Finite Group Databasebeta

← #34111675

Subgroups — #34111675

Order 20096.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2C2×#19097792C211yesmaximal, characteristic, complemented
3#23757943C211yesmaximal, characteristic, complemented
4#23757916C211yesmaximal, characteristic, complemented
5#23757516C211yesmaximal, characteristic, complemented
6#23757506C211yesmaximal, characteristic, complemented
7#23757492C211yesmaximal, characteristic, complemented
8#23757465C211yesmaximal, characteristic, complemented
9#23757432C211yesmaximal, characteristic, complemented
10#23757417C211yesmaximal, characteristic, complemented
11#23757415C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757375C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#496×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578953C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic
20#19097795—24no—
21#19097792—24no—
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578939C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic, complemented
29#19097767—21no—
30#19097753—21no—
31#19097747C2211yescharacteristic
32#19097744—21no—
33#19097743—21no—
34#19097743—21no—
35#19097742C2211yescharacteristic
36#19097742C2211yescharacteristic
37#19097742C2211yescharacteristic, complemented
38#19097741—21no—
39C4×Dic314C2211yescharacteristic, complemented
40C4×D628C2211yescharacteristic
41C4×D628C2211yescharacteristic
42D4×Dic157—21no—
43C22⋊C4×D157—21no—
44#19097705C2211yescharacteristic, complemented
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097696C2211yescharacteristic, complemented
47#19097695C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097692C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic
53#19097683C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic, complemented
56C2×D8×C157C2211yescharacteristic
57C2×D8×C157C2211yescharacteristic
58#127×C157C2211yescharacteristic, complemented
59#124×C157C2211yescharacteristic
60#97×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578955—22no—
63#14578955—41no—
64#14578953—22no—
65#14578953—22no—
66#14578953—41no—
67#14578953—41no—
68C2×Dic314C2311yescharacteristic
69C2×D628—21no—
70C2×D628D411yescharacteristic
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—
78D4×D157—22no—
79D4×D157—41no—
80C22×Dic157—21no—
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157—21no—
84C2×C4×D157C2311yescharacteristic
85C2×C4×D157C2311yescharacteristic
86#14578944—22no—
87#14578943D414yescomplemented
88#14578942—22no—
89#14578939—22no—
90#14578939—22no—
91#14578937—22no—
92C2×#11929434C2311yescharacteristic
93C2×#11929434C2311yescharacteristic
94C8×D157—22no—
95C4⋊Dic157D411yescharacteristic
96C4⋊Dic157C2311yescharacteristic
97C4⋊Dic157C2311yescharacteristic
98#14578933—21no—
99#14578933—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931D411yescharacteristic
106#14578931D411yescharacteristic
107C4×Dic157—21no—
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157—21no—
110C2×D4×C157—21no—
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114SD16×C157—22no—
115SD16×C157—41no—
116D8×C157—22no—
117D8×C157—41no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157—81no—
127C22×D157C2×D411yescharacteristic
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314C2×D412yes—
131D628—21no—
132D628—41no—
133D628—42no—
134D628C2×D412yes—
135C22⋊D157—42no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C22⋊D157—81no—
139C22⋊D157—81no—
140C22⋊D157—81no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157—42no—
145C2×Dic157—42no—
146C2×Dic157C2×D411yescharacteristic
147C2×Dic157C2×D411yescharacteristic
148C4×D157—21no—
149C4×D157—21no—
150C4×D157—21no—
151C4×D157—21no—
152C4×D157—42no—
153C4×D157—81no—
154#11929434—21no—
155#11929434—21no—
156C23×C157—21no—
157C23×C157—21no—
158C23×C157—21no—
159Q8×C157—21no—
160Q8×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—21no—
164D4×C157—21no—
165D4×C157—21no—
166D4×C157—21no—
167D4×C157—42no—
168D4×C157—81no—
169C2×C4×C157—21no—
170C2×C4×C157C2×D411yescharacteristic
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157C2×D412yes—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
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—22no—
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—21no—
191C2×D157—21no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—84no—
195C4×C157—21no—
196C4×C157—21no—
197C4×C157—41no—
198C4×C157C22×D411yescharacteristic
199C4×C157H(5,𝔽2)11yescharacteristic
200Dic157—21no—
201Dic157—21no—
202Dic157—22no—
203Dic157—41no—
204Dic157—41no—
205C2×C157—81no—
206C2×C157—81no—
207C2×C157—81no—
208C2×C157D4211yescharacteristic
209C2×C157#57511yescharacteristic
210C2×C157#57611yescharacteristic
211D157—41no—
212D157—41no—
213D157—42no—
214C157#326811yescharacteristic, complemented, Sylow
