Finite Group Databasebeta

← #27537329

Subgroups — #27537329

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811041C211yesmaximal, characteristic, complemented
2C2×#16811040C211yesmaximal, characteristic, complemented
3#20283616C211yesmaximal, characteristic
4#20276554C214yesmaximal, complemented
5#20276549C214yesmaximal, complemented
6#20276480C211yesmaximal, characteristic, complemented
7#20276342C211yesmaximal, characteristic, complemented
8#20276314C211yesmaximal, characteristic, complemented
9C53×#2117C211yesmaximal, characteristic, complemented
10#16811421C411yescharacteristic
11C2×#13133223C2211yescharacteristic
12#16811041—21no—
13#16811041—21no—
14#16811041C2212yescomplemented
15#16811041C2212yescomplemented
16#16811040—21no—
17#16811040—21no—
18#16811040C2212yescomplemented
19#16811040C2212yescomplemented
20C2×#13133157—21no—
21C2×#13133157C2211yescharacteristic
22C2×#13133156—21no—
23C22×#11719908—21no—
24C22×#11719908C2211yescharacteristic
25C2×C16×D53—21no—
26#16810743C414yes—
27#16810738C2214yes—
28#16810671C411yescharacteristic
29#16810640C411yescharacteristic
30#16810639C411yescharacteristic
31#16810578C2211yescharacteristic
32#16810443C2214yescomplemented
33#16810406C2214yescomplemented
34#16810398C2212yescomplemented
35C16×Dic53C2212yescomplemented
36C2×#131×C53C2211yescharacteristic
37C2×M32×C53C2211yescharacteristic
38C2×M32×C53C2211yescharacteristic
39C53×#350C2214yescomplemented
40C2×#11719945C2×C411yescharacteristic
41#13133248C812yes—
42#13133247C812yes—
43#13133243C812yes—
44#13133242C812yes—
45#13133237C2×C414yes—
46#13133223—21no—
47#13133223—21no—
48#13133223C2×C412yes—
49#13133223C2312yes—
50C2×#11719917C2×C411yescharacteristic
51C2×#11719917C2×C411yescharacteristic
52C2×#11719912—21no—
53C2×C8×D53—21no—
54#13133176C2×C411yescharacteristic
55#13133157—21no—
56#13133157—22no—
57#13133157—22no—
58#13133157C2312yes—
59#13133156—22no—
60#13133156—22no—
61C2×#11719908—21no—
62C2×#11719908—22no—
63C2×#11719908—22no—
64C2×#11719908D411yescharacteristic
65C2×#11719908D411yescharacteristic
66C2×#11719908D411yescharacteristic
67C2×#11719908D411yescharacteristic
68C2×#11719908C2311yescharacteristic
69C2×#11719908C2311yescharacteristic
70C16×D53—22no—
71C16×D53—22no—
72#13133152C2×C414yes—
73#13133144C2×C414yes—
74#13133143C2×C414yes—
75#13133127C2×C412yes—
76C8×Dic53C2312yes—
77C2×#41×C53C2×C411yescharacteristic
78#131×C53—21no—
79#131×C53—21no—
80#131×C53C2×C412yes—
81#131×C53C2312yes—
82C2×M16×C53—21no—
83C2×M16×C53C2×C411yescharacteristic
84C22×C8×C53—21no—
85C22×C8×C53C2311yescharacteristic
86M32×C53—21no—
87M32×C53—21no—
88M32×C53—21no—
89M32×C53C2312yes—
90C2×C16×C53—21no—
91C2×C16×C53C2312yes—
92#41×C53—21no—
93#41×C53—21no—
94#41×C53C2×C812yes—
95#41×C53C22×C412yes—
96C2×Q8×C53C2×C811yescharacteristic
97C2×D4×C53—21no—
98C2×D4×C53C2×C811yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53C22×C411yescharacteristic
101C2×C22⋊D53—21no—
102#11719945—21no—
103#11719945—21no—
104#11719945C2×C812yes—
105#11719945C22×C412yes—
106C2×D212C2×C811yescharacteristic
107C2×C4×D53—21no—
108C2×Dic106C2×C811yescharacteristic
109M16×C53—21no—
110M16×C53—22no—
111M16×C53—22no—
112M16×C53C22×C412yes—
113C2×C8×C53—21no—
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C2×C8×C53C22×C411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C2×C8×C53#4111yescharacteristic
121C2×C8×C53C2411yescharacteristic
