Finite Group Databasebeta

← #34080871

Subgroups — #34080871

Order 19328.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23753023C211yesmaximal, characteristic, complemented
2#23753022C211yesmaximal, characteristic, complemented
3#23753016C211yesmaximal, characteristic, complemented
4#23753015C211yesmaximal, characteristic, complemented
5#23752599C211yesmaximal, characteristic, complemented
6#23752481C211yesmaximal, characteristic, complemented
7#23752480C211yesmaximal, characteristic, complemented
8#23752474C211yesmaximal, characteristic, complemented
9#23752473C211yesmaximal, characteristic, complemented
10#23752391C211yesmaximal, characteristic, complemented
11#23752386C211yesmaximal, characteristic, complemented
12#23752379C211yesmaximal, characteristic, complemented
13#23752331C211yesmaximal, characteristic, complemented
14#23752331C211yesmaximal, characteristic, complemented
15#453×C151C211yesmaximal, characteristic, complemented
16#18012786C2211yescharacteristic, complemented
17#18012785C2211yescharacteristic
18#18012784C2211yescharacteristic, complemented
19#18012772C2211yescharacteristic, complemented
20#18012769—21no—
21#18012769C2211yescharacteristic, complemented
22#18012768—21no—
23#18012767C2211yescharacteristic, complemented
24#18012766—21no—
25#18012765—21no—
26#18012765—21no—
27#18012765C2211yescharacteristic
28#18012765C2211yescharacteristic, complemented
29#18012764—21no—
30#18012764C2211yescharacteristic, complemented
31#18012763—21no—
32#18012763—21no—
33#18012760—21no—
34#18012759—21no—
35#18012759—21no—
36#18012759C2211yescharacteristic
37#18012759C2211yescharacteristic
38#18012758—21no—
39C2×#14558992—21no—
40C2×#14558992C2211yescharacteristic, complemented
41C2×#14558992C2211yescharacteristic, complemented
42SD16×D151—21no—
43C2×#14558989—21no—
44C2×#14558989C2211yescharacteristic, complemented
45C2×#14558989C2211yescharacteristic, complemented
46D8×D151—21no—
47#18012747C2211yescharacteristic, complemented
48#18012747C2211yescharacteristic, complemented
49#18012746C2211yescharacteristic, complemented
50#18012746C2211yescharacteristic, complemented
51#18012727C2211yescharacteristic, complemented
52#18012674C2211yescharacteristic, complemented
53#18012674C2211yescharacteristic, complemented
54#18012667C2211yescharacteristic, complemented
55#18012667C2211yescharacteristic, complemented
56#18012657C2211yescharacteristic, complemented
57#18012656C2211yescharacteristic, complemented
58#18012655C2211yescharacteristic, complemented
59#18012650C2211yescharacteristic, complemented
60H(5,𝔽2)×C151C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C151C2211yescharacteristic
62AGL(1,ℤ/8)×C151C2211yescharacteristic, complemented
63#127×C151C2211yescharacteristic, complemented
64#104×C151C2211yescharacteristic, complemented
65#104×C151C2211yescharacteristic, complemented
66#100×C151C2211yescharacteristic, complemented
67#14559005—21no—
68#14559004—21no—
69#14559004—21no—
70#14559004C2311yescharacteristic
71#14559004C2311yescharacteristic
72#14559003—21no—
73#14559003—21no—
74#14559003—21no—
75#14559003—41no—
76C2×Dic302—21no—
77C2×Dic302C2311yescharacteristic
78Q8×D151—21no—
79C2×D604—21no—
