Finite Group Databasebeta

← #34109634

Subgroups — #34109634

Order 20096.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097795C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757836C211yesmaximal, characteristic, complemented
4#23757745C211yesmaximal, characteristic, complemented
5#23757489C211yesmaximal, characteristic, complemented
6#23757488C211yesmaximal, characteristic, complemented
7#23757487C211yesmaximal, characteristic, complemented
8#23757486C211yesmaximal, characteristic, complemented
9#23757467C211yesmaximal, characteristic, complemented
10#23757466C211yesmaximal, characteristic, complemented
11#23757465C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757276C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#459×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097795—24no—
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578942C2211yescharacteristic, complemented
22C2×D1256C2211yescharacteristic
23C2×#14578939C2211yescharacteristic
24C2×#14578939C2211yescharacteristic
25C2×#14578939C2211yescharacteristic, complemented
26D8×D157—24no—
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097767C2211yescharacteristic, complemented
30#19097759C2211yescharacteristic, complemented
31#19097758C2211yescharacteristic, complemented
32#19097743—21no—
33#19097743—21no—
34#19097739—21no—
35#19097739—21no—
36C4×D628C2211yescharacteristic, complemented
37D4×Dic157C2211yescharacteristic, complemented
38#19097733C2211yescharacteristic, complemented
39C4×C22⋊D157—21no—
40#19097731—21no—
41#19097729C2211yescharacteristic
42#19097728C2211yescharacteristic
43C2×#14578931—21no—
44C22⋊C4×D157—21no—
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097700C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097694C2211yescharacteristic, complemented
50#19097693C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097692C2211yescharacteristic, complemented
53#19097691C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic, complemented
56#127×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578953—22no—
64#14578953—22no—
65C2×Dic314C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73D4×D157—22no—
74D4×D157—22no—
75D4×D157—22no—
76D4×D157D414yescomplemented
77C22×Dic157—21no—
78C22×Dic157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157C2311yescharacteristic
82#14578943—22no—
83#14578942—22no—
84D1256—22no—
85#14578939—22no—
86#14578939—22no—
87#14578939—22no—
88#14578937—22no—
89C2×#11929434C2311yescharacteristic
90C2×#11929434C2311yescharacteristic
91C8×D157—22no—
92C4⋊Dic157C2311yescharacteristic
93#14578933D411yescharacteristic
94#14578933D411yescharacteristic
95C22⋊Dic157—21no—
96C22⋊Dic157—21no—
97C22⋊Dic157—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931D411yescharacteristic
104#14578931D411yescharacteristic
105C4×Dic157—21no—
106C4×Dic157C2311yescharacteristic
107C2×D4×C157—21no—
108C2×D4×C157—21no—
109C2×D4×C157C2311yescharacteristic
110C2×D4×C157C2311yescharacteristic
111C2×D4×C157C2311yescharacteristic
112C22×C4×C157—21no—
113D8×C157—22no—
114D8×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C4⋊C4×C157C2311yescharacteristic
118C22⋊C4×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157—24no—
122C22×D157—21no—
123C22×D157—42no—
124C22×D157—81no—
125C22×D157—81no—
126C22×D157—81no—
127C22×D157—81no—
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131D628—21no—
132D628—21no—
133D628—21no—
134D628C2×D412yes—
135C22⋊D157—24no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C22⋊D157—81no—
139C22⋊D157—81no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157—42no—
143C2×Dic157—81no—
144C2×Dic157C2×D411yescharacteristic
145C2×Dic157C2×D411yescharacteristic
146C2×Dic157C2×D411yescharacteristic
147C4×D157—81no—
148C4×D157—81no—
149C4×D157C2×D412yes—
150C4×D157#4112yes—
151#11929434—21no—
152#11929434—21no—
153C23×C157—21no—
154C23×C157—21no—
155C23×C157—21no—
156D4×C157—21no—
157D4×C157—21no—
158D4×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—81no—
162D4×C157—81no—
163D4×C157C2×D412yes—
164C2×C4×C157—21no—
165C2×C4×C157—81no—
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2411yescharacteristic
169C8×C157—21no—
170C8×C157—21no—
171C22×C157—22no—
172C22×C157—41no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157—41no—
179C22×C157—41no—
180C22×C157C22×D411yescharacteristic
181C2×D157—21no—
182C2×D157—21no—
183C2×D157—22no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—41no—
189C2×D157—84no—
190C2×D157—84no—
191C2×D157#12112yes—
192C4×C157—21no—
193C4×C157—21no—
194C4×C157—41no—
195C4×C157C22×D411yescharacteristic
196C4×C157C2×#4111yescharacteristic
197Dic157—21no—
198Dic157—21no—
199Dic157—41no—
200Dic157—41no—
201Dic157#12112yes—
202C2×C157—42no—
203C2×C157—81no—
204C2×C157—81no—
205C2×C157C2×#12111yescharacteristic
206C2×C157#57611yescharacteristic
207C2×C157#57611yescharacteristic
208D157—22no—
209D157—42no—
210D157—81no—
211C157#304111yescharacteristic, complemented, Sylow
212#3041—1571nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1571no—
