Finite Group Databasebeta

← #34111730

Subgroups — #34111730

Order 20096.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2#23757922C211yesmaximal, characteristic, complemented
3D4×C22⋊D157C211yesmaximal, characteristic, complemented
4#23757644C211yesmaximal, characteristic, complemented
5#23757519C211yesmaximal, characteristic, complemented
6#23757516C211yesmaximal, characteristic, complemented
7#23757476C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757424C211yesmaximal, characteristic, complemented
10#23757422C211yesmaximal, characteristic, complemented
11#23757418C211yesmaximal, characteristic, complemented
12#23757415C211yesmaximal, characteristic, complemented
13D8×Dic157C211yesmaximal, characteristic, complemented
14#23757349C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic, complemented
17C22×C22⋊D157—21no—
18C2×D4×D157C2211yescharacteristic
19#19097795—24no—
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic
22C2×#14578942C2211yescharacteristic, complemented
23C2×#14578941C2211yescharacteristic, complemented
24C2×#14578939C2211yescharacteristic, complemented
25C2×#14578937C2211yescharacteristic, complemented
26#19097769C2211yescharacteristic, complemented
27#19097767C2211yescharacteristic
28#19097754—21no—
29#19097752—21no—
30#19097750C2211yescharacteristic, complemented
31#19097747C2211yescharacteristic, complemented
32#19097745C2211yescharacteristic
33#19097743—21no—
34#19097743—21no—
35#19097742C2211yescharacteristic
36#19097740—21no—
37#19097739—21no—
38D4×Dic157C2211yescharacteristic
39D4×Dic157C2211yescharacteristic, complemented
40C4×C22⋊D157C2211yescharacteristic
41C2×#14578933—21no—
42#19097705C2211yescharacteristic, complemented
43#19097700C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50C22⋊#11929434C2211yescharacteristic, complemented
51#19097683C2211yescharacteristic, complemented
52C8×Dic157C2211yescharacteristic, complemented
53C2×#41×C157C2211yescharacteristic, complemented
54C22×D4×C157C2211yescharacteristic
55AGL(1,ℤ/8)×C157—24no—
56C2×SD16×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic, complemented
58C2×D8×C157C2211yescharacteristic
59C2×D8×C157C2211yescharacteristic, complemented
60C2×M16×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578953—22no—
63#14578953—41no—
64C2×Dic314C2311yescharacteristic
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—44no—
73C2×C22⋊D157D411yescharacteristic
74D4×D157—22no—
75D4×D157—41no—
76C22×Dic157—21no—
77C22×Dic157—21no—
78C22×Dic157—21no—
79C2×C4×D157C2311yescharacteristic
80#14578943—22no—
81#14578942—22no—
82#14578942—41no—
83#14578941—41no—
84#14578939—22no—
85#14578937—22no—
86C2×#11929434C2311yescharacteristic
87C2×#11929434C2311yescharacteristic
88C4⋊Dic157C2311yescharacteristic
89C4⋊Dic157C2311yescharacteristic
90#14578933—21no—
91#14578933—21no—
92#14578933—21no—
93#14578933—21no—
94#14578933D411yescharacteristic
95C22⋊Dic157—21no—
96C22⋊Dic157—21no—
97C22⋊Dic157—21no—
98C22⋊Dic157—21no—
99C22⋊Dic157D411yescharacteristic
100#14578931—21no—
101#14578931—21no—
102#14578931D411yescharacteristic
103C4×Dic157C2311yescharacteristic
104C24×C157—21no—
105#41×C157—22no—
106#41×C157—41no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157—22no—
109C2×D4×C157—21no—
110C2×D4×C157—21no—
111C2×D4×C157—41no—
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157—21no—
116C22×C4×C157C2311yescharacteristic
117SD16×C157—22no—
118SD16×C157—22no—
119D8×C157—22no—
120D8×C157D414yescomplemented
121M16×C157—22no—
122C2×C8×C157C2311yescharacteristic
123C2×C8×C157C2311yescharacteristic
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—81no—
