Finite Group Databasebeta

← #34111701

Subgroups — #34111701

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×#19097800C211yesmaximal, characteristic, complemented
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3Q8×D628C211yesmaximal, characteristic, complemented
4#23757931C211yesmaximal, characteristic
5#23757668C211yesmaximal, characteristic, complemented
6#23757530C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757484C211yesmaximal, characteristic, complemented
9#23757436C211yesmaximal, characteristic, complemented
10#23757430C211yesmaximal, characteristic, complemented
11#23757427C211yesmaximal, characteristic, complemented
12#23757410C211yesmaximal, characteristic, complemented
13#23757378C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#464×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097800—24no—
21C2×Dic628C2211yescharacteristic
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578944C2211yescharacteristic
26Q16×D157—24no—
27C2×#14578942C2211yescharacteristic
28C2×#14578942C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic
31#19097771—21no—
32#19097771C2211yescharacteristic
33#19097757—21no—
34#19097751—21no—
35#19097751—21no—
36#19097751C2211yescharacteristic
37#19097751C2211yescharacteristic
38#19097748—21no—
39#19097747C2211yescharacteristic
40#19097744—21no—
41#19097742C2211yescharacteristic
42C4×Dic314C2211yescharacteristic
43C4×D628—21no—
44C4×D628C2211yescharacteristic
45#19097731—21no—
46#19097702C2211yescharacteristic
47#19097696C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097695C2211yescharacteristic
50#19097694C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097682C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—41no—
63C2×Dic314—21no—
64C2×Dic314—21no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—22no—
71Q8×D157—22no—
72Q8×D157—41no—
73Q8×D157—41no—
74Q8×D157—41no—
75C2×D628D411yescharacteristic
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C22×Dic157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157C2311yescharacteristic
83C2×C4×D157C2311yescharacteristic
84Dic628—22no—
85Dic628—22no—
86#14578944—22no—
87#14578944—41no—
88#14578944D414yescomplemented
89#14578942—22no—
90#14578942—41no—
91#14578937—22no—
92C2×#11929434C2311yescharacteristic
93C2×#11929434C2311yescharacteristic
94C8×D157—22no—
95C4⋊Dic157—21no—
96C4⋊Dic157—21no—
97C4⋊Dic157—21no—
98C4⋊Dic157—21no—
99C4⋊Dic157D411yescharacteristic
100C4⋊Dic157C2311yescharacteristic
101#14578933—21no—
102#14578933—21no—
103C22⋊Dic157—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
113Q16×C157—22no—
114SD16×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C4⋊C4×C157—21no—
118C4⋊C4×C157C2311yescharacteristic
119C22⋊C4×C157—21no—
120C22⋊C4×C157—21no—
121C42×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157C2×D411yescharacteristic
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—21no—
130Dic314—42no—
131Dic314—42no—
132Dic314—42no—
133Dic314C2×D412yes—
134D628—22no—
135C22⋊D157—42no—
136C22⋊D157—42no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157—42no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C4×D157—21no—
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—21no—
148C4×D157—42no—
149C4×D157—42no—
150C4×D157—42no—
151#11929434—21no—
152#11929434C2×D412yes—
153C23×C157—21no—
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157C2×D412yes—
158D4×C157—21no—
159D4×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×C157—41no—
169C22×C157—41no—
170C22×C157—41no—
171C22×C157C22×D411yescharacteristic
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157#13712yescomplemented
178C4×C157—22no—
179C4×C157—21no—
180C4×C157—21no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157C22×D411yescharacteristic
184C4×C157H(5,𝔽2)11yescharacteristic
185Dic157—21no—
186Dic157—21no—
187Dic157—22no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191C2×C157—81no—
192C2×C157D4211yescharacteristic
193C2×C157C2×#13711yescharacteristic
194C2×C157#57811yescharacteristic
195D157—22no—
196D157—41no—
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#464D15711yescharacteristic, complemented
207#462—1571no—
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×Q16—1571no—
225C2×Q16C2×D15711yescharacteristic
226C2×SD16—1571no—
227C2×SD16—1571no—
228C2×SD16C2×D15711yescharacteristic
229C2×M16—1571no—
230C22×C8—1571no—
231C4⋊Q8—3141no—
232C4⋊Q8—1571no—
233#124—3141no—
234#124C2×D15711yescharacteristic
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#103—1571no—
254#103C2×D15711yescharacteristic
255#102C2×D15711yescharacteristic
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×Q8—1571no—
273C2×Q8C22×D15711yescharacteristic
274C2×Q8C22×D15711yescharacteristic
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4C22×D15711yescharacteristic
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4—1571no—
284C22×C4—1571no—
285Q16—22no—
286Q16—3142no—
287Q16—3142no—
288Q16—3142no—
289Q16—6281no—
290Q16—1574no—
291SD16—22no—
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—3141no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—1571no—
308C4⋊C4—1571no—
309C4⋊C4C22×D15711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
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—21no—
322C23—1571no—
323C23—1571no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—3141no—
328Q8—3141no—
329Q8—3141no—
330Q8—3141no—
331Q8—3141no—
332Q8—6282no—
333Q8—6282no—
334Q8—6282no—
335Q8—1572no—
336Q8D4×D15712yes—
337D4—21no—
338D4—21no—
339D4—3142no—
340D4—6282no—
341D4—6282no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3141no—
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—41no—
367C22—41no—
368C22—41no—
369C22—3141no—
370C22—3141no—
371C22—3141no—
372C22—3141no—
373C22—3141no—
374C22—1572no—
375C22C2×D4×D15711yescharacteristic
376C4—21no—
377C4—22no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—3141no—
382C4—3141no—
383C4—3142no—
384C4—6281no—
385C4—6281no—
386C4—6281no—
387C4C2×D4×D15711yescharacteristic
388C4#1909781711yescharacteristic
389C2—81no—
390C2—3142no—
391C2—6281no—
392C2#2375790711yescharacteristic
393C2#137×D15711yescharacteristic
394C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#464×C157C2
socleC22×C157C22×D4
prime radical#23757931C2
hypercenter#464D157
hypocenterC157#3270
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
2. C2×Q16×D157C2semidirect
3. Q8×D628C2semidirect
5. #23757668C2semidirect
6. #23757530C2semidirect
7. #23757501C2semidirect
8. #23757484C2semidirect
9. #23757436C2semidirect
10. #23757430C2semidirect
11. #23757427C2semidirect
12. #23757410C2semidirect
13. #23757378C2semidirect
14. #23757343C2semidirect
15. #464×C157C2semidirect
88. #14578944D4semidirect
177. C2×D157#137semidirect
197. C157#3270semidirect
206. #464D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C2×Q16×D1572
Q8×D6282
#237579312
#237576682
#237575302
#237575012
#237574842
#237574362
#237574302
#237574272
#237574102
#237573782
#237573432
#464×C1572
#3270157