Finite Group Databasebeta

← #34111741

Subgroups — #34111741

Order 20096.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2#23757917C211yesmaximal, characteristic, complemented
3#23757911C211yesmaximal, characteristic, complemented
4#23757654C211yesmaximal, characteristic, complemented
5#23757519C211yesmaximal, characteristic, complemented
6#23757516C211yesmaximal, characteristic, complemented
7#23757490C211yesmaximal, characteristic, complemented
8#23757486C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757407C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13D8×Dic157C211yesmaximal, characteristic, complemented
14#23757349C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic, complemented
18#19097795—24no—
19C2×#14578944C2211yescharacteristic, complemented
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic, complemented
22C2×#14578939C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×#14578937C2211yescharacteristic, complemented
25#19097760C2211yescharacteristic, complemented
26#19097758C2211yescharacteristic, complemented
27#19097754—21no—
28#19097752—21no—
29#19097750C2211yescharacteristic, complemented
30#19097747C2211yescharacteristic, complemented
31#19097745C2211yescharacteristic
32#19097743—21no—
33#19097743—21no—
34#19097742C2211yescharacteristic
35#19097740—21no—
36#19097739—21no—
37D4×Dic157C2211yescharacteristic, complemented
38D4×Dic157C2211yescharacteristic, complemented
39C4×C22⋊D157C2211yescharacteristic
40C2×C22⋊Dic157—21no—
41C2×#14578931—21no—
42#19097705C2211yescharacteristic, complemented
43#19097700C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic, complemented
48#19097692C2211yescharacteristic, complemented
49#19097692C2211yescharacteristic, complemented
50C22⋊#11929434C2211yescharacteristic, complemented
51#19097683C2211yescharacteristic, complemented
52C8×Dic157C2211yescharacteristic, complemented
53C2×#41×C157C2211yescharacteristic
54C22×D4×C157C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C157—24no—
56C2×SD16×C157C2211yescharacteristic, complemented
57C2×SD16×C157C2211yescharacteristic, complemented
58C2×D8×C157C2211yescharacteristic
59C2×D8×C157C2211yescharacteristic
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⋊D157D411yescharacteristic
71D4×D157—22no—
72D4×D157—41no—
73C22×Dic157—21no—
74C22×Dic157—21no—
75C22×Dic157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578944—41no—
78#14578943—22no—
79#14578942—22no—
80#14578939—22no—
81#14578939—41no—
82#14578937—22no—
83C2×#11929434C2311yescharacteristic
84C2×#11929434C2311yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578933D411yescharacteristic
90C22⋊Dic157—21no—
91C22⋊Dic157—21no—
92C22⋊Dic157—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96C22⋊Dic157D411yescharacteristic
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931—21no—
101#14578931D411yescharacteristic
102C4×Dic157C2311yescharacteristic
103C24×C157—21no—
104#41×C157—22no—
105#41×C157—41no—
106C2×Q8×C157C2311yescharacteristic
107C2×D4×C157—22no—
108C2×D4×C157—21no—
109C2×D4×C157—21no—
110C2×D4×C157—41no—
111C2×D4×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116SD16×C157—22no—
117SD16×C157—22no—
118D8×C157—22no—
119D8×C157D414yescomplemented
120M16×C157—22no—
121C2×C8×C157C2311yescharacteristic
122C2×C8×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157—81no—
127C22×D157C2×D411yescharacteristic
