Finite Group Databasebeta

← #27537549

Subgroups — #27537549

Order 13568.

646 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#20283633C211yesmaximal, characteristic, complemented
3#20276611C214yesmaximal, complemented
4#20276604C214yesmaximal, complemented
5C2×#16810445C211yesmaximal, characteristic, complemented
6#20276464C211yesmaximal, characteristic, complemented
7C2×#16810432C211yesmaximal, characteristic, complemented
8#20276320C211yesmaximal, characteristic, complemented
9C53×#2952C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11#16811389—21no—
12#16811064C2212yes—
13#16811063C412yescomplemented
14#16811059C2212yes—
15#16811058C412yescomplemented
16#16811053C414yescomplemented
17C2×#13133157C2211yescharacteristic
18C2×#13133157C2211yescharacteristic
19#16810746C2214yes—
20#16810741—22no—
21C2×#13133153C2211yescharacteristic
22#16810641—21no—
23#16810445C2214yescomplemented
24#16810439C2212yescomplemented
25#16810436C2212yescomplemented
26#16810432C2214yes—
27#16810401C2214yescomplemented
28C2×#135×C53C2211yescharacteristic
29C53×#520—21no—
30C53×#442C2212yescomplemented
31C53×#441C2212yescomplemented
32C4×SD16×C53—22no—
33C2×#106×C53C2211yescharacteristic
34C53×#417—21no—
35C53×#405C2211yescharacteristic
36C2×#11719945—21no—
37#13133239—21no—
38#13133239—21no—
39#13133239C2×C412yescomplemented
40#13133239C2312yes—
41C2×Dic212C2×C411yescharacteristic, complemented
42C2×#11719913—21no—
43C2×D424C2×C411yescharacteristic, complemented
44#13133225—42no—
45#13133223—24no—
46C2×#11719917—21no—
47M16×D53—42no—
48#13133163C2×C414yescomplemented
49#13133161C2×C414yescomplemented
50#13133160C2×C414yescomplemented
51#13133158C2×C414yescomplemented
52#13133157—21no—
53#13133157C2×C412yescomplemented
54#13133157C2312yes—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908C2×C412yescomplemented
58#13133153—21no—
59#13133153C2312yes—
60#13133145—24no—
61C2×#41×C53—21no—
62#135×C53—21no—
63#135×C53—21no—
64#135×C53C2×C412yes—
65#135×C53C2×C412yes—
66C2×Q16×C53C2311yescharacteristic
67C2×SD16×C53—21no—
68C2×D8×C53C2311yescharacteristic
69C2×M16×C53D411yescharacteristic
70C22×C8×C53C2311yescharacteristic
71C4×Q8×C53—22no—
72C4×D4×C53—22no—
73C4×D4×C53—42no—
74#117×C53—41no—
75#117×C53D411yescharacteristic
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—41no—
78#106×C53—21no—
79#106×C53C2312yes—
80#103×C53—22no—
81#102×C53—22no—
82#97×C53C2312yes—
83C4×C8×C53D412yes—
84#41×C53—22no—
85#41×C53—22no—
86#41×C53—41no—
87#41×C53—41no—
88C2×Q8×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—41no—
91C22×C4×C53—42no—
92C22×C4×C53—41no—
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×Dic106—21no—
103Q16×C53—21no—
104Q16×C53C22×C412yes—
105SD16×C53—22no—
106SD16×C53—22no—
107D8×C53—21no—
108D8×C53C22×C412yes—
109M16×C53—21no—
110M16×C53—21no—
111C2×C8×C53C22×C411yescharacteristic
112C2×C8×C53C22×C411yescharacteristic
113C2×C8×C53C2×D412yes—
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53#4111yescharacteristic
116C2×C8×C53#4111yescharacteristic
117C2×C8×C53C2411yescharacteristic
118C4⋊C4×C53—21no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—42no—
121C4⋊C4×C53—41no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—42no—
