Finite Group Databasebeta

← #34111652

Subgroups — #34111652

Order 20096.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097800C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757938C211yesmaximal, characteristic, complemented
4#23757918C211yesmaximal, characteristic
5#23757665C211yesmaximal, characteristic, complemented
6#23757522C211yesmaximal, characteristic, complemented
7#23757513C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757432C211yesmaximal, characteristic, complemented
10#23757430C211yesmaximal, characteristic, complemented
11#23757422C211yesmaximal, characteristic, complemented
12#23757413C211yesmaximal, characteristic, complemented
13#23757373C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#461×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic
20#19097800—24no—
21C2×Dic628C2211yescharacteristic, complemented
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578944C2211yescharacteristic, complemented
24C2×#14578943C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28SD16×D157—24no—
29C2×#14578937C2211yescharacteristic, complemented
30C2×C8×D157C2211yescharacteristic
31#19097770—21no—
32#19097759C2211yescharacteristic
33#19097755—21no—
34#19097751C2211yescharacteristic
35#19097749—21no—
36#19097749—21no—
37#19097747C2211yescharacteristic
38#19097747C2211yescharacteristic, complemented
39#19097746C2211yescharacteristic
40#19097745—21no—
41#19097741—21no—
42C4×Dic314C2211yescharacteristic, complemented
43C4×D628C2211yescharacteristic
44#19097733—21no—
45C2×#14578931—21no—
46#19097704C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097692C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic, complemented
53#19097682C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C4⋊Q8×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#103×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578953—22no—
63#14578953—41no—
64C2×Dic314—21no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68Q8×D157—22no—
69Q8×D157—22no—
70Q8×D157—41no—
71Q8×D157—41no—
72C2×D628—21no—
73C2×D628D411yescharacteristic
74C2×D628C2311yescharacteristic
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77D4×D157—22no—
78D4×D157—41no—
79C22×Dic157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157C2311yescharacteristic
83C2×C4×D157C2311yescharacteristic
84Dic628—22no—
85#14578944—22no—
86#14578944—41no—
87#14578943—22no—
88#14578942D414yescomplemented
89#14578941—22no—
90#14578941—41no—
91#14578937—22no—
92C2×#11929434C2311yescharacteristic
93C2×#11929434C2311yescharacteristic
94C8×D157—22no—
95C4⋊Dic157—21no—
96C4⋊Dic157D411yescharacteristic
97C4⋊Dic157C2311yescharacteristic
98#14578933—21no—
99#14578933—21no—
100#14578933—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931D411yescharacteristic
108#14578931D411yescharacteristic
109C4×Dic157—21no—
110C2×Q8×C157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C22×C4×C157—21no—
114SD16×C157—22no—
115SD16×C157—22no—
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157—21no—
119C4⋊C4×C157—21no—
120C4⋊C4×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157—81no—
127C22×D157C2×D411yescharacteristic
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—42no—
133Dic314—42no—
134Dic314C2×D412yes—
135D628—22no—
136D628—21no—
137D628—21no—
138D628—81no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157—42no—
145C2×Dic157C2×D411yescharacteristic
146C2×Dic157C2×D411yescharacteristic
147C4×D157—21no—
148C4×D157—21no—
149C4×D157—21no—
150C4×D157—21no—
151C4×D157—42no—
152C4×D157—42no—
153#11929434—21no—
154#11929434C2×D412yes—
155C23×C157—21no—
156Q8×C157—21no—
157Q8×C157—21no—
158Q8×C157—21no—
159Q8×C157—21no—
160D4×C157—21no—
161D4×C157C2×D412yes—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—81no—
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2411yescharacteristic
169C8×C157—21no—
170C8×C157—21no—
171C22×C157—22no—
172C22×C157—41no—
173C22×C157—41no—
174C22×C157C22×D411yescharacteristic
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—84no—
184C2×D157#13712yescomplemented
185C4×C157—21no—
186C4×C157—21no—
187C4×C157—41no—
188C4×C157—41no—
189C4×C157—41no—
190C4×C157C22×D411yescharacteristic
191C4×C157H(5,𝔽2)11yescharacteristic
192Dic157—21no—
193Dic157—21no—
194Dic157—22no—
195Dic157—41no—
196Dic157—41no—
197C2×C157—42no—
198C2×C157D4211yescharacteristic
199C2×C157C2×#13711yescharacteristic
200C2×C157#57711yescharacteristic
201D157—22no—
202D157—41no—
203D157—81no—
204C157#326611yescharacteristic, complemented, Sylow
205#3266—1571nomaximal, Sylow
206C2×#137—1571no—
207C22×SD16—1571no—
208D4×Q8—1571no—
209#546—1571no—
210#497—1571no—
211#468—1571no—
212#467—1571no—
213#464—1571no—
214#461D15711yescharacteristic, complemented