128Dic314—21no—
129Dic314—21no—
130D628—21no—
131D628—21no—
132C22⋊D157—41no—
133C22⋊D157—42no—
134C22⋊D157—42no—
135C22⋊D157—81no—
136C22⋊D157—81no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—42no—
141C2×Dic157—81no—
142C2×Dic157—81no—
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C2×Dic157C2×D411yescharacteristic
146C4×D157—21no—
147C4×D157—21no—
148#11929434—21no—
149#11929434—21no—
150C23×C157—21no—
151C23×C157—21no—
152C23×C157—21no—
153C23×C157—42no—
154C23×C157—42no—
155C23×C157—81no—
156C23×C157C2×D411yescharacteristic
157Q8×C157—21no—
158Q8×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—42no—
164D4×C157—42no—
165D4×C157C2×D412yes—
166D4×C157C2×D412yes—
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×C157C2×D412yes—
175C22×C157—21no—
176C22×C157—22no—
177C22×C157—22no—
178C22×C157—21no—
179C22×C157—21no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—84no—
188C22×C157C22×D411yescharacteristic
189C2×D157—21no—
190C2×D157—21no—
191C2×D157—21no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—84no—
196C4×C157—21no—
197C4×C157—41no—
198C4×C157C22×D411yescharacteristic
199C4×C157H(5,𝔽2)11yescharacteristic
200Dic157—21no—
201Dic157—21no—
202Dic157—21no—
203Dic157—41no—
204Dic157—41no—
205Dic157—41no—
206C2×C157—42no—
207C2×C157—41no—
208C2×C157—42no—
209C2×C157—81no—
210C2×C157D4211yescharacteristic
211C2×C157#57511yescharacteristic
212C2×C157#57611yescharacteristic
213D157—41no—
214D157—81no—
215C157#327311yescharacteristic, complemented, Sylow
216#3273—1571nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1571no—
218C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
219#546—1571no—
220#546—1571no—
221#495—1571no—
222#468—1571no—
223#468—1571no—
224#463—1571no—
225#463—1571no—
226#452—1571no—
227#452—1571no—
228#447—1571no—
229#447—1571no—
230C4×D8—1571no—
231#436—1571no—
232C2×#41—1571no—
233C2×#41C2×D15711yescharacteristic
234C22×D4—1571no—
235C22×D4C2×D15711yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—3144no—
238C2×Q16—1571no—
239C2×SD16—1571no—
240C2×SD16—1571no—
241C2×SD16C2×D15711yescharacteristic, complemented
242C2×SD16C2×D15711yescharacteristic, complemented
243C2×D8—1571no—
244C2×D8—1571no—
245C2×D8C2×D15711yescharacteristic
246C2×D8C2×D15711yescharacteristic
247C2×M16—1571no—
248C2×M16C2×D15711yescharacteristic, complemented
249#124—1571no—
250#124—1571no—
251#123—3141no—
252#123—3141no—
253C22⋊Q8—1571no—
254C22⋊Q8—1571no—
255#121—3141no—
256#121—3141no—
257#121—1571no—
258#121—1571no—
259#120—3141no—
260#120—3141no—
261C4×D4—1571no—
262C4×D4—1571no—
263C4×D4—1571no—
264C2×C22⋊C4—3141no—
265C2×C22⋊C4—3141no—
266#107—1571no—
267C4⋊C8—1571no—
268#103—1571no—
269#103—1571no—
270#102—1571no—
271#102—1571no—
272#102—1571no—
273#102—1571no—
274C22⋊C8—1571no—
275C22⋊C8—1571no—
276C4×C8—1571no—
277C24—21no—
278C24—3141no—
279#41—22no—
280#41—41no—
281#41—3142no—
282#41—6281no—
283C2×Q8—1571no—
284C2×Q8C22×D15711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3142no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—6281no—
295C2×D4—1571no—
296C2×D4—1571no—
297C2×D4C22×D15711yescharacteristic
298C2×D4C22×D15711yescharacteristic
299C2×D4C22×D15711yescharacteristic
300C22×C4—21no—
301C22×C4—3141no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—1571no—
305C22×C4C22×D15711yescharacteristic
306Q16—6281no—
307SD16—22no—
308SD16—22no—
309SD16—3142no—
