Finite Group Databasebeta

← #27287275

Subgroups — #27287275

Order 13568.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#20280358C211yesmaximal, characteristic
2#20273266C211yesmaximal, characteristic
3#20273227C211yesmaximal, characteristic
4#20272609C211yesmaximal, characteristic
5#20272580C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#1986C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810562—21no—
11#16810543—21no—
12#16810529—21no—
13#16810527—21no—
14#16810507—21no—
15#16810506—21no—
16#16810491C2211yescharacteristic
17#16810469C2211yescharacteristic
18#16810360—21no—
19#16810360—21no—
20#16810360—21no—
21#16810360—21no—
22C53×#533C2211yescharacteristic
23C4×M16×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C53×#339—21no—
26C53×#339—21no—
27C22×Dic106—21no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C4×Dic53—21no—
32C4⋊#88322—21no—
33C4⋊#88322—21no—
34C22⋊#88322—21no—
35C22⋊#88322—21no—
36#13133128—21no—
37#13133128—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124D411yescharacteristic
46#13133124D411yescharacteristic
47#13133124D411yescharacteristic
48#13133124D411yescharacteristic
49C2×M16×C53—21no—
50C4⋊Q8×C53Q811yescharacteristic, complemented
51C4⋊Q8×C53Q811yescharacteristic, complemented
52#125×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57#117×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C42×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C2×C42×C53C2311yescharacteristic
62#97×C53—21no—
63C4×C8×C53—21no—
64#95×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×Dic106—42no—
77C2×Dic106—42no—
78C2×Dic106—81no—
79M16×C53—22no—
80M16×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—21no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—42no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—41no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53C2×Q811yescharacteristic
94C42×C53#4111yescharacteristic
95C42×C53#4111yescharacteristic
96C42×C53#4111yescharacteristic
97C4⋊Dic53—42no—
98#11719919—42no—
99#11719919—81no—
100C4×Dic53—42no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53#12011yescharacteristic
104Q8×C53—81no—
105Q8×C53—81no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53C22⋊Q811yescharacteristic
119C2×C4×C53C22⋊Q811yescharacteristic
120C2×C4×C53C22⋊Q811yescharacteristic
121C2×C4×C53#12411yescharacteristic
122C2×C4×C53#12411yescharacteristic
123C2×C4×C53#12411yescharacteristic
124C2×Dic53—22no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136Dic106—84no—
137C8×C53—42no—
138#88322—81no—
139#88322—81no—
140#10379—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53#39311yescharacteristic
144C22×C53#45511yescharacteristic
145C22×C53#45611yescharacteristic
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—42no—
157Dic53—81no—
158Dic53—81no—
159Dic53—81no—
160#2605—531no—
161#1986D5311yescharacteristic
162#1515—531no—
163#1506—531no—
164#1300—531no—
165#1288—531no—
166#1285—531no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53#202811yescharacteristic
170C2×C53#203111yescharacteristic
171C2×C53#203111yescharacteristic
172#533C2×D5311yescharacteristic
173#432—531no—
174#432—531no—
175C4×M16C2×D5311yescharacteristic
176#400—1061no—
177#395—1061no—
178#391—1061no—
179#391—1061no—
180#387—1061no—
181#387—1061no—
182#383—531no—
183C4×C4⋊C4C2×D5311yescharacteristic
184#376—531no—
185#339—21no—
186#339—21no—
187#328—1061no—
188#328—1061no—
189#328—1061no—
190#328—1061no—
191C53#1037911yescharacteristic, complemented, Sylow
192C22×Q8—1061no—
193C2×M16—21no—
194C4⋊Q8Dic10611yescharacteristic, complemented
195C4⋊Q8Dic10611yescharacteristic, complemented
196#125—21no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201#117—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C42—1061no—
207C2×C42C22⋊D5311yescharacteristic
208C2×C42C22⋊D5311yescharacteristic
209C2×C42C22×D5311yescharacteristic
210C4⋊C8—1061no—
211C4⋊C8—1061no—
212C22⋊C8—1061no—
213C22⋊C8—1061no—
214#97—21no—
215#97—1061no—
216#97—1061no—
217C4×C8—21no—
218#95—21no—
219#95—531no—
220#95—531no—
221#95—531no—
222#95—531no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—4241no—
233C2×Q8—2122no—
234C2×Q8—2122no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1061no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4D4×D5311yescharacteristic
242C22×C4D4×D5311yescharacteristic
243C22×C4C2×C22⋊D5311yescharacteristic
244M16—22no—
245M16—41no—
246C2×C8—41no—
247C2×C8—2121no—
248C2×C8—2121no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—4241no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C22⋊C4—41no—
260C42—41no—
261C42—41no—
262C42—41no—
263C42—2122no—
264C42C2×Dic10611yescharacteristic
265C42#1171994511yescharacteristic
266C42#1171994711yescharacteristic
267C42#1171994711yescharacteristic
268C23#1313318511yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—4244no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—1062no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4#1313319411yescharacteristic
297C2×C4#1313319411yescharacteristic
298C2×C4#1313319611yescharacteristic
299C2×C4#1313320311yescharacteristic
300C2×C4#1313320311yescharacteristic
301C2×C4#1313320411yescharacteristic
302C8—42no—
303C8—4241no—
304C8—4241no—
305C22—21no—
306C22—21no—
307C22#1681053511yescharacteristic
308C22#1681080011yescharacteristic
309C22#1681080511yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—4241no—
321C4—4241no—
322C4—4241no—
323C4—2122no—
324C2—21no—
325C2—21no—
326C2#2027587611yescharacteristic
327C2#2027588511yescharacteristic
328C2#2027588711yescharacteristic
subgroupG/H
centerC22#16810535
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1986C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#1986D53
hypocenterC53#10379
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
50. C4⋊Q8×C53Q8semidirect
51. C4⋊Q8×C53Q8semidirect
191. C53#10379semidirect
194. C4⋊Q8Dic106semidirect
195. C4⋊Q8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202803582
#202732662
#202732272
#202726092
#202725802
#202725712
C53×#19862
#1037953