Finite Group Databasebeta

← #34080902

Subgroups — #34080902

Order 19328.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23753024C211yesmaximal, characteristic, complemented
2#23753022C211yesmaximal, characteristic, complemented
3#23753021C211yesmaximal, characteristic, complemented
4#23753018C211yesmaximal, characteristic, complemented
5#23752583C211yesmaximal, characteristic, complemented
6#23752478C211yesmaximal, characteristic, complemented
7#23752476C211yesmaximal, characteristic, complemented
8#23752474C211yesmaximal, characteristic, complemented
9#23752397C211yesmaximal, characteristic, complemented
10#23752393C211yesmaximal, characteristic, complemented
11#23752382C211yesmaximal, characteristic, complemented
12#23752378C211yesmaximal, characteristic, complemented
13#23752328C211yesmaximal, characteristic, complemented
14#23752327C211yesmaximal, characteristic, complemented
15#471×C151C211yesmaximal, characteristic, complemented
16#18012787C2211yescharacteristic
17#18012786C2211yescharacteristic
18#18012785C2211yescharacteristic, complemented
19#18012784C2211yescharacteristic, complemented
20#18012772—21no—
21#18012772C2211yescharacteristic, complemented
22#18012772C2211yescharacteristic, complemented
23#18012771—21no—
24#18012770C2211yescharacteristic, complemented
25#18012768—21no—
26#18012768—21no—
27#18012767—21no—
28#18012767C2211yescharacteristic
29#18012767C2211yescharacteristic
30#18012766—21no—
31#18012766—21no—
32#18012765C2211yescharacteristic
33#18012763—21no—
34#18012762—21no—
35#18012761—21no—
36#18012761C2211yescharacteristic
37#18012760—21no—
38#18012759C2211yescharacteristic
39#18012758—21no—
40C2×#14558993—21no—
41C2×#14558993C2211yescharacteristic, complemented
42C2×#14558992C2211yescharacteristic, complemented
43C2×#14558991C2211yescharacteristic, complemented
44SD16×D151—21no—
45SD16×D151—21no—
46SD16×D151—21no—
47#18012747C2211yescharacteristic, complemented
48#18012747C2211yescharacteristic, complemented
49#18012745C2211yescharacteristic, complemented
50#18012737C2211yescharacteristic, complemented
51#18012735C2211yescharacteristic
52#18012674C2211yescharacteristic, complemented
53#18012666C2211yescharacteristic, complemented
54#18012665C2211yescharacteristic, complemented
55#18012665C2211yescharacteristic, complemented
56#18012665C2211yescharacteristic, complemented
57#18012657C2211yescharacteristic, complemented
58#18012655C2211yescharacteristic, complemented
59C8×Dic151C2211yescharacteristic, complemented
60#137×C151C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C151C2211yescharacteristic
62C2×SD16×C151C2211yescharacteristic, complemented
63#131×C151C2211yescharacteristic, complemented
64#108×C151C2211yescharacteristic, complemented
65#101×C151C2211yescharacteristic, complemented
66#100×C151C2211yescharacteristic, complemented
67#14559005—21no—
68#14559005—21no—
69#14559005—21no—
70#14559004—21no—
71#14559004—21no—
72#14559004—41no—
73#14559004C2311yescharacteristic
74#14559004C2311yescharacteristic
75#14559004C2311yescharacteristic
76#14559003—21no—
77#14559003—21no—
78#14559003—21no—
79#14559003—41no—