310SD16—3142no—
311D8—22no—
312D8—3142no—
313D8—6281no—
314D8C22⋊D15714yescomplemented
315M16—22no—
316M16—3142no—
317C2×C8—1571no—
318C2×C8—1571no—
319C2×C8C22×D15711yescharacteristic
320C2×C8C22×D15711yescharacteristic
321C4⋊C4—3141no—
322C4⋊C4—3141no—
323C4⋊C4—1571no—
324C4⋊C4—1571no—
325C4⋊C4—1571no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—3141no—
330C22⋊C4—3141no—
331C22⋊C4—3141no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—3141no—
335C22⋊C4—3141no—
336C22⋊C4—1571no—
337C22⋊C4—1571no—
338C42—1571no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—3141no—
346C23—6282no—
347C23—6282no—
348C23—1571no—
349C23—12561no—
350C23D4×D15711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3141no—
354Q8—3141no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—3141no—
362D4—3141no—
363D4—6281no—
364D4—6282no—
365D4—6282no—
366D4—12561no—
367D4—12561no—
368D4C2×C22⋊D15712yes—
369D4C2×C22⋊D15712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—6282no—
380C2×C4—1571no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4—12561no—
384C2×C4—12561no—
385C2×C4D4×D15711yescharacteristic
386C2×C4C23×D15711yescharacteristic
387C8—21no—
388C8—3141no—
389C8—3141no—
390C8C2×C22⋊D15712yes—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3141no—
405C22—3141no—
406C22—3141no—
407C22—6281no—
408C22—6281no—
409C22—6281no—
410C22—12564no—
411C22C2×D4×D15711yescharacteristic
412C4—21no—
413C4—41no—
414C4—3141no—
415C4—3141no—
416C4—3141no—
417C4—6281no—
418C4—6281no—
419C4—6281no—
420C4C22×C22⋊D15711yescharacteristic
421C4#1909781611yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—6281no—
427C2—12561no—
428C2D4×C22⋊D15711yescharacteristic
429C2#2375806911yescharacteristic
430C2#2375807711yescharacteristic
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#3273
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. #23757917C2semidirect
3. #23757911C2semidirect
4. #23757654C2semidirect
5. #23757519C2semidirect
6. #23757516C2semidirect
7. #23757490C2semidirect
8. #23757486C2semidirect
9. #23757435C2semidirect
10. #23757432C2semidirect
11. #23757407C2semidirect
12. #23757405C2semidirect
13. D8×Dic157C2semidirect
14. #23757349C2semidirect
15. C2×AGL(1,ℤ/8)×C157C2semidirect
17. C2×D4×D157C22semidirect
19. C2×#14578944C22semidirect
20. C2×#14578943C22semidirect
21. C2×#14578942C22semidirect
24. C2×#14578937C22semidirect
25. #19097760C22semidirect
26. #19097758C22semidirect
29. #19097750C22semidirect
30. #19097747C22semidirect
37. D4×Dic157C22semidirect
38. D4×Dic157C22semidirect
42. #19097705C22semidirect
43. #19097700C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097693C22semidirect
47. #19097692C22semidirect
48. #19097692C22semidirect
49. #19097692C22semidirect
50. C22⋊#11929434C22semidirect
51. #19097683C22semidirect
52. C8×Dic157C22semidirect
54. C22×D4×C157C22semidirect
56. C2×SD16×C157C22semidirect
57. C2×SD16×C157C22semidirect
60. C2×M16×C157C22semidirect
119. D8×C157D4semidirect
215. C157#3273semidirect
218. C2×AGL(1,ℤ/8)D157semidirect
235. C22×D4C2×D157semidirect
241. C2×SD16C2×D157semidirect
242. C2×SD16C2×D157semidirect
248. C2×M16C2×D157semidirect
314. D8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
#237579172
#237579112
#237576542
#237575192
#237575162
#237574902
#237574862
#237574352
#237574322
#237574072
#237574052
D8×Dic1572
#237573492
C2×AGL(1,ℤ/8)×C1572
#3273157