Finite Group Databasebeta

← #34111688

Subgroups — #34111688

Order 20096.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×#19097793C211yesmaximal, characteristic, complemented
3#23757933C211yesmaximal, characteristic, complemented
4#23757925C211yesmaximal, characteristic, complemented
5#23757666C211yesmaximal, characteristic, complemented
6#23757520C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757482C211yesmaximal, characteristic, complemented
9#23757433C211yesmaximal, characteristic, complemented
10#23757433C211yesmaximal, characteristic, complemented
11#23757430C211yesmaximal, characteristic, complemented
12#23757421C211yesmaximal, characteristic, complemented
13#23757372C211yesmaximal, characteristic, complemented
14#23757344C211yesmaximal, characteristic, complemented
15#467×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18C2×#14578954C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097803—24no—
21#19097793—24no—
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578943C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28C2×#14578941C2211yescharacteristic
29C2×D1256C2211yescharacteristic
30C2×#14578937C2211yescharacteristic
31#19097769C2211yescharacteristic
32#19097760C2211yescharacteristic
33#19097757—21no—
34#19097756—21no—
35#19097753—21no—
36#19097751C2211yescharacteristic
37#19097749—21no—
38#19097746C2211yescharacteristic
39#19097744—21no—
40#19097744—21no—
41#19097736C2211yescharacteristic
42#19097733—21no—
43#19097733C2211yescharacteristic
44C4×C22⋊D157C2211yescharacteristic
45C4⋊C4×D157—21no—
46#19097700C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097691C2211yescharacteristic
51C22⋊#11929434C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097681C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#107×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#103×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63#14578954—22no—
64#14578954—22no—
65#14578954—41no—
66#14578954—41no—
67C2×Dic314—21no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70Q8×D157—22no—
71Q8×D157—41no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628C2311yescharacteristic
75C2×D628C2311yescharacteristic
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157D411yescharacteristic
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85Dic628—22no—
86#14578944—22no—
87#14578944—41no—
88#14578943—22no—
89#14578943D414yescomplemented
90#14578941—22no—
91#14578941—41no—
92D1256—22no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C4⋊Dic157—21no—
97#14578933—21no—
98#14578933—21no—
99#14578933—21no—
100#14578933D411yescharacteristic
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931D411yescharacteristic
108C4×Dic157—21no—
109C4×Dic157C2311yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C22×C4×C157C2311yescharacteristic
113Q16×C157—22no—
114C2×C8×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C4⋊C4×C157—21no—
118C4⋊C4×C157—21no—
119C4⋊C4×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130Dic314C2×D412yes—
131D628—21no—
132D628—21no—
133D628—21no—
134D628—42no—
135D628—42no—
136D628C2×D412yes—
137C22⋊D157—41no—
138C22⋊D157—42no—
139C22⋊D157—42no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—42no—
148C4×D157—42no—
149C4×D157—42no—
150C4×D157—42no—
151C4×D157—42no—
152#11929434—21no—
153#11929434—21no—
154C23×C157C2×D411yescharacteristic
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157—21no—
158Q8×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2411yescharacteristic
167C8×C157—21no—
168C8×C157C2×D412yes—
169C22×C157—21no—
170C22×C157—21no—
171C22×C157—21no—
172C22×C157C22×D411yescharacteristic
173C2×D157—21no—
174C2×D157—22no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—41no—
178C2×D157—41no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C4×C157—21no—
183C4×C157—41no—
184C4×C157—41no—
185C4×C157—41no—
186C4×C157—41no—
187C4×C157C22×D411yescharacteristic
188C4×C157H(5,𝔽2)11yescharacteristic
189Dic157—21no—
190Dic157—22no—
191Dic157—21no—
192Dic157—21no—
193Dic157—41no—
194C2×C157—41no—
195C2×C157D4211yescharacteristic
196C2×C157#57511yescharacteristic
197C2×C157#57811yescharacteristic
198D157—41no—
199D157—42no—
200D157—81no—
201C157#326911yescharacteristic, complemented, Sylow
