Finite Group Databasebeta

← #34109770

Subgroups — #34109770

Order 20096.

492 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×SD16×D157C211yesmaximal, characteristic, complemented
2C2×#19097777C211yesmaximal, characteristic, complemented
3#23757846C211yesmaximal, characteristic, complemented
4#23757839C211yesmaximal, characteristic
5#23757522C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757518C211yesmaximal, characteristic, complemented
8#23757517C211yesmaximal, characteristic, complemented
9#23757503C211yesmaximal, characteristic, complemented
10#23757502C211yesmaximal, characteristic, complemented
11#23757499C211yesmaximal, characteristic, complemented
12#23757308C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757276C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578942C2211yescharacteristic
21C2×#14578941C2211yescharacteristic
22SD16×D157—24no—
23#19097777—24no—
24C2×#14578938C2211yescharacteristic, complemented
25C2×#14578937C2211yescharacteristic
26C2×#14578937C2211yescharacteristic, complemented
27C2×C8×D157C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic, complemented
29#19097770—21no—
30#19097767—21no—
31#19097757—21no—
32#19097757—21no—
33#19097756—21no—
34#19097753—21no—
35#19097751C2211yescharacteristic, complemented
36#19097750C2211yescharacteristic, complemented
37#19097747C2211yescharacteristic, complemented
38#19097746C2211yescharacteristic
39#19097745C2211yescharacteristic
40#19097742C2211yescharacteristic
41#19097741—21no—
42#19097739—21no—
43#19097729C2211yescharacteristic
44#19097729C2211yescharacteristic
45#19097704C2211yescharacteristic, complemented
46#19097704C2211yescharacteristic, complemented
47#19097703C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097693C2211yescharacteristic
52#19097692C2211yescharacteristic
53#19097691C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic, complemented
56C22⋊Q8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578954—22no—
63#14578954—22no—
64C2×Dic314—21no—
65C2×Dic314D411yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—22no—
69C2×D628—21no—
70C2×D628D411yescharacteristic
71C2×D628C2311yescharacteristic
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157D411yescharacteristic
76C2×C22⋊D157D411yescharacteristic
77D4×D157—22no—
78D4×D157—22no—
79C2×C4×D157—21no—
80C2×C4×D157C2311yescharacteristic
81C2×C4×D157C2311yescharacteristic
82#14578943—22no—
83#14578942—22no—
84#14578941—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—
112C2×C8×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—21no—
121C22×D157—42no—
122C22×D157—81no—
123C22×D157—81no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—22no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130D628—22no—
131D628—21no—
132D628—21no—
133D628—81no—
134C22⋊D157—22no—
135C22⋊D157—22no—
136C22⋊D157—42no—
137C22⋊D157—81no—
138C22⋊D157—81no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C4×D157—21no—
145C4×D157—21no—
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—81no—
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—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—84no—
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#57711yescharacteristic
199C2×C157#57711yescharacteristic
200D157—22no—
201D157—41no—
202D157—81no—
203C157#305711yescharacteristic, complemented, Sylow
204#3057—1571nomaximal, Sylow
205C22×SD16—1571no—
206C2×#131—1571no—
207#536—1571no—
208#535—1571no—
209#468—1571no—
210#468—1571no—
211#468—1571no—
212#468—1571no—
213#465—1571no—
214#465—1571no—
215#465—1571no—
216#465D15711yescharacteristic, complemented
217#427—1571no—
218#419—1571no—
219#418—1571no—
220C2×#41—1571no—
221C22×Q8—1571no—
222C22×D4—1571no—
223C2×SD16—3144no—
224C2×SD16—1571no—
225C2×SD16—1571no—
226C2×SD16—1571no—
227C2×SD16C2×D15711yescharacteristic
228#131—3144no—
229C2×M16—1571no—
230C2×M16—1571no—
231C2×M16—1571no—
232C22×C8—1571no—
233C22×C8—1571no—
234C22×C8C2×D15711yescharacteristic, complemented
235C4⋊Q8—3141no—
236#127—3141no—
237#124—3141no—
238#124—3141no—
239#123—3141no—
240#123—3141no—
241C22⋊Q8—1571no—
242C22⋊Q8—1571no—
243C22⋊Q8—1571no—
244C22⋊Q8C2×D15711yescharacteristic, complemented
245#121—1571no—
246#121—1571no—
247#121—1571no—
248#121C2×D15711yescharacteristic
249#120—3141no—
250#120—3141no—
251#117—1571no—
252#117—1571no—
253#107—1571no—
254#107—1571no—
255#106—1571no—
256#106C2×D15711yescharacteristic, complemented
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
265C24—3141no—
266#41—3142no—
267#41—3142no—
268C2×Q8—3141no—
269C2×Q8—3142no—
270C2×Q8—3142no—
271C2×Q8—1571no—
272C2×Q8—1571no—
273C2×Q8C22×D15711yescharacteristic
274C2×D4—21no—
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—3141no—
278C2×D4—3142no—
279C2×D4—3142no—
280C2×D4—3141no—
281C2×D4—1571no—
282C2×D4—1571no—
283C2×D4—1571no—
284C2×D4—1571no—
285C2×D4C22×D15711yescharacteristic
286C22×C4—3141no—
287C22×C4—1571no—
288C22×C4—1571no—
289C22×C4C22×D15711yescharacteristic
290SD16—22no—
291SD16—3142no—
292SD16—3142no—
293SD16—3142no—
294M16—3142no—
295M16—3142no—
296M16—3142no—
297C2×C8—22no—
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—3141no—
326C23—6282no—
327C23—1571no—
328C23—1571no—
329C23—12561no—
330C23—12561no—
331C23D4×D15711yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3142no—
335Q8—3141no—
336Q8—3141no—
337Q8—6282no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—3142no—
342D4—3142no—
343D4—3142no—
344D4—3141no—
345D4—3141no—
346D4—6282no—
347D4—12561no—
348D4—12561no—
349D4—12561no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
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—6281no—
383C22—6281no—
384C22—6281no—
385C22—1572no—
386C22—12564no—
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#2375807911yescharacteristic
406C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757839C2
hypercenter#465D157
hypocenterC157#3057
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D157C2semidirect
2. C2×#19097777C2semidirect
3. #23757846C2semidirect
5. #23757522C2semidirect
6. #23757521C2semidirect
7. #23757518C2semidirect
8. #23757517C2semidirect
9. #23757503C2semidirect
10. #23757502C2semidirect
11. #23757499C2semidirect
12. #23757308C2semidirect
13. #23757280C2semidirect
14. #23757276C2semidirect
15. #465×C157C2semidirect
24. C2×#14578938C22semidirect
26. C2×#14578937C22semidirect
28. C2×C8×D157C22semidirect
35. #19097751C22semidirect
36. #19097750C22semidirect
37. #19097747C22semidirect
45. #19097704C22semidirect
46. #19097704C22semidirect
47. #19097703C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097694C22semidirect
55. C22×C8×C157C22semidirect
56. C22⋊Q8×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
91. C8×D157D4semidirect
203. C157#3057semidirect
216. #465D157semidirect
234. C22×C8C2×D157semidirect
244. C22⋊Q8C2×D157semidirect
256. #106C2×D157semidirect
260. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1572
C2×#190977772
#237578462
#237578392
#237575222
#237575212
#237575182
#237575172
#237575032
#237575022
#237574992
#237573082
#237572802
#237572762
#465×C1572
#3057157