Finite Group Databasebeta

← #27367208

Subgroups — #27367208

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285845C211yesmaximal, characteristic
2#20285843C211yesmaximal, characteristic
3C2×#16811026C211yesmaximal, characteristic, complemented
4#20283426C211yesmaximal, characteristic
5#20274237C214yesmaximal, complemented
6#20274113C214yesmaximal, complemented
7#20273461C211yesmaximal, characteristic, complemented
8#20273397C211yesmaximal, characteristic, complemented
9C53×#1548C211yesmaximal, characteristic, complemented
10C2×#13133225—21no—
11#16811173C2211yescharacteristic
12C2×#13133217C2211yescharacteristic
13#16811026—21no—
14#16811026—21no—
15#16811026C2212yes—
16#16811026C2212yes—
17#16811021—21no—
18#16811019—21no—
19#16810971—21no—
20#16810970—21no—
21#16810942—22no—
22#16810941—22no—
23#16810914C2214yes—
24#16810910C2214yes—
25#16810700—21no—
26#16810700C2212yes—
27#16810688—21no—
28#16810688C2212yes—
29#16810676C2211yescharacteristic
30C2×#13133150—21no—
31C2×#13133150—21no—
32#16810591—21no—
33C2×C4×#88322C2211yescharacteristic
34#16810375C2212yes—
35#16810373C2212yes—
36#16810371C2214yes—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#423C2211yescharacteristic
40C53×#333C2214yes—
41C2×#11719945—21no—
42C22×Dic106—21no—
43#13133225—44no—
44C2×#11719912—22no—
45C22×#88322D411yescharacteristic
46C22×#88322D411yescharacteristic
47C2×C8×D53—22no—
48#13133217—21no—
49#13133217—21no—
50#13133217—21no—
51#13133217D411yescharacteristic
52#13133217C2312yes—
53#13133214D411yescharacteristic
54#13133205—21no—
55#13133191—21no—
56#13133191D412yes—
57C4×D212—21no—
58C4×D212C2312yes—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61#13133150—22no—
62#13133150—22no—
63#13133150—22no—
64#13133150—22no—
65#13133149—24no—
66#13133148—24no—
67C4⋊#88322C2312yes—
68#13133139—24no—
69#13133137—24no—
70C22⋊#88322—21no—
71#13133129—22no—
72#13133129—22no—
73C4×#88322—21no—
74C4×#88322D411yescharacteristic
75C4×#88322D411yescharacteristic
76C4×#88322C2311yescharacteristic
77C4×#88322C2311yescharacteristic
78C22×Q8×C53—21no—
79C2×M16×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C42×C53C2311yescharacteristic
87C4⋊C8×C53—21no—
88C4⋊C8×C53C2312yes—
89C4⋊C8×C53C2312yes—
90C22⋊C8×C53—21no—
91C4×C8×C53—21no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—82no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×C22⋊D53—22no—
100C22×Dic53—41no—
101#11719945—42no—
102#11719945—42no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212C2×D412yes—
106C2×C4×D53—22no—
107C2×Dic106—21no—
108C2×Dic106—22no—
109C2×Dic106—41no—
110C2×Dic106—82no—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122#11719921—22no—
123#11719921—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53C2×D412yes—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322SD1614yes—
133C2×#88322C2×D411yescharacteristic
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137#11719912—44no—
138C8×D53—44no—
139C23×C53#12111yescharacteristic
140C22×D53—22no—
141Q8×C53—82no—
142Q8×C53—82no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—42no—
146C2×C4×C53—82no—
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53C2×SD1612yes—
151C2×C4×C53#13712yes—
152C2×C4×C53C22×D411yescharacteristic
153C2×C4×C53C22×D411yescharacteristic
154C2×C4×C53C2×#4111yescharacteristic
155C22⋊D53—44no—
156C2×Dic53—22no—
157C2×Dic53—42no—
158C2×Dic53—82no—
159D212—24no—
160D212—42no—
161D212—42no—
162C4×D53—42no—
163C4×D53—42no—
164Dic106—42no—
165Dic106—82no—
166Dic106—82no—
167C8×C53—42no—
168C8×C53—42no—
169#88322—21no—
170#88322—21no—
171#88322—42no—
172#88322C2×SD1612yes—
173#88322C2×SD1612yes—
174#19193—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53#46112yes—
178C22×C53C2×#12111yescharacteristic
179C2×D53—42no—
180C2×D53—42no—
181C2×D53—42no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53#46112yes—
186C4×C53#47212yes—
187C4×C53C22×SD1611yescharacteristic
188C4×C53C2×#13711yescharacteristic
189C4×C53#57812yes—
190Dic53—42no—
191Dic53—82no—
192#3227—531no—
193#3227—531no—
194C2×#500—531no—
195#2931—531no—
196#1695—534no—
197#1675—534no—
198#1564—531no—
199#1548—531no—