122C22⋊Dic53—21no—
123#11719925C2×C814yes—
124#11719924C2×C814yes—
125#11719923C2×C814yes—
126#11719922C2×C814yes—
127C4⋊Dic53—21no—
128C4×Dic53C22×C412yes—
129#11719917—21no—
130#11719917C2×C812yes—
131#11719917C2×C812yes—
132#11719917C22×C412yes—
133C2×#88322—21no—
134C2×#88322C22×C412yes—
135#11719912—22no—
136#11719912—22no—
137C8×D53—22no—
138C8×D53—22no—
139C16×C53—22no—
140C16×C53—22no—
141#11719908—22no—
142#11719908C2×D411yescharacteristic
143#11719908C2×D411yescharacteristic
144#11719908C2×D411yescharacteristic
145#11719908C2×D411yescharacteristic
146C23×C53—21no—
147C23×C53#13111yescharacteristic
148C22×D53—21no—
149Q8×C53—21no—
150Q8×C53C22×C812yes—
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—21no—
154D4×C53C22×C812yes—
155C2×C4×C53—21no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53C22×C811yescharacteristic
161C2×C4×C53C22×C811yescharacteristic
162C2×C4×C53#13111yescharacteristic
163C2×C4×C53C23×C411yescharacteristic
164C22⋊D53—22no—
165C22⋊D53—22no—
166C2×Dic53—21no—
167C2×Dic53—22no—
168D212—21no—
169D212C22×C812yes—
170C4×D53—22no—
171C4×D53—22no—
172Dic106—21no—
173Dic106C22×C812yes—
174C8×C53—22no—
175C8×C53C4×D411yescharacteristic
176C8×C53C4×D411yescharacteristic
177C8×C53C22×D411yescharacteristic
178C8×C53C2×#4111yescharacteristic
179#88322—22no—
180#88322C22×C814yes—
181#33441—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—41no—
184C22×C53—42no—
185C22×C53C8×D411yescharacteristic
186C22×C53C8×D411yescharacteristic
187C22×C53C2×#13111yescharacteristic
188C2×D53—22no—
189C2×D53—42no—
190C4×C53—22no—
191C4×C53C8×D411yescharacteristic
192C4×C53C8×D411yescharacteristic
193C4×C53C2×C4×D411yescharacteristic
194C4×C53C23×C811yescharacteristic
195Dic53—22no—
196Dic53—42no—
197C2×#504—531no—
198C2×#504—531no—
199#2955—531no—
200#2171—534no—
201#2170—534no—
202#2153—531no—
203#2117—531no—
204#2117D5311yescharacteristic, complemented
205#2108—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—21no—
209C2×C53—42no—
210C2×C53C2×C8×D411yescharacteristic
211D53—42no—
212#575—531no—
213C2×#131—531no—
214C2×#131C2×D5311yescharacteristic
215#504—532no—
216#504—532no—
217#504—532no—
218#504—532no—
219#504—1061no—
220#504—1061no—
221#504—1061no—
222#504—1061no—
223C2×M32—531no—
224C2×M32—1061no—
225C2×M32—1061no—
226C2×M32C2×D5311yescharacteristic
227C2×M32C2×D5311yescharacteristic
228C22×C16—531no—
229C22×C16—1061no—
230C22×C16—1061no—
231#444—534no—
232#443—534no—
233#430—531no—
234#421—531no—
235#421—531no—
236#405—531no—
237#364—534no—
238#350—534no—
239#350C2×D5314yescomplemented
240#346—532no—
241C4×C16—532no—
242C53#3344111yescharacteristic, complemented, Sylow
243C2×#41—531no—
244C2×#41C4×D5311yescharacteristic
245#137—532no—
246#137—532no—
247AGL(1,ℤ/8)—532no—
248AGL(1,ℤ/8)—532no—
249#135—534no—
250#131—21no—
251#131—21no—
252#131—532no—
253#131—532no—
254#131—1061no—
255#131—1061no—
256#131C4×D5312yes—
257#131C22×D5312yes—
258C2×M16—21no—
259C2×M16—531no—
260C2×M16—531no—
261C2×M16—1061no—
262C2×M16C4×D5311yescharacteristic
263C22×C8—21no—
264C22×C8—1061no—
265C22×C8C22×D5311yescharacteristic
266#117—531no—
267M32—21no—
268M32—21no—
269M32—21no—
270M32—532no—
271M32—1062no—
272M32—1062no—
273M32—1062no—
274M32—1062no—
275M32—1061no—
276M32C22×D5312yes—
277C2×C16—21no—
278C2×C16—531no—
279C2×C16—531no—
280C2×C16—531no—
281C2×C16—531no—
282C2×C16—531no—
