Finite Group Databasebeta

← #34112099

Subgroups — #34112099

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#23757927C211yesmaximal, characteristic, complemented
2#23757919C211yesmaximal, characteristic, complemented
3#23757633C211yesmaximal, characteristic, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757592C211yesmaximal, characteristic, complemented
6#23757564C211yesmaximal, characteristic, complemented
7#23757526C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757489C211yesmaximal, characteristic, complemented
10#23757420C211yesmaximal, characteristic, complemented
11#23757409C211yesmaximal, characteristic, complemented
12#23757386C211yesmaximal, characteristic, complemented
13#23757275C211yesmaximal, characteristic, complemented
14C2×#19097693C211yesmaximal, characteristic, complemented
15#435×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×#14578953—21no—
18C22×D628C2211yescharacteristic
19C2×#14578943C2211yescharacteristic, complemented
20C2×D1256C2211yescharacteristic
21C2×D1256C2211yescharacteristic
22#19097771C2211yescharacteristic, complemented
23#19097759C2211yescharacteristic
24#19097756—21no—
25#19097755—21no—
26#19097755—21no—
27#19097751—21no—
28#19097750C2211yescharacteristic
29#19097745C2211yescharacteristic
30#19097745C2211yescharacteristic
31#19097745C2211yescharacteristic, complemented
32#19097744—21no—
33#19097739—21no—
34C4×D628C2211yescharacteristic
35C4×D628C2211yescharacteristic, complemented
36C2×C4⋊Dic157C2211yescharacteristic
37C2×C4⋊Dic157C2211yescharacteristic, complemented
38C22⋊C4×D157—21no—
39#19097703C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097703C2211yescharacteristic, complemented
42#19097696C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097693C2211yescharacteristic, complemented
52C2×M16×C157C2211yescharacteristic, complemented
53C22×C8×C157C2211yescharacteristic
54C4×D4×C157C2211yescharacteristic
55C4⋊C8×C157C2211yescharacteristic, complemented
56C22⋊C8×C157C2211yescharacteristic
57C22⋊C8×C157C2211yescharacteristic, complemented
58#97×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62#14578953—44no—
63C2×Dic314—21no—
64C2×Dic314C2311yescharacteristic
65C2×D628—22no—
66C2×D628—21no—
67C2×D628—41no—
68C2×D628C2311yescharacteristic
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C22×Dic157—21no—
76C22×Dic157—21no—
77C2×C4×D157—21no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80#14578943—41no—
81D1256—22no—
82D1256—41no—
83C4⋊Dic157—21no—
84C4⋊Dic157—21no—
85C4⋊Dic157—21no—
86C4⋊Dic157C2311yescharacteristic
87C4⋊Dic157C2311yescharacteristic
88C4⋊Dic157C2311yescharacteristic
89C4⋊Dic157C2311yescharacteristic
90#14578933—21no—
91C22⋊Dic157—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931—21no—
99C2×D4×C157D411yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C22×C4×C157C2311yescharacteristic
102M16×C157—41no—
103C2×C8×C157—41no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C4⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C22⋊C4×C157D411yescharacteristic
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—21no—
115C22×D157—42no—
116C22×D157—42no—
117C22×D157—81no—
118Dic314—21no—
119Dic314—21no—
120Dic314—42no—
121D628—21no—
122D628—21no—
123D628—21no—
124D628—21no—
125D628—21no—
126D628—42no—
127D628#4112yescomplemented
128C22⋊D157—42no—
129C22⋊D157—42no—
130C22⋊D157—42no—
131C22⋊D157—81no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—42no—
138C2×Dic157—42no—
139C4×D157—42no—
140C4×D157—42no—
141C4×D157—81no—
142C23×C157C2×D411yescharacteristic
143C23×C157C2×D411yescharacteristic
144D4×C157—22no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157C2×D411yescharacteristic
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2×D411yescharacteristic
153C2×C4×C157C2411yescharacteristic
154C8×C157—21no—
155C8×C157—21no—
156C8×C157—21no—
157C8×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—21no—
161C22×C157—21no—
162C22×C157—21no—
163C22×C157AGL(1,ℤ/8)12yes—
164C22×C157C22×D411yescharacteristic
165C2×D157—22no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
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×C157C2×#4111yescharacteristic
180C4×C157H(5,𝔽2)11yescharacteristic
181Dic157—41no—
182Dic157—41no—
183Dic157—41no—
184Dic157—41no—
185Dic157—41no—
186C2×C157—22no—
187C2×C157—41no—
188C2×C157#54611yescharacteristic
