Finite Group Databasebeta

← #27297016

Subgroups — #27297016

Order 13568.

324 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 59 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#20276060C211yesmaximal, characteristic
2#20273756C211yesmaximal, characteristic
3#20273722C211yesmaximal, characteristic
4#20272742C211yesmaximal, characteristic
5#20272721C211yesmaximal, characteristic
6#20272661C211yesmaximal, characteristic
7C53×#2794C211yesmaximal, characteristic
8#16811254C2211yescharacteristic
9#16810958—21no—
10#16810953—21no—
11#16810663—21no—
12#16810392C2211yescharacteristic
13#16810383—21no—
14#16810359—21no—
15#16810359—21no—
16#16810349C2211yescharacteristic
17#16810349C2211yescharacteristic
18C53×#400—21no—
19C53×#400—21no—
20C53×#400C2211yescharacteristic
21C53×#398C2211yescharacteristic
22C53×#387—21no—
23C53×#387—21no—
24C53×#387—21no—
25C53×#387—21no—
26C53×#384C2211yescharacteristic
27C2×#11719917—21no—
28#13133216—41no—
29#13133208—41no—
30#13133196—21no—
31#13133196—21no—
32#13133176—21no—
33#13133150—41no—
34#13133150—41no—
35#13133148—41no—
36#13133148—41no—
37#13133140—41no—
38#13133140—41no—
39C22⋊#88322—21no—
40C22⋊#88322—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C4⋊C4×C53C2311yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53D411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C2×Dic106—21no—
72C2×Dic106—21no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—81no—
85C22⋊Dic53—41no—
86C4⋊Dic53—21no—
87C4⋊Dic53—21no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90#11719919—41no—
91#11719919—41no—
92C4×Dic53—41no—
93#11719917—22no—
94#11719917—41no—
95C2×#88322—41no—
96C2×#88322—41no—
97C2×#88322—41no—
98C23×C53#12011yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
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#13511yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×Dic53—41no—
125C2×Dic53—41no—
126Dic106—81no—
127Dic106—81no—
128#88322—42no—
129#88322—81no—
130#88322—81no—
131#12236—531nomaximal, Sylow
132C22×C53—21no—
133C22×C53—21no—
134C22×C53#45211yescharacteristic
135C22×C53#45611yescharacteristic
136C22×C53B(4,𝔽2)11yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—81no—
143C4×C53—81no—
144C4×C53—81no—
145C4×C53—81no—
146C4×C53—81no—
147Dic53—81no—
148Dic53—81no—
149#2794D5311yescharacteristic
150#2060—531no—
151#1617—531no—
152#1611—531no—
153#1353—531no—
154#1346—531no—
155#1325—531no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53#160311yescharacteristic
159C2×C53#219811yescharacteristic
160C2×C53#220311yescharacteristic
161#544—531no—
162#484—1061no—
163#483—1061no—
164#428—1061no—
165#400—21no—
166#400—21no—
167#400C2×D5311yescharacteristic
168#398C2×D5311yescharacteristic
169#387—21no—
170#387—21no—
171#387—21no—
172#387—21no—
173#384C2×D5311yescharacteristic
174#343—531no—
175#339—1061no—
176#328—1061no—
177#328—1061no—
178#324—531no—
179#324—531no—
180C53#1223611yescharacteristic, complemented, Sylow
181C2×M16—1061no—
182C4⋊Q8—2121no—
183#125—2121no—
184C22⋊Q8—1061no—
185C22⋊Q8—1061no—
186#117—1061no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊D5311yescharacteristic
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C42—21no—
196C2×C42C22⋊D5311yescharacteristic
197#107—2121no—
198#107—2121no—
199#106—2121no—
200#106—2121no—
201#103—2121no—
202#103—2121no—
203C22⋊C8—1061no—
204C22⋊C8—1061no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C22⋊D5311yescharacteristic
215#95C22⋊D5311yescharacteristic
216C2×Q8—1061no—
217C2×Q8—1061no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4C2×C22⋊D5311yescharacteristic
225C22×C4C2×C22⋊D5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—1062no—
228M16—2121no—
229C2×C8—2121no—
230C2×C8—2121no—
231C2×C8—2121no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—1061no—
242C4⋊C4—1061no—
243C4⋊C4—1061no—
244C4⋊C4—1061no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C22⋊C4—2121no—
248C42—21no—
249C42—21no—
250C42—81no—
251C42—2121no—
252C23#1313318611yescharacteristic
253Q8—4241no—
254Q8—4241no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4#1313323711yescharacteristic
281C2×C4#1313324711yescharacteristic
282C8—4241no—
283C8—4241no—
284C8—2122no—
285C22—21no—
286C22—21no—
287C22#1681078511yescharacteristic
288C22#1681080611yescharacteristic
289C22#1681081011yescharacteristic
290C4—21no—
291C4—21no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—4241no—
301C4—4241no—
302C2—21no—
303C2—21no—
304C2#2027367811yescharacteristic
305C2#2027671111yescharacteristic
306C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2794C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2794D53
hypocenterC53#12236
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2794

Complements

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

NH ≅ G/Ntype
180. C53#12236semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202737562
#202737222
#202727422
#202727212
#202726612
C53×#27942
#1223653