Finite Group Databasebeta

← #27280996

Subgroups — #27280996

Order 13568.

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

normal subgroups only

This group has 536 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#20277080C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic
3#20273227C211yesmaximal, characteristic
4#20273162C211yesmaximal, characteristic
5#20272607C211yesmaximal, characteristic
6#20272572C211yesmaximal, characteristic
7C53×#3113C211yesmaximal, characteristic
8C2×Q8×Dic53—21no—
9#16810680C2211yescharacteristic
10C4×#11719917C2211yescharacteristic
11C4⋊C4×Dic53—22no—
12C4⋊C4×Dic53—21no—
13C4×C4⋊Dic53—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810469—21no—
16#16810469C2211yescharacteristic
17#16810382—22no—
18#16810360—21no—
19#16810360—21no—
20Q8⋊#88322—22no—
21C53×#533C2211yescharacteristic
22C2×C4×Q8×C53C2211yescharacteristic
23C53×#483—21no—
24C53×#475C412yescomplemented
25C53×#462C412yescomplemented
26C53×#451—21no—
27C53×#441—22no—
28C53×#432C2211yescharacteristic
29C2×#11719917—21no—
30Q8×Dic53—28no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53—21no—
33C2×C4×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×Q8×C53D411yescharacteristic
48C2×Q16×C53—22no—
49C4⋊Q8×C53C2×C411yescharacteristic
50C4⋊Q8×C53C2×C411yescharacteristic
51#125×C53—21no—
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C4×Q8×C53C2×C412yes—
59C4×Q8×C53D412yes—
60#117×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C42×C53—21no—
64C2×C42×C53C2311yescharacteristic
65#106×C53—22no—
66C4⋊C8×C53C2×C412yes—
67#103×C53—22no—
68#103×C53—22no—
69C22⋊C8×C53—21no—
70#97×C53—21no—
71C2×Q8×C53—24no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53C22⋊C412yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—22no—
83C22×Dic53—21no—
84Q16×C53—44no—
85C2×C8×C53—22no—
86C4⋊C4×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53C22⋊C412yes—
96C22⋊C4×C53—41no—
97C42×C53—22no—
98C42×C53—22no—
99C42×C53—41no—
100C42×C53C22×C411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104C4⋊Dic53—28no—
105C4⋊Dic53—24no—
106C4×Dic53—28no—
107C4×Dic53—24no—
108C4×Dic53—42no—
109C4×Dic53—82no—
110#11719917—22no—
111#11719917—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C23×C53#12011yescharacteristic
115Q8×C53—24no—
116Q8×C53—22no—
117Q8×C53—42no—
118Q8×C53—81no—
119Q8×C53#10414yes—
120C2×C4×C53—24no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
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×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12311yescharacteristic
137C2×Dic53—22no—
138C2×Dic53—24no—
139C2×Dic53—22no—
140C2×Dic53—41no—
141C2×Dic53—42no—
142C2×Dic53—41no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C8×C53—42no—
147#88322—42no—
148#88322—81no—
149#9324—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53#38611yescharacteristic
153C22×C53#45511yescharacteristic
154C22×C53#45611yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—24no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—42no—
162C4×C53—81no—
163C4×C53—81no—
164C4×C53C2×#10412yes—
165Dic53—44no—
166Dic53—42no—
167Dic53—42no—
168#3113D5311yescharacteristic
169#2276—531no—
170#1986—531no—
171#1506—531no—
172#1491—531no—
173#1300—531no—
174#1285—531no—
175C2×C53—21no—
176C2×C53—21no—
177C2×C53#188211yescharacteristic
178C2×C53#189611yescharacteristic
179C2×C53#190511yescharacteristic
180#533C2×D5311yescharacteristic
181C2×C4×Q8—1061no—
182C2×C4×Q8C2×D5311yescharacteristic
183#483—21no—
184#475Dic5312yescomplemented
185#462Dic5312yescomplemented
186#451—21no—
187#441—22no—
188#432—531no—
189#432C2×D5311yescharacteristic
190C4×M16—531no—
191C4×C4⋊C4—531no—
192C4×C4⋊C4—1061no—
193C4×C4⋊C4—1062no—
194C4×C4⋊C4—1061no—
195#376—531no—
196#376—1061no—
197#339—1062no—
198#328—1061no—
199#328—1061no—
200Q8⋊C8—1062no—
201C53#932411yescharacteristic, complemented, Sylow
202C22×Q8C22⋊D5311yescharacteristic
203C2×Q16—22no—
204C2×M16—1061no—
205C4⋊Q8C2×Dic5311yescharacteristic
206C4⋊Q8C2×Dic5311yescharacteristic
207#125—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—1068no—
215C4×Q8C2×Dic5312yes—
216C4×Q8C22⋊D5312yes—
217#117—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1062no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4C22⋊D5311yescharacteristic
222C2×C42—21no—
223C2×C42—532no—
224C2×C42—1062no—
225C2×C42—1061no—
226C2×C42—1062no—
227C2×C42C22×D5311yescharacteristic
228#106—22no—
229C4⋊C8—1061no—
230C4⋊C8C2×Dic5312yes—
231#103—22no—
232#103—22no—
233C22⋊C8—21no—
234C22⋊C8—1061no—
235#97—21no—
236#97—1061no—
237#97—1061no—
238C4×C8—1061no—
239#95—531no—
240#95—531no—
241#95—1062no—
242#95—1061no—
243#95—1061no—
244C2×Q8—24no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8C22⋊Dic5312yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1062no—
254C22×C4D4×D5311yescharacteristic
255C22×C4D4×D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257Q16—44no—
258M16—1062no—
259M16—2121no—
260C2×C8—22no—
261C2×C8—2121no—
262C2×C8—2121no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—1068no—
273C4⋊C4—1064no—
274C4⋊C4C22⋊Dic5312yes—
275C22⋊C4—41no—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—1068no—
280C42—1064no—
281C42—4242no—
282C42—2122no—
283C42#1171994711yescharacteristic
284C42#1171994711yescharacteristic
285C42C22×Dic5311yescharacteristic
286C42C2×C22⋊D5311yescharacteristic
287C23#1313318511yescharacteristic
288Q8—24no—
289Q8—22no—
290Q8—42no—
291Q8—81no—
292Q8#1313314414yes—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—1064no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2122no—
310C2×C4—2121no—
311C2×C4—2122no—
312C2×C4—2121no—
313C2×C4C2×C22⋊Dic5311yescharacteristic
314C2×C4D4×Dic5311yescharacteristic
315C2×C4D4×Dic5311yescharacteristic
316C2×C4#1313318911yescharacteristic
317C2×C4#1313318911yescharacteristic
318C2×C4#1313319811yescharacteristic
319C8—42no—
320C8—4241no—
321C8—2122no—
322C22—21no—
323C22—21no—
324C22#1681050311yescharacteristic
325C22#1681079911yescharacteristic
326C22#1681080411yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—24no—
332C4—21no—
333C4—42no—
334C4—81no—
335C4—81no—
336C4—2122no—
337C4—2124no—
338C4—2122no—
339C4C2×#1313314412yes—
340C2—21no—
341C2—21no—
342C2#2027508611yescharacteristic
343C2#2027515911yescharacteristic
344C2#2027521111yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3113C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#3113D53
hypocenterC53#9324
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
24. C53×#475C4semidirect
25. C53×#462C4semidirect
184. #475Dic53semidirect
185. #462Dic53semidirect
201. C53#9324semidirect

Maximal subgroups

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

subgroupindex
#202770802
#202756602
#202732272
#202731622
#202726072
#202725722
C53×#31132
#932453