80C2×Dic302—21no—
81C2×Dic302C2311yescharacteristic
82Q8×D151—21no—
83Q8×D151—21no—
84Q8×D151—21no—
85C2×D604—21no—
86C2×D604C2311yescharacteristic
87C2×C22⋊D151—41no—
88C2×C22⋊D151—41no—
89D4×D151—21no—
90D4×D151—21no—
91D4×D151—21no—
92D4×D151—41no—
93Dic604—21no—
94Dic604—21no—
95Dic604D411yescharacteristic, complemented
96#14558994—21no—
97#14558994D411yescharacteristic, complemented
98#14558993—21no—
99#14558993—21no—
100#14558993—21no—
101#14558993—21no—
102#14558993—21no—
103#14558993—21no—
104#14558993D411yescharacteristic, complemented
105#14558993D411yescharacteristic, complemented
106#14558992—21no—
107#14558992—21no—
108#14558992—21no—
109#14558992D411yescharacteristic, complemented
110#14558991—21no—
111#14558991—21no—
112#14558991—21no—
113#14558991D411yescharacteristic, complemented
114D1208—21no—
115D1208—21no—
116D1208D411yescharacteristic, complemented
117#14558989—21no—
118#14558989D411yescharacteristic, complemented
119#14558988C2311yescharacteristic
120#14558987—21no—
121#14558987—21no—
122#14558987—21no—
123C2×C151⋊C8C2311yescharacteristic
124C8×D151—21no—
125C8×D151—21no—
126C8×D151—21no—
127Dic151⋊C4—41no—
128C4×Dic151C2311yescharacteristic
129#41×C151C2311yescharacteristic
130C2×Q8×C151C2311yescharacteristic
131C2×D4×C151C2311yescharacteristic
132Q16×C151—21no—
133SD16×C151—21no—
134SD16×C151—21no—
135SD16×C151—21no—
136SD16×C151—21no—
137D8×C151—21no—
138M16×C151—21no—
139M16×C151C2311yescharacteristic
140M16×C151C2311yescharacteristic
141M16×C151C2311yescharacteristic
142C2×C8×C151—21no—
143C2×C8×C151C2311yescharacteristic
144C22×D151—21no—
145C22×D151—41no—
146C22×D151—41no—
147Dic302—21no—
148Dic302—21no—
149Dic302—41no—
150Dic302—41no—
151Dic302C2×D411yescharacteristic
152Dic302C2×D411yescharacteristic
153Dic302C2×D411yescharacteristic
154D604—21no—
155D604—21no—
156D604—41no—
157D604—41no—
158D604C2×D411yescharacteristic
159D604C2×D411yescharacteristic
160D604C2×D411yescharacteristic
161C22⋊D151—21no—
162C22⋊D151—21no—
163C22⋊D151—21no—
164C22⋊D151—41no—
165C22⋊D151—41no—
166C22⋊D151—41no—
167C22⋊D151—41no—
168C22⋊D151—82no—
169C2×Dic151—21no—
170C2×Dic151—21no—
171C2×Dic151—41no—
172C2×Dic151—41no—
173C4×D151—21no—
174C4×D151—21no—
175C4×D151—21no—
176C4×D151—41no—
177C4×D151—41no—
178C4×D151—41no—
179C4×D151—41no—
180C151⋊C8—21no—
181C151⋊C8C2×D411yescharacteristic
182C151⋊C8C2×D411yescharacteristic
183C23×C151—21no—
184Q8×C151—21no—
185Q8×C151—21no—
186Q8×C151C2×D411yescharacteristic
187D4×C151—21no—
188D4×C151—21no—
189D4×C151—21no—
190D4×C151C2×D411yescharacteristic
191C2×C4×C151—21no—
192C2×C4×C151—21no—
193C2×C4×C151C2411yescharacteristic
194C8×C151—21no—
195C8×C151—21no—
196C8×C151—21no—
197C8×C151C2×D411yescharacteristic
198C8×C151C2×D411yescharacteristic
199C22×C151—21no—
200C22×C151—41no—
201C22×C151—81no—
202C22×C151H(5,𝔽2)11yescharacteristic
203C2×D151—21no—
204C2×D151—21no—
205C2×D151—21no—
206C2×D151—41no—
207C2×D151—81no—
208C2×D151—161no—
209C2×D151—161no—
210C4×C151—21no—
211C4×C151—41no—
212C4×C151C22×D411yescharacteristic
213C4×C151C22×D411yescharacteristic
214Dic151—22no—
215Dic151—21no—
216Dic151—21no—
