Finite Group Databasebeta

← #27280989

Subgroups — #27280989

Order 13568.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277076C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic, complemented
3#20273227C211yesmaximal, characteristic, complemented
4#20273158C211yesmaximal, characteristic
5#20272607C211yesmaximal, characteristic, complemented
6#20272572C211yesmaximal, characteristic, complemented
7C53×#3110C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9#16810680C2211yescharacteristic
10C4×#11719917C2211yescharacteristic
11C4⋊C4×Dic53—21no—
12C4×C4⋊Dic53C2211yescharacteristic
13C22⋊C4×Dic53—22no—
14C4×C22⋊Dic53—21no—
15#16810469C2211yescharacteristic
16C2×#13133124—21no—
17#16810382—22no—
18#16810360—21no—
19#16810360—21no—
20D4⋊#88322—22no—
21C53×#533C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#484—21no—
24C53×#474C412yescomplemented
25C53×#461C412yescomplemented
26C53×#450—21no—
27C53×#440—22no—
28C53×#432C2211yescharacteristic
29C23×Dic53—22no—
30C2×#11719917—21no—
31D4×Dic53—28no—
32C2×C4⋊Dic53—21no—
33C2×C22⋊Dic53—22no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53D412yes—
37C4⋊#88322—21no—
38C22⋊#88322—21no—
39#13133128—21no—
40#13133128—21no—
41C4×#88322—21no—
42#13133124—22no—
43#13133124—21no—
44#13133124—21no—
45#13133124D411yescharacteristic
46#13133124D411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53—21no—
49C2×SD16×C53—22no—
50C4⋊Q8×C53C2×C411yescharacteristic, complemented
51C4⋊Q8×C53C2×C411yescharacteristic, complemented
52#125×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C4×D4×C53C2×C412yes—
60C4×D4×C53D412yes—
61#117×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65#107×C53—22no—
66C4⋊C8×C53C2×C412yescomplemented
67#103×C53—22no—
68#102×C53—22no—
69C22⋊C8×C53—21no—
70#97×C53—21no—
71C24×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×D4×C53—24no—
75C2×D4×C53—21no—
76C2×D4×C53C22⋊C412yes—
77C22×C4×C53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—82no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—22no—
86C22×Dic53—21no—
87C22×Dic53—28no—
88C22×Dic53—82no—
89C22×Dic53—82no—
90SD16×C53—44no—
91C2×C8×C53—22no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53C22⋊C412yes—
100C22⋊C4×C53—22no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—41no—
103C42×C53—41no—
104C42×C53C22×C411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53#4111yescharacteristic
107C42×C53#4111yescharacteristic
108C22⋊Dic53—28no—
109C4⋊Dic53—24no—
110C4×Dic53—24no—
111C4×Dic53—42no—
112C4×Dic53—82no—
113#11719917—22no—
114#11719917—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C23×C53—24no—
118C23×C53—22no—
119C23×C53—42no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53#12011yescharacteristic
123Q8×C53—42no—
124Q8×C53—81no—
125D4×C53—24no—
126D4×C53—22no—
127D4×C53#10414yes—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—82no—
137C2×C4×C53—82no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×Dic53—22no—
145C2×Dic53—24no—
146C2×Dic53—22no—
147C2×Dic53—41no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—41no—
153C2×Dic53—88no—
154C8×C53—42no—
155#88322—42no—
156#88322—81no—
157#9323—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—24no—
161C22×C53—44no—
162C22×C53—44no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53#38611yescharacteristic
166C22×C53#45511yescharacteristic
167C22×C53#45611yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53C2×#10412yes—
177Dic53—42no—
178Dic53—44no—
179Dic53—42no—
180#3110D5311yescharacteristic
181#2275—531no—
182#1986—531no—
183#1506—531no—
184#1490—531no—
185#1300—531no—
186#1285—531no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—44no—
190C2×C53#188211yescharacteristic
191C2×C53#189611yescharacteristic
192C2×C53#190411yescharacteristic