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131D628—21no—
132D628—21no—
133C22⋊D157—22no—
134C22⋊D157—42no—
135C22⋊D157—42no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C22⋊D157—42no—
139C22⋊D157—81no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—42no—
144C2×Dic157—42no—
145C2×Dic157—42no—
146C2×Dic157C2×D411yescharacteristic
147C2×Dic157C2×D411yescharacteristic
148C2×Dic157C2×D411yescharacteristic
149C4×D157—21no—
150C4×D157—21no—
151#11929434—21no—
152#11929434—21no—
153C23×C157—21no—
154C23×C157—21no—
155C23×C157—21no—
156C23×C157—42no—
157C23×C157—42no—
158C23×C157—81no—
159C23×C157C2×D411yescharacteristic
160Q8×C157—21no—
161Q8×C157—21no—
162D4×C157—21no—
163D4×C157—21no—
164D4×C157—21no—
165D4×C157—21no—
166D4×C157—42no—
167D4×C157—42no—
168D4×C157C2×D412yes—
169D4×C157C2×D412yescomplemented
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—21no—
173C2×C4×C157—42no—
174C2×C4×C157C2×D411yescharacteristic
175C2×C4×C157C2411yescharacteristic
176C8×C157—21no—
177C8×C157C2×D412yescomplemented
178C22×C157—21no—
179C22×C157—22no—
180C22×C157—22no—
181C22×C157—21no—
182C22×C157—21no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—41no—
188C22×C157—41no—
189C22×C157—41no—
190C22×C157—84no—
191C22×C157C22×D411yescharacteristic
192C2×D157—21no—
193C2×D157—21no—
194C2×D157—21no—
195C2×D157—41no—
196C2×D157—41no—
197C2×D157—41no—
198C2×D157—84no—
199C4×C157—21no—
200C4×C157—41no—
201C4×C157C22×D411yescharacteristic
202C4×C157H(5,𝔽2)11yescharacteristic
203Dic157—21no—
204Dic157—21no—
205Dic157—41no—
206Dic157—41no—
207Dic157—41no—
208Dic157AGL(1,ℤ/8)12yescomplemented
209C2×C157—42no—
210C2×C157—41no—
211C2×C157—42no—
212C2×C157—81no—
213C2×C157D4211yescharacteristic
214C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C157#57711yescharacteristic
216D157—41no—
217D157—81no—
218C157#327211yescharacteristic, complemented, Sylow
219#3272—1571nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1571no—
221C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
222#547—1571no—
223D42—1571no—
224#492—1571no—
225#468—1571no—
226#468—1571no—
227#461—1571no—
228#459—1571no—
229#450—1571no—
230#450—1571no—
231#449—1571no—
232#449—1571no—
233C4×D8—1571no—
234#436—1571no—
235C2×#41—1571no—
236C2×#41C2×D15711yescharacteristic, complemented
237C22×D4—3141no—
238C22×D4—1571no—
239C22×D4C2×D15711yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—3144no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16—1571no—
246C2×SD16C2×D15711yescharacteristic
247C2×SD16C2×D15711yescharacteristic, complemented
248C2×D8—1571no—
249C2×D8C2×D15711yescharacteristic
250C2×D8C2×D15711yescharacteristic, complemented
251C2×M16—1571no—
252C2×M16C2×D15711yescharacteristic, complemented
253C4⋊Q8—1571no—
254#127—1571no—
255#123—3141no—
256#123—3141no—
257C22⋊Q8—1571no—
258C22⋊Q8—1571no—
259#121—3141no—
260#121—3141no—
261#121—1571no—
262#121—1571no—
263#120—3141no—
264#120—3141no—
265C4×D4—1571no—
266C4×D4—1571no—
267C4×D4—1571no—
268C2×C4⋊C4—3141no—
269#107—1571no—
270C4⋊C8—1571no—
271#103—1571no—
272#103—1571no—
273#102—1571no—
274#102—1571no—
275#102—1571no—
276#102—1571no—
277C22⋊C8—1571no—
278C22⋊C8—1571no—
279C4×C8—1571no—
280C24—21no—
281C24—3141no—
282#41—22no—
283#41—41no—
284#41—3142no—
285#41—6281no—
286C2×Q8—1571no—
287C2×Q8C22×D15711yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3142no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3141no—
299C2×D4—6281no—
300C2×D4—6284no—
301C2×D4—1571no—
302C2×D4—1571no—
303C2×D4C22×D15711yescharacteristic
304C2×D4C22×D15711yescharacteristic
305C2×D4C22×D15711yescharacteristic
306C22×C4—21no—
