Finite Group Databasebeta

← #27367255

Subgroups — #27367255

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285843C211yesmaximal, characteristic
2#20285836C211yesmaximal, characteristic
3#20283428C211yesmaximal, characteristic
4#20274287C214yesmaximal, complemented
5#20274163C214yesmaximal, complemented
6#20273463C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273387C211yesmaximal, characteristic, complemented
9C2×C53×#487C211yesmaximal, characteristic, complemented
10C2×#13133223—21no—
11#16811173C2211yescharacteristic
12#16811026—21no—
13#16811019—21no—
14#16810971—21no—
15#16810971—21no—
16#16810941—22no—
17#16810936—22no—
18#16810910C2214yes—
19#16810900C2214yes—
20#16810702—21no—
21#16810702C2212yes—
22C8×D212—21no—
23C8×D212C2212yes—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26#16810647C2211yescharacteristic
27#16810589—21no—
28#16810376C2212yes—
29#16810374C2212yes—
30#16810372C2214yes—
31#16810369C2214yes—
32C2×C4⋊Q8×C53C2211yescharacteristic
33C2×C4⋊Q8×C53C2211yescharacteristic
34C53×#487—21no—
35C53×#487—21no—
36C53×#487C2212yes—
37C53×#487C2212yes—
38C2×#103×C53—21no—
39C2×#103×C53—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C2×#11719945—21no—
42#13133223—44no—
43C2×#11719917—21no—
44C2×#11719912—22no—
45C2×C8×D53—22no—
46#13133217D411yescharacteristic
47#13133214D411yescharacteristic
48#13133205—21no—
49#13133191—21no—
50#13133191D412yes—
51C4×D212—21no—
52C4×D212C2312yes—
53#13133150—24no—
54#13133148—24no—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58C4⋊#88322C2312yes—
59#13133137—24no—
60#13133137—24no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63#13133129—22no—
64#13133129—22no—
65C4×#88322—21no—
66C4×#88322—21no—
67C22×Q8×C53—21no—
68C22×Q8×C53—21no—
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C4⋊Q8×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53—21no—
77C4⋊Q8×C53C2312yes—
78C4⋊Q8×C53C2312yes—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C42×C53C2311yescharacteristic
84#103×C53—22no—
85#103×C53—22no—
86#103×C53—22no—
87#103×C53—22no—
88C4×C8×C53D411yescharacteristic
89C4×C8×C53D411yescharacteristic
90C4×C8×C53C2311yescharacteristic
91C4×C8×C53C2311yescharacteristic
92C2×Q8×C53—22no—
93C2×Q8×C53—22no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53—42no—
97C2×Q8×C53—42no—
98C2×Q8×C53—82no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C2×C22⋊D53—22no—
105#11719945—42no—
106#11719945—42no—
107C2×D212—21no—
108C2×D212—21no—
109C2×D212C2×D412yes—
110C2×C4×D53—22no—
111C2×Dic106—21no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53C2×D411yescharacteristic
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129C42×C53C2411yescharacteristic
130#11719921—22no—
131#11719921—22no—
132C4⋊Dic53—21no—
133C4⋊Dic53C2×D412yes—
134#11719917—42no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—22no—
138#11719912—44no—
139C8×D53—44no—
140C23×C53#12111yescharacteristic
141C22×D53—22no—
142Q8×C53—42no—
143Q8×C53—42no—
144Q8×C53—82no—
145Q8×C53—82no—
146Q8×C53—82no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—82no—
152C2×C4×C53—82no—
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#13512yes—
157C2×C4×C53AGL(1,ℤ/8)12yes—
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C2×#4111yescharacteristic
161C22⋊D53—44no—
162C2×Dic53—22no—
163D212—24no—
164D212—42no—
165D212—42no—
166C4×D53—42no—
167C4×D53—42no—
168Dic106—42no—
169C8×C53—21no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—21no—
173#88322—42no—
174#88322—42no—
175#88322—42no—
176#19199—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53#46312yes—
180C22×C53C2×#12111yescharacteristic
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—42no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53—82no—
188C4×C53#46312yes—
189C4×C53#47112yes—
190C4×C53C2×#13511yescharacteristic
191C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
192C4×C53#57812yes—
193Dic53—42no—
194#3227—531no—
195#3226—531no—
196C2×#487D5311yescharacteristic, complemented
197#2931—531no—
198#1703—534no—
199#1683—534no—
200#1564—531no—
201#1548—531no—
202#1546—531no—
203C2×C53—22no—