80C2×D604C2311yescharacteristic
81C2×C22⋊D151—41no—
82D4×D151—21no—
83D4×D151—21no—
84D4×D151—21no—
85D4×D151—41no—
86Dic604—21no—
87#14558994—21no—
88#14558994D411yescharacteristic, complemented
89#14558994D411yescharacteristic, complemented
90#14558993—21no—
91#14558993—21no—
92#14558992—21no—
93#14558992—21no—
94#14558992—21no—
95#14558992—21no—
96#14558992—21no—
97#14558992—21no—
98#14558992—21no—
99#14558992D411yescharacteristic, complemented
100#14558992D411yescharacteristic, complemented
101#14558991—21no—
102#14558991D411yescharacteristic, complemented
103#14558991D411yescharacteristic, complemented
104D1208—21no—
105#14558989—21no—
106#14558989—21no—
107#14558989—21no—
108#14558989—21no—
109#14558989—21no—
110#14558989—21no—
111#14558989—21no—
112#14558989D411yescharacteristic, complemented
113#14558989D411yescharacteristic, complemented
114#14558988—21no—
115#14558988—21no—
116#14558988C2311yescharacteristic
117#14558988C2311yescharacteristic
118#14558987—21no—
119#14558987—21no—
120C2×C151⋊C8—21no—
121C2×C151⋊C8—21no—
122C2×C151⋊C8C2311yescharacteristic
123C2×C151⋊C8C2311yescharacteristic
124C8×D151—21no—
125C8×D151—21no—
126(C2×D151)⋊C4—41no—
127#41×C151—21no—
128#41×C151—21no—
129#41×C151C2311yescharacteristic
130#41×C151C2311yescharacteristic
131C2×D4×C151—21no—
132C2×D4×C151—21no—
133C2×D4×C151—41no—
134C2×D4×C151—41no—
135C2×D4×C151C2311yescharacteristic
136C2×D4×C151C2311yescharacteristic
137SD16×C151—21no—
138SD16×C151—21no—
139D8×C151—21no—
140D8×C151—21no—
141M16×C151C2311yescharacteristic
142M16×C151C2311yescharacteristic
143C42×C151C2311yescharacteristic
144C22×D151—21no—
145C22×D151—41no—
146C22×D151—41no—
147Dic302—21no—
148Dic302—21no—
149Dic302—41no—
150Dic302C2×D411yescharacteristic
151Dic302C2×D411yescharacteristic
152D604—21no—
153D604—21no—
154D604—41no—
155D604C2×D411yescharacteristic
156D604C2×D411yescharacteristic
157C22⋊D151—21no—
158C22⋊D151—21no—
159C22⋊D151—41no—
160C22⋊D151—41no—
161C22⋊D151—41no—
162C22⋊D151—41no—
163C2×Dic151—21no—
164C2×Dic151—41no—
165C2×Dic151—41no—
166C4×D151—21no—
167C4×D151—21no—
168C4×D151—41no—
169C4×D151—41no—
170C151⋊C8—21no—
171C151⋊C8—21no—
172C151⋊C8C2×D411yescharacteristic
173C151⋊C8C2×D411yescharacteristic
174C151⋊C8C2×D411yescharacteristic
175C151⋊C8C2×D411yescharacteristic
176C23×C151—21no—
177C23×C151—21no—
178C23×C151—41no—
179Q8×C151C2×D411yescharacteristic
180Q8×C151C2×D411yescharacteristic
181D4×C151—21no—
182D4×C151—21no—
183D4×C151—21no—
184D4×C151—21no—
185D4×C151—21no—
186D4×C151—21no—
187D4×C151—41no—
188D4×C151—41no—
189D4×C151—82no—
190D4×C151C2×D411yescharacteristic
191D4×C151C2×D411yescharacteristic
192C2×C4×C151—21no—
193C2×C4×C151—21no—
194C2×C4×C151—21no—
195C2×C4×C151—41no—
196C2×C4×C151C2411yescharacteristic
197C8×C151—21no—
198C8×C151—21no—
199C22×C151—21no—
200C22×C151—21no—
201C22×C151—41no—
202C22×C151—41no—
203C22×C151—81no—
204C22×C151—81no—
205C22×C151—161no—
206C22×C151H(5,𝔽2)11yescharacteristic
207C2×D151—21no—
208C2×D151—21no—
209C2×D151—41no—
210C2×D151—81no—
211C2×D151—161no—
212C2×D151—161no—
213C4×C151—21no—
214C4×C151—21no—
215C4×C151—41no—
216C4×C151C22×D411yescharacteristic
217C4×C151C22×D411yescharacteristic
