Finite Group Databasebeta

← #34111670

Subgroups — #34111670

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×#19097792C211yesmaximal, characteristic, complemented
2#23757942C211yesmaximal, characteristic, complemented
3#23757917C211yesmaximal, characteristic, complemented
4#23757661C211yesmaximal, characteristic, complemented
5#23757519C211yesmaximal, characteristic, complemented
6#23757506C211yesmaximal, characteristic, complemented
7#23757493C211yesmaximal, characteristic, complemented
8#23757466C211yesmaximal, characteristic, complemented
9#23757435C211yesmaximal, characteristic, complemented
10#23757417C211yesmaximal, characteristic, complemented
11#23757415C211yesmaximal, characteristic, complemented
12#23757407C211yesmaximal, characteristic, complemented
13#23757374C211yesmaximal, characteristic, complemented
14#23757342C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic, complemented
17C2×#14578953C2211yescharacteristic
18#19097792—24no—
19C2×#14578944C2211yescharacteristic, complemented
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic, complemented
22C2×#14578939C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×C8×D157C2211yescharacteristic, complemented
25#19097767—21no—
26#19097767C2211yescharacteristic, complemented
27#19097758C2211yescharacteristic
28#19097752—21no—
29#19097750C2211yescharacteristic
30#19097745—21no—
31#19097745C2211yescharacteristic
32#19097743—21no—
33#19097742—21no—
34#19097742C2211yescharacteristic
35#19097742C2211yescharacteristic, complemented
36#19097740—21no—
37Q8×Dic157C2211yescharacteristic, complemented
38D4×Dic157C2211yescharacteristic
39C4×C22⋊D157—21no—
40C4×C22⋊D157C2211yescharacteristic
41C2×C22⋊Dic157—21no—
42#19097705C2211yescharacteristic, complemented
43#19097700C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097692C2211yescharacteristic, complemented
50C22⋊#11929434C2211yescharacteristic
51#19097683C2211yescharacteristic, complemented
52#19097681C2211yescharacteristic, complemented
53C2×#41×C157C2211yescharacteristic
54C22×D4×C157C2211yescharacteristic
55AGL(1,ℤ/8)×C157—24no—
56C2×SD16×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic
58C2×D8×C157C2211yescharacteristic
59C2×D8×C157C2211yescharacteristic
60C2×M16×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63#14578953—22no—
64#14578953—41no—
65C2×Dic314C2311yescharacteristic
66C2×D628—21no—
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157D411yescharacteristic
75C22×Dic157—21no—
76C22×Dic157—21no—
77C2×C4×D157—21no—
78C2×C4×D157C2311yescharacteristic
79#14578944—22no—
80#14578943—22no—
81#14578942—41no—
82#14578939—22no—
83#14578939—41no—
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C8×D157—22no—
87C4⋊Dic157—21no—
88C4⋊Dic157C2311yescharacteristic
89C4⋊Dic157C2311yescharacteristic
90#14578933—21no—
91#14578933D411yescharacteristic
92C22⋊Dic157—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96C22⋊Dic157—21no—
97C22⋊Dic157—21no—
98C22⋊Dic157D411yescharacteristic
99#14578931—21no—
100#14578931D411yescharacteristic
101C4×Dic157—21no—
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×C157D414yescomplemented
118D8×C157—22no—
119D8×C157—22no—
120M16×C157—22no—
121C2×C8×C157C2311yescharacteristic
122C2×C8×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127D628—21no—
128D628—21no—
129D628—42no—
130C22⋊D157—41no—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C22⋊D157—81no—
134C22⋊D157—81no—
135C22⋊D157—81no—
136C22⋊D157—81no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—42no—
141C2×Dic157—81no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—42no—
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×C157C2×D412yes—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—21no—
163D4×C157—21no—
164D4×C157—42no—
165D4×C157—42no—
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—21no—
178C22×C157—21no—
179C22×C157—41no—
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—
195C4×C157—21no—
196C4×C157—22no—
197C4×C157C22×D411yescharacteristic
198C4×C157H(5,𝔽2)11yescharacteristic
199Dic157—21no—
200Dic157—21no—
201Dic157—21no—
202Dic157—41no—
203Dic157—41no—