200#1548D5311yescharacteristic, complemented
201C2×C53—22no—
202C2×C53C2×#46111yescharacteristic
203C2×C53#305911yescharacteristic
204C2×C53C2×#47211yescharacteristic
205D53—82no—
206C2×#131—1061no—
207#532—531no—
208C2×C4⋊Q8—531no—
209C2×C4⋊Q8C2×D5311yescharacteristic
210#500—532no—
211#500—532no—
212#500—1061no—
213#500—1061no—
214#498—1061no—
215#498—1061no—
216#486—1061no—
217#486—1061no—
218#481—1062no—
219#481—1062no—
220#476—534no—
221#476—534no—
222#436—532no—
223#436—1061no—
224C8×D4—532no—
225C8×D4—1061no—
226#431—531no—
227#431C2×D5311yescharacteristic
228C2×#107—1061no—
229C2×#107—1061no—
230#423C2×D5311yescharacteristic
231#408—1061no—
232C2×C4×C8—531no—
233#335—532no—
234#334—532no—
235#333—534no—
236#333C2×D5314yes—
237C53#1919311yescharacteristic, complemented, Sylow
238C2×#41—1061no—
239C22×Q8—21no—
240C22×Q8—1061no—
241#131—2124no—
242C2×M16—21no—
243C2×M16—1062no—
244C22×C8—531no—
245C22×C8—531no—
246C22×C8—1062no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—531no—
251C4⋊Q8—532no—
252C4⋊Q8—1061no—
253C4⋊Q8—1061no—
254C4⋊Q8—1061no—
255C4⋊Q8C22×D5312yes—
256#127—531no—
257#124—1061no—
258#121—532no—
259#121—1061no—
260C4×D4—532no—
261C4×D4—1061no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—1062no—
268#107—1062no—
269#107—1062no—
270#107—1062no—
271#106—1064no—
272#106—1064no—
273C4⋊C8—21no—
274C4⋊C8—532no—
275C4⋊C8C22×D5312yes—
276C4⋊C8C22×D5312yes—
277#102—1064no—
278#102—1064no—
279C22⋊C8—21no—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282C22⋊C8—1061no—
283C4×C8—21no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—1061no—
289#41—2122no—
290#41—2122no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—1061no—
295C2×Q8—1062no—
296C2×Q8—4242no—
297C2×Q8—2121no—
298C2×D4—532no—
299C2×D4—1062no—
300C2×D4—1061no—
301C2×D4—1061no—
302C22×C4—41no—
303C22×C4—1062no—
304C22×C4—2121no—
305C22×C4#1171994511yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4D4×D5311yescharacteristic
308M16—42no—
309M16—2124no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—534no—
313C2×C8—531no—
314C2×C8—531no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8—2124no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—532no—
325C4⋊C4—1061no—
326C4⋊C4—1062no—
327C4⋊C4—1062no—
328C4⋊C4—2121no—
329C4⋊C4—2121no—
330C22⋊C4—1062no—
331C22⋊C4—1062no—
332C42#1171994511yescharacteristic
333C42D4×D5311yescharacteristic
334C42D4×D5311yescharacteristic
335C42C23×D5311yescharacteristic
336C23—1062no—
337C23#1313319011yescharacteristic
338Q8—82no—
339Q8—82no—
340Q8—4242no—
341Q8—4242no—
342Q8—2122no—
343D4—1064no—
344D4—2124no—
345D4—2122no—
346D4—2122no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—1062no—
352C2×C4—4242no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4#1313319011yescharacteristic
357C2×C4#1313319011yescharacteristic
358C2×C4#1313319011yescharacteristic
359C2×C4SD16×D5312yes—
360C2×C4#1313324912yes—
361C2×C4C2×D4×D5311yescharacteristic
362C2×C4C2×D4×D5311yescharacteristic
363C2×C4C2×#1171994511yescharacteristic
364C8—42no—
365C8—42no—
366C8—532no—
367C8—532no—
368C8—1061no—
369C8—1061no—
370C8—2122no—
371C22—21no—
372C22—21no—
373C22—2122no—
374C22—2122no—
375C22—2122no—
376C22#1681082712yes—
377C22C2×#1313319011yescharacteristic
378C4—21no—
379C4—21no—
380C4—82no—
381C4—4242no—
382C4—2122no—
383C4#1681082712yes—
384C4#1681089212yes—
385C4C2×SD16×D5311yescharacteristic
386C4C2×#1313324911yescharacteristic
387C4#1681143912yes—
388C2—22no—
389C2—4242no—
390C2C2×#1681082711yescharacteristic
391C2#2028436911yescharacteristic
392C2C2×#1681089211yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1548C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#1548D53
hypocenterC53#19193
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C2×#16811026C2semidirect
5. #20274237C2semidirect
6. #20274113C2semidirect
7. #20273461C2semidirect
8. #20273397C2semidirect
9. C53×#1548C2semidirect
200. #1548D53semidirect
237. C53#19193semidirect

Maximal subgroups

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

subgroupindex
#202858452
#202858432
C2×#168110262
#202834262
#202742372
#202741132
#202734612
#202733972
C53×#15482
#1919353