Finite Group Databasebeta

← #27537547

Subgroups — #27537547

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#20283632C211yesmaximal, characteristic, complemented
3#20283598C211yesmaximal, characteristic, complemented
4#20276610C214yesmaximal, complemented
5#20276603C214yesmaximal, complemented
6C2×#16810445C211yesmaximal, characteristic, complemented
7#20276465C211yesmaximal, characteristic, complemented
8#20276319C211yesmaximal, characteristic, complemented
9C2×C53×#360C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11#16811390—21no—
12#16811104—21no—
13#16811064C412yescomplemented
14#16811063C2212yes—
15#16811059C412yescomplemented
16#16811058C2212yes—
17#16811053C414yescomplemented
18C2×#13133157C2211yescharacteristic
19#16810745C2214yes—
20#16810740—22no—
21#16810734C2212yescomplemented
22#16810729C2212yescomplemented
23#16810710—22no—
24C2×#13133153C2211yescharacteristic
25#16810640—21no—
26#16810622—21no—
27#16810578C2211yescharacteristic
28#16810445C2214yescomplemented
29#16810440C2212yescomplemented
30#16810437C2212yescomplemented
31#16810400C2214yescomplemented
32C2×#135×C53C2211yescharacteristic
33C2×M32×C53C2211yescharacteristic
34C2×#106×C53C2211yescharacteristic
35C53×#360C2214yes—
36C2×#11719945—21no—
37#13133239—21no—
38#13133239—21no—
39#13133239C2×C412yes—
40#13133239C2×C412yes—
41C2×Dic212C2311yescharacteristic
42C2×#11719913—21no—
43C2×D424C2311yescharacteristic
44#13133224—24no—
45#13133224—42no—
46C2×#11719917—21no—
47C2×#11719917—42no—
48C2×#11719917D411yescharacteristic
49#13133182—22no—
50#13133179—22no—
51C4×C22⋊D53—42no—
52#13133176D411yescharacteristic
53#13133175—41no—
54C4⋊C4×D53—41no—
55#13133163C2×C414yescomplemented
56#13133161C2×C414yescomplemented
57#13133160C2×C414yescomplemented
58#13133158C2×C414yescomplemented
59#13133157C2×C412yescomplemented
60#13133157C2312yes—
61C2×#11719908C2×C412yescomplemented
62#13133153—21no—
63#13133153C2312yes—
64#13133144—24no—
65#13133140—22no—
66#13133137—22no—
67#13133127C2312yes—
68C8×Dic53D412yes—
69C2×#41×C53—21no—
70#135×C53—21no—
71#135×C53—21no—
72#135×C53C2×C412yescomplemented
73#135×C53C2312yes—
74C2×Q16×C53C2×C411yescharacteristic, complemented
75C2×SD16×C53—21no—
76C2×D8×C53C2×C411yescharacteristic, complemented
77C22×C8×C53C2311yescharacteristic
78C2×C4⋊C4×C53—21no—
79M32×C53—21no—
80M32×C53C2312yes—
81C2×C16×C53—21no—
82#106×C53—21no—
83#106×C53C2312yes—
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—41no—
92C22×C4×C53—41no—
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—41no—
95#11719945—22no—
96#11719945—22no—
97#11719945—41no—
98#11719945—41no—
99C2×D212—21no—
100C2×C4×D53—41no—
101C2×C4×D53—42no—
102C2×Dic106—21no—
103Q16×C53—21no—
104Q16×C53C22×C412yes—
105SD16×C53—22no—
106SD16×C53—22no—
107D8×C53—21no—
108D8×C53C22×C412yes—
109C2×C8×C53C22×C411yescharacteristic
110C2×C8×C53C22×C411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C2×C8×C53C2411yescharacteristic
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C22⋊Dic53—21no—
118#11719921—42no—
119C4⋊Dic53—21no—
120#11719919—42no—
121C4×Dic53—42no—
122C4×Dic53C2×D412yes—
123#11719917—21no—
124#11719917—21no—
125#11719917—22no—
126#11719917—44no—
127#11719917—41no—
128C2×#88322—44no—
129C2×#88322—41no—
130C2×#88322C2×D412yes—
131Dic212—21no—
132Dic212C22×C412yes—
133D424—21no—
134D424C22×C412yes—
135#11719913—22no—
136#11719913—22no—
137C16×C53—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—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—161no—
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53C22×D411yescharacteristic
159C22⋊D53—42no—
160C22⋊D53—42no—
161C2×Dic53—22no—
162C2×Dic53—41no—
163D212—22no—
164D212—41no—
165C4×D53—42no—
166C4×D53—42no—
167C4×D53—162no—
168Dic106—22no—
169Dic106—41no—
170C8×C53C4×D411yescharacteristic
171C8×C53C4×D411yescharacteristic
172C8×C53C23×C411yescharacteristic
173C8×C53C2×#4111yescharacteristic
174#88322—22no—
175#88322—42no—
176#33469—531nomaximal, Sylow
177C22×C53—41no—
178C22×C53—42no—
179C22×C53—161no—
180C22×C53#44211yescharacteristic
181C22×C53#44211yescharacteristic
182C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D53—42no—
184C2×D53—81no—
185C2×D53—81no—
186C4×C53—42no—
187C4×C53—82no—
188C4×C53#44211yescharacteristic
189C4×C53#44211yescharacteristic
190C4×C53C2×C4×D411yescharacteristic
191C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