204Dic157—41no—
205C2×C157—41no—
206C2×C157—42no—
207C2×C157—81no—
208C2×C157—81no—
209C2×C157D4211yescharacteristic
210C2×C157#57511yescharacteristic
211C2×C157#57611yescharacteristic
212D157—41no—
213D157—81no—
214C157#326811yescharacteristic, complemented, Sylow
215#3268—1571nomaximal, Sylow
216C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
217C2×#135—1571no—
218#550—1571no—
219#546—1571no—
220#496—1571no—
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×#41C2×D15711yescharacteristic
234C22×D4C2×D15711yescharacteristic
235AGL(1,ℤ/8)—24no—
236#135—3144no—
237C2×Q16—1571no—
238C2×SD16—1571no—
239C2×SD16—1571no—
240C2×SD16C2×D15711yescharacteristic
241C2×SD16C2×D15711yescharacteristic
242C2×D8—1571no—
243C2×D8—1571no—
244C2×D8C2×D15711yescharacteristic
245C2×D8C2×D15711yescharacteristic
246C2×M16C2×D15711yescharacteristic
247C22×C8—1571no—
248#127—3141no—
249#127—1571no—
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×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#97—1571no—
276C24—21no—
277#41—22no—
278#41—41no—
279#41—3142no—
280#41—3142no—
281#41—6281no—
282#41—6281no—
283C2×Q8—1571no—
284C2×Q8C22×D15711yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3141no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—3141no—
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×C4C22×D15711yescharacteristic
307Q16—3142no—
308SD16—22no—
309SD16—3142no—
310SD16—6281no—
311SD16C22⋊D15714yescomplemented
312D8—22no—
313D8—22no—
314D8—3142no—
315D8—6281no—
316M16—22no—
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—3141no—
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—
337C42—1571no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3141no—
345C23—1571no—
346C23D4×D15711yescharacteristic
347Q8—21no—
348Q8—3141no—
349Q8—3141no—
350Q8C2×C22⋊D15712yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—3141no—
359D4—3141no—
360D4—6282no—
361D4—6281no—
362D4—6282no—
363D4—6282no—
364D4—12561no—
365D4—12561no—
366D4—12561no—
367D4—12561no—
368D4C2×C22⋊D15712yes—
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—3141no—
378C2×C4—6282no—
379C2×C4—6282no—
380C2×C4—1571no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4—12561no—
384C2×C4D4×D15711yescharacteristic
385C2×C4C23×D15711yescharacteristic
386C8—21no—
387C8—3141no—
388C8—3141no—
389C8C2×C22⋊D15712yes—
390C22—21no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—84no—
403C22—3141no—
404C22—3141no—
405C22—3141no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22C2×D4×D15711yescharacteristic
410C4—21no—
411C4—22no—
412C4—3141no—
413C4—3141no—
414C4—3141no—
415C4—6281no—
416C4—6281no—
417C4—6281no—
418C4C22×C22⋊D15711yescharacteristic
419C4#1909781611yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—81no—
424C2—6281no—
425C2—12561no—
426C2D4×C22⋊D15711yescharacteristic
427C2#2375806911yescharacteristic
428C2#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#3268
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×#19097792C2semidirect
2. #23757942C2semidirect
3. #23757917C2semidirect
4. #23757661C2semidirect
5. #23757519C2semidirect
6. #23757506C2semidirect
7. #23757493C2semidirect
8. #23757466C2semidirect
9. #23757435C2semidirect
10. #23757417C2semidirect
11. #23757415C2semidirect
12. #23757407C2semidirect
13. #23757374C2semidirect
14. #23757342C2semidirect
15. C2×AGL(1,ℤ/8)×C157C2semidirect
16. C2×#14578955C22semidirect
19. C2×#14578944C22semidirect
20. C2×#14578943C22semidirect
21. C2×#14578942C22semidirect
24. C2×C8×D157C22semidirect
26. #19097767C22semidirect
35. #19097742C22semidirect
37. Q8×Dic157C22semidirect
42. #19097705C22semidirect
43. #19097700C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097693C22semidirect
49. #19097692C22semidirect
51. #19097683C22semidirect
52. #19097681C22semidirect
117. SD16×C157D4semidirect
214. C157#3268semidirect
216. C2×AGL(1,ℤ/8)D157semidirect
311. SD16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977922
#237579422
#237579172
#237576612
#237575192
#237575062
#237574932
#237574662
#237574352
#237574172
#237574152
#237574072
#237573742
#237573422
C2×AGL(1,ℤ/8)×C1572
#3268157