Finite Group Databasebeta

← #34111712

Subgroups — #34111712

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3#23757938C211yesmaximal, characteristic, complemented
4#23757935C211yesmaximal, characteristic
5#23757665C211yesmaximal, characteristic, complemented
6#23757530C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757434C211yesmaximal, characteristic, complemented
10#23757430C211yesmaximal, characteristic, complemented
11#23757427C211yesmaximal, characteristic, complemented
12#23757413C211yesmaximal, characteristic, complemented
13#23757378C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#462×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097803—24no—
21C2×Dic628C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25Q16×D157—24no—
26C2×#14578943C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28C2×#14578941C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic
31#19097770—21no—
32#19097759—21no—
33#19097759C2211yescharacteristic
34#19097751—21no—
35#19097751C2211yescharacteristic
36#19097749—21no—
37#19097749—21no—
38#19097747C2211yescharacteristic
39#19097747C2211yescharacteristic
40#19097746—21no—
41#19097746C2211yescharacteristic
42C4×Dic314C2211yescharacteristic
43C4×D628—21no—
44C4×D628C2211yescharacteristic
45#19097733—21no—
46#19097702C2211yescharacteristic
47#19097696C2211yescharacteristic
48#19097695C2211yescharacteristic
49#19097694C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097682C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic
56C4⋊Q8×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#103×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63C2×Dic314—21no—
64C2×Dic314—21no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68Q8×D157—22no—
69Q8×D157—22no—
70Q8×D157—22no—
71Q8×D157—41no—
72Q8×D157—41no—
73Q8×D157—41no—
74C2×D628—21no—
75C2×D628—21no—
76C2×D628D411yescharacteristic
77C2×D628C2311yescharacteristic
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157C2311yescharacteristic
83C2×C4×D157C2311yescharacteristic
84Dic628—22no—
85#14578944—22no—
86#14578944—41no—
87#14578944D414yescomplemented
88#14578943—22no—
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×Q8×C157C2311yescharacteristic
113Q16×C157—22no—
114Q16×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C4⋊C4×C157—21no—
118C4⋊C4×C157—21no—
119C4⋊C4×C157—21no—
120C4⋊C4×C157C2311yescharacteristic
121C42×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130Dic314—42no—
131Dic314—42no—
132Dic314C2×D412yes—
133D628—22no—
134D628—21no—
135D628—21no—
136D628—42no—
137D628—42no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C4×D157—21no—
144C4×D157—21no—
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—42no—
148C4×D157—42no—
149C4×D157—42no—
150C4×D157—42no—
151#11929434—21no—
152#11929434C2×D412yes—
153Q8×C157—21no—
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157—21no—
158Q8×C157C2×D412yes—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2411yescharacteristic
166C8×C157—21no—
167C8×C157—21no—
168C22×C157C22×D411yescharacteristic
169C2×D157—21no—
170C2×D157—21no—
171C2×D157—21no—
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157#13712yescomplemented
178C4×C157—22no—
179C4×C157—21no—
180C4×C157—21no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157C22×D411yescharacteristic
185C4×C157H(5,𝔽2)11yescharacteristic
186Dic157—21no—
187Dic157—21no—
188Dic157—22no—
189Dic157—41no—
190Dic157—41no—
191C2×C157D4211yescharacteristic
192C2×C157C2×#13711yescharacteristic
193C2×C157#57811yescharacteristic
194D157—22no—
195D157—41no—
196D157—81no—
197C157#327011yescharacteristic, complemented, Sylow
198#3270—1571nomaximal, Sylow
199C2×#137—1571no—
200C22×Q16—1571no—
201D4×Q8—1571no—
202#548—1571no—
203#497—1571no—
204#470—1571no—
205#465—1571no—
206#464—1571no—
207#462D15711yescharacteristic, complemented
208#452—1571no—
209#451—1571no—
210#451—1571no—
211#448—1571no—
212#441—1571no—
