Finite Group Databasebeta

← #34109740

Subgroups — #34109740

Order 20096.

500 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097792C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757786C211yesmaximal, characteristic, complemented
4#23757783C211yesmaximal, characteristic, complemented
5#23757530C211yesmaximal, characteristic, complemented
6#23757529C211yesmaximal, characteristic, complemented
7#23757525C211yesmaximal, characteristic, complemented
8#23757524C211yesmaximal, characteristic, complemented
9#23757520C211yesmaximal, characteristic, complemented
10#23757519C211yesmaximal, characteristic, complemented
11#23757516C211yesmaximal, characteristic, complemented
12#23757310C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#468×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19#19097792—24no—
20C2×#14578944C2211yescharacteristic, complemented
21C2×#14578943C2211yescharacteristic, complemented
22C2×#14578939C2211yescharacteristic, complemented
23C2×#14578938C2211yescharacteristic
24C2×#14578937C2211yescharacteristic
25C2×#14578937C2211yescharacteristic
26M16×D157—24no—
27C2×C8×D157C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097756—21no—
30#19097753—21no—
31#19097751C2211yescharacteristic
32#19097750C2211yescharacteristic
33#19097747C2211yescharacteristic
34#19097746C2211yescharacteristic
35#19097745C2211yescharacteristic
36#19097742C2211yescharacteristic
37#19097741—21no—
38#19097739—21no—
39D4×Dic157—21no—
40#19097733—21no—
41#19097731—21no—
42#19097731—21no—
43#19097729C2211yescharacteristic
44#19097729C2211yescharacteristic
45#19097705C2211yescharacteristic, complemented
46#19097702C2211yescharacteristic, complemented
47#19097702C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097693C2211yescharacteristic, complemented
52#19097692C2211yescharacteristic, complemented
53#19097691C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×M16×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic
57#121×C157C2211yescharacteristic
58#107×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157D411yescharacteristic
62#14578955—22no—
63#14578955—22no—
64#14578953—22no—
65#14578953—22no—
66C2×Dic314C2311yescharacteristic
67C2×D628—21no—
68C2×D628—21no—
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C22×Dic157—21no—
74C22×Dic157D411yescharacteristic
75C2×C4×D157—22no—
76C2×C4×D157—22no—
77C2×C4×D157—21no—
78C2×C4×D157D411yescharacteristic
79C2×C4×D157D411yescharacteristic
80C2×C4×D157C2311yescharacteristic
81C2×C4×D157C2311yescharacteristic
82#14578944—22no—
83#14578943—22no—
84#14578939—22no—
85#14578938—22no—
86#14578937—22no—
87#14578937—22no—
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C8×D157—22no—
91C8×D157D414yescomplemented
92C4⋊Dic157—21no—
93C4⋊Dic157C2311yescharacteristic
94C4⋊Dic157C2311yescharacteristic
95#14578933—21no—
96#14578933—21no—
97C22⋊Dic157—21no—
98C22⋊Dic157—21no—
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105C4×Dic157—21no—
106C4×Dic157—21no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157—21no—
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111SD16×C157—22no—
112M16×C157—22no—
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
115C4⋊C4×C157—21no—
116C4⋊C4×C157C2311yescharacteristic
117C4⋊C4×C157C2311yescharacteristic
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C22×D157—22no—
121C22×D157—21no—
122C22×D157—41no—
123C22×D157—41no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—21no—
128D628—21no—
129D628—21no—
130D628—42no—
131D628—42no—
132C22⋊D157—42no—
133C22⋊D157—42no—
134C22⋊D157—42no—
135C2×Dic157—22no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—42no—
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C4×D157—22no—
143C4×D157—21no—
144C4×D157—21no—
145C4×D157—22no—
146C4×D157—42no—
147C4×D157C2×D412yes—
148C4×D157#4112yes—
149#11929434—21no—
150#11929434C2×D412yes—
151C23×C157—21no—
152C23×C157C2×D411yescharacteristic
153Q8×C157—21no—
154Q8×C157—21no—
155D4×C157—21no—
156D4×C157—21no—
157D4×C157—42no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2411yescharacteristic
165C8×C157—21no—
166C8×C157C2×D412yes—
167C22×C157—21no—
168C22×C157—21no—
169C22×C157—21no—
170C22×C157—41no—
171C22×C157—41no—
172C22×C157—41no—
173C22×C157C22×D411yescharacteristic
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—44no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157#12112yes—
184C4×C157—21no—
185C4×C157—41no—
186C4×C157—41no—
187C4×C157—41no—
188C4×C157C22×D411yescharacteristic
189C4×C157C2×#4111yescharacteristic
190Dic157—21no—
191Dic157—41no—
192Dic157—41no—
193Dic157—41no—
194Dic157#12112yes—
195C2×C157—41no—
196C2×C157—81no—
197C2×C157C2×#12111yescharacteristic
198C2×C157#57511yescharacteristic
199C2×C157#57511yescharacteristic
200D157—22no—
201D157—41no—
202D157—81no—
203C157#305411yescharacteristic, complemented, Sylow