204C2×C53C2×#46311yescharacteristic
205C2×C53#305911yescharacteristic
206C2×C53C2×#47111yescharacteristic
207D53—82no—
208C2×#131—1061no—
209#532—531no—
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×C4⋊Q8C2×D5311yescharacteristic
212#500—1061no—
213#498—1061no—
214#487—21no—
215#487—21no—
216#487C2×D5312yes—
217#487C2×D5312yes—
218#486—1061no—
219#486—1061no—
220#481—1062no—
221#480—1062no—
222#476—534no—
223#474—534no—
224#436—532no—
225#436—1061no—
226C8×D4—532no—
227C8×D4—1061no—
228#431—531no—
229#431—531no—
230#423—531no—
231C2×#103—21no—
232C2×#103—21no—
233#408—1061no—
234C2×C4×C8C2×D5311yescharacteristic
235#335—532no—
236#334—532no—
237#333—534no—
238#332—534no—
239C53#1919911yescharacteristic, complemented, Sylow
240C2×#41—1061no—
241C22×Q8—21no—
242C22×Q8—21no—
243#131—2124no—
244C2×M16—1061no—
245C2×M16—1062no—
246C22×C8—1062no—
247C22×C8C22⋊D5311yescharacteristic
248C22×C8C22⋊D5311yescharacteristic
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—531no—
256C4⋊Q8C22×D5312yes—
257C4⋊Q8C22×D5312yes—
258#127—531no—
259#124—1061no—
260#121—532no—
261#121—1061no—
262C4×D4—532no—
263C4×D4—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D5311yescharacteristic
269#107—1064no—
270#106—1064no—
271C4⋊C8—532no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—1061no—
275#103—22no—
276#103—22no—
277#103—22no—
278#103—22no—
279#102—1064no—
280#102—1064no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283C22⋊C8—1062no—
284C22⋊C8—1062no—
285C4×C8—1061no—
286C4×C8—1061no—
287C4×C8C22⋊D5311yescharacteristic
288C4×C8C22⋊D5311yescharacteristic
289C4×C8C22×D5311yescharacteristic
290C4×C8C22×D5311yescharacteristic
291#41—2122no—
292#41—2122no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—82no—
300C2×Q8—1061no—
301C2×D4—532no—
302C2×D4—1062no—
303C2×D4—1061no—
304C2×D4—1061no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—1062no—
308C22×C4D4×D5311yescharacteristic
309C22×C4#1171994711yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311M16—2124no—
312M16—2122no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—1062no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—2124no—
319C2×C8C2×C22⋊D5311yescharacteristic
320C2×C8C2×C22⋊D5311yescharacteristic
321C2×C8C2×C22⋊D5311yescharacteristic
322C2×C8C2×C22⋊D5311yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—532no—
332C4⋊C4—1061no—
333C22⋊C4—1062no—
334C22⋊C4—1062no—
335C42D4×D5311yescharacteristic
336C42#1171994711yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C42C23×D5311yescharacteristic
339C23—1062no—
340C23#1313318811yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346Q8—2122no—
347D4—1064no—
348D4—2124no—
349D4—2122no—
350D4—2122no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—1062no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313318811yescharacteristic
361C2×C4#1313318811yescharacteristic
362C2×C4#1313318811yescharacteristic
363C2×C4#1313323712yes—
364C2×C4#1313324212yes—
365C2×C4C2×D4×D5311yescharacteristic
366C2×C4C22×C22⋊D5311yescharacteristic
367C2×C4C2×#1171994711yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—2122no—
373C8—2122no—
374C8—2122no—
375C22—21no—
376C22—21no—
377C22—2122no—
378C22—2122no—
379C22—2122no—
380C22#1681083512yes—
381C22C2×#1313318811yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—2122no—
387C4#1681083512yes—
388C4#1681088312yes—
389C4C2×#1313323711yescharacteristic
390C4C2×#1313324211yescharacteristic
391C4#1681143612yes—
392C2—22no—
393C2—4242no—
394C2C2×#1681083511yescharacteristic
395C2#2028436611yescharacteristic
396C2C2×#1681088311yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#487C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#487D53
hypocenterC53#19199
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #20274287C2semidirect
5. #20274163C2semidirect
6. #20273463C2semidirect
7. #20273396C2semidirect
8. #20273387C2semidirect
9. C2×C53×#487C2semidirect
196. C2×#487D53semidirect
239. C53#19199semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202858362
#202834282
#202742872
#202741632
#202734632
#202733962
#202733872
C2×C53×#4872
#1919953