Finite Group Databasebeta

← #34111889

Subgroups — #34111889

Order 20096.

504 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097792C211yesmaximal, characteristic, complemented
2C2×#19097791C211yesmaximal, characteristic, complemented
3#23757942C211yesmaximal, characteristic, complemented
4#23757941C211yesmaximal, characteristic, complemented
5#23757648C211yesmaximal, characteristic, complemented
6#23757508C211yesmaximal, characteristic, complemented
7#23757507C211yesmaximal, characteristic, complemented
8#23757472C211yesmaximal, characteristic, complemented
9#23757471C211yesmaximal, characteristic, complemented
10#23757421C211yesmaximal, characteristic, complemented
11#23757420C211yesmaximal, characteristic, complemented
12#23757415C211yesmaximal, characteristic, complemented
13#23757365C211yesmaximal, characteristic, complemented
14#23757336C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18C2×#14578953C2211yescharacteristic
19#19097792—24no—
20#19097791—24no—
21C2×#14578944C2211yescharacteristic
22C2×#14578943C2211yescharacteristic
23C2×#14578942C2211yescharacteristic
24C2×#14578941C2211yescharacteristic
25C2×D1256C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C22×#11929434C2211yescharacteristic
29C2×C8×D157C2211yescharacteristic
30#19097767—21no—
31#19097767—21no—
32#19097767C2211yescharacteristic
33#19097767C2211yescharacteristic
34#19097746C2211yescharacteristic
35#19097745—21no—
36#19097745C2211yescharacteristic
37#19097744—21no—
38#19097743—21no—
39#19097742—21no—
40#19097742C2211yescharacteristic
41Q8×Dic157C2211yescharacteristic
42#19097736C2211yescharacteristic
43C4×C22⋊D157—21no—
44C4×C22⋊D157C2211yescharacteristic
45#19097731—21no—
46#19097704C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097692C2211yescharacteristic
51#19097691C2211yescharacteristic
52#19097683C2211yescharacteristic
53C8×Dic157C2211yescharacteristic
54C2×#41×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C2×D8×C157C2211yescharacteristic
57#121×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic
60C22⋊C8×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63#14578954—22no—
64#14578954—41no—
65#14578953—22no—
66#14578953—41no—
67C2×Dic314C2311yescharacteristic
68C2×D628—21no—
69C2×D628—21no—
70C2×D628C2311yescharacteristic
71C2×D628C2311yescharacteristic
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157D411yescharacteristic
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85#14578944D414yescomplemented
86#14578943—22no—
87#14578942—22no—
88#14578941—22no—
89D1256—41no—
90#14578939—22no—
91#14578939—22no—
92C2×#11929434—22no—
93C2×#11929434C2311yescharacteristic
94C2×#11929434C2311yescharacteristic
95C8×D157—22no—
96C4⋊Dic157—21no—
97C4⋊Dic157C2311yescharacteristic
98#14578933—21no—
99#14578933D411yescharacteristic
100C22⋊Dic157—21no—
101C22⋊Dic157D411yescharacteristic
102#14578931—21no—
103#14578931—21no—
104#14578931D411yescharacteristic
105C4×Dic157—21no—
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108#41×C157—22no—
109#41×C157—41no—
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—21no—
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157—21no—
116C22×C4×C157C2311yescharacteristic
117SD16×C157—22no—
118D8×C157—41no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314C2×D412yes—
128D628—21no—
129D628—21no—
130D628—21no—
131D628—21no—
132D628—42no—
133D628—81no—
134C22⋊D157—22no—
135C22⋊D157—42no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C22⋊D157—42no—
139C22⋊D157—81no—
140C22⋊D157—81no—
141C22⋊D157—81no—
142C22⋊D157—81no—
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—42no—
146C2×Dic157C2×D411yescharacteristic
147C2×Dic157C2×D411yescharacteristic
148C2×Dic157C2×D411yescharacteristic
149C4×D157—21no—
150C4×D157—21no—
151C4×D157—42no—
152C4×D157—42no—
153#11929434—21no—
154#11929434C2×D412yes—
155C23×C157—21no—
156C23×C157—21no—
157C23×C157C2×D411yescharacteristic
158Q8×C157—21no—
159Q8×C157C2×D412yes—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—21no—
164D4×C157—42no—
165D4×C157—81no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—42no—
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157—21no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—21no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157C22×D411yescharacteristic
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—21no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C4×C157—21no—
195C4×C157—22no—
196C4×C157—41no—
197C4×C157C22×D411yescharacteristic
198C4×C157H(5,𝔽2)11yescharacteristic
199Dic157—21no—
200Dic157—22no—
201Dic157—21no—
202Dic157—41no—
203Dic157#13512yescomplemented
