Finite Group Databasebeta

← #27281022

Subgroups — #27281022

Order 13568.

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

normal subgroups only

This group has 444 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#20284845C211yesmaximal, characteristic
2#20275658C211yesmaximal, characteristic
3#20273226C211yesmaximal, characteristic
4#20273162C211yesmaximal, characteristic
5#20272608C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#2320C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810953—21no—
10#16810908C412yescomplemented
11#16810833C412yescomplemented
12#16810779—21no—
13#16810728—22no—
14#16810680C2211yescharacteristic
15#16810680C2211yescharacteristic
16C4×#11719917C2211yescharacteristic
17#16810383—22no—
18#16810359—21no—
19#16810359—21no—
20Q8⋊#88322—22no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#391—21no—
23C53×#389—21no—
24C53×#387—22no—
25C53×#383—21no—
26C53×#382—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C53×#376C2211yescharacteristic
29C2×#11719925—22no—
30C2×#11719917—21no—
31#13133217C2×C411yescharacteristic
32#13133217C2×C411yescharacteristic
33#13133209—21no—
34#13133196—21no—
35#13133196—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38#13133176—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322C2×C412yes—
42#13133141—22no—
43#13133140—22no—
44C22⋊#88322—21no—
45C22⋊#88322—21no—
46#13133128—21no—
47#13133128—21no—
48#13133128—21no—
49C4×#88322—21no—
50C22×Q8×C53D411yescharacteristic
51C4×Q8×C53—21no—
52C4×Q8×C53—21no—
53C4×Q8×C53C2×C412yes—
54C4×Q8×C53D412yes—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53—21no—
61C2×C42×C53D412yes—
62C2×C42×C53C2311yescharacteristic
63#95×C53—22no—
64#95×C53—22no—
65#95×C53—22no—
66#95×C53—21no—
67#95×C53—21no—
68#95×C53D411yescharacteristic
69#95×C53D411yescharacteristic
70C2×Q8×C53—21no—
71C2×Q8×C53—42no—
72C2×Q8×C53C22⋊C412yes—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×Dic106—21no—
82C2×Dic106—21no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—82no—
88C4⋊C4×C53C22⋊C412yes—
89C42×C53—22no—
90C42×C53—22no—
91C42×C53—42no—
92C42×C53—42no—
93C42×C53C22×C411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53#4111yescharacteristic
96C42×C53#4111yescharacteristic
97C22⋊Dic53—41no—
98#11719925—44no—
99C4⋊Dic53—21no—
100C4⋊Dic53—21no—
101C4⋊Dic53—21no—
102C4⋊Dic53—21no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—41no—
106#11719917—22no—
107#11719917—41no—
108C2×#88322—22no—
109C2×#88322—41no—
110C2×#88322—41no—
111C23×C53#12011yescharacteristic
112Q8×C53—22no—
113Q8×C53—22no—
114Q8×C53—42no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×Dic53—41no—
140C2×Dic53—41no—
141Dic106—42no—
142Dic106—81no—
143#88322—42no—
144#88322—42no—
145#88322—81no—
146#9328—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53#38611yescharacteristic
150C22×C53#45511yescharacteristic
151C22×C53#45611yescharacteristic
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—42no—
158C4×C53—42no—
159C4×C53—81no—
160C4×C53—82no—
161C4×C53—81no—
162C4×C53#42112yes—
163Dic53—81no—
164Dic53—81no—
165#3113—531no—
166#2320D5311yescharacteristic
167#1986—531no—
168#1506—531no—
169#1491—531no—
170#1300—531no—
171#1285—531no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53#188211yescharacteristic
175C2×C53#189711yescharacteristic
176C2×C53#190511yescharacteristic
177#533—531no—
178C2×C4×Q8C2×D5311yescharacteristic
179#483—1061no—
180#475—532no—
181#462—532no—
182#451—1061no—
183#441—1062no—
184#432—531no—
185#432—531no—
186C4×M16—531no—
187#391—21no—
188#389—21no—
189#387—22no—
190#383—21no—
191#382—21no—
192C4×C4⋊C4C2×D5311yescharacteristic
193#376C2×D5311yescharacteristic
194#339—1062no—
195#328—1061no—
196#328—1061no—
197Q8⋊C8—1062no—
198C53#932811yescharacteristic, complemented, Sylow
199C22×Q8C22⋊D5311yescharacteristic
200C2×Q16—1062no—
201C2×M16—1061no—
202C4⋊Q8—531no—
203C4⋊Q8—531no—
204#125—1061no—
205C22⋊Q8—1061no—
206C22⋊Q8—1061no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—1061no—
210C4×Q8—1061no—
211C4×Q8C4×D5312yes—
212C4×Q8D21212yes—
213#117—1061no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22⋊D5311yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C22⋊D5312yes—
221C2×C42C22×D5311yescharacteristic
222#106—1062no—
223C4⋊C8—532no—
224C4⋊C8—1061no—
225#103—1062no—
226#103—1062no—
227C22⋊C8—1061no—
228C22⋊C8—1061no—
229#97—1061no—
230#97—1061no—
231#97—1061no—
232C4×C8—1061no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95C22⋊D5311yescharacteristic
239#95C22⋊D5311yescharacteristic
240C2×Q8—21no—
241C2×Q8—42no—
242C2×Q8—1061no—
243C2×Q8—1061no—
244C2×Q8#1171992112yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×C22⋊D5311yescharacteristic
251C22×C4C2×C22⋊D5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253Q16—2124no—
254M16—1062no—
255M16—2121no—
256C2×C8—1062no—
257C2×C8—2121no—
258C2×C8—2121no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—42no—
263C4⋊C4—82no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4#1171992112yes—
271C22⋊C4—2121no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—42no—
276C42—2121no—
277C42C2×C4×D5311yescharacteristic
278C42C2×D21211yescharacteristic
279C42#1171994511yescharacteristic
280C42#1171994511yescharacteristic
281C23#1313318611yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—42no—
285Q8—4241no—
286Q8—2122no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4C2×#1171992111yescharacteristic
308C2×C4C4×C22⋊D5311yescharacteristic
309C2×C4C4×C22⋊D5311yescharacteristic
310C2×C4#1313319111yescharacteristic
311C2×C4#1313319111yescharacteristic
312C2×C4#1313320011yescharacteristic
313C8—4241no—
314C8—2122no—
315C8—2122no—
316C22—21no—
317C22—21no—
318C22#1681050211yescharacteristic
319C22#1681080111yescharacteristic
320C22#1681080611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—82no—
330C4—81no—
331C4—4241no—
332C4—4241no—
333C4#1681064112yes—
334C2—21no—
335C2—21no—
336C2#2027508411yescharacteristic
337C2#2027516311yescharacteristic
338C2#2027520911yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2320C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#2320D53
hypocenterC53#9328
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2320

Complements

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

NH ≅ G/Ntype
10. #16810908C4semidirect
11. #16810833C4semidirect
198. C53#9328semidirect

Maximal subgroups

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

subgroupindex
#202848452
#202756582
#202732262
#202731622
#202726082
#202725712
C53×#23202
#932853