Finite Group Databasebeta

← #34111716

Subgroups — #34111716

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097800C211yesmaximal, characteristic, complemented
2C2×#19097792C211yesmaximal, characteristic, complemented
3#23757943C211yesmaximal, characteristic, complemented
4#23757935C211yesmaximal, characteristic, complemented
5#23757660C211yesmaximal, characteristic, complemented
6#23757525C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757436C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757417C211yesmaximal, characteristic, complemented
12#23757410C211yesmaximal, characteristic, complemented
13#23757378C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#460×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578953C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097800—24no—
21#19097792—24no—
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic
31#19097767—21no—
32#19097759—21no—
33#19097759C2211yescharacteristic
34#19097751—21no—
35#19097751C2211yescharacteristic
36#19097746—21no—
37#19097746C2211yescharacteristic
38#19097743—21no—
39#19097743—21no—
40#19097742C2211yescharacteristic
41#19097742C2211yescharacteristic
42C4×Dic314C2211yescharacteristic
43C4×D628—21no—
44C4×D628C2211yescharacteristic
45D4×Dic157—21no—
46#19097702C2211yescharacteristic
47#19097696C2211yescharacteristic
48#19097695C2211yescharacteristic
49#19097694C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097682C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56#127×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63#14578953—22no—
64#14578953—22no—
65#14578953—41no—
66#14578953—41no—
67C2×Dic314—21no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70Q8×D157—22no—
71Q8×D157—41no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628D411yescharacteristic
75C2×D628C2311yescharacteristic
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C22×Dic157—21no—
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85C2×C4×D157C2311yescharacteristic
86Dic628—22no—
87#14578944D414yescomplemented
88#14578943—22no—
89#14578942—22no—
90#14578942—41no—
91#14578939—22no—
92#14578939—41no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C8×D157—22no—
97C4⋊Dic157—21no—
98C4⋊Dic157D411yescharacteristic
99C4⋊Dic157C2311yescharacteristic
100#14578933—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931D411yescharacteristic
108#14578931D411yescharacteristic
109C4×Dic157—21no—
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115SD16×C157—22no—
116SD16×C157—22no—
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
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—
130Dic314C2×D412yes—
131D628—21no—
132D628—21no—
133D628—41no—
134D628—42no—
135D628—42no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C22⋊D157—81no—
139C22⋊D157—81no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—42no—
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—
156C23×C157—21no—
157Q8×C157—21no—
158Q8×C157C2×D412yes—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—42no—
164D4×C157—81no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157—21no—
172C22×C157—41no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157C22×D411yescharacteristic
179C2×D157—21no—
180C2×D157—21no—
181C2×D157—21no—
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C4×C157—22no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—41no—
192C4×C157C22×D411yescharacteristic
193C4×C157H(5,𝔽2)11yescharacteristic
194Dic157—21no—
195Dic157—21no—
196Dic157—22no—
197Dic157—41no—
198Dic157—41no—
199C2×C157—81no—
200C2×C157—81no—
201C2×C157D4211yescharacteristic
202C2×C157#57511yescharacteristic
203C2×C157#57711yescharacteristic
204D157—41no—
205D157—41no—
206D157—81no—
207C157#327111yescharacteristic, complemented, Sylow
208#3271—1571nomaximal, Sylow
209C2×#137—1571no—
210C2×#135—1571no—
211#550—1571no—
212#548—1571no—
213#496—1571no—
214#469—1571no—
215#465—1571no—
216#464—1571no—
217#460D15711yescharacteristic, complemented