192Dic53—42no—
193Dic53—42no—
194#3414—531no—
195#2957—531no—
196#2952—531no—
197AGL(1,ℤ/16)—534no—
198#2180—534no—
199C2×#365—531no—
200#2148—531no—
201C2×#360D5311yescharacteristic, complemented
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—1061no—
213#510—532no—
214#510—532no—
215#509—532no—
216#509—532no—
217#508—534no—
218C2×M32—531no—
219C2×M32C2×D5311yescharacteristic
220#444—534no—
221#443—1062no—
222#442—532no—
223#441—532no—
224C4×SD16—1062no—
225C2×#108—531no—
226C2×#106C2×D5311yescharacteristic
227#421—1061no—
228#417—1061no—
229#405—531no—
230#365—534no—
231#362—532no—
232#361—532no—
233#360C2×D5314yes—
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×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—1064no—
253#131—2122no—
254C2×M16—531no—
255C2×M16—1061no—
256C2×M16—2122no—
257C22×C8C22×D5311yescharacteristic
258C4×Q8—1062no—
259C4×D4—1062no—
260C4×D4—2122no—
261#117—531no—
262#117—2121no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—2121no—
265Q32—534no—
266SD32—534no—
267SD32—534no—
268D16—534no—
269M32—21no—
270M32—532no—
271M32—532no—
272M32C22×D5312yes—
273C2×C16—21no—
274C2×C16—532no—
275#108—532no—
276#108—1061no—
277#106—21no—
278#106C22×D5312yes—
279#104—1064no—
280#103—1062no—
281#102—1062no—
282#97—532no—
283C4×C8—532no—
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—41no—
299C22×C4—41no—
300C22×C4—2122no—
301C22×C4—2121no—
302C22×C4D4×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—1061no—
316M16—1061no—
317M16—1062no—
318M16—2124no—
319M16—2121no—
320C2×C8—532no—
321C2×C8—2124no—
322C2×C8—2121no—
323C2×C8C2×C4×D5311yescharacteristic
324C2×C8C2×C4×D5311yescharacteristic
325C2×C8D4×D5311yescharacteristic
326C2×C8#1171994711yescharacteristic
327C2×C8#1171994711yescharacteristic
328C2×C8C23×D5311yescharacteristic
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—1061no—
332C4⋊C4—2122no—
333C22⋊C4—1061no—
334C22⋊C4—2122no—
335C42—532no—
336C42—2122no—
337C16—22no—
338C16—534no—
339C23—41no—
340C23—2121no—
341C23#1313324111yescharacteristic
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—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—161no—
359C2×C4—8482no—
360C2×C4—1062no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4#1313317711yescharacteristic
365C2×C4#1313317711yescharacteristic
366C2×C4#1313324111yescharacteristic
367C2×C4#1313324411yescharacteristic
368C2×C4#1313324411yescharacteristic
369C2×C4C2×D4×D5311yescharacteristic
370C8—1062no—
371C8—2122no—
372C8#1313317711yescharacteristic
373C8#1313317711yescharacteristic
374C8C22×C4×D5311yescharacteristic
375C8C2×#1171994711yescharacteristic
376C22—41no—
377C22—42no—
378C22—161no—
379C22—4241no—
380C22—4241no—
381C22—2122no—
382C22#1681073211yescharacteristic
383C22#1681073211yescharacteristic
384C22C2×#1313324111yescharacteristic
385C4—42no—
386C4—82no—
387C4—2122no—
388C4—2122no—
389C4#1681073211yescharacteristic
390C4#1681073211yescharacteristic
391C4C2×#1313317711yescharacteristic
392C4C2×#1313324411yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4242no—
398C2C2×#1681073211yescharacteristic
subgroupG/H
centerC2C2×#16810732
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC2×C53×#360C2
socleC2×C53C2×#442
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenterC2×#360D53
hypocenterC53#33469
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#360

Complements

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

NH ≅ G/Ntype
2. #20283632C2semidirect
3. #20283598C2semidirect
4. #20276610C2semidirect
5. #20276603C2semidirect
6. C2×#16810445C2semidirect
7. #20276465C2semidirect
8. #20276319C2semidirect
9. C2×C53×#360C2semidirect
13. #16811064C4semidirect
15. #16811059C4semidirect
17. #16811053C4semidirect
21. #16810734C22semidirect
22. #16810729C22semidirect
28. #16810445C22semidirect
29. #16810440C22semidirect
30. #16810437C22semidirect
31. #16810400C22semidirect
55. #13133163C2×C4semidirect
56. #13133161C2×C4semidirect
57. #13133160C2×C4semidirect
58. #13133158C2×C4semidirect
59. #13133157C2×C4semidirect
61. C2×#11719908C2×C4semidirect
72. #135×C53C2×C4semidirect
74. C2×Q16×C53C2×C4semidirect
76. C2×D8×C53C2×C4semidirect
201. C2×#360D53semidirect
235. C53#33469semidirect
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
#202836322
#202835982
#202766102
#202766032
C2×#168104452
#202764652
#202763192
C2×C53×#3602
#3346953