283C2×C16—531no—
284C2×C16—1062no—
285C2×C16—1062no—
286C2×C16—1062no—
287C2×C16—1062no—
288C2×C16—1061no—
289C2×C16C22×D5312yes—
290#108—534no—
291#104—534no—
292#104—534no—
293#97—532no—
294C4×C8—532no—
295#41—21no—
296#41—21no—
297#41—532no—
298#41—532no—
299#41—1061no—
300#41—1061no—
301#41C8×D5312yes—
302#41C2×C4×D5312yes—
303C2×Q8—531no—
304C2×Q8C8×D5311yescharacteristic
305C2×D4—21no—
306C2×D4—531no—
307C2×D4—1061no—
308C2×D4C8×D5311yescharacteristic
309C22×C4—21no—
310C22×C4—1061no—
311C22×C4C2×C4×D5311yescharacteristic
312Q16—534no—
313SD16—534no—
314SD16—534no—
315D8—534no—
316M16—21no—
317M16—22no—
318M16—22no—
319M16—532no—
320M16—532no—
321M16—532no—
322M16—1061no—
323M16—1062no—
324M16—1062no—
325M16C2×C4×D5312yes—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—532no—
330C2×C8—1061no—
331C2×C8—1062no—
332C2×C8—1062no—
333C2×C8C2×C4×D5311yescharacteristic
334C2×C8D4×D5311yescharacteristic
335C2×C8D4×D5311yescharacteristic
336C2×C8#1171994711yescharacteristic
337C2×C8#1171994711yescharacteristic
338C2×C8C23×D5311yescharacteristic
339C4⋊C4—1061no—
340C22⋊C4—1061no—
341C42—532no—
342C16—22no—
343C16—22no—
344C16—531no—
345C16—531no—
346C16—531no—
347C16—531no—
348C16—1062no—
349C23—21no—
350C23—1061no—
351C23#1313322511yescharacteristic
352Q8—21no—
353Q8—532no—
354Q8—1061no—
355Q8C2×C8×D5312yes—
356D4—22no—
357D4—22no—
358D4—21no—
359D4—532no—
360D4—1062no—
361D4—1062no—
362D4—1061no—
363D4C2×C8×D5312yes—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—1061no—
368C2×C4—1062no—
369C2×C4—1062no—
370C2×C4—1062no—
371C2×C4#1313317711yescharacteristic
372C2×C4#1313317711yescharacteristic
373C2×C4C2×C8×D5311yescharacteristic
374C2×C4C2×C8×D5311yescharacteristic
375C2×C4#1313322511yescharacteristic
376C2×C4C22×C4×D5311yescharacteristic
377C8—22no—
378C8—534no—
379C8—1062no—
380C8#1313317711yescharacteristic
381C8#1313317711yescharacteristic
382C8C2×D4×D5311yescharacteristic
383C8C2×#1171994711yescharacteristic
384C22—22no—
385C22—41no—
386C22—42no—
387C22—1062no—
388C22—2122no—
389C22#1681068611yescharacteristic
390C22#1681068611yescharacteristic
391C22C2×#1313322511yescharacteristic
392C4—22no—
393C4—1062no—
394C4—2122no—
395C4#1681068611yescharacteristic
396C4#1681068611yescharacteristic
397C4C2×#1313317711yescharacteristic
398C4C22×C8×D5311yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—2122no—
404C2C2×#1681068611yescharacteristic
subgroupG/H
centerC8C2×D4×D53
commutatorC4×C53C23×C8
FrattiniC2×C8C23×D53
FittingC53×#2117C2
socleC2×C53C2×C8×D4
prime radical#16811421C4
cosocleC2×C8×C53C24
hypercenter#2117D53
hypocenterC53#33441
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. C2×#16811041C2semidirect
2. C2×#16811040C2semidirect
4. #20276554C2semidirect
5. #20276549C2semidirect
6. #20276480C2semidirect
7. #20276342C2semidirect
8. #20276314C2semidirect
9. C53×#2117C2semidirect
14. #16811041C22semidirect
15. #16811041C22semidirect
18. #16811040C22semidirect
19. #16811040C22semidirect
32. #16810443C22semidirect
33. #16810406C22semidirect
34. #16810398C22semidirect
35. C16×Dic53C22semidirect
39. C53×#350C22semidirect
204. #2117D53semidirect
239. #350C2×D53semidirect
242. C53#33441semidirect

Maximal subgroups

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

subgroupindex
C2×#168110412
C2×#168110402
#202836162
#202765542
#202765492
#202764802
#202763422
#202763142
C53×#21172
#3344153