Finite Group Databasebeta

← #27367246

Subgroups — #27367246

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285640C211yesmaximal, characteristic, complemented
2#20285633C211yesmaximal, characteristic
3#20283428C211yesmaximal, characteristic, complemented
4#20274273C214yesmaximal, complemented
5#20274149C214yesmaximal, complemented
6#20273463C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273383C211yesmaximal, characteristic, complemented
9C2×C53×#486C211yesmaximal, characteristic, complemented
10C2×#13133223—21no—
11#16811173C2211yescharacteristic
12#16810999—21no—
13#16810997—21no—
14#16810973—21no—
15#16810969—21no—
16#16810825C2214yescomplemented
17#16810815C2214yes—
18#16810762—22no—
19#16810757—22no—
20#16810702—21no—
21#16810702C2212yescomplemented
22C8×D212—21no—
23C8×D212C2212yescomplemented
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810589—21no—
28#16810376C2212yescomplemented
29#16810374C2212yescomplemented
30#16810369C2214yescomplemented
31#16810367C2214yes—
32C2×C4⋊Q8×C53C2211yescharacteristic
33C2×#127×C53C2211yescharacteristic
34C53×#486—21no—
35C53×#486—21no—
36C53×#486C2212yes—
37C53×#486C2212yes—
38C2×#102×C53—21no—
39C2×#102×C53—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C2×#11719945—21no—
42C2×#11719924—24no—
43C2×#11719922—24no—
44#13133223—44no—
45C2×#11719917—21no—
46C2×#11719912—22no—
47C2×C8×D53—22no—
48#13133217D411yescharacteristic
49#13133214D411yescharacteristic
50#13133205—21no—
51#13133191—21no—
52#13133191D412yes—
53C4×D212—21no—
54C4×D212C2312yes—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58C4⋊#88322C2312yes—
59#13133141—24no—
60#13133138—24no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63#13133129—22no—
64#13133129—22no—
65C4×#88322—21no—
66C4×#88322—21no—
67C22×Q8×C53—21no—
68C22×D4×C53—21no—
69C22×D4×C53—21no—
70C22×D4×C53—21no—
71C22×C8×C53D411yescharacteristic
72C22×C8×C53D411yescharacteristic
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53C2312yes—
77#127×C53—21no—
78#127×C53—21no—
79#127×C53—21no—
80#127×C53C2312yes—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C42×C53C2311yescharacteristic
84#102×C53—22no—
85#102×C53—22no—
86#102×C53—22no—
87#102×C53—22no—
88C4×C8×C53D411yescharacteristic
89C4×C8×C53D411yescharacteristic
90C4×C8×C53C2311yescharacteristic
91C4×C8×C53C2311yescharacteristic
92C24×C53—41no—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—82no—
96C2×D4×C53—22no—
97C2×D4×C53—22no—
98C2×D4×C53—22no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C2×D4×C53—82no—
105C2×D4×C53—82no—
106C22×C4×C53—41no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×C22⋊D53—22no—
111#11719945—42no—
112#11719945—42no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212C2×D412yes—
116C2×C4×D53—22no—
117C2×Dic106—21no—
118C2×C8×C53—22no—
119C2×C8×C53—22no—
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132#11719924—84no—
133#11719922—84no—
134#11719921—22no—
135#11719921—22no—
136C4⋊Dic53—21no—
137C4⋊Dic53C2×D412yes—
138#11719917—42no—
139C2×#88322—22no—
140C2×#88322—22no—
141C2×#88322—22no—
142#11719912—44no—
143C8×D53—44no—
144C23×C53—42no—
145C23×C53—82no—
146C23×C53—82no—
147C23×C53—82no—
148C23×C53#12111yescharacteristic
149C22×D53—22no—
150Q8×C53—44no—
151Q8×C53—82no—
152D4×C53—42no—
153D4×C53—42no—
154D4×C53—44no—
155D4×C53—82no—
156D4×C53—82no—
157D4×C53—82no—
158D4×C53—82no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—42no—
162C2×C4×C53—82no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#13512yes—
167C2×C4×C53AGL(1,ℤ/8)12yes—
168C2×C4×C53C22×D411yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×C4×C53C2×#4111yescharacteristic
171C22⋊D53—44no—
172C2×Dic53—22no—
173D212—24no—
174D212—42no—
175D212—42no—
176C4×D53—42no—
177C4×D53—42no—
178Dic106—42no—
179C8×C53—21no—
180C8×C53—21no—
181C8×C53—21no—
182C8×C53—21no—
183#88322—42no—
184#88322—42no—
185#88322—42no—
186#19198—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—82no—
190C22×C53—82no—
191C22×C53—82no—
192C22×C53—164no—
193C22×C53#46312yes—
194C22×C53C2×#12111yescharacteristic
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—42no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—82no—
201C4×C53#46312yes—
202C4×C53#47112yes—
203C4×C53C2×#13511yescharacteristic
204C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C53#57612yes—
206Dic53—42no—
207#3206—531no—
208#3205—531no—
209C2×#486D5311yescharacteristic, complemented
210#2931—531no—
211#1701—534no—
212#1681—534no—
213#1564—531no—
214#1549—531no—
215#1545—531no—
216C2×C53—22no—
