Finite Group Databasebeta

← #27287286

Subgroups — #27287286

Order 13568.

352 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 77 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#20275658C211yesmaximal, characteristic
2#20273265C211yesmaximal, characteristic
3#20273226C211yesmaximal, characteristic
4#20272608C211yesmaximal, characteristic
5#20272579C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#2606C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810680C2211yescharacteristic
11C4×#11719917C2211yescharacteristic
12#16810383—21no—
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810359—21no—
17#16810359—21no—
18C53×#398—21no—
19C53×#398—21no—
20C53×#389—21no—
21C53×#389—21no—
22C53×#387—21no—
23C53×#387—21no—
24C53×#383C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#376C2211yescharacteristic
27C2×#11719917—21no—
28#13133217Q811yescharacteristic
29#13133217Q811yescharacteristic
30#13133209—21no—
31#13133196—21no—
32#13133196—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35#13133176—21no—
36C4⋊#88322—21no—
37C4⋊#88322—21no—
38C22⋊#88322—21no—
39C22⋊#88322—21no—
40#13133128—21no—
41#13133128—21no—
42#13133128—21no—
43C4×#88322—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C42×C53—21no—
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53D411yescharacteristic
62#95×C53D411yescharacteristic
63#95×C53D411yescharacteristic
64#95×C53D411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—81no—
82C4⋊C4×C53—81no—
83C42×C53—42no—
84C42×C53—41no—
85C42×C53—41no—
86C42×C53C2×Q811yescharacteristic
87C42×C53#4111yescharacteristic
88C42×C53#4111yescharacteristic
89C42×C53#4111yescharacteristic
90C22⋊Dic53—41no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95#11719919—41no—
96#11719919—41no—
97C4×Dic53—41no—
98#11719917—22no—
99#11719917—41no—
100C2×#88322—41no—
101C2×#88322—41no—
102C2×#88322—41no—
103C23×C53#12011yescharacteristic
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—41no—
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×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53C22⋊Q811yescharacteristic
127C2×C4×C53C22⋊Q811yescharacteristic
128C2×C4×C53C22⋊Q811yescharacteristic
129C2×C4×C53#12411yescharacteristic
130C2×C4×C53#12411yescharacteristic
131C2×C4×C53#12411yescharacteristic
132C2×Dic53—41no—
133C2×Dic53—41no—
134Dic106—81no—
135Dic106—81no—
136#88322—42no—
137#88322—81no—
138#88322—81no—
139#10381—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53#39311yescharacteristic
143C22×C53#45511yescharacteristic
144C22×C53#45611yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53—81no—
156C4×C53—81no—
157Dic53—81no—
158Dic53—81no—
159#2606D5311yescharacteristic
160#1986—531no—
161#1515—531no—
162#1506—531no—
163#1300—531no—
164#1288—531no—
165#1285—531no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53#202811yescharacteristic
169C2×C53#203111yescharacteristic
170C2×C53#203111yescharacteristic
171#533—531no—
172#432—531no—
173#432—531no—
174C4×M16—531no—
175#398—21no—
176#398—21no—
177#389—21no—
178#389—21no—
179#387—21no—
180#387—21no—
181#383C2×D5311yescharacteristic
182C4×C4⋊C4C2×D5311yescharacteristic
183#376C2×D5311yescharacteristic
184#339—1061no—
185#339—1061no—
186#328—1061no—
187#328—1061no—
188#328—1061no—
189#328—1061no—
190C53#1038111yescharacteristic, complemented, Sylow
191C2×M16—1061no—
192C4⋊Q8—531no—
193C4⋊Q8—531no—
194#125—1061no—
195C22⋊Q8—1061no—
196C22⋊Q8—1061no—
197C4×Q8—1061no—
198C4×Q8—1061no—
199#117—1061no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C42—21no—
208C2×C42C22⋊D5311yescharacteristic
209C2×C42C22⋊D5311yescharacteristic
210C2×C42C22×D5311yescharacteristic
211C4⋊C8—1061no—
212C4⋊C8—1061no—
213C22⋊C8—1061no—
214C22⋊C8—1061no—
215#97—1061no—
216#97—1061no—
217#97—1061no—
218C4×C8—1061no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C22⋊D5311yescharacteristic
226#95C22⋊D5311yescharacteristic
227#95C22⋊D5311yescharacteristic
228#95C22⋊D5311yescharacteristic
229C2×Q8—1061no—
230C2×Q8—1061no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×C22⋊D5311yescharacteristic
238C22×C4C2×C22⋊D5311yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240M16—1062no—
241M16—2121no—
242C2×C8—2121no—
243C2×C8—2121no—
244C2×C8—2121no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—1061no—
253C4⋊C4—1061no—
254C4⋊C4—1061no—
255C4⋊C4—1061no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C22⋊C4—2121no—
259C42—41no—
260C42—41no—
261C42—42no—
262C42—2121no—
263C42Q8×D5311yescharacteristic
264C42#1171994911yescharacteristic
265C42#1171994911yescharacteristic
266C42#1171994911yescharacteristic
267C23#1313318611yescharacteristic
268Q8—4241no—
269Q8—4241no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
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—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4#1313319311yescharacteristic
295C2×C4#1313319311yescharacteristic
296C2×C4#1313319311yescharacteristic
297C2×C4#1313320611yescharacteristic
298C2×C4#1313320611yescharacteristic
299C2×C4#1313320611yescharacteristic
300C8—4241no—
301C8—4241no—
302C8—2122no—
303C22—21no—
304C22—21no—
305C22#1681053411yescharacteristic
306C22#1681080111yescharacteristic
307C22#1681080611yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—4241no—
321C4—4241no—
322C2—21no—
323C2—21no—
324C2#2027587311yescharacteristic
325C2#2027588411yescharacteristic
326C2#2027588411yescharacteristic
subgroupG/H
centerC22#16810534
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2606C2
socleC22×C53#393
prime radicalC23×C53#120
hypercenter#2606D53
hypocenterC53#10381
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2606

Complements

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

NH ≅ G/Ntype
190. C53#10381semidirect

Maximal subgroups

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

subgroupindex
#202756582
#202732652
#202732262
#202726082
#202725792
#202725712
C53×#26062
#1038153