Finite Group Databasebeta

← #27537542

Subgroups — #27537542

Order 13568.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20283517C211yesmaximal, characteristic, complemented
3#20276613C214yesmaximal, complemented
4#20276606C214yesmaximal, complemented
5#20276489C211yesmaximal, characteristic, complemented
6#20276462C211yesmaximal, characteristic, complemented
7#20276451C211yesmaximal, characteristic, complemented
8C2×#16810401C211yesmaximal, characteristic, complemented
9C2×C53×#444C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11C4×#11719945—21no—
12#16811064C412yescomplemented
13#16811063C2212yes—
14#16811059C412yescomplemented
15#16811058C2212yes—
16#16811053C414yescomplemented
17C2×#13133157C2211yescharacteristic
18C2×#13133157C2211yescharacteristic
19#16810736C2212yescomplemented
20#16810731C2212yescomplemented
21#16810725—22no—
22#16810661C2211yescharacteristic
23#16810620—21no—
24#16810447C2214yescomplemented
25#16810438C2212yescomplemented
26#16810435C2212yescomplemented
27#16810433C2214yes—
28#16810401C2214yescomplemented
29C2×#135×C53C2211yescharacteristic
30C2×#131×C53—21no—
31C53×#444—22no—
32C53×#444C2214yes—
33C2×#108×C53C2211yescharacteristic
34C2×#104×C53—21no—
35C2×#97×C53C2211yescharacteristic
36C2×#11719945—21no—
37#13133239—21no—
38#13133239—21no—
39#13133239C2×C412yes—
40#13133239C2×C412yes—
41C2×Dic212C2311yescharacteristic
42C2×#11719913—21no—
43C2×D424C2311yescharacteristic
44C4×Dic106—22no—
45C4×D212—22no—
46C4×C22⋊D53—42no—
47#13133176—21no—
48#13133174—41no—
49C42×D53—41no—
50#13133163C2×C414yescomplemented
51#13133161C2×C414yescomplemented
52#13133160C2×C414yescomplemented
53#13133158C2×C414yescomplemented
54#13133157—21no—
55#13133157C2×C412yescomplemented
56#13133157C2312yes—
57#13133157C2312yes—
58C2×#11719908—21no—
59C2×#11719908C2×C412yescomplemented
60#13133151—21no—
61#13133149C2312yes—
62#13133142—22no—
63#13133139—22no—
64C2×#41×C53—21no—
65#135×C53—21no—
66#135×C53—21no—
67#135×C53C2×C412yescomplemented
68#135×C53C2312yes—
69C2×Q16×C53C2×C411yescharacteristic, complemented
70C2×SD16×C53—21no—
71C2×D8×C53C2×C411yescharacteristic, complemented
72#131×C53—24no—
73#131×C53—42no—
74C2×M16×C53—21no—
75C2×M16×C53—41no—
76C22×C8×C53—41no—
77C22×C8×C53D411yescharacteristic
78C22×C8×C53C2311yescharacteristic
79C2×C42×C53D411yescharacteristic
80#108×C53—21no—
81#108×C53C2312yes—
82#104×C53—24no—
83#97×C53D412yes—
84#97×C53C2312yes—
85#41×C53—22no—
86#41×C53—22no—
87#41×C53—41no—
88#41×C53—41no—
89C2×Q8×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C2×C22⋊D53—41no—
96#11719945—22no—
97#11719945—22no—
98#11719945—41no—
99#11719945—41no—
100C2×D212—21no—
101C2×C4×D53—41no—
102C2×C4×D53—42no—
103C2×Dic106—21no—
104Q16×C53—21no—
105Q16×C53C22×C412yes—
106SD16×C53—22no—
107SD16×C53—22no—
108D8×C53—21no—
109D8×C53C22×C412yes—
110M16×C53—22no—
111M16×C53—44no—
112M16×C53—41no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53—44no—
116C2×C8×C53—41no—
117C2×C8×C53C22×C411yescharacteristic
118C2×C8×C53C22×C411yescharacteristic
119C2×C8×C53C2×D412yes—
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53#4111yescharacteristic
122C2×C8×C53#4111yescharacteristic
123C2×C8×C53C2411yescharacteristic
124C42×C53—21no—
125C42×C53C2×D412yes—
126C22⋊Dic53—41no—
127#11719921—42no—
