Finite Group Databasebeta

← #27281386

Subgroups — #27281386

Order 13568.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285921C211yesmaximal, characteristic
2#20275658C211yesmaximal, characteristic
3#20273317C211yesmaximal, characteristic
4#20273265C211yesmaximal, characteristic
5#20272579C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#2336C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811033C412yescomplemented
10#16810993C412yescomplemented
11#16810958—21no—
12#16810953—21no—
13#16810927—22no—
14#16810680C2211yescharacteristic
15#16810680C2211yescharacteristic
16C4×#11719917C2211yescharacteristic
17#16810383—21no—
18#16810383—21no—
19#16810365—22no—
20#16810359—21no—
21#16810359—21no—
22C2×#125×C53C2211yescharacteristic
23C53×#389—21no—
24C53×#387—21no—
25C53×#387—21no—
26C53×#387—21no—
27C53×#387—21no—
28C53×#383C2211yescharacteristic
29C4×C4⋊C4×C53—21no—
30C4×C4⋊C4×C53C2211yescharacteristic
31C2×#11719917—21no—
32#13133217C2×C411yescharacteristic
33#13133217C2×C411yescharacteristic
34#13133209—21no—
35#13133196—21no—
36#13133196—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39#13133176—21no—
40#13133150—22no—
41#13133148—22no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133140—22no—
45#13133140—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133128—21no—
49#13133128—21no—
50#13133128C2×C412yes—
51C4×#88322—21no—
52#125×C53—21no—
53#125×C53—21no—
54#125×C53C2×C412yes—
55#125×C53D412yes—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C42×C53—21no—
64C2×C42×C53—21no—
65C2×C42×C53D411yescharacteristic
66C2×C42×C53D411yescharacteristic
67C2×C42×C53C2311yescharacteristic
68#95×C53—21no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53D411yescharacteristic
76#95×C53D411yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C2×Dic106—21no—
87C2×Dic106—21no—
88C4⋊C4×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—81no—
97C4⋊C4×C53C22⋊C412yes—
98C4⋊C4×C53C22⋊C412yes—
99C42×C53—42no—
100C42×C53—41no—
101C42×C53—41no—
102C42×C53—81no—
103C42×C53C22×C411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107C22⋊Dic53—41no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112#11719919—41no—
113#11719919—41no—
114C4×Dic53—41no—
115#11719917—22no—
116#11719917—41no—
117C2×#88322—22no—
118C2×#88322—41no—
119C2×#88322—41no—
120C23×C53#12011yescharacteristic
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×Dic53—41no—
150C2×Dic53—41no—
151Dic106—42no—
152Dic106—81no—
153#88322—42no—
154#88322—42no—
155#88322—81no—
156#9380—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53#38611yescharacteristic
160C22×C53#45511yescharacteristic
161C22×C53#45611yescharacteristic
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—42no—
169C4×C53—42no—
170C4×C53—42no—
171C4×C53—81no—
172C4×C53—81no—
173C4×C53—81no—
174Dic53—81no—
175Dic53—81no—
176#3236—531no—
177#2336D5311yescharacteristic
178#1986—531no—
179#1528—531no—
180#1515—531no—
181#1288—531no—
182#1285—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53#188211yescharacteristic
186C2×C53#190611yescharacteristic
187C2×C53#191211yescharacteristic
188#533—531no—
189C2×#125C2×D5311yescharacteristic
190#501—532no—
191#491—532no—
192#484—1061no—
193#483—1061no—
194#479—1062no—
195#432—531no—
196#432—531no—
197C4×M16—531no—
198#389—21no—
199#387—21no—
200#387—21no—
201#387—21no—
202#387—21no—
203#383C2×D5311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2×D5311yescharacteristic
206#339—1061no—
207#339—1061no—
208#330—1062no—
209#328—1061no—
210#328—1061no—
211C53#938011yescharacteristic, complemented, Sylow
212C2×M16—1061no—
213C4⋊Q8—531no—
214C4⋊Q8—531no—
215#125—21no—
216#125—21no—
217#125—1061no—
218#125C4×D5312yes—
219#125D21212yes—
220C22⋊Q8—1061no—
221C22⋊Q8—1061no—
222C4×Q8—1061no—
223C4×Q8—1061no—
224#117—1061no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22⋊D5311yescharacteristic
235C2×C42C22⋊D5311yescharacteristic
236C2×C42C22×D5311yescharacteristic
237#107—1062no—
238#106—1062no—
239C4⋊C8—1061no—
240C4⋊C8—1061no—
241#103—1062no—
242#103—1062no—
243C22⋊C8—1061no—
244C22⋊C8—1061no—
245#97—532no—
246#97—1061no—
247#97—1061no—
248C4×C8—1061no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C22⋊D5311yescharacteristic
257#95C22⋊D5311yescharacteristic
258C2×Q8—1061no—
259C2×Q8—1061no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4C2×C22⋊D5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269M16—1062no—
270M16—2121no—
271C2×C8—1062no—
272C2×C8—2121no—
273C2×C8—2121no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—81no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—2121no—
288C4⋊C4—2121no—
289C4⋊C4#1171992112yes—
290C4⋊C4#1171992112yes—
291C22⋊C4—2121no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—81no—
296C42—2121no—
297C42C2×C4×D5311yescharacteristic
298C42C2×D21211yescharacteristic
299C42#1171994511yescharacteristic
300C42#1171994511yescharacteristic
301C23#1313318611yescharacteristic
302Q8—4241no—
303Q8—2122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4C2×#1171992111yescharacteristic
329C2×C4C4×C22⋊D5311yescharacteristic
330C2×C4C4×C22⋊D5311yescharacteristic
331C2×C4#1313319111yescharacteristic
332C2×C4#1313319111yescharacteristic
333C2×C4#1313320011yescharacteristic
334C8—4241no—
335C8—2122no—
336C8—2122no—
337C22—21no—
338C22—21no—
339C22#1681050211yescharacteristic
340C22#1681080111yescharacteristic
341C22#1681080611yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—4241no—
355C4—4241no—
356C2—21no—
357C2—21no—
358C2#2027508411yescharacteristic
359C2#2027521611yescharacteristic
360C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2336C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#2336D53
hypocenterC53#9380
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
9. #16811033C4semidirect
10. #16810993C4semidirect
211. C53#9380semidirect

Maximal subgroups

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

subgroupindex
#202859212
#202756582
#202733172
#202732652
#202725792
#202725712
C53×#23362
#938053