Finite Group Databasebeta

← #27281388

Subgroups — #27281388

Order 13568.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277521C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic
3#20273317C211yesmaximal, characteristic
4#20273266C211yesmaximal, characteristic
5#20272581C211yesmaximal, characteristic
6#20272570C211yesmaximal, characteristic
7C53×#3236C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C4×#11719917C2211yescharacteristic
10#16810522—21no—
11#16810509—21no—
12#16810509—21no—
13#16810509—21no—
14#16810509—21no—
15#16810491C2211yescharacteristic
16C4⋊C4×Dic53—21no—
17C4×C4⋊Dic53C2211yescharacteristic
18#16810382—21no—
19#16810382—21no—
20#16810365—22no—
21#16810360—21no—
22#16810360—21no—
23C53×#533C2211yescharacteristic
24C2×#125×C53C2211yescharacteristic
25C53×#501C412yescomplemented
26C53×#491C412yescomplemented
27C53×#484—21no—
28C53×#483—21no—
29C53×#479—22no—
30C53×#432C2211yescharacteristic
31C2×#11719917—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C4⋊#88322—21no—
40C22⋊#88322—21no—
41#13133128—21no—
42#13133128—21no—
43C4×#88322—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124D411yescharacteristic
52#13133124D411yescharacteristic
53C4⋊Q8×C53C2×C411yescharacteristic
54C4⋊Q8×C53C2×C411yescharacteristic
55#125×C53—21no—
56#125×C53—21no—
57#125×C53—21no—
58#125×C53C2×C412yes—
59#125×C53D412yes—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64#117×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C42×C53C2311yescharacteristic
70#107×C53—22no—
71#106×C53—22no—
72C4⋊C8×C53—21no—
73#103×C53—22no—
74#103×C53—22no—
75C22⋊C8×C53—21no—
76#97×C53C2×C412yes—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C2×C8×C53—22no—
89C4⋊C4×C53—22no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53C22⋊C412yes—
102C4⋊C4×C53C22⋊C412yes—
103C22⋊C4×C53—41no—
104C42×C53—41no—
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C42×C53#4111yescharacteristic
109C4⋊Dic53—42no—
110C4⋊Dic53—42no—
111C4⋊Dic53—81no—
112C4×Dic53—42no—
113C4×Dic53—41no—
114C4×Dic53—41no—
115C4×Dic53—81no—
116#11719917—22no—
117#11719917—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C23×C53#12011yescharacteristic
121Q8×C53—42no—
122Q8×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
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×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×Dic53—22no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C8×C53—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—81no—
171C4×C53—81no—
172Dic53—42no—
173Dic53—81no—
174Dic53—81no—
175Dic53—81no—
176#3236D5311yescharacteristic
177#2336—531no—
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#533C2×D5311yescharacteristic
189C2×#125C2×D5311yescharacteristic
190#501Dic5312yescomplemented
191#491Dic5312yescomplemented
192#484—21no—
193#483—21no—
194#479—22no—
195#432—531no—
196#432C2×D5311yescharacteristic
197C4×M16—531no—
198#389—1061no—
199#387—1061no—
200#387—1061no—
201#387—1061no—
202#387—1061no—
203#383—531no—
204C4×C4⋊C4—531no—
205C4×C4⋊C4—1061no—
206#339—1061no—
207#339—1061no—
208#330—1062no—
209#328—1061no—
210#328—1061no—
211C53#938011yescharacteristic, complemented, Sylow
212C2×M16—1061no—
213C4⋊Q8C2×Dic5311yescharacteristic
214C4⋊Q8C2×Dic5311yescharacteristic
215#125—21no—
216#125—21no—
217#125—21no—
218#125C2×Dic5312yes—
219#125C22⋊D5312yes—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224#117—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1061no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4C22⋊D5311yescharacteristic
231C2×C4⋊C4C22⋊D5311yescharacteristic
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—1061no—
235C2×C42—1061no—
236C2×C42C22×D5311yescharacteristic
237#107—22no—
238#106—22no—
239C4⋊C8—21no—
240C4⋊C8—1061no—
241#103—22no—
242#103—22no—
243C22⋊C8—21no—
244C22⋊C8—1061no—
245#97—1061no—
246#97—1061no—
247#97C2×Dic5312yes—
248C4×C8—1061no—
249#95—531no—
250#95—531no—
251#95—1061no—
252#95—1061no—
253#95—1061no—
254#95—1061no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258C2×Q8—21no—
259C2×Q8—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4D4×D5311yescharacteristic
267C22×C4D4×D5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269M16—1062no—
270M16—2121no—
271C2×C8—22no—
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—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—4241no—
287C4⋊C4—2122no—
288C4⋊C4—2122no—
289C4⋊C4C22⋊Dic5312yes—
290C4⋊C4C22⋊Dic5312yes—
291C22⋊C4—41no—
292C42—41no—
293C42—4241no—
294C42—2121no—
295C42—2121no—
296C42—2122no—
297C42#1171994711yescharacteristic
298C42#1171994711yescharacteristic
299C42C22×Dic5311yescharacteristic
300C42C2×C22⋊D5311yescharacteristic
301C23#1313318511yescharacteristic
302Q8—42no—
303Q8—81no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—1062no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4C2×C22⋊Dic5311yescharacteristic
329C2×C4D4×Dic5311yescharacteristic
330C2×C4D4×Dic5311yescharacteristic
331C2×C4#1313318911yescharacteristic
332C2×C4#1313318911yescharacteristic
333C2×C4#1313319811yescharacteristic
334C8—42no—
335C8—4241no—
336C8—2122no—
337C22—21no—
338C22—21no—
339C22#1681050311yescharacteristic
340C22#1681079911yescharacteristic
341C22#1681080411yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—4241no—
353C4—4241no—
354C4—4241no—
355C4—2122no—
356C2—21no—
357C2—21no—
358C2#2027508611yescharacteristic
359C2#2027521811yescharacteristic
360C2#2027525311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3236C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#3236D53
hypocenterC53#9380
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3236

Complements

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

NH ≅ G/Ntype
25. C53×#501C4semidirect
26. C53×#491C4semidirect
190. #501Dic53semidirect
191. #491Dic53semidirect
211. C53#9380semidirect

Maximal subgroups

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

subgroupindex
#202775212
#202756602
#202733172
#202732662
#202725812
#202725702
C53×#32362
#938053