217Dic151—21no—
218Dic151—41no—
219C2×C151—21no—
220C2×C151—41no—
221C2×C151—81no—
222C2×C151D4211yescharacteristic
223D151—41no—
224D151—41no—
225D151—41no—
226D151—81no—
227C151#329111yescharacteristic, complemented, Sylow
228#3291—1511nomaximal, Sylow
229#577—1511no—
230#577—1511no—
231#577—1511no—
232#577—1511no—
233#492—1511no—
234#471—1511no—
235#471—1511no—
236#471—1511no—
237#471D15111yescharacteristic, complemented
238#456—1511no—
239#456—1511no—
240#453—1511no—
241#453—1511no—
242#443—1511no—
243#443—1511no—
2442−1+4—1511no—
2452−1+4—1511no—
246H(5,𝔽2)—1511no—
247H(5,𝔽2)—1511no—
248#137—3021no—
249#137—3021no—
250#137—3021no—
251#137—3021no—
252#137—1511no—
253#137—1511no—
254#137—1511no—
255#137C2×D15111yescharacteristic, complemented
256AGL(1,ℤ/8)—3021no—
257AGL(1,ℤ/8)—3021no—
258AGL(1,ℤ/8)—3021no—
259AGL(1,ℤ/8)—3021no—
260AGL(1,ℤ/8)—1511no—
261AGL(1,ℤ/8)—1511no—
262AGL(1,ℤ/8)—1511no—
263AGL(1,ℤ/8)C2×D15111yescharacteristic
264#135—3021no—
265#135—3021no—
266#135—3021no—
267#135—3021no—
268#135—1511no—
269#135—1511no—
270C2×SD16—3021no—
271C2×SD16—3021no—
272C2×SD16—3021no—
273C2×SD16—3021no—
274C2×SD16—1511no—
275C2×SD16—1511no—
276C2×SD16—1511no—
277C2×SD16C2×D15111yescharacteristic, complemented
278#131—1511no—
279#131—1511no—
280#131—1511no—
281#131C2×D15111yescharacteristic, complemented
282C4⋊Q8—1511no—
283#127—1511no—
284#108—1511no—
285#108C2×D15111yescharacteristic, complemented
286#104—1511no—
287#104—1511no—
288#104—1511no—
289#104—1511no—
290#101—1511no—
291#101C2×D15111yescharacteristic, complemented
292#100—1511no—
293#100C2×D15111yescharacteristic, complemented
294C4×C8—1511no—
295#41—3021no—
296#41—3021no—
297#41—3021no—
298#41—3021no—
299#41—3021no—
300#41—3021no—
301#41—3021no—
302#41—3021no—
303#41—6041no—
304#41—6041no—
305#41—1511no—
306#41—1511no—
307#41—1511no—
308#41C22×D15111yescharacteristic
309C2×Q8—3021no—
310C2×Q8—3021no—
311C2×Q8—3021no—
312C2×Q8—3021no—
313C2×Q8—1511no—
314C2×Q8C22×D15111yescharacteristic
315C2×D4—3021no—
316C2×D4—3021no—
317C2×D4—3021no—
318C2×D4—3021no—
319C2×D4—6041no—
320C2×D4—6041no—
321C2×D4—6041no—
322C2×D4—1511no—
323C2×D4C22×D15111yescharacteristic
324Q16—21no—
325Q16—3021no—
326Q16—3021no—
327Q16—3021no—
328Q16—1511no—
329Q16—1511no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—3021no—
335SD16—3021no—
336SD16—3021no—
337SD16—3021no—
338SD16—3021no—
339SD16—3021no—
340SD16—3021no—
341SD16—3021no—
342SD16—3021no—
343SD16—3021no—
344SD16—3021no—
345SD16—3021no—
346SD16—1511no—
347SD16—1511no—
348SD16—1511no—
349SD16—1511no—
350D8—21no—
351D8—3021no—
352D8—3021no—
353D8—3021no—
354D8—1511no—
355D8—1511no—
356M16—21no—
357M16—3021no—
358M16—3021no—
359M16—3021no—
360M16—1511no—
361M16C22×D15111yescharacteristic
362M16C22×D15111yescharacteristic
363M16C22×D15111yescharacteristic
364C2×C8—21no—
365C2×C8—3021no—
366C2×C8—3021no—
367C2×C8—3021no—
368C2×C8—1511no—
369C2×C8C22×D15111yescharacteristic
370C4⋊C4—6041no—
371C42—1511no—
372C23—21no—
373C23—3021no—
374C23—6041no—
375C23—6041no—
376Q8—21no—
377Q8—21no—
378Q8—3021no—
379Q8—3021no—
380Q8—6041no—
381Q8—6041no—
382Q8—1511no—