124C42×C53—42no—
125C42×C53C2×D412yes—
126#11719917—22no—
127#11719917—41no—
128C2×#88322—41no—
129Dic212—21no—
130Dic212C22×C412yes—
131D424—21no—
132D424C22×C412yes—
133#11719913—22no—
134#11719913—22no—
135#11719912—44no—
136C8×D53—44no—
137#11719908—22no—
138#11719908C22×C414yes—
139C23×C53—41no—
140C23×C53AGL(1,ℤ/8)11yescharacteristic
141C22×D53—41no—
142Q8×C53—22no—
143Q8×C53—41no—
144D4×C53—22no—
145D4×C53—42no—
146D4×C53—42no—
147D4×C53—41no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—162no—
154C2×C4×C53—161no—
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C22⋊D53—42no—
162C22⋊D53—42no—
163C2×Dic53—41no—
164D212—22no—
165D212—41no—
166C4×D53—42no—
167C4×D53—42no—
168Dic106—22no—
169Dic106—41no—
170C8×C53—22no—
171C8×C53C4×D411yescharacteristic
172C8×C53C4×D411yescharacteristic
173C8×C53C23×C411yescharacteristic
174C8×C53C2×#4111yescharacteristic
175#88322—42no—
176#33469—531nomaximal, Sylow
177C22×C53—41no—
178C22×C53—42no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53#44211yescharacteristic
182C22×C53#44211yescharacteristic
183C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
184C2×D53—42no—
185C2×D53—161no—
186C4×C53—42no—
187C4×C53—42no—
188C4×C53—82no—
189C4×C53#44211yescharacteristic
190C4×C53#44211yescharacteristic
191C4×C53C2×C4×D411yescharacteristic
192C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
193Dic53—42no—
194#3414—531no—
195#2957—531no—
196#2952D5311yescharacteristic, complemented
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199C2×#365—531no—
200#2148—531no—
201C2×#360—531no—
202#2110—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—82no—
207C2×C53C2×#44211yescharacteristic
208D53—82no—
209C2×#135—531no—
210C2×#135C2×D5311yescharacteristic
211#568—1061no—
212#520—21no—
213#510—532no—
214#510—532no—
215#509—532no—
216#509—532no—
217#508—534no—
218C2×M32—531no—
219C2×M32—531no—
220#444—534no—
221#443—1062no—
222#442C2×D5312yescomplemented
223#441C2×D5312yescomplemented
224C4×SD16—22no—
225C2×#108—531no—
226C2×#106C2×D5311yescharacteristic
227#421—1061no—
228#417—21no—
229#405C2×D5311yescharacteristic
230#365—534no—
231#362—532no—
232#361—532no—
233#360—534no—
234#347—534no—
235C53#3346911yescharacteristic, 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×D5312yes—
245#135C4×D5312yes—
246C2×Q16—531no—
247C2×Q16C22×D5311yescharacteristic
248C2×SD16—21no—
249C2×SD16—1061no—
250C2×D8—531no—
251C2×D8C22×D5311yescharacteristic
252#131—1064no—
253#131—2122no—
254C2×M16—1061no—
255C2×M16—2122no—
256C2×M16C22⋊D5311yescharacteristic
257C22×C8C22×D5311yescharacteristic
258C4×Q8—22no—
259C4×D4—22no—
260C4×D4—42no—
261#117—41no—
262#117C22⋊D5311yescharacteristic
263C2×C4⋊C4—21no—
264C2×C4⋊C4—41no—
265Q32—534no—
266SD32—534no—
267SD32—534no—
268D16—534no—
269M32—532no—
270M32—532no—
271M32—532no—
272M32—1061no—
273C2×C16—532no—
274C2×C16—1061no—
275#108—532no—
276#108—1061no—
277#106—21no—
278#106C22×D5312yes—
279#104—1064no—
280#103—22no—
281#102—22no—
282#97C22×D5312yes—
283C4×C8C22⋊D5312yes—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—1062no—
289#41—1062no—
290#41—2121no—
291#41—2121no—
292C2×Q8—21no—
293C2×Q8—1061no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1061no—