214C22×D8—1571no—
215#535—1571no—
216#518—1571no—
217#463—1571no—
218#463—1571no—
219#463—1571no—
220#463—1571no—
221#459—1571no—
222#459—1571no—
223#459—1571no—
224#459D15711yescharacteristic, complemented
225#424—1571no—
226#418—1571no—
227#416—1571no—
228C2×#41—1571no—
229C22×D4—1571no—
230C22×D4—1571no—
231AGL(1,ℤ/8)—3144no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×D8—3144no—
235C2×D8—1571no—
236C2×D8—1571no—
237C2×D8—1571no—
238C2×D8—1571no—
239C2×D8C2×D15711yescharacteristic
240C2×D8C2×D15711yescharacteristic, complemented
241C2×M16—1571no—
242C22×C8—1571no—
243#127—1571no—
244#127C2×D15711yescharacteristic, complemented
245#124—1571no—
246#124—1571no—
247#121—3141no—
248#121—3141no—
249#120—3141no—
250#120—3141no—
251C4×D4—3141no—
252C4×D4—3141no—
253C4×D4—1571no—
254C4×D4—1571no—
255C4×D4—1571no—
256C4×D4C2×D15711yescharacteristic, complemented
257#117—1571no—
258#117—1571no—
259C2×C22⋊C4—3141no—
260C2×C22⋊C4—3141no—
261C4⋊C8—1571no—
262C4⋊C8—1571no—
263C4⋊C8—1571no—
264C4⋊C8C2×D15711yescharacteristic, complemented
265#103—1571no—
266#103—1571no—
267#102—1571no—
268#102—1571no—
269#102—1571no—
270#102—1571no—
271#102C2×D15711yescharacteristic, complemented
272#102C2×D15711yescharacteristic, complemented
273C24—3141no—
274C24—3141no—
275#41—3142no—
276#41—3142no—
277C2×Q8—1571no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3141no—
281C2×D4—3142no—
282C2×D4—3142no—
283C2×D4—3142no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—3141no—
288C2×D4—1574no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4C22×D15711yescharacteristic
292C2×D4C22×D15711yescharacteristic
293C2×D4C22×D15711yescharacteristic
294C22×C4—21no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4—3141no—
298C22×C4—3141no—
299C22×C4—1571no—
300SD16—3142no—
301SD16—3142no—
302D8—22no—
303D8—22no—
304D8—3142no—
305D8—3142no—
306D8—3142no—
307D8—3142no—
308M16—3142no—
309C2×C8—3142no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8C22×D15711yescharacteristic
313C2×C8C22×D15711yescharacteristic
314C4⋊C4—1571no—
315C4⋊C4—1571no—
316C4⋊C4—1571no—
317C4⋊C4C22×D15711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—1571no—
328C22⋊C4—1571no—
329C42—3141no—
330C42—1571no—
331C42C22×D15711yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—3144no—
336C23—3141no—
337C23—3141no—
338C23—6282no—
339C23—1571no—
340C23—12561no—
341C23—12561no—
342C23—12561no—
343C23—12561no—
344Q8—3141no—
345Q8—3141no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—3144no—
354D4—3141no—
355D4—3141no—
356D4—3141no—
357D4—6282no—
358D4—6282no—
359D4—1572no—
360D4—12561no—
361D4—12561no—
362D4D4×D15712yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—6282no—
368C2×C4—1571no—
369C2×C4—1571no—
370C2×C4—1571no—
371C2×C4—1572no—
372C2×C4—1572no—
373C2×C4—12561no—
374C2×C4—12561no—
375C2×C4—12561no—
376C2×C4D4×D15711yescharacteristic
377C2×C4D4×D15711yescharacteristic
378C2×C4C23×D15711yescharacteristic
379C8—21no—
380C8—21no—
381C8—3141no—
382C8—3141no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—3141no—
393C22—3141no—
394C22—3142no—
395C22—6281no—
396C22—6281no—
397C22—6281no—
398C22—6281no—
399C22—6281no—
400C22—1572no—
401C22—12564no—
402C22—12564no—
403C22C2×D4×D15711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—3141no—
408C4—3141no—
409C4—6281no—
410C4—6281no—
411C4—1572no—
412C4C2×D4×D15711yescharacteristic
413C4#41×D15711yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—3142no—
418C2—6282no—
419C2—12561no—
420C2#121×D15711yescharacteristic
421C2#2375807411yescharacteristic
422C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3041
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×D8×D157C2semidirect
3. #23757836C2semidirect
4. #23757745C2semidirect
5. #23757489C2semidirect
6. #23757488C2semidirect
7. #23757487C2semidirect
8. #23757486C2semidirect
9. #23757467C2semidirect
10. #23757466C2semidirect
11. #23757465C2semidirect
12. #23757298C2semidirect
13. #23757276C2semidirect
14. #23757266C2semidirect
15. #459×C157C2semidirect
20. C2×#14578943C22semidirect
21. C2×#14578942C22semidirect
25. C2×#14578939C22semidirect
29. #19097767C22semidirect
30. #19097759C22semidirect
31. #19097758C22semidirect
36. C4×D628C22semidirect
37. D4×Dic157C22semidirect
38. #19097733C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097700C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097694C22semidirect
50. #19097693C22semidirect
51. #19097692C22semidirect
52. #19097692C22semidirect
53. #19097691C22semidirect
55. C2×D8×C157C22semidirect
56. #127×C157C22semidirect
57. C4×D4×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #102×C157C22semidirect
60. #102×C157C22semidirect
76. D4×D157D4semidirect
211. C157#3041semidirect
224. #459D157semidirect
240. C2×D8C2×D157semidirect
244. #127C2×D157semidirect
256. C4×D4C2×D157semidirect
264. 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×D8×D1572
#237578362
#237577452
#237574892
#237574882
#237574872
#237574862
#237574672
#237574662
#237574652
#237572982
#237572762
#237572662
#459×C1572
#3041157