218Dic151—21no—
219Dic151—21no—
220Dic151—41no—
221C2×C151—21no—
222C2×C151—41no—
223C2×C151—41no—
224C2×C151—81no—
225C2×C151—81no—
226C2×C151D4211yescharacteristic
227D151—41no—
228D151—41no—
229D151—81no—
230C151#328811yescharacteristic, complemented, Sylow
231#3288—1511nomaximal, Sylow
232#577—1511no—
233#577—1511no—
234#576—1511no—
235#576—1511no—
236#496—1511no—
237#472—1511no—
238#472—1511no—
239#471—1511no—
240#471—1511no—
241#455—1511no—
242#454—1511no—
243#453—1511no—
244#453D15111yescharacteristic, complemented
245#444—1511no—
246#444—1511no—
2472−1+4—1511no—
248H(5,𝔽2)—1511no—
249H(5,𝔽2)—1511no—
250H(5,𝔽2)C2×D15111yescharacteristic, complemented
251#137—3021no—
252#137—3021no—
253#137—1511no—
254#137—1511no—
255AGL(1,ℤ/8)—3021no—
256AGL(1,ℤ/8)—3021no—
257AGL(1,ℤ/8)—3021no—
258AGL(1,ℤ/8)—3021no—
259AGL(1,ℤ/8)—3021no—
260AGL(1,ℤ/8)—3021no—
261AGL(1,ℤ/8)—1511no—
262AGL(1,ℤ/8)—1511no—
263AGL(1,ℤ/8)—1511no—
264AGL(1,ℤ/8)—1511no—
265AGL(1,ℤ/8)C2×D15111yescharacteristic
266AGL(1,ℤ/8)C2×D15111yescharacteristic, complemented
267#135—3021no—
268#135—3021no—
269#135—3021no—
270#135—3021no—
271#135—1511no—
272#135—1511no—
273C2×SD16—3021no—
274C2×SD16—3021no—
275C2×SD16—1511no—
276C2×SD16—1511no—
277C2×D8—3021no—
278C2×D8—3021no—
279C2×D8—1511no—
280C2×D8—1511no—
281#131—1511no—
282#131—1511no—
283#131—1511no—
284#131—1511no—
285#127C2×D15111yescharacteristic, complemented
286#124—1511no—
287#108—1511no—
288#108—1511no—
289#104—1511no—
290#104—1511no—
291#104C2×D15111yescharacteristic, complemented
292#104C2×D15111yescharacteristic, complemented
293#101—1511no—
294#100—1511no—
295#100—1511no—
296#100C2×D15111yescharacteristic, complemented
297#97—1511no—
298#41—21no—
299#41—21no—
300#41—3021no—
301#41—3021no—
302#41—3021no—
303#41—3021no—
304#41—3021no—
305#41—3021no—
306#41—6041no—
307#41—1511no—
308#41—1511no—
309#41C22×D15111yescharacteristic
310#41C22×D15111yescharacteristic
311C2×Q8—3021no—
312C2×Q8—3021no—
313C2×Q8—1511no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—3021no—
319C2×D4—3021no—
320C2×D4—3021no—
321C2×D4—3021no—
322C2×D4—6041no—
323C2×D4—6041no—
324C2×D4—1511no—
325C2×D4C22×D15111yescharacteristic
326C2×D4C22×D15111yescharacteristic
327Q16—3021no—
328Q16—3021no—
329Q16—1511no—
330Q16—1511no—
331SD16—21no—
332SD16—21no—
333SD16—3021no—
334SD16—3021no—
335SD16—3021no—
336SD16—3021no—
337SD16—3021no—
338SD16—3021no—
339SD16—3021no—
340SD16—3021no—
341SD16—3021no—
342SD16—3021no—
343SD16—1511no—
344SD16—1511no—
345SD16—1511no—
346SD16—1511no—
347D8—21no—
348D8—21no—
349D8—3021no—
350D8—3021no—
351D8—3021no—
352D8—3021no—
353D8—3021no—
354D8—3021no—
355D8—3021no—
356D8—3021no—
357D8—1511no—
358D8—1511no—
359M16—3021no—
360M16—3021no—
361M16—3021no—
362M16—3021no—
363M16—1511no—
364M16—1511no—
365M16C22×D15111yescharacteristic
366M16C22×D15111yescharacteristic
367C2×C8—3021no—
368C2×C8—3021no—
369C2×C8—3021no—
370C2×C8—3021no—
371C2×C8—1511no—
372C2×C8—1511no—
373C22⋊C4—6041no—
374C42C22×D15111yescharacteristic
375C23—21no—
376C23—21no—
377C23—41no—
378C23—3021no—
379C23—6041no—
380C23—6041no—
381Q8—3021no—
382Q8—3021no—
383Q8—6041no—
384Q8—1511no—
385Q8—1511no—
