Finite Group Databasebeta

← #34109589

Subgroups — #34109589

Order 20096.

496 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×#19097793C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757819C211yesmaximal, characteristic, complemented
4#23757741C211yesmaximal, characteristic, complemented
5#23757498C211yesmaximal, characteristic, complemented
6#23757494C211yesmaximal, characteristic, complemented
7#23757491C211yesmaximal, characteristic, complemented
8#23757487C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757477C211yesmaximal, characteristic, complemented
11#23757472C211yesmaximal, characteristic, complemented
12#23757467C211yesmaximal, characteristic, complemented
13C2×#19097700C211yesmaximal, characteristic, complemented
14#23757267C211yesmaximal, characteristic, complemented
15#416×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18#19097793—24no—
19#19097791—24no—
20C2×Dic628C2211yescharacteristic
21C2×#14578944C2211yescharacteristic
22C2×#14578943C2211yescharacteristic
23C2×#14578943C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578941C2211yescharacteristic
26C2×D1256C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C22×#11929434C2211yescharacteristic
29#19097769C2211yescharacteristic
30#19097767C2211yescharacteristic
31#19097760C2211yescharacteristic
32#19097758C2211yescharacteristic
33#19097743—21no—
34#19097743—21no—
35#19097736C2211yescharacteristic
36#19097736C2211yescharacteristic
37#19097733C2211yescharacteristic
38#19097733C2211yescharacteristic
39C4×C22⋊D157—21no—
40C4×C22⋊D157—21no—
41#19097731—21no—
42#19097731—21no—
43#19097728—21no—
44C2×C4×Dic157C2211yescharacteristic
45C42×D157—21no—
46#19097700C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097700C2211yescharacteristic
49#19097700C2211yescharacteristic
50#19097694C2211yescharacteristic
51#19097694C2211yescharacteristic
52#19097691C2211yescharacteristic
53#19097691C2211yescharacteristic
54C2×#41×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic
56#117×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578954—22no—
62#14578954—22no—
63#14578954—22no—
64#14578954D414yescomplemented
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C22×Dic157D411yescharacteristic
74C22×Dic157D411yescharacteristic
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157—21no—
79Dic628—22no—
80#14578944—22no—
81#14578943—22no—
82#14578943—22no—
83#14578942—22no—
84#14578941—22no—
85D1256—22no—
86#14578939—22no—
87C2×#11929434—22no—
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90#14578933—21no—
91#14578933—21no—
92#14578933—21no—
93#14578933—21no—
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100C4×Dic157—21no—
101C4×Dic157—21no—
102C4×Dic157D411yescharacteristic
103C4×Dic157D411yescharacteristic
104C4×Dic157C2311yescharacteristic
105C4×Dic157C2311yescharacteristic
106#41×C157—22no—
107#41×C157—22no—
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157—21no—
110C2×D4×C157C2311yescharacteristic
111C22×C4×C157—21no—
112C22×C4×C157C2311yescharacteristic
113C2×C8×C157—22no—
114C2×C8×C157C2311yescharacteristic
115C2×C8×C157C2311yescharacteristic
116C4⋊C4×C157C2311yescharacteristic
117C4⋊C4×C157C2311yescharacteristic
118C22⋊C4×C157—21no—
119C42×C157—21no—
120C22×D157—21no—
121C22×D157—21no—
122Dic314—21no—
123Dic314—21no—
124Dic314—21no—
125Dic314C2×D412yes—
126D628—21no—
127D628—21no—
128D628—21no—
129D628C2×D412yes—
130C22⋊D157—24no—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C22⋊D157—42no—
134C2×Dic157—21no—
135C2×Dic157—22no—
136C2×Dic157—21no—
137C2×Dic157—41no—
138C2×Dic157C2×D411yescharacteristic
139C2×Dic157C2×D411yescharacteristic
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C4×D157—24no—
143C4×D157—42no—
144C4×D157—42no—
145C4×D157—42no—
146#11929434—21no—
147#11929434—21no—
148C23×C157—21no—
149C23×C157C2×D411yescharacteristic
150Q8×C157—21no—
151Q8×C157—21no—
152D4×C157—21no—
153D4×C157—21no—
154D4×C157—42no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—42no—
159C2×C4×C157C2×D412yes—
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157#4112yes—
162C2×C4×C157C2411yescharacteristic
163C8×C157—21no—
164C8×C157—21no—
165C22×C157—21no—
166C22×C157—21no—
167C22×C157—41no—
168C22×C157—41no—
169C22×C157—41no—
170C22×C157#12112yes—
171C22×C157C22×D411yescharacteristic
172C2×D157—22no—
173C2×D157—41no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—41no—
178C4×C157—41no—
179C4×C157—41no—
180C4×C157—41no—
181C4×C157#12112yes—
182C4×C157C22×D411yescharacteristic
183C4×C157C2×#4111yescharacteristic
184Dic157—22no—
185Dic157—21no—
186Dic157—21no—
187Dic157—21no—
188Dic157—41no—
189Dic157#13512yescomplemented
190C2×C157—22no—