202#3269—1571nomaximal, Sylow
203C2×#137—1571no—
204C2×#135—1571no—
205#548—1571no—
206#547—1571no—
207#497—1571no—
208#468—1571no—
209#467D15711yescharacteristic, complemented
210#463—1571no—
211#462—1571no—
212#452—1571no—
213#452—1571no—
214#451—1571no—
215#449—1571no—
216#440—1571no—
217#435—1571no—
218C2×#41—1571no—
219C2×#41—1571no—
220C2×#41—1571no—
221C22×Q8—1571no—
222#137—3144no—
223#135—3144no—
224C2×Q16—1571no—
225C2×Q16—1571no—
226C2×Q16—1571no—
227C2×Q16C2×D15711yescharacteristic
228C2×SD16—1571no—
229C2×SD16—1571no—
230C2×SD16—1571no—
231C2×SD16—1571no—
232C2×D8—1571no—
233C2×M16—1571no—
234C22×C8C2×D15711yescharacteristic
235C4⋊Q8—1571no—
236#124—3141no—
237#124—1571no—
238#123—3141no—
239#123—3141no—
240C22⋊Q8—3141no—
241C22⋊Q8—1571no—
242C22⋊Q8C2×D15711yescharacteristic
243C22⋊Q8C2×D15711yescharacteristic
244#121—3141no—
245#121—3141no—
246#121—1571no—
247C4×Q8—1571no—
248C4×D4—3141no—
249C4×D4—1571no—
250C4×D4—1571no—
251C2×C4⋊C4—3141no—
252#107C2×D15711yescharacteristic
253C4⋊C8—1571no—
254#103—1571no—
255#103—1571no—
256#103C2×D15711yescharacteristic
257#103C2×D15711yescharacteristic
258#102—1571no—
259#102—1571no—
260C22⋊C8—1571no—
261C22⋊C8—1571no—
262#97—1571no—
263#41—3142no—
264#41—3142no—
265#41—3142no—
266#41—6281no—
267#41—6281no—
268#41—6281no—
269C2×Q8—3141no—
270C2×Q8—3142no—
271C2×Q8—6281no—
272C2×Q8—1571no—
273C2×Q8—1571no—
274C2×Q8C22×D15711yescharacteristic
275C2×Q8C22×D15711yescharacteristic
276C2×D4—3141no—
277C2×D4—3141no—
278C2×D4—3141no—
279C2×D4—3141no—
280C2×D4—1571no—
281C2×D4—1571no—
282C2×D4—1571no—
283C22×C4—3141no—
284C22×C4—3141no—
285C22×C4—3141no—
286C22×C4—3141no—
287C22×C4—3141no—
288C22×C4—1571no—
289C22×C4C22×D15711yescharacteristic
290Q16—22no—
291Q16—3142no—
292Q16—3142no—
293Q16—6281no—
294SD16—3142no—
295SD16—3142no—
296SD16—6281no—
297SD16—1574no—
298D8—3142no—
299M16—3142no—
300C2×C8—22no—
301C2×C8—1571no—
302C2×C8—1571no—
303C2×C8C22×D15711yescharacteristic
304C2×C8C22×D15711yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3141no—
308C4⋊C4—3141no—
309C4⋊C4—3141no—
310C4⋊C4—3141no—
311C4⋊C4—1571no—
312C4⋊C4C22×D15711yescharacteristic
313C4⋊C4C22×D15711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—1571no—
322C22⋊C4—1571no—
323C42—3141no—
324C42—1571no—
325C23—3141no—
326C23—3141no—
327C23—1571no—
328C23D4×D15711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—3141no—
334Q8—3141no—
335Q8—3141no—
336Q8—6282no—
337Q8—1572no—
338D4—3141no—
339D4—3141no—
340D4—3141no—
341D4—6282no—
342D4—6282no—
343D4—6281no—
344D4—6282no—
345D4—6282no—
346D4—1572no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—6282no—
358C2×C4—6282no—
359C2×C4—6282no—
360C2×C4—6282no—
361C2×C4—6282no—
362C2×C4—1571no—
363C2×C4—1571no—
364C2×C4—1571no—
365C2×C4D4×D15711yescharacteristic
366C2×C4C23×D15711yescharacteristic
367C8—21no—
368C8—3141no—
369C8—3141no—
370C8D4×D15712yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—3141no—
375C22—3141no—
376C22—3142no—
377C22—3141no—
378C22—6281no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22—6281no—
383C22C2×D4×D15711yescharacteristic
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—3141no—
390C4—3141no—
391C4—3142no—
392C4—3141no—
393C4—6281no—
394C4C2×D4×D15711yescharacteristic
395C4#1909781711yescharacteristic
396C2—41no—
397C2—6281no—
398C2—6282no—
399C2—12561no—
400C2#2375790711yescharacteristic
401C2#2375807011yescharacteristic
402C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#467×C157C2
socleC22×C157C22×D4
hypercenter#467D157
hypocenterC157#3269
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×#19097793C2semidirect
3. #23757933C2semidirect
4. #23757925C2semidirect
5. #23757666C2semidirect
6. #23757520C2semidirect
7. #23757491C2semidirect
8. #23757482C2semidirect
9. #23757433C2semidirect
10. #23757433C2semidirect
11. #23757430C2semidirect
12. #23757421C2semidirect
13. #23757372C2semidirect
14. #23757344C2semidirect
15. #467×C157C2semidirect
89. #14578943D4semidirect
201. C157#3269semidirect
209. #467D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×#190977932
#237579332
#237579252
#237576662
#237575202
#237574912
#237574822
#237574332
#237574302
#237574212
#237573722
#237573442
#467×C1572
#3269157