307C22×C4—3141no—
308C22×C4—3141no—
309C22×C4—3141no—
310C22×C4—1571no—
311C22×C4C22×D15711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—3142no—
315SD16—3142no—
316SD16—6281no—
317SD16—6281no—
318D8—22no—
319D8—3142no—
320D8C22⋊D15714yescomplemented
321M16—22no—
322M16—3142no—
323C2×C8—1571no—
324C2×C8—1571no—
325C2×C8C22×D15711yescharacteristic
326C2×C8C22×D15711yescharacteristic
327C4⋊C4—3141no—
328C4⋊C4—3141no—
329C4⋊C4—3141no—
330C4⋊C4—3141no—
331C4⋊C4—1571no—
332C4⋊C4—1571no—
333C4⋊C4—1571no—
334C22⋊C4—3141no—
335C22⋊C4—3141no—
336C22⋊C4—3141no—
337C22⋊C4—3141no—
338C22⋊C4—3141no—
339C22⋊C4—3141no—
340C22⋊C4—1571no—
341C22⋊C4—1571no—
342C42—1571no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—3141no—
350C23—6282no—
351C23—6282no—
352C23—1571no—
353C23—12561no—
354C23D4×D15711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3141no—
358Q8—3141no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—3142no—
366D4—3141no—
367D4—3141no—
368D4—6282no—
369D4—6282no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—12561no—
374D4C2×C22⋊D15712yes—
375D4C2×C22⋊D15712yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—3141no—
381C2×C4—3141no—
382C2×C4—3141no—
383C2×C4—3141no—
384C2×C4—3141no—
385C2×C4—6282no—
386C2×C4—6282no—
387C2×C4—6282no—
388C2×C4—1571no—
389C2×C4—1571no—
390C2×C4—1571no—
391C2×C4D4×D15711yescharacteristic
392C2×C4C23×D15711yescharacteristic
393C8—21no—
394C8—3141no—
395C8—3141no—
396C8C2×C22⋊D15712yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—3141no—
411C22—3141no—
412C22—3141no—
413C22—6281no—
414C22—6281no—
415C22—6281no—
416C22—12564no—
417C22C2×D4×D15711yescharacteristic
418C4—21no—
419C4—41no—
420C4—3141no—
421C4—3141no—
422C4—6281no—
423C4—6281no—
424C4—6281no—
425C4—1572no—
426C4C22×C22⋊D15711yescharacteristic
427C4#1909781611yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—6281no—
433C2—12561no—
434C2D4×C22⋊D15711yescharacteristic
435C2AGL(1,ℤ/8)×D15711yescharacteristic
436C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. #23757922C2semidirect
3. D4×C22⋊D157C2semidirect
4. #23757644C2semidirect
5. #23757519C2semidirect
6. #23757516C2semidirect
7. #23757476C2semidirect
8. #23757466C2semidirect
9. #23757424C2semidirect
10. #23757422C2semidirect
11. #23757418C2semidirect
12. #23757415C2semidirect
13. D8×Dic157C2semidirect
14. #23757349C2semidirect
15. C2×AGL(1,ℤ/8)×C157C2semidirect
16. C2×#14578953C22semidirect
20. C2×#14578943C22semidirect
22. C2×#14578942C22semidirect
23. C2×#14578941C22semidirect
24. C2×#14578939C22semidirect
25. C2×#14578937C22semidirect
26. #19097769C22semidirect
30. #19097750C22semidirect
31. #19097747C22semidirect
39. D4×Dic157C22semidirect
42. #19097705C22semidirect
43. #19097700C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097693C22semidirect
48. #19097692C22semidirect
49. #19097692C22semidirect
50. C22⋊#11929434C22semidirect
51. #19097683C22semidirect
52. C8×Dic157C22semidirect
53. C2×#41×C157C22semidirect
57. C2×SD16×C157C22semidirect
59. C2×D8×C157C22semidirect
60. C2×M16×C157C22semidirect
120. D8×C157D4semidirect
169. D4×C157C2×D4semidirect
177. C8×C157C2×D4semidirect
208. Dic157AGL(1,ℤ/8)semidirect
218. C157#3272semidirect
221. C2×AGL(1,ℤ/8)D157semidirect
236. C2×#41C2×D157semidirect
247. C2×SD16C2×D157semidirect
250. C2×D8C2×D157semidirect
252. C2×M16C2×D157semidirect
320. D8C22⋊D157semidirect
375. D4C2×C22⋊D157semidirect
396. C8C2×C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
#237579222
D4×C22⋊D1572
#237576442
#237575192
#237575162
#237574762
#237574662
#237574242
#237574222
#237574182
#237574152
D8×Dic1572
#237573492
C2×AGL(1,ℤ/8)×C1572
#3272157