128C4⋊Dic53—22no—
129#11719919—42no—
130C4×Dic53—42no—
131C4×Dic53—41no—
132Dic212—21no—
133Dic212C22×C412yes—
134D424—21no—
135D424C22×C412yes—
136#11719913—22no—
137#11719913—22no—
138#11719908—22no—
139#11719908C22×C414yes—
140C23×C53—41no—
141C23×C53AGL(1,ℤ/8)11yescharacteristic
142C22×D53—41no—
143Q8×C53—22no—
144Q8×C53—41no—
145D4×C53—22no—
146D4×C53—42no—
147D4×C53—42no—
148D4×C53—41no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—81no—
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53C22×D411yescharacteristic
160C22⋊D53—42no—
161C22⋊D53—42no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164D212—22no—
165D212—41no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—84no—
169Dic106—22no—
170Dic106—41no—
171C8×C53—22no—
172C8×C53—42no—
173C8×C53C4×D411yescharacteristic
174C8×C53C4×D411yescharacteristic
175C8×C53C23×C411yescharacteristic
176C8×C53C2×#4111yescharacteristic
177#33468—531nomaximal, Sylow
178C22×C53—41no—
179C22×C53—42no—
180C22×C53—82no—
181C22×C53#44211yescharacteristic
182C22×C53#44211yescharacteristic
183C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D53—42no—
185C2×D53—82no—
186C4×C53—42no—
187C4×C53—42no—
188C4×C53#44211yescharacteristic
189C4×C53#44211yescharacteristic
190C4×C53C2×C4×D411yescharacteristic
191C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
192Dic53—42no—
193Dic53—82no—
194#3414—531no—
195C2×#444D5311yescharacteristic, complemented
196#2943—531no—
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199#2155—531no—
200#2148—531no—
201#2145—531no—
202C2×#347—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—82no—
207C2×C53C2×#44211yescharacteristic
208D53—82no—
209C2×#135—531no—
210C2×#135C2×D5311yescharacteristic
211C2×#131—21no—
212C4×#41—1061no—
213#510—532no—
214#510—532no—
215#509—532no—
216#509—532no—
217#508—534no—
218C2×M32—531no—
219C2×M32—531no—
220#444—22no—
221#444C2×D5314yes—
222#442—532no—
223#441—532no—
224#440—1062no—
225C2×#108C2×D5311yescharacteristic
226#427—531no—
227C2×#104—21no—
228#416—1061no—
229C2×#97C2×D5311yescharacteristic
230#365—534no—
231#362—532no—
232#361—532no—
233#360—534no—
234#347—534no—
235C53#3346811yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1061no—
238#135—21no—
239#135—21no—
240#135—532no—
241#135—532no—
242#135—1061no—
243#135—1061no—
244#135C4×D5312yescomplemented
245#135C22×D5312yes—
246C2×Q16—531no—
247C2×Q16C4×D5311yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—1061no—
250C2×D8—531no—
251C2×D8C4×D5311yescharacteristic, complemented
252#131—24no—
253#131—42no—
254C2×M16—21no—
255C2×M16—41no—
256C22×C8—41no—
257C22×C8C22⋊D5311yescharacteristic
258C22×C8C22×D5311yescharacteristic
259C4×Q8—1062no—
260C4×D4—1062no—
261C4×D4—2122no—
262#117—1061no—
263#117—2121no—
264C2×C42—2121no—
265C2×C42C22⋊D5311yescharacteristic
266Q32—534no—
267SD32—534no—
268SD32—534no—
269D16—534no—
270M32—532no—
271M32—532no—
272M32—532no—
273M32—1061no—
274C2×C16—532no—
275C2×C16—1061no—
276#108—21no—
277#108C22×D5312yes—
278#107—1061no—
279#106—532no—
280#104—24no—
281#103—1062no—
282#102—1062no—
283#97C22⋊D5312yes—
284#97C22×D5312yes—
285#41—22no—
286#41—22no—
287#41—41no—
288#41—41no—
289#41—1062no—
290#41—1062no—
291#41—2121no—
292#41—2121no—
293C2×Q8—21no—
294C2×Q8—1061no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1061no—