204#3054—1571nomaximal, Sylow
205C2×#135—1571no—
206C22×M16—1571no—
207#525—1571no—
208#525—1571no—
209#470—1571no—
210#470—1571no—
211#469—1571no—
212#469—1571no—
213#468—1571no—
214#468—1571no—
215#468—1571no—
216#468D15711yescharacteristic, complemented
217#427—1571no—
218#416—1571no—
219#416—1571no—
220C2×#41—1571no—
221C2×#41—1571no—
222C23×C4—1571no—
223#135—3144no—
224C2×Q16—1571no—
225C2×SD16—1571no—
226C2×SD16C2×D15711yescharacteristic, complemented
227C2×D8—1571no—
228C2×M16—3144no—
229C2×M16—1571no—
230C2×M16—1571no—
231C2×M16—1571no—
232C2×M16C2×D15711yescharacteristic
233C22×C8—1571no—
234C22×C8—1571no—
235#123—3141no—
236#123—3141no—
237C22⋊Q8—1571no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240C22⋊Q8C2×D15711yescharacteristic
241#121—1571no—
242#121—1571no—
243#121—1571no—
244#121C2×D15711yescharacteristic
245#120—3141no—
246#120—3141no—
247C4×D4—3141no—
248C4×D4—3141no—
249C4×D4—3141no—
250C4×D4—3141no—
251#117—1571no—
252#117—1571no—
253#107—1571no—
254#107C2×D15711yescharacteristic, complemented
255#106—1571no—
256#106—1571no—
257#103—1571no—
258#103—1571no—
259#103—1571no—
260#103C2×D15711yescharacteristic, complemented
261#102—1571no—
262#102—1571no—
263#102—1571no—
264#102C2×D15711yescharacteristic, complemented
265C24—1571no—
266#41—3142no—
267#41—3142no—
268#41—3142no—
269#41—3142no—
270C2×Q8—1571no—
271C2×Q8C22×D15711yescharacteristic
272C2×D4—21no—
273C2×D4—3141no—
274C2×D4—3141no—
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—3141no—
278C2×D4—1571no—
279C2×D4C22×D15711yescharacteristic
280C22×C4—3142no—
281C22×C4—3142no—
282C22×C4—3141no—
283C22×C4—3141no—
284C22×C4—1571no—
285C22×C4—1571no—
286C22×C4—1571no—
287C22×C4—1571no—
288C22×C4—1571no—
289C22×C4C22×D15711yescharacteristic
290Q16—3142no—
291SD16—22no—
292SD16—3142no—
293D8—3142no—
294M16—22no—
295M16—3142no—
296M16—3142no—
297M16—3142no—
298C2×C8—3142no—
299C2×C8—1571no—
300C2×C8—1571no—
301C2×C8—1574no—
302C2×C8C22×D15711yescharacteristic
303C2×C8C22×D15711yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—3141no—
308C4⋊C4—1571no—
309C4⋊C4—1571no—
310C4⋊C4C22×D15711yescharacteristic
311C4⋊C4C22×D15711yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C42—3141no—
323C42—3141no—
324C23—21no—
325C23—3142no—
326C23—3141no—
327C23—6281no—
328C23—6281no—
329C23—1571no—
330C23—1571no—
331C23D4×D15711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3141no—
335Q8—3141no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—3141no—
340D4—3141no—
341D4—6282no—
342D4—6282no—
343D4—6282no—
344D4—6282no—
345D4—6282no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3142no—
352C2×C4—3142no—
353C2×C4—3142no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—6282no—
360C2×C4—6282no—
361C2×C4—1571no—
362C2×C4—1571no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4D4×D15711yescharacteristic
366C2×C4C23×D15711yescharacteristic
367C8—21no—
368C8—3141no—
369C8—1572no—
370C8D4×D15712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3141no—
378C22—3141no—
379C22—3141no—
380C22—3141no—
381C22—3141no—
382C22—6284no—
383C22—6281no—
384C22—6281no—
385C22—6281no—
386C22—1572no—
387C22C2×D4×D15711yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3141no—
393C4—6281no—
394C4—6281no—
395C4—6281no—
396C4—1572no—
397C4C2×D4×D15711yescharacteristic
398C4#41×D15711yescharacteristic
399C2—41no—
400C2—81no—
401C2—3142no—
402C2—6281no—
403C2—12561no—
404C2#121×D15711yescharacteristic
405C2#2375806911yescharacteristic
406C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#468×C157C2
socleC22×C157C22×D4
hypercenter#468D157
hypocenterC157#3054
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19097792C2semidirect
2. C2×M16×D157C2semidirect
3. #23757786C2semidirect
4. #23757783C2semidirect
5. #23757530C2semidirect
6. #23757529C2semidirect
7. #23757525C2semidirect
8. #23757524C2semidirect
9. #23757520C2semidirect
10. #23757519C2semidirect
11. #23757516C2semidirect
12. #23757310C2semidirect
13. #23757268C2semidirect
14. #23757266C2semidirect
15. #468×C157C2semidirect
20. C2×#14578944C22semidirect
21. C2×#14578943C22semidirect
22. C2×#14578939C22semidirect
45. #19097705C22semidirect
46. #19097702C22semidirect
47. #19097702C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097694C22semidirect
51. #19097693C22semidirect
52. #19097692C22semidirect
53. #19097691C22semidirect
54. C2×SD16×C157C22semidirect
58. #107×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
91. C8×D157D4semidirect
203. C157#3054semidirect
216. #468D157semidirect
226. C2×SD16C2×D157semidirect
254. #107C2×D157semidirect
260. #103C2×D157semidirect
264. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977922
C2×M16×D1572
#237577862
#237577832
#237575302
#237575292
#237575252
#237575242
#237575202
#237575192
#237575162
#237573102
#237572682
#237572662
#468×C1572
#3054157