386Q8D4×D15111yescharacteristic
387Q8D4×D15111yescharacteristic
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—21no—
393D4—21no—
394D4—41no—
395D4—41no—
396D4—82no—
397D4—3021no—
398D4—3021no—
399D4—3021no—
400D4—3021no—
401D4—6041no—
402D4—6041no—
403D4—6041no—
404D4—6041no—
405D4—6041no—
406D4—1511no—
407D4—1511no—
408D4D4×D15111yescharacteristic
409D4D4×D15111yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—3021no—
415C2×C4—3021no—
416C2×C4—3021no—
417C2×C4—6041no—
418C2×C4—6041no—
419C2×C4—6041no—
420C2×C4—6041no—
421C2×C4C23×D15111yescharacteristic
422C8—21no—
423C8—21no—
424C8—3021no—
425C8—3021no—
426C8—1511no—
427C8—1511no—
428C8—1511no—
429C8—1511no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22—161no—
437C22—3021no—
438C22—3021no—
439C22—6041no—
440C22—24161no—
441C22—24161no—
442C22—12081no—
443C22#1801278411yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3021no—
448C4—3021no—
449C4—6041no—
450C4C2×D4×D15111yescharacteristic
451C4C2×D4×D15111yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2—81no—
457C2—6041no—
458C2—6041no—
459C2—12081no—
460C2#2375284411yescharacteristic
subgroupG/H
centerC2#23752844
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#453×C151C2
socleC2×C151D42
hypercenter#453D151
hypocenterC151#3288
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23753023C2semidirect
2. #23753022C2semidirect
3. #23753016C2semidirect
4. #23753015C2semidirect
5. #23752599C2semidirect
6. #23752481C2semidirect
7. #23752480C2semidirect
8. #23752474C2semidirect
9. #23752473C2semidirect
10. #23752391C2semidirect
11. #23752386C2semidirect
12. #23752379C2semidirect
13. #23752331C2semidirect
14. #23752331C2semidirect
15. #453×C151C2semidirect
16. #18012786C22semidirect
18. #18012784C22semidirect
19. #18012772C22semidirect
21. #18012769C22semidirect
23. #18012767C22semidirect
28. #18012765C22semidirect
30. #18012764C22semidirect
40. C2×#14558992C22semidirect
41. C2×#14558992C22semidirect
44. C2×#14558989C22semidirect
45. C2×#14558989C22semidirect
47. #18012747C22semidirect
48. #18012747C22semidirect
49. #18012746C22semidirect
50. #18012746C22semidirect
51. #18012727C22semidirect
52. #18012674C22semidirect
53. #18012674C22semidirect
54. #18012667C22semidirect
55. #18012667C22semidirect
56. #18012657C22semidirect
57. #18012656C22semidirect
58. #18012655C22semidirect
59. #18012650C22semidirect
60. H(5,𝔽2)×C151C22semidirect
62. AGL(1,ℤ/8)×C151C22semidirect
63. #127×C151C22semidirect
64. #104×C151C22semidirect
65. #104×C151C22semidirect
66. #100×C151C22semidirect
88. #14558994D4semidirect
89. #14558994D4semidirect
99. #14558992D4semidirect
100. #14558992D4semidirect
102. #14558991D4semidirect
103. #14558991D4semidirect
112. #14558989D4semidirect
113. #14558989D4semidirect
230. C151#3288semidirect
244. #453D151semidirect
250. H(5,𝔽2)C2×D151semidirect
266. AGL(1,ℤ/8)C2×D151semidirect
285. #127C2×D151semidirect
291. #104C2×D151semidirect
292. #104C2×D151semidirect
296. #100C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530232
#237530222
#237530162
#237530152
#237525992
#237524812
#237524802
#237524742
#237524732
#237523912
#237523862
#237523792
#237523312
#453×C1512
#3288151