204C2×C157—41no—
205C2×C157—81no—
206C2×C157—81no—
207C2×C157D4211yescharacteristic
208C2×C157C2×#13511yescharacteristic
209C2×C157#57611yescharacteristic
210D157—41no—
211D157—81no—
212D157—81no—
213C157#328711yescharacteristic, complemented, Sylow
214#3287—1571nomaximal, Sylow
215C2×#135—1571no—
216C2×#135—1571no—
217#550—1571no—
218#550—1571no—
219#493—1571no—
220#466—1571no—
221#466—1571no—
222#460—1571no—
223#460—1571no—
224#449—1571no—
225#449—1571no—
226#449—1571no—
227#449D15711yescharacteristic, complemented
228C4×Q16—1571no—
229C8×D4—1571no—
230C2×#41—1571no—
231C2×#41—1571no—
232C2×#41—1571no—
233C2×#41C2×D15711yescharacteristic
234#135—3144no—
235#135—3144no—
236C2×Q16—1571no—
237C2×SD16—1571no—
238C2×SD16—1571no—
239C2×SD16—1571no—
240C2×SD16C2×D15711yescharacteristic
241C2×D8—1571no—
242C2×D8—1571no—
243C2×D8—1571no—
244C2×D8C2×D15711yescharacteristic
245C22×C8—1571no—
246C22×C8—1571no—
247#127—3141no—
248#127—3141no—
249#127—1571no—
250#127—1571no—
251#121—3141no—
252#121—3141no—
253#121—3141no—
254#121—3141no—
255#121—1571no—
256#121—1571no—
257#121—1571no—
258#121C2×D15711yescharacteristic
259C4×Q8—1571no—
260C4×Q8—1571no—
261C4×D4—3141no—
262C4×D4—3141no—
263C4×D4—1571no—
264#107—1571no—
265C4⋊C8—1571no—
266#103—1571no—
267#103C2×D15711yescharacteristic
268#102—1571no—
269#102—1571no—
270#102—1571no—
271#102C2×D15711yescharacteristic
272C22⋊C8—1571no—
273C22⋊C8C2×D15711yescharacteristic
274C4×C8—1571no—
275#41—22no—
276#41—41no—
277#41—3142no—
278#41—3142no—
279#41—3142no—
280#41—6281no—
281#41—6281no—
282#41—6281no—
283C2×Q8—1571no—
284C2×Q8C22×D15711yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—3141no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—3141no—
297C2×D4—1571no—
298C2×D4—1571no—
299C2×D4—1571no—
300C2×D4C22×D15711yescharacteristic
301C2×D4C22×D15711yescharacteristic
302C22×C4—21no—
303C22×C4—3141no—
304C22×C4—3141no—
305C22×C4—3141no—
306C22×C4—1571no—
307C22×C4C22×D15711yescharacteristic
308Q16—1574no—
309SD16—22no—
310SD16—3142no—
311SD16—3142no—
312SD16—3142no—
313D8—41no—
314D8—3142no—
315D8—3142no—
316D8—6281no—
317C2×C8—3142no—
318C2×C8—3142no—
319C2×C8—1571no—
320C2×C8—1571no—
321C2×C8C22×D15711yescharacteristic
322C2×C8C22×D15711yescharacteristic
323C4⋊C4—3141no—
324C4⋊C4—3141no—
325C4⋊C4—1571no—
326C4⋊C4—1571no—
327C4⋊C4C22×D15711yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—3141no—
330C22⋊C4—3141no—
331C22⋊C4—3141no—
332C22⋊C4—1571no—
333C22⋊C4—1571no—
334C42—3141no—
335C42—3141no—
336C42—1571no—
337C23—21no—
338C23—21no—
339C23—3141no—
340C23—3141no—
341C23—1571no—
342C23D4×D15711yescharacteristic
343Q8—21no—
344Q8—3141no—
345Q8—1572no—
346Q8D4×D15712yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—3142no—
354D4—3141no—
355D4—3141no—
356D4—3141no—
357D4—3141no—
358D4—6282no—
359D4—6282no—
360D4—6282no—
361D4—6282no—
362D4—6282no—
363D4—12561no—
364D4—12561no—
365D4—12561no—
366D4—12561no—
367D4—12561no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—3141no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—6282no—
378C2×C4—6282no—
379C2×C4—6282no—
380C2×C4—1571no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4D4×D15711yescharacteristic
384C2×C4C23×D15711yescharacteristic
385C8—21no—
386C8—21no—
387C8—3141no—
388C8—1572no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—3141no—
399C22—3141no—
400C22—3141no—
401C22—6281no—
402C22—6281no—
403C22—6281no—
404C22—6281no—
405C22—6281no—
406C22—6281no—
407C22C2×D4×D15711yescharacteristic
408C4—21no—
409C4—22no—
410C4—41no—
411C4—3141no—
412C4—3142no—
413C4—3141no—
414C4—6281no—
415C4—1572no—
416C4C2×D4×D15711yescharacteristic
417C4#1909781611yescharacteristic
418C2—41no—
419C2—81no—
420C2—81no—
421C2—6281no—
422C2—12561no—
423C2—12561no—
424C2#2375790511yescharacteristic
425C2#135×D15711yescharacteristic
426C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3287
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19097792C2semidirect
2. C2×#19097791C2semidirect
3. #23757942C2semidirect
4. #23757941C2semidirect
5. #23757648C2semidirect
6. #23757508C2semidirect
7. #23757507C2semidirect
8. #23757472C2semidirect
9. #23757471C2semidirect
10. #23757421C2semidirect
11. #23757420C2semidirect
12. #23757415C2semidirect
13. #23757365C2semidirect
14. #23757336C2semidirect
15. #449×C157C2semidirect
85. #14578944D4semidirect
203. Dic157#135semidirect
213. C157#3287semidirect
227. #449D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977922
C2×#190977912
#237579422
#237579412
#237576482
#237575082
#237575072
#237574722
#237574712
#237574212
#237574202
#237574152
#237573652
#237573362
#449×C1572
#3287157