215#3268—1571nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1571no—
217C2×#135—1571no—
218#550—1571no—
219#546—1571no—
220#496D15711yescharacteristic, complemented
221#468—1571no—
222#466—1571no—
223#464—1571no—
224#459—1571no—
225#452—1571no—
226#449—1571no—
227#449—1571no—
228#447—1571no—
229#440—1571no—
230#435—1571no—
231C2×#41—1571no—
232C2×#41—1571no—
233C2×#41—1571no—
234C22×D4—1571no—
235AGL(1,ℤ/8)—3144no—
236#135—3144no—
237C2×Q16—1571no—
238C2×SD16—1571no—
239C2×SD16—1571no—
240C2×SD16C2×D15711yescharacteristic, complemented
241C2×SD16C2×D15711yescharacteristic, complemented
242C2×D8—1571no—
243C2×D8—1571no—
244C2×D8C2×D15711yescharacteristic
245C2×D8C2×D15711yescharacteristic
246C2×M16—1571no—
247C22×C8—1571no—
248#127—3141no—
249#127C2×D15711yescharacteristic, complemented
250#124C2×D15711yescharacteristic
251#123—3141no—
252C22⋊Q8—1571no—
253#121—3141no—
254#121—3141no—
255#121—3141no—
256#121—1571no—
257#121—1571no—
258#121—1571no—
259#120—3141no—
260C4×Q8—1571no—
261C4×D4—3141no—
262C4×D4—1571no—
263C4×D4—1571no—
264C2×C22⋊C4—3141no—
265#107—1571no—
266C4⋊C8—1571no—
267#103—1571no—
268#103—1571no—
269#102—1571no—
270#102—1571no—
271#102—1571no—
272#102—1571no—
273C22⋊C8—1571no—
274C22⋊C8—1571no—
275#97C2×D15711yescharacteristic, complemented
276C24—3141no—
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—3142no—
289C2×D4—3141no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—6281no—
296C2×D4—1571no—
297C2×D4—1571no—
298C2×D4C22×D15711yescharacteristic
299C2×D4C22×D15711yescharacteristic
300C2×D4C22×D15711yescharacteristic
301C22×C4—3141no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—3141no—
305C22×C4—1571no—
306C22×C4—1571no—
307Q16—3142no—
308SD16—22no—
309SD16—41no—
310SD16—3142no—
311SD16—1574no—
312D8—22no—
313D8—41no—
314D8—3142no—
315D8—3142no—
316M16—3142no—
317C2×C8—3142no—
318C2×C8—1571no—
319C2×C8—1571no—
320C2×C8C22×D15711yescharacteristic
321C2×C8C22×D15711yescharacteristic
322C4⋊C4—3141no—
323C4⋊C4—3141no—
324C4⋊C4—1571no—
325C4⋊C4—1571no—
326C4⋊C4—1571no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3141no—
330C22⋊C4—3141no—
331C22⋊C4—3141no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—1571no—
335C22⋊C4—1571no—
336C42—3141no—
337C42C22×D15711yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—3141no—
342C23—6282no—
343C23—6282no—
344C23—1571no—
345C23—1571no—
346C23—12561no—
347Q8—21no—
348Q8—21no—
349Q8—3141no—
350Q8—1572no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—3141no—
360D4—6281no—
361D4—6282no—
362D4—6282no—
363D4—6282no—
364D4—6282no—
365D4—1572no—
366D4—12561no—
367D4—12561no—
368D4—12561no—
369C2×C4—21no—
370C2×C4—3141no—
371C2×C4—3141no—
372C2×C4—3141no—
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—12561no—
383C2×C4D4×D15711yescharacteristic
384C2×C4D4×D15711yescharacteristic
385C2×C4C23×D15711yescharacteristic
386C8—21no—
387C8—3141no—
388C8—3141no—
389C8D4×D15712yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3141no—
400C22—3141no—
401C22—3142no—
402C22—3141no—
403C22—3141no—
404C22—3141no—
405C22—3141no—
406C22—6281no—
407C22—6281no—
408C22—12564no—
409C22C2×D4×D15711yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—3141no—
414C4—3141no—
415C4—3142no—
416C4—6281no—
417C4—6281no—
418C4C2×D4×D15711yescharacteristic
419C4#1909781611yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—6281no—
424C2—6281no—
425C2—6282no—
426C2#2375790811yescharacteristic
427C2#2375806911yescharacteristic
428C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157#3268
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×#19097792C2semidirect
3. #23757943C2semidirect
4. #23757916C2semidirect
5. #23757516C2semidirect
6. #23757506C2semidirect
7. #23757492C2semidirect
8. #23757465C2semidirect
9. #23757432C2semidirect
10. #23757417C2semidirect
11. #23757415C2semidirect
12. #23757405C2semidirect
13. #23757375C2semidirect
14. #23757346C2semidirect
15. #496×C157C2semidirect
18. C2×#14578953C22semidirect
22. C2×#14578944C22semidirect
28. C2×C8×D157C22semidirect
37. #19097742C22semidirect
39. C4×Dic314C22semidirect
44. #19097705C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097696C22semidirect
47. #19097695C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
53. #19097683C22semidirect
54. C2×SD16×C157C22semidirect
55. C2×SD16×C157C22semidirect
58. #127×C157C22semidirect
60. #97×C157C22semidirect
87. #14578943D4semidirect
214. C157#3268semidirect
220. #496D157semidirect
240. C2×SD16C2×D157semidirect
241. C2×SD16C2×D157semidirect
249. #127C2×D157semidirect
275. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×#190977922
#237579432
#237579162
#237575162
#237575062
#237574922
#237574652
#237574322
#237574172
#237574152
#237574052
#237573752
#237573462
#496×C1572
#3268157