383Q8—1511no—
384Q8—1511no—
385Q8D4×D15111yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—3021no—
390D4—3021no—
391D4—3021no—
392D4—3021no—
393D4—3021no—
394D4—6041no—
395D4—6041no—
396D4—6041no—
397D4—6041no—
398D4—6041no—
399D4—6041no—
400D4—1511no—
401D4—1511no—
402D4—1511no—
403D4—12082no—
404D4D4×D15111yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—3021no—
408C2×C4—3021no—
409C2×C4—3021no—
410C2×C4—3021no—
411C2×C4—3021no—
412C2×C4—6041no—
413C2×C4—6041no—
414C2×C4—6041no—
415C2×C4—6041no—
416C2×C4—6041no—
417C2×C4—6041no—
418C2×C4C23×D15111yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—3021no—
423C8—1511no—
424C8—1511no—
425C8D4×D15111yescharacteristic
426C8D4×D15111yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—3021no—
431C22—3021no—
432C22—3021no—
433C22—6041no—
434C22—24161no—
435C22—24161no—
436C22—12081no—
437C22#1801278511yescharacteristic
438C4—21no—
439C4—41no—
440C4—3022no—
441C4—3021no—
442C4—3021no—
443C4—3021no—
444C4—6041no—
445C4C2×D4×D15111yescharacteristic
446C4C2×D4×D15111yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—6041no—
451C2—6041no—
452C2—6041no—
453C2—12081no—
454C2#2375284611yescharacteristic
subgroupG/H
centerC2#23752846
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#471×C151C2
socleC2×C151D42
hypercenter#471D151
hypocenterC151#3291
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #23753024C2semidirect
2. #23753022C2semidirect
3. #23753021C2semidirect
4. #23753018C2semidirect
5. #23752583C2semidirect
6. #23752478C2semidirect
7. #23752476C2semidirect
8. #23752474C2semidirect
9. #23752397C2semidirect
10. #23752393C2semidirect
11. #23752382C2semidirect
12. #23752378C2semidirect
13. #23752328C2semidirect
14. #23752327C2semidirect
15. #471×C151C2semidirect
18. #18012785C22semidirect
19. #18012784C22semidirect
21. #18012772C22semidirect
22. #18012772C22semidirect
24. #18012770C22semidirect
41. C2×#14558993C22semidirect
42. C2×#14558992C22semidirect
43. C2×#14558991C22semidirect
47. #18012747C22semidirect
48. #18012747C22semidirect
49. #18012745C22semidirect
50. #18012737C22semidirect
52. #18012674C22semidirect
53. #18012666C22semidirect
54. #18012665C22semidirect
55. #18012665C22semidirect
56. #18012665C22semidirect
57. #18012657C22semidirect
58. #18012655C22semidirect
59. C8×Dic151C22semidirect
60. #137×C151C22semidirect
62. C2×SD16×C151C22semidirect
63. #131×C151C22semidirect
64. #108×C151C22semidirect
65. #101×C151C22semidirect
66. #100×C151C22semidirect
95. Dic604D4semidirect
97. #14558994D4semidirect
104. #14558993D4semidirect
105. #14558993D4semidirect
109. #14558992D4semidirect
113. #14558991D4semidirect
116. D1208D4semidirect
118. #14558989D4semidirect
227. C151#3291semidirect
237. #471D151semidirect
255. #137C2×D151semidirect
277. C2×SD16C2×D151semidirect
281. #131C2×D151semidirect
285. #108C2×D151semidirect
291. #101C2×D151semidirect
293. #100C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237530242
#237530222
#237530212
#237530182
#237525832
#237524782
#237524762
#237524742
#237523972
#237523932
#237523822
#237523782
#237523282
#237523272
#471×C1512
#3291151