Finite Group Databasebeta

← #34111682

Subgroups — #34111682

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×#19097790C211yesmaximal, characteristic, complemented
3#23757943C211yesmaximal, characteristic, complemented
4#23757918C211yesmaximal, characteristic, complemented
5#23757660C211yesmaximal, characteristic, complemented
6#23757518C211yesmaximal, characteristic, complemented
7#23757508C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757436C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757415C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757373C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×#14578953C2211yescharacteristic, complemented
19C2×D4×D157C2211yescharacteristic
20#19097795—24no—
21#19097790—24no—
22C2×Dic628C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic, complemented
26C2×#14578942C2211yescharacteristic, complemented
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30C2×C8×D157C2211yescharacteristic, complemented
31#19097767—21no—
32#19097759C2211yescharacteristic
33#19097755—21no—
34#19097751C2211yescharacteristic
35#19097746C2211yescharacteristic
36#19097745—21no—
37#19097743—21no—
38#19097743—21no—
39#19097742C2211yescharacteristic
40#19097742C2211yescharacteristic, complemented
41#19097741—21no—
42C4×Dic314C2211yescharacteristic, complemented
43C4×D628C2211yescharacteristic
44D4×Dic157—21no—
45C2×#14578931—21no—
46#19097704C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097692C2211yescharacteristic
51#19097683C2211yescharacteristic
52#19097683C2211yescharacteristic, complemented
53#19097682C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic
56#127×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578953—22no—
63#14578953—22no—
64#14578953—22no—
65#14578953—41no—
66#14578953—41no—
67#14578953—41no—
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70C2×D628—21no—
71C2×D628D411yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79D4×D157—22no—
80D4×D157—41no—
81C22×Dic157—21no—
82C22×Dic157—21no—
83C22×Dic157—21no—
84C2×C4×D157C2311yescharacteristic
85C2×C4×D157C2311yescharacteristic
86Dic628—22no—
87#14578943—22no—
88#14578942—22no—
89#14578942—41no—
90#14578942D414yescomplemented
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×D4×C157—21no—
111C2×D4×C157—21no—
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157—21no—
116D8×C157—22no—
117D8×C157—22no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157C2311yescharacteristic
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—
130Dic314—21no—
131Dic314—21no—
132Dic314C2×D412yes—
133D628—21no—
134D628—21no—
135D628—41no—
136D628—81no—
137C22⋊D157—42no—
138C22⋊D157—42no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C22⋊D157—81no—
142C22⋊D157—81no—
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—42no—
147C2×Dic157—42no—
148C2×Dic157—42no—
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—21no—
154C4×D157—21no—
155#11929434—21no—
156#11929434C2×D412yes—
157C23×C157—21no—
158C23×C157—21no—
159C23×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—21no—
164D4×C157—21no—
165D4×C157—42no—
166D4×C157—81no—
167D4×C157C2×D412yes—
168C2×C4×C157—21no—
169C2×C4×C157—81no—
170C2×C4×C157C2×D411yescharacteristic
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157—21no—
175C22×C157—22no—
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—21no—
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—21no—
191C2×D157—41no—
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—42no—
206C2×C157—81no—
207C2×C157—81no—
208C2×C157D4211yescharacteristic
209C2×C157#57511yescharacteristic
210C2×C157#57611yescharacteristic
211D157—41no—
212D157—41no—
213D157—81no—
214C157#326811yescharacteristic, complemented, Sylow
215#3268—1571nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1571no—
217C2×#135—1571no—
218#550—1571no—
219#546—1571no—
220#496—1571no—
221#468—1571no—
222#466—1571no—
223#464—1571no—
224#459D15711yescharacteristic, complemented
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×SD16—1571no—
241C2×SD16—1571no—
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#124—1571no—
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×D4C2×D15711yescharacteristic
264C2×C22⋊C4—3141no—
265#107—1571no—
266C4⋊C8C2×D15711yescharacteristic, complemented
267#103—1571no—
268#103—1571no—
269#102—1571no—
270#102—1571no—
271#102C2×D15711yescharacteristic, complemented
272#102C2×D15711yescharacteristic, complemented
273C22⋊C8—1571no—
274C22⋊C8—1571no—
275#97—1571no—
276C24—3141no—
277#41—3142no—
278#41—3142no—
279#41—3142no—
280#41—6281no—
281#41—6281no—
282#41—6281no—
283C2×Q8—1571no—
284C2×Q8—1571no—
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—21no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—3141no—
305C22×C4—1571no—
306C22×C4—1571no—
307Q16—3142no—
308SD16—3142no—
309SD16—3142no—
310SD16—6281no—
311SD16—1574no—
312D8—22no—
313D8—22no—
314D8—3142no—
315D8—6281no—
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⋊C4C22×D15711yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—3141no—
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—3141no—
348Q8—3141no—
349Q8—3141no—
350Q8—1572no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—3141no—
359D4—3141no—
360D4—6281no—
361D4—6282no—
362D4—6282no—
363D4—6282no—
364D4—6282no—
365D4—12561no—
366D4—12561no—
367D4—12561no—
368D4D4×D15712yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—3141no—
372C2×C4—3141no—
373C2×C4—3141no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—6282no—
379C2×C4—6282no—
380C2×C4—6282no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4D4×D15711yescharacteristic
384C2×C4D4×D15711yescharacteristic
385C2×C4C23×D15711yescharacteristic
386C8—21no—
387C8—21no—
388C8—3141no—
389C8—1572no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3141no—
400C22—3141no—
401C22—3141no—
402C22—3141no—
403C22—3141no—
404C22—3141no—
405C22—6281no—
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#1909781711yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—6281no—
424C2—6281no—
425C2—12561no—
426C2#2375790711yescharacteristic
427C2#2375806911yescharacteristic
428C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3268
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×#19097790C2semidirect
3. #23757943C2semidirect
4. #23757918C2semidirect
5. #23757660C2semidirect
6. #23757518C2semidirect
7. #23757508C2semidirect
8. #23757496C2semidirect
9. #23757436C2semidirect
10. #23757419C2semidirect
11. #23757415C2semidirect
12. #23757405C2semidirect
13. #23757373C2semidirect
14. #23757343C2semidirect
15. #459×C157C2semidirect
18. C2×#14578953C22semidirect
22. C2×Dic628C22semidirect
25. C2×#14578942C22semidirect
26. C2×#14578942C22semidirect
30. C2×C8×D157C22semidirect
40. #19097742C22semidirect
42. C4×Dic314C22semidirect
46. #19097704C22semidirect
47. #19097696C22semidirect
48. #19097694C22semidirect
52. #19097683C22semidirect
53. #19097682C22semidirect
56. #127×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #102×C157C22semidirect
60. #102×C157C22semidirect
90. #14578942D4semidirect
214. C157#3268semidirect
224. #459D157semidirect
249. #127C2×D157semidirect
266. C4⋊C8C2×D157semidirect
271. #102C2×D157semidirect
272. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×#190977902
#237579432
#237579182
#237576602
#237575182
#237575082
#237574962
#237574362
#237574192
#237574152
#237574052
#237573732
#237573432
#459×C1572
#3268157