217C2×C53—162no—
218C2×C53C2×#46311yescharacteristic
219C2×C53#305811yescharacteristic
220C2×C53C2×#47111yescharacteristic
221D53—82no—
222C2×#131—1061no—
223#532—531no—
224C2×C4⋊Q8C2×D5311yescharacteristic
225C2×#127C2×D5311yescharacteristic
226#493—1061no—
227#492—1061no—
228#487—1061no—
229#486—21no—
230#486—21no—
231#486—1061no—
232#486C2×D5312yes—
233#486C2×D5312yes—
234#461—534no—
235#459—534no—
236#448—1062no—
237#447—1062no—
238#436—532no—
239#436—1061no—
240C8×D4—532no—
241C8×D4—1061no—
242#431—531no—
243#431—531no—
244#423—531no—
245C2×#102—21no—
246C2×#102—21no—
247#408—1061no—
248C2×C4×C8C2×D5311yescharacteristic
249#335—532no—
250#334—532no—
251#332—534no—
252#331—534no—
253C53#1919811yescharacteristic, complemented, Sylow
254C2×#41—1061no—
255C22×Q8—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C2×SD16—1064no—
260C2×D8—1064no—
261#131—2124no—
262C2×M16—1061no—
263C2×M16—1062no—
264C22×C8—1062no—
265C22×C8C22⋊D5311yescharacteristic
266C22×C8C22⋊D5311yescharacteristic
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—531no—
271C4⋊Q8C22×D5312yes—
272#127—21no—
273#127—21no—
274#127—21no—
275#127—531no—
276#127C22×D5312yes—
277#124—1061no—
278#121—532no—
279#121—1061no—
280C4×D4—532no—
281C4×D4—1061no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C42C22×D5311yescharacteristic
285C4⋊C8—532no—
286C4⋊C8—532no—
287C4⋊C8—532no—
288C4⋊C8—1061no—
289#103—1064no—
290#102—22no—
291#102—22no—
292#102—22no—
293#102—22no—
294#102—1064no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297C22⋊C8—1062no—
298C22⋊C8—1062no—
299C4×C8—1061no—
300C4×C8—1061no—
301C4×C8C22⋊D5311yescharacteristic
302C4×C8C22⋊D5311yescharacteristic
303C4×C8C22×D5311yescharacteristic
304C4×C8C22×D5311yescharacteristic
305C24—41no—
306#41—2122no—
307#41—2122no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×Q8—1061no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—532no—
323C2×D4—1062no—
324C2×D4—1061no—
325C2×D4—1061no—
326C22×C4—41no—
327C22×C4—1062no—
328C22×C4D4×D5311yescharacteristic
329C22×C4#1171994711yescharacteristic
330C22×C4C2×C22⋊D5311yescharacteristic
331SD16—4244no—
332D8—4244no—
333M16—2124no—
334M16—2122no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8—1062no—
340C2×C8—2124no—
341C2×C8C2×C22⋊D5311yescharacteristic
342C2×C8C2×C22⋊D5311yescharacteristic
343C2×C8C2×C22⋊D5311yescharacteristic
344C2×C8C2×C22⋊D5311yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—532no—
350C4⋊C4—1061no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C23×D5311yescharacteristic
357C23—42no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—1062no—
362C23#1313318811yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—2122no—
366D4—42no—
367D4—42no—
368D4—44no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373D4—1064no—
374D4—2124no—
375D4—2122no—
376D4—2122no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4—1062no—
382C2×C4—2122no—
383C2×C4—2122no—
384C2×C4#1313318811yescharacteristic
385C2×C4#1313318811yescharacteristic
386C2×C4#1313318811yescharacteristic
387C2×C4#1313323712yes—
388C2×C4#1313324212yes—
389C2×C4C2×D4×D5311yescharacteristic
390C2×C4C22×C22⋊D5311yescharacteristic
391C2×C4C2×#1171994711yescharacteristic
392C8—21no—
393C8—21no—
394C8—21no—
395C8—21no—
396C8—2122no—
397C8—2122no—
398C8—2122no—
399C22—21no—
400C22—21no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—164no—
405C22—2122no—
406C22—2122no—
407C22—2122no—
408C22#1681083512yes—
409C22C2×#1313318811yescharacteristic
410C4—21no—
411C4—21no—
412C4—82no—
413C4—2122no—
414C4#1681083512yes—
415C4#1681088312yes—
416C4C2×#1313323711yescharacteristic
417C4C2×#1313324211yescharacteristic
418C4#1681142412yes—
419C2—22no—
420C2—162no—
421C2—4242no—
422C2C2×#1681083511yescharacteristic
423C2#2028435911yescharacteristic
424C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#486C2
socleC22×C53C2×#121
prime radical#20285633C2
hypercenterC2×#486D53
hypocenterC53#19198
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #20285640C2semidirect
3. #20283428C2semidirect
4. #20274273C2semidirect
5. #20274149C2semidirect
6. #20273463C2semidirect
7. #20273400C2semidirect
8. #20273383C2semidirect
9. C2×C53×#486C2semidirect
16. #16810825C22semidirect
21. #16810702C22semidirect
23. C8×D212C22semidirect
28. #16810376C22semidirect
29. #16810374C22semidirect
30. #16810369C22semidirect
209. C2×#486D53semidirect
253. C53#19198semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202856332
#202834282
#202742732
#202741492
#202734632
#202734002
#202733832
C2×C53×#4862
#1919853