C4—3141no—
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—3141no—
276C22×C4—1571no—
277C22×C4C22×D15711yescharacteristic
278SD16—6281no—
279D8—22no—
280D8—41no—
281M16—6281no—
282C2×C8—6281no—
283C2×C8—1571no—
284C2×C8—1571no—
285C2×C8C22×D15711yescharacteristic
286C2×C8C22×D15711yescharacteristic
287C4⋊C4—3141no—
288C4⋊C4—3141no—
289C4⋊C4—3141no—
290C4⋊C4—3141no—
291C4⋊C4—1571no—
292C4⋊C4—1571no—
293C4⋊C4—1571no—
294C4⋊C4—1571no—
295C4⋊C4C22×D15711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—3141no—
304C22⋊C4—1571no—
305C22⋊C4—1571no—
306C42—1571no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1571no—
314C23D4×D15711yescharacteristic
315Q8—3141no—
316Q8—3141no—
317Q8—6282no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—3142no—
326D4—6282no—
327D4—6282no—
328D4—12561no—
329D4#1457895312yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—3141no—
336C2×C4—3141no—
337C2×C4—3141no—
338C2×C4—3141no—
339C2×C4—6282no—
340C2×C4—6282no—
341C2×C4—6282no—
342C2×C4—6282no—
343C2×C4—1571no—
344C2×C4—1571no—
345C2×C4—1571no—
346C2×C4—12561no—
347C2×C4D4×D15711yescharacteristic
348C2×C4C23×D15711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3141no—
352C8—3141no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3141no—
366C22—3141no—
367C22—3141no—
368C22#1909779512yes—
369C22C2×D4×D15711yescharacteristic
370C4—21no—
371C4—41no—
372C4—3141no—
373C4—3141no—
374C4—3141no—
375C4—6281no—
376C4—6281no—
377C4—6281no—
378C4—6281no—
379C4C2×#1457895311yescharacteristic
380C4#1909781611yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—6281no—
386C2#2375791011yescharacteristic
387C2C2×#1909779511yescharacteristic
388C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#447×C157C2
socleC22×C157C22×D4
hypercenter#447D157
hypocenterC157#3306
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23757923C2semidirect
2. #23757917C2semidirect
3. #23757631C2semidirect
4. #23757604C2semidirect
5. #23757590C2semidirect
6. #23757561C2semidirect
7. #23757523C2semidirect
8. #23757500C2semidirect
9. #23757486C2semidirect
10. #23757415C2semidirect
11. #23757385C2semidirect
12. #23757341C2semidirect
13. #23757271C2semidirect
14. C2×#19097692C2semidirect
15. #447×C157C2semidirect
18. C2×#14578942C22semidirect
19. C2×#14578937C22semidirect
21. #19097770C22semidirect
30. #19097742C22semidirect
33. D4×Dic157C22semidirect
36. C4⋊C4×D157C22semidirect
38. #19097703C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097696C22semidirect
43. #19097696C22semidirect
44. #19097694C22semidirect
49. #19097692C22semidirect
50. #19097683C22semidirect
51. #19097681C22semidirect
150. D4×C157#41semidirect
194. C157#3306semidirect
206. #447D157semidirect
329. D4#14578953semidirect

Maximal subgroups

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

subgroupindex
#237579232
#237579172
#237576312
#237576042
#237575902
#237575612
#237575232
#237575002
#237574862
#237574152
#237573852
#237573412
#237572712
C2×#190976922
#447×C1572
#3306157