298C2×D4—2121no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2122no—
302C22×C4—2121no—
303C22×C4C2×C22⋊D5311yescharacteristic
304Q16—21no—
305Q16—532no—
306Q16—1061no—
307Q16C2×C4×D5312yes—
308SD16—22no—
309SD16—22no—
310SD16—1062no—
311SD16—1062no—
312D8—21no—
313D8—532no—
314D8—1061no—
315D8C2×C4×D5312yes—
316M16—22no—
317M16—44no—
318M16—41no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—44no—
322C2×C8—41no—
323C2×C8C2×C4×D5311yescharacteristic
324C2×C8C2×C4×D5311yescharacteristic
325C2×C8#1171994511yescharacteristic
326C2×C8#1171994511yescharacteristic
327C2×C8C2×C22⋊D5312yes—
328C2×C8C2×C22⋊D5311yescharacteristic
329C2×C8C23×D5311yescharacteristic
330C4⋊C4—1062no—
331C4⋊C4—2122no—
332C22⋊C4—2122no—
333C22⋊C4—2121no—
334C42—21no—
335C42—2122no—
336C42—2121no—
337C42C2×C22⋊D5312yes—
338C16—534no—
339C16—1062no—
340C23—41no—
341C23—2121no—
342C23#1313324211yescharacteristic
343Q8—22no—
344Q8—41no—
345Q8—1062no—
346Q8—2121no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—1062no—
352D4—2122no—
353D4—2122no—
354D4—2121no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—4244no—
361C2×C4—2122no—
362C2×C4—2121no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4C4×C22⋊D5311yescharacteristic
366C2×C4C4×C22⋊D5311yescharacteristic
367C2×C4#1313324211yescharacteristic
368C2×C4#1313324311yescharacteristic
369C2×C4#1313324311yescharacteristic
370C2×C4C22×C22⋊D5311yescharacteristic
371C8—22no—
372C8—42no—
373C8C4×C22⋊D5311yescharacteristic
374C8C4×C22⋊D5311yescharacteristic
375C8C22×C4×D5311yescharacteristic
376C8C2×#1171994511yescharacteristic
377C22—41no—
378C22—42no—
379C22—82no—
380C22—4242no—
381C22—2122no—
382C22#1681073311yescharacteristic
383C22#1681073311yescharacteristic
384C22C2×#1313324211yescharacteristic
385C4—42no—
386C4—42no—
387C4—4242no—
388C4—2122no—
389C4#1681073311yescharacteristic
390C4#1681073311yescharacteristic
391C4C2×C4×C22⋊D5311yescharacteristic
392C4C2×#1313324311yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4242no—
398C2C2×#1681073311yescharacteristic
subgroupG/H
centerC4#16810733
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#444C2
socleC2×C53C2×#442
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenterC2×#444D53
hypocenterC53#33468
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #20283517C2semidirect
3. #20276613C2semidirect
4. #20276606C2semidirect
5. #20276489C2semidirect
6. #20276462C2semidirect
7. #20276451C2semidirect
8. C2×#16810401C2semidirect
9. C2×C53×#444C2semidirect
12. #16811064C4semidirect
14. #16811059C4semidirect
16. #16811053C4semidirect
19. #16810736C22semidirect
20. #16810731C22semidirect
24. #16810447C22semidirect
25. #16810438C22semidirect
26. #16810435C22semidirect
28. #16810401C22semidirect
50. #13133163C2×C4semidirect
51. #13133161C2×C4semidirect
52. #13133160C2×C4semidirect
53. #13133158C2×C4semidirect
55. #13133157C2×C4semidirect
59. C2×#11719908C2×C4semidirect
67. #135×C53C2×C4semidirect
69. C2×Q16×C53C2×C4semidirect
71. C2×D8×C53C2×C4semidirect
195. C2×#444D53semidirect
235. C53#33468semidirect
244. #135C4×D53semidirect
247. C2×Q16C4×D53semidirect
251. C2×D8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202835172
#202766132
#202766062
#202764892
#202764622
#202764512
C2×#168104012
C2×C53×#4442
#3346853