191C2×C157—81no—
192C2×C157C2×#12111yescharacteristic
193C2×C157C2×#13511yescharacteristic
194C2×C157#57511yescharacteristic
195D157—42no—
196D157—81no—
197C157#303511yescharacteristic, complemented, Sylow
198#3035—1571nomaximal, Sylow
199C2×#135—1571no—
200C2×#135—1571no—
201#532—1571no—
202C4×#41—1571no—
203#464—1571no—
204#464—1571no—
205#463—1571no—
206#463—1571no—
207#462—1571no—
208#461—1571no—
209#460—1571no—
210#459—1571no—
211C2×C4⋊C8—1571no—
212#416—1571no—
213#416D15711yescharacteristic, complemented
214C2×#41—1571no—
215C2×#41—1571no—
216C2×#41C2×D15711yescharacteristic
217#135—3144no—
218#135—3144no—
219C2×Q16—1571no—
220C2×Q16—1571no—
221C2×SD16—1571no—
222C2×SD16—1571no—
223C2×SD16—1571no—
224C2×SD16—1571no—
225C2×D8—1571no—
226C2×D8—1571no—
227C22×C8—1571no—
228C22×C8C2×D15711yescharacteristic
229C4⋊Q8—1571no—
230#127—1571no—
231#124—1571no—
232#124—1571no—
233#121—3141no—
234#121—3141no—
235C4×Q8—1571no—
236C4×Q8—1571no—
237C4×D4—3141no—
238C4×D4—3141no—
239C4×D4—3141no—
240C4×D4—3141no—
241C4×D4—1571no—
242C4×D4—1571no—
243#117—3141no—
244#117C2×D15711yescharacteristic
245C2×C42—3141no—
246C2×C42—1571no—
247C4⋊C8—1571no—
248C4⋊C8—1571no—
249C4⋊C8—1571no—
250C4⋊C8—1571no—
251#103—1571no—
252#103—1571no—
253#103C2×D15711yescharacteristic
254#103C2×D15711yescharacteristic
255#102—1571no—
256#102—1571no—
257#102C2×D15711yescharacteristic
258#102C2×D15711yescharacteristic
259#41—22no—
260#41—22no—
261#41—3142no—
262#41—3142no—
263#41—3142no—
264#41—1574no—
265C2×Q8—1571no—
266C2×Q8—1571no—
267C2×Q8C22×D15711yescharacteristic
268C2×D4—21no—
269C2×D4—3141no—
270C2×D4—3141no—
271C2×D4—3141no—
272C2×D4—3141no—
273C2×D4—1571no—
274C2×D4—1571no—
275C2×D4C22×D15711yescharacteristic
276C22×C4—21no—
277C22×C4—3141no—
278C22×C4—3141no—
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—1571no—
282C22×C4—1571no—
283C22×C4C22×D15711yescharacteristic
284Q16—3142no—
285Q16—3142no—
286SD16—3142no—
287SD16—3142no—
288SD16—3142no—
289SD16—3142no—
290D8—3142no—
291D8—3142no—
292C2×C8—22no—
293C2×C8—3142no—
294C2×C8—1571no—
295C2×C8—1571no—
296C2×C8C22×D15711yescharacteristic
297C2×C8C22×D15711yescharacteristic
298C4⋊C4—3141no—
299C4⋊C4—3141no—
300C4⋊C4—3141no—
301C4⋊C4—3141no—
302C4⋊C4C22×D15711yescharacteristic
303C4⋊C4C22×D15711yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—3141no—
306C22⋊C4—3141no—
307C22⋊C4—3141no—
308C22⋊C4—3141no—
309C22⋊C4—3141no—
310C22⋊C4—3141no—
311C42—21no—
312C42—3141no—
313C42—3141no—
314C42—1571no—
315C42—1571no—
316C42—1571no—
317C42—1571no—
318C23—21no—
319C23—3141no—
320C23—3141no—
321C23D4×D15711yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—3141no—
325Q8—3141no—
326Q8—3141no—
327Q8—1572no—
328D4—21no—
329D4—21no—
330D4—42no—
331D4—3144no—
332D4—3141no—
333D4—3141no—
334D4—3141no—
335D4—6282no—
336D4—6282no—
337D4—6282no—
338D4—1572no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3141no—
344C2×C4—3142no—
345C2×C4—3144no—
346C2×C4—3141no—
347C2×C4—6282no—
348C2×C4—6282no—
349C2×C4—6281no—
350C2×C4—6282no—
351C2×C4—1571no—
352C2×C4—1571no—
353C2×C4—1571no—
354C2×C4—1571no—
355C2×C4D4×D15712yes—
356C2×C4D4×D15711yescharacteristic
357C2×C4#1457895412yes—
358C2×C4C23×D15711yescharacteristic
359C8—21no—
360C8—21no—
361C8—3141no—
362C8—3141no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—3142no—
369C22—6281no—
370C22—6281no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22#1909774412yes—
375C22C2×D4×D15711yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—3142no—
380C4—3141no—
381C4—3141no—
382C4—3141no—
383C4—6281no—
384C4—1572no—
385C4#1909774412yes—
386C4C2×D4×D15711yescharacteristic
387C4C2×#1457895411yescharacteristic
388C2—22no—
389C2—81no—
390C2—6282no—
391C2—12561no—
392C2C2×#1909774411yescharacteristic
393C2#135×D15711yescharacteristic
394C2#2375807011yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
hypercenter#416D157
hypocenterC157#3035
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#19097793C2semidirect
2. C2×#19097791C2semidirect
3. #23757819C2semidirect
4. #23757741C2semidirect
5. #23757498C2semidirect
6. #23757494C2semidirect
7. #23757491C2semidirect
8. #23757487C2semidirect
9. #23757482C2semidirect
10. #23757477C2semidirect
11. #23757472C2semidirect
12. #23757467C2semidirect
13. C2×#19097700C2semidirect
14. #23757267C2semidirect
15. #416×C157C2semidirect
64. #14578954D4semidirect
189. Dic157#135semidirect
197. C157#3035semidirect
213. #416D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977932
C2×#190977912
#237578192
#237577412
#237574982
#237574942
#237574912
#237574872
#237574822
#237574772
#237574722
#237574672
C2×#190977002
#237572672
#416×C1572
#3035157