218#452—1571no—
219#449—1571no—
220#449—1571no—
221#448—1571no—
222#441—1571no—
223#435—1571no—
224C2×#41—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C22×Q8—1571no—
228#137—3144no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16—1571no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16C2×D15711yescharacteristic
236C2×SD16C2×D15711yescharacteristic
237C2×D8—1571no—
238C2×D8—1571no—
239C2×M16—1571no—
240C22×C8—1571no—
241#127—3141no—
242#127C2×D15711yescharacteristic
243#124—3141no—
244#124—1571no—
245C22⋊Q8—3141no—
246C22⋊Q8—1571no—
247#121—3141no—
248#121—3141no—
249#121—3141no—
250#121—1571no—
251#121—1571no—
252#121—1571no—
253C4×Q8—1571no—
254C4×Q8C2×D15711yescharacteristic
255C4×D4—3141no—
256C4×D4—3141no—
257C4×D4—1571no—
258#106—1571no—
259C4⋊C8C2×D15711yescharacteristic
260#103—1571no—
261#103—1571no—
262#103—1571no—
263#102—1571no—
264#102C2×D15711yescharacteristic
265#102C2×D15711yescharacteristic
266C22⋊C8—1571no—
267C22⋊C8—1571no—
268#97—1571no—
269#41—3142no—
270#41—3142no—
271#41—3142no—
272#41—6281no—
273#41—6281no—
274#41—6281no—
275C2×Q8—3141no—
276C2×Q8—3142no—
277C2×Q8—6281no—
278C2×Q8—1571no—
279C2×Q8—1571no—
280C2×Q8C22×D15711yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4C22×D15711yescharacteristic
292C2×D4C22×D15711yescharacteristic
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4—1571no—
298C22×C4—1571no—
299Q16—3142no—
300Q16—1574no—
301SD16—22no—
302SD16—22no—
303SD16—3142no—
304SD16—3142no—
305SD16—6281no—
306D8—3142no—
307D8—6281no—
308M16—3142no—
309C2×C8—3142no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8C22×D15711yescharacteristic
313C2×C8C22×D15711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—1571no—
318C4⋊C4—1571no—
319C4⋊C4C22×D15711yescharacteristic
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—1571no—
327C22⋊C4—1571no—
328C42—21no—
329C42—3141no—
330C42C22×D15711yescharacteristic
331C23—21no—
332C23—21no—
333C23—3141no—
334C23—1571no—
335C23—1571no—
336Q8—21no—
337Q8—3141no—
338Q8—3141no—
339Q8—3141no—
340Q8—6282no—
341Q8—1572no—
342Q8D4×D15712yes—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—81no—
349D4—3141no—
350D4—3141no—
351D4—6281no—
352D4—6282no—
353D4—6282no—
354D4—6282no—
355D4—6282no—
356D4—12561no—
357D4—12561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—3141no—
361C2×C4—3141no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—6282no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4D4×D15711yescharacteristic
374C2×C4D4×D15711yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—21no—
377C8—21no—
378C8—3141no—
379C8—1572no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3141no—
387C22—3141no—
388C22—3141no—
389C22—3141no—
390C22—3141no—
391C22—3141no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22C2×D4×D15711yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—3141no—
401C4—3141no—
402C4—3142no—
403C4—6281no—
404C4—6281no—
405C4C2×D4×D15711yescharacteristic
406C4#1909781711yescharacteristic
407C2—81no—
408C2—81no—
409C2—6281no—
410C2—6281no—
411C2—12561no—
412C2#2375790711yescharacteristic
413C2#2375806911yescharacteristic
414C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#460×C157C2
socleC22×C157C22×D4
hypercenter#460D157
hypocenterC157#3271
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
2. C2×#19097792C2semidirect
3. #23757943C2semidirect
4. #23757935C2semidirect
5. #23757660C2semidirect
6. #23757525C2semidirect
7. #23757501C2semidirect
8. #23757496C2semidirect
9. #23757436C2semidirect
10. #23757419C2semidirect
11. #23757417C2semidirect
12. #23757410C2semidirect
13. #23757378C2semidirect
14. #23757343C2semidirect
15. #460×C157C2semidirect
87. #14578944D4semidirect
207. C157#3271semidirect
217. #460D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C2×#190977922
#237579432
#237579352
#237576602
#237575252
#237575012
#237574962
#237574362
#237574192
#237574172
#237574102
#237573782
#237573432
#460×C1572
#3271157