189C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C157#57711yescharacteristic
191D157—42no—
192D157—81no—
193D157—81no—
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#447—1571no—
207#442—1571no—
208#435D15711yescharacteristic, complemented
209#417—1571no—
210C2×#102—1571no—
211C2×#41—3141no—
212C2×#41—1571no—
213C22×D4—1571no—
214C2×SD16—1571no—
215C2×D8—1571no—
216C2×D8—1571no—
217C2×M16C2×D15711yescharacteristic, complemented
218C22×C8C2×D15711yescharacteristic
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#121—1571no—
230#120—3141no—
231C4×D4—1571no—
232C4×D4—1571no—
233C4×D4C2×D15711yescharacteristic
234C2×C4⋊C4—1571no—
235C2×C4⋊C4—1571no—
236C2×C22⋊C4—3141no—
237#106—1571no—
238#106—1571no—
239#106—1571no—
240C4⋊C8C2×D15711yescharacteristic, complemented
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⋊C8C2×D15711yescharacteristic
252C22⋊C8C2×D15711yescharacteristic, complemented
253#97C2×D15711yescharacteristic, complemented
254C24—3141no—
255#41—3142no—
256#41—6281no—
257#41—6284no—
258C2×Q8—3141no—
259C2×Q8—1571no—
260C2×D4—3142no—
261C2×D4—3141no—
262C2×D4—3141no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—6281no—
267C2×D4—1571no—
268C2×D4—1571no—
269C2×D4—1571no—
270C2×D4D62811yescharacteristic
271C22×C4—3141no—
272C22×C4—3141no—
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—3141no—
276C22×C4C22×D15711yescharacteristic
277C22×C4C22×D15711yescharacteristic
278SD16—6281no—
279D8—3142no—
280D8—6281no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D15711yescharacteristic
284C2×C8C22×D15711yescharacteristic
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⋊C4D62811yescharacteristic
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—3141no—
304C22⋊C4D62811yescharacteristic
305C22⋊C4D62811yescharacteristic
306C42C22×D15711yescharacteristic
307C23—3141no—
308C23—3141no—
309C23—3141no—
310C23—6282no—
311C23—6282no—
312C23—12561no—
313C23C2×D62811yescharacteristic
314C23C2×D62811yescharacteristic
315Q8—3141no—
316Q8—3141no—
317Q8—6282no—
318D4—22no—
319D4—3141no—
320D4—3141no—
321D4—3141no—
322D4—3141no—
323D4—3141no—
324D4—6282no—
325D4—6282no—
326D4—6282no—
327D4—6282no—
328D4—1572no—
329D4—12561no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
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—12561no—
344C2×C4C2×D62811yescharacteristic
345C2×C4C2×D62811yescharacteristic
346C2×C4C2×D62811yescharacteristic
347C2×C4C2×D62811yescharacteristic
348C2×C4C23×D15711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—3142no—
359C22—6281no—
360C22—6281no—
361C22—6281no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—12564no—
368C22#1909779912yes—
369C22C22×D62811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—6281no—
375C4—6281no—
376C4—6281no—
377C4—6281no—
378C4—6281no—
379C4#41×D15711yescharacteristic
380C4#1909781611yescharacteristic
381C2—22no—
382C2—41no—
383C2—6282no—
384C2—12561no—
385C2—12561no—
386C2#2375791811yescharacteristic
387C2C2×#1909779911yescharacteristic
388C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#435×C157C2
socleC22×C157C22×D4
hypercenter#435D157
hypocenterC157#3306
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #23757927C2semidirect
2. #23757919C2semidirect
3. #23757633C2semidirect
4. #23757604C2semidirect
5. #23757592C2semidirect
6. #23757564C2semidirect
7. #23757526C2semidirect
8. #23757502C2semidirect
9. #23757489C2semidirect
10. #23757420C2semidirect
11. #23757409C2semidirect
12. #23757386C2semidirect
13. #23757275C2semidirect
14. C2×#19097693C2semidirect
15. #435×C157C2semidirect
19. C2×#14578943C22semidirect
22. #19097771C22semidirect
31. #19097745C22semidirect
35. C4×D628C22semidirect
37. C2×C4⋊Dic157C22semidirect
39. #19097703C22semidirect
40. #19097703C22semidirect
41. #19097703C22semidirect
42. #19097696C22semidirect
43. #19097696C22semidirect
44. #19097696C22semidirect
51. #19097693C22semidirect
52. C2×M16×C157C22semidirect
55. C4⋊C8×C157C22semidirect
57. C22⋊C8×C157C22semidirect
58. #97×C157C22semidirect
127. D628#41semidirect
194. C157#3306semidirect
208. #435D157semidirect
217. C2×M16C2×D157semidirect
240. C4⋊C8C2×D157semidirect
252. C22⋊C8C2×D157semidirect
253. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579272
#237579192
#237576332
#237576042
#237575922
#237575642
#237575262
#237575022
#237574892
#237574202
#237574092
#237573862
#237572752
C2×#190976932
#435×C1572
#3306157