297C2×D4—2121no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—2121no—
302C22×C4C2×C22⋊D5311yescharacteristic
303Q16—21no—
304Q16—532no—
305Q16—1061no—
306Q16C2×C4×D5312yes—
307SD16—22no—
308SD16—22no—
309SD16—1062no—
310SD16—1062no—
311D8—21no—
312D8—532no—
313D8—1061no—
314D8C2×C4×D5312yes—
315M16—21no—
316M16—21no—
317M16—1062no—
318M16—2124no—
319M16—2121no—
320C2×C8—2124no—
321C2×C8—2121no—
322C2×C8C2×C4×D5311yescharacteristic
323C2×C8C2×C4×D5311yescharacteristic
324C2×C8#1171994511yescharacteristic
325C2×C8#1171994511yescharacteristic
326C2×C8C2×C22⋊D5312yes—
327C2×C8C2×C22⋊D5311yescharacteristic
328C2×C8C23×D5311yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C22⋊C4—21no—
334C22⋊C4—42no—
335C42—42no—
336C42C2×C22⋊D5312yes—
337C16—534no—
338C16—1062no—
339C23—41no—
340C23—2121no—
341C23#1313324311yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—1062no—
345Q8—2121no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—1062no—
351D4—2122no—
352D4—2122no—
353D4—2121no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—162no—
360C2×C4—161no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4C4×C22⋊D5311yescharacteristic
365C2×C4C4×C22⋊D5311yescharacteristic
366C2×C4#1313324211yescharacteristic
367C2×C4#1313324211yescharacteristic
368C2×C4#1313324311yescharacteristic
369C2×C4C22×C22⋊D5311yescharacteristic
370C8—22no—
371C8—2122no—
372C8C4×C22⋊D5311yescharacteristic
373C8C4×C22⋊D5311yescharacteristic
374C8C22×C4×D5311yescharacteristic
375C8C2×#1171994511yescharacteristic
376C22—41no—
377C22—42no—
378C22—81no—
379C22—81no—
380C22—8481no—
381C22—2122no—
382C22#1681073311yescharacteristic
383C22#1681073311yescharacteristic
384C22C2×#1313324311yescharacteristic
385C4—42no—
386C4—42no—
387C4—82no—
388C4—2122no—
389C4#1681073311yescharacteristic
390C4#1681073311yescharacteristic
391C4C2×C4×C22⋊D5311yescharacteristic
392C4C2×#1313324211yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4242no—
398C2C2×#1681073311yescharacteristic
subgroupG/H
centerC2C2×#16810733
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2952C2
socleC2×C53C2×#442
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenter#2952D53
hypocenterC53#33469
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. #20283633C2semidirect
3. #20276611C2semidirect
4. #20276604C2semidirect
5. C2×#16810445C2semidirect
6. #20276464C2semidirect
7. C2×#16810432C2semidirect
8. #20276320C2semidirect
9. C53×#2952C2semidirect
13. #16811063C4semidirect
15. #16811058C4semidirect
16. #16811053C4semidirect
23. #16810445C22semidirect
24. #16810439C22semidirect
25. #16810436C22semidirect
27. #16810401C22semidirect
30. C53×#442C22semidirect
31. C53×#441C22semidirect
39. #13133239C2×C4semidirect
41. C2×Dic212C2×C4semidirect
43. C2×D424C2×C4semidirect
48. #13133163C2×C4semidirect
49. #13133161C2×C4semidirect
50. #13133160C2×C4semidirect
51. #13133158C2×C4semidirect
53. #13133157C2×C4semidirect
57. C2×#11719908C2×C4semidirect
196. #2952D53semidirect
222. #442C2×D53semidirect
223. #441C2×D53semidirect
235. C53#33469semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202836332
#202766112
#202766042
C2×#168104452
#202764642
C2×#168104322
#202763202
C53×#29522
#3346953