215#452—1571no—
216#451—1571no—
217#450—1571no—
218#448—1571no—
219#440—1571no—
220#435—1571no—
221C2×#41—1571no—
222C22×Q8—1571no—
223C22×Q8—1571no—
224C22×D4—1571no—
225#137—3144no—
226C2×Q16—1571no—
227C2×Q16—1571no—
228C2×Q16—1571no—
229C2×SD16—3144no—
230C2×SD16—1571no—
231C2×SD16—1571no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16C2×D15711yescharacteristic
235C2×SD16C2×D15711yescharacteristic
236C2×M16—1571no—
237C22×C8—1571no—
238C4⋊Q8—3141no—
239C4⋊Q8C2×D15711yescharacteristic, complemented
240#124—1571no—
241#123—3141no—
242C22⋊Q8—3141no—
243C22⋊Q8—3141no—
244C22⋊Q8—1571no—
245C22⋊Q8—1571no—
246C22⋊Q8—1571no—
247#121—3141no—
248#121—1571no—
249#120—3141no—
250C4×Q8—1571no—
251C4×D4—3141no—
252C4×D4—1571no—
253C4×D4C2×D15711yescharacteristic
254C2×C22⋊C4—3141no—
255#107—1571no—
256C4⋊C8C2×D15711yescharacteristic, complemented
257#103—1571no—
258#103—1571no—
259#103C2×D15711yescharacteristic, complemented
260#103C2×D15711yescharacteristic, complemented
261#102—1571no—
262#102—1571no—
263C22⋊C8—1571no—
264C22⋊C8—1571no—
265#97—1571no—
266C24—3141no—
267#41—3142no—
268#41—6281no—
269C2×Q8—3141no—
270C2×Q8—3141no—
271C2×Q8—3142no—
272C2×Q8—3142no—
273C2×Q8—6281no—
274C2×Q8—6281no—
275C2×Q8—1571no—
276C2×Q8—1571no—
277C2×Q8C22×D15711yescharacteristic
278C2×Q8C22×D15711yescharacteristic
279C2×D4—3141no—
280C2×D4—3142no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—6281no—
284C2×D4—1571no—
285C2×D4—1571no—
286C2×D4C22×D15711yescharacteristic
287C22×C4—21no—
288C22×C4—3141no—
289C22×C4—3141no—
290C22×C4—3141no—
291C22×C4—1571no—
292C22×C4—1571no—
293Q16—3142no—
294Q16—3142no—
295Q16—6281no—
296SD16—22no—
297SD16—22no—
298SD16—3142no—
299SD16—3142no—
300SD16—6281no—
301SD16—1574no—
302M16—3142no—
303C2×C8—3142no—
304C2×C8—1571no—
305C2×C8—1571no—
306C2×C8C22×D15711yescharacteristic
307C2×C8C22×D15711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—3141no—
311C4⋊C4—3141no—
312C4⋊C4—3141no—
313C4⋊C4—3141no—
314C4⋊C4—1571no—
315C4⋊C4—1571no—
316C4⋊C4C22×D15711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—1571no—
325C22⋊C4—1571no—
326C42—3141no—
327C42C22×D15711yescharacteristic
328C23—21no—
329C23—3141no—
330C23—6282no—
331C23—6282no—
332C23—1571no—
333C23—1571no—
334C23—12561no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—3141no—
340Q8—3141no—
341Q8—3141no—
342Q8—6282no—
343Q8—6282no—
344Q8—1572no—
345D4—21no—
346D4—3142no—
347D4—3141no—
348D4—3141no—
349D4—6282no—
350D4—6282no—
351D4—12561no—
352D4D4×D15712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—3141no—
361C2×C4—3141no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—6282no—
365C2×C4—6282no—
366C2×C4—6282no—
367C2×C4—1571no—
368C2×C4—1571no—
369C2×C4D4×D15711yescharacteristic
370C2×C4D4×D15711yescharacteristic
371C2×C4C23×D15711yescharacteristic
372C8—21no—
373C8—21no—
374C8—3141no—
375C8—1572no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—3141no—
380C22—3141no—
381C22—3141no—
382C22—3141no—
383C22—3141no—
384C22—6281no—
385C22—6281no—
386C22—6281no—
387C22—1572no—
388C22—12564no—
389C22C2×D4×D15711yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3141no—
396C4—3141no—
397C4—3142no—
398C4—6281no—
399C4—6281no—
400C4C2×D4×D15711yescharacteristic
401C4#1909781711yescharacteristic
402C2—42no—
403C2—3142no—
404C2—6281no—
405C2—12561no—
406C2#2375790711yescharacteristic
407C2#137×D15711yescharacteristic
408C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
prime radical#23757918C2
hypercenter#461D157
hypocenterC157#3266
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757938C2semidirect
5. #23757665C2semidirect
6. #23757522C2semidirect
7. #23757513C2semidirect
8. #23757496C2semidirect
9. #23757432C2semidirect
10. #23757430C2semidirect
11. #23757422C2semidirect
12. #23757413C2semidirect
13. #23757373C2semidirect
14. #23757343C2semidirect
15. #461×C157C2semidirect
18. C2×Q8×D157C22semidirect
21. C2×Dic628C22semidirect
22. C2×#14578944C22semidirect
23. C2×#14578944C22semidirect
29. C2×#14578937C22semidirect
38. #19097747C22semidirect
42. C4×Dic314C22semidirect
46. #19097704C22semidirect
47. #19097696C22semidirect
48. #19097694C22semidirect
52. #19097683C22semidirect
53. #19097682C22semidirect
56. C4⋊Q8×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
60. #103×C157C22semidirect
88. #14578942D4semidirect
184. C2×D157#137semidirect
204. C157#3266semidirect
214. #461D157semidirect
239. C4⋊Q8C2×D157semidirect
256. C4⋊C8C2×D157semidirect
259. #103C2×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×#190978002
C2×SD16×D1572
#237579382
#237579182
#237576652
#237575222
#237575132
#237574962
#237574322
#237574302
#237574222
#237574132
#237573732
#237573432
#461×C1572
#3266157