213#435—1571no—
214C2×#41—1571no—
215C22×Q8—1571no—
216C22×Q8—1571no—
217C22×Q8—1571no—
218#137—3144no—
219C2×Q16—3144no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×Q16—1571no—
223C2×Q16—1571no—
224C2×Q16C2×D15711yescharacteristic
225C2×Q16C2×D15711yescharacteristic
226C2×SD16—1571no—
227C2×SD16—1571no—
228C2×SD16—1571no—
229C2×M16—1571no—
230C22×C8—1571no—
231C4⋊Q8—3141no—
232C4⋊Q8C2×D15711yescharacteristic
233#124—3141no—
234#124—1571no—
235C22⋊Q8—3141no—
236C22⋊Q8—3141no—
237C22⋊Q8—3141no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240C22⋊Q8—1571no—
241#121—3141no—
242#121—1571no—
243C4×Q8—1571no—
244C4×Q8C2×D15711yescharacteristic
245C4×D4—3141no—
246C4×D4—3141no—
247C4×D4—1571no—
248#106—1571no—
249C4⋊C8C2×D15711yescharacteristic
250#103—1571no—
251#103—1571no—
252#103—1571no—
253#103C2×D15711yescharacteristic
254#103C2×D15711yescharacteristic
255#102—1571no—
256C22⋊C8—1571no—
257C22⋊C8—1571no—
258#97—1571no—
259#41—3142no—
260#41—6281no—
261C2×Q8—3141no—
262C2×Q8—3141no—
263C2×Q8—3141no—
264C2×Q8—3142no—
265C2×Q8—3142no—
266C2×Q8—3142no—
267C2×Q8—6281no—
268C2×Q8—6281no—
269C2×Q8—6281no—
270C2×Q8—1571no—
271C2×Q8—1571no—
272C2×Q8C22×D15711yescharacteristic
273C2×Q8C22×D15711yescharacteristic
274C2×Q8C22×D15711yescharacteristic
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4—1571no—
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4—1571no—
284C22×C4—1571no—
285Q16—22no—
286Q16—22no—
287Q16—3142no—
288Q16—3142no—
289Q16—6281no—
290Q16—1574no—
291SD16—3142no—
292SD16—3142no—
293SD16—6281no—
294M16—3142no—
295C2×C8—3142no—
296C2×C8—1571no—
297C2×C8—1571no—
298C2×C8C22×D15711yescharacteristic
299C2×C8C22×D15711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—1571no—
308C4⋊C4—1571no—
309C4⋊C4C22×D15711yescharacteristic
310C22⋊C4—3141no—
311C22⋊C4—3141no—
312C22⋊C4—3141no—
313C22⋊C4—3141no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4—1571no—
317C22⋊C4—1571no—
318C42—21no—
319C42—3141no—
320C42C22×D15711yescharacteristic
321C23—3141no—
322C23—1571no—
323C23—1571no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—3141no—
330Q8—3141no—
331Q8—3141no—
332Q8—6282no—
333Q8—6282no—
334Q8—6282no—
335Q8—1572no—
336Q8D4×D15712yes—
337D4—3142no—
338D4—3141no—
339D4—3141no—
340D4—6282no—
341D4—6282no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—6282no—
354C2×C4—6282no—
355C2×C4—6282no—
356C2×C4—6282no—
357C2×C4—1571no—
358C2×C4—1571no—
359C2×C4D4×D15711yescharacteristic
360C2×C4D4×D15711yescharacteristic
361C2×C4C23×D15711yescharacteristic
362C8—21no—
363C8—21no—
364C8—3141no—
365C8—1572no—
366C22—3141no—
367C22—3141no—
368C22—3141no—
369C22—3141no—
370C22—3141no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22—1572no—
375C22C2×D4×D15711yescharacteristic
376C4—21no—
377C4—22no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—3141no—
383C4—3141no—
384C4—3142no—
385C4—6281no—
386C4—6281no—
387C4C2×D4×D15711yescharacteristic
388C4#1909781711yescharacteristic
389C2—3142no—
390C2—6281no—
391C2—12561no—
392C2#2375790711yescharacteristic
393C2#137×D15711yescharacteristic
394C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#462×C157C2
socleC22×C157C22×D4
prime radical#23757935C2
hypercenter#462D157
hypocenterC157#3270
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×Q16×D157C2semidirect
3. #23757938C2semidirect
5. #23757665C2semidirect
6. #23757530C2semidirect
7. #23757505C2semidirect
8. #23757496C2semidirect
9. #23757434C2semidirect
10. #23757430C2semidirect
11. #23757427C2semidirect
12. #23757413C2semidirect
13. #23757378C2semidirect
14. #23757343C2semidirect
15. #462×C157C2semidirect
87. #14578944D4semidirect
177. C2×D157#137semidirect
197. C157#3270semidirect
207. #462D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×Q16×D1572
#237579382
#237579352
#237576652
#237575302
#237575052
#237574962
#237574342
#237574302
#237574272
#237574132
#237573782
#237573432
#462×C1572
#3270157