193#533C2×D5311yescharacteristic
194C2×C4×D4—1061no—
195C2×C4×D4C2×D5311yescharacteristic
196#484—21no—
197#474Dic5312yescomplemented
198#461Dic5312yescomplemented
199#450—21no—
200#440—22no—
201#432—531no—
202#432C2×D5311yescharacteristic
203C4×M16—531no—
204C4×C4⋊C4—531no—
205C4×C4⋊C4—1061no—
206C4×C22⋊C4—1061no—
207C4×C22⋊C4—1062no—
208#376—531no—
209C2×#95—1061no—
210#339—1062no—
211#328—1061no—
212#328—1061no—
213D4⋊C8—1062no—
214C53#932311yescharacteristic, complemented, Sylow
215C22×D4C22⋊D5311yescharacteristic
216C23×C4—21no—
217C23×C4—1062no—
218C2×SD16—22no—
219C2×M16—1061no—
220C4⋊Q8C2×Dic5311yescharacteristic, complemented
221C4⋊Q8C2×Dic5311yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—1068no—
230C4×D4C2×Dic5312yes—
231C4×D4C22⋊D5312yes—
232#117—21no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4C22⋊D5311yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—1062no—
237C2×C42—532no—
238C2×C42—1061no—
239C2×C42—1062no—
240C2×C42C22×D5311yescharacteristic
241#107—22no—
242C4⋊C8—1061no—
243C4⋊C8C2×Dic5312yescomplemented
244#103—22no—
245#102—22no—
246C22⋊C8—21no—
247C22⋊C8—1061no—
248#97—21no—
249#97—1061no—
250#97—1061no—
251C4×C8—1061no—
252#95—531no—
253#95—531no—
254#95—1062no—
255#95—1061no—
256#95—1061no—
257C24—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—24no—
261C2×D4—21no—
262C2×D4C22⋊Dic5312yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—82no—
267C22×C4—1068no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1062no—
271C22×C4—4242no—
272C22×C4—4242no—
273C22×C4D4×D5311yescharacteristic
274C22×C4D4×D5311yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276SD16—44no—
277M16—1062no—
278M16—2121no—
279C2×C8—22no—
280C2×C8—2121no—
281C2×C8—2121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—1064no—
290C4⋊C4C22⋊Dic5312yes—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—1068no—
295C42—41no—
296C42—1064no—
297C42—4242no—
298C42—2122no—
299C42#1171994711yescharacteristic
300C42#1171994711yescharacteristic
301C42C22×Dic5311yescharacteristic
302C42C2×C22⋊D5311yescharacteristic
303C23—24no—
304C23—22no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23#1313318511yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—24no—
312D4—22no—
313D4#1313314414yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1064no—
327C2×C4—4248no—
328C2×C4—2121no—
329C2×C4—2122no—
330C2×C4—2121no—
331C2×C4—2122no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4C2×C22⋊Dic5311yescharacteristic
335C2×C4D4×Dic5311yescharacteristic
336C2×C4D4×Dic5311yescharacteristic
337C2×C4#1313318911yescharacteristic
338C2×C4#1313318911yescharacteristic
339C2×C4#1313319811yescharacteristic
340C8—42no—
341C8—4241no—
342C8—2122no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22#1681050311yescharacteristic
351C22#1681079911yescharacteristic
352C22#1681080411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—2122no—
362C4—2124no—
363C4—2122no—
364C4C2×#1313314412yes—
365C2—21no—
366C2—21no—
367C2—44no—
368C2#2027508611yescharacteristic
369C2#2027515911yescharacteristic
370C2#2027520411yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3110C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#3110D53
hypocenterC53#9323
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #20275660C2semidirect
3. #20273227C2semidirect
5. #20272607C2semidirect
6. #20272572C2semidirect
24. C53×#474C4semidirect
25. C53×#461C4semidirect
50. C4⋊Q8×C53C2×C4semidirect
51. C4⋊Q8×C53C2×C4semidirect
66. C4⋊C8×C53C2×C4semidirect
197. #474Dic53semidirect
198. #461Dic53semidirect
214. C53#9323semidirect
220. C4⋊Q8C2×Dic53semidirect
221. C4⋊Q8C2×Dic53semidirect
243. C4⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202770762
#202756602
#202732272
#202731582
#202726072
#202725722
C53×#31102
#932353