Finite Group Databasebeta

← #27297051

Subgroups — #27297051

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#20276065C211yesmaximal, characteristic
2#20273756C211yesmaximal, characteristic
3#20273722C211yesmaximal, characteristic
4#20272742C211yesmaximal, characteristic
5#20272721C211yesmaximal, characteristic
6#20272659C211yesmaximal, characteristic
7C53×#2769C211yesmaximal, 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×#400C2211yescharacteristic
20C53×#398—21no—
21C53×#398C2211yescharacteristic
22C53×#389—21no—
23C53×#387—21no—
24C53×#387—21no—
25C53×#383—21no—
26C53×#382C2211yescharacteristic
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—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—81no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—42no—
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—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#13511yescharacteristic
124C2×C4×C53#13711yescharacteristic
125C2×Dic53—41no—
126C2×Dic53—41no—
127Dic106—81no—
128Dic106—81no—
129#88322—42no—
130#88322—81no—
131#88322—81no—
132#12241—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53#45211yescharacteristic
136C22×C53#45611yescharacteristic
137C22×C53B(4,𝔽2)11yescharacteristic
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—42no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—81no—
144C4×C53—81no—
145C4×C53—81no—
146C4×C53—81no—
147C4×C53—81no—
148Dic53—81no—
149Dic53—81no—
150#2769D5311yescharacteristic
151#2061—531no—
152#1617—531no—
153#1611—531no—
154#1353—531no—
155#1346—531no—
156#1324—531no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53#160311yescharacteristic
160C2×C53#220111yescharacteristic
161C2×C53#220411yescharacteristic
162#544—531no—
163#484—1061no—
164#483—1061no—
165#428—1061no—
166#400—21no—
167#400C2×D5311yescharacteristic
168#398—21no—
169#398C2×D5311yescharacteristic
170#389—21no—
171#387—21no—
172#387—21no—
173#383—21no—
174#382C2×D5311yescharacteristic
175#343—531no—
176#339—1061no—
177#328—1061no—
178#328—1061no—
179#324—531no—
180#324—531no—
181C53#1224111yescharacteristic, complemented, Sylow
182C2×M16—1061no—
183C4⋊Q8—2121no—
184#125—2121no—
185C22⋊Q8—1061no—
186C22⋊Q8—1061no—
187#117—1061no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22⋊D5311yescharacteristic
195C2×C4⋊C4C22×D5311yescharacteristic
196C2×C42—21no—
197C2×C42C22⋊D5311yescharacteristic
198#107—2121no—
199#107—2121no—
200#106—2121no—
201#106—2121no—
202#103—2121no—
203#103—2121no—
204C22⋊C8—1061no—
205C22⋊C8—1061no—
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#95—21no—
215#95C22⋊D5311yescharacteristic
216#95C22⋊D5311yescharacteristic
217C2×Q8—1061no—
218C2×Q8—1061no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C2×C22⋊D5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228M16—1062no—
229M16—2121no—
230C2×C8—2121no—
231C2×C8—2121no—
232C2×C8—2121no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—81no—
242C4⋊C4—1061no—
243C4⋊C4—1061no—
244C4⋊C4—1061no—
245C4⋊C4—1061no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C22⋊C4—2121no—
249C42—21no—
250C42—21no—
251C42—42no—
252C42—2121no—
253C23#1313318611yescharacteristic
254Q8—4241no—
255Q8—4241no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
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—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4#1313323711yescharacteristic
283C2×C4#1313324711yescharacteristic
284C8—4241no—
285C8—4241no—
286C8—2122no—
287C22—21no—
288C22—21no—
289C22#1681078511yescharacteristic
290C22#1681080611yescharacteristic
291C22#1681081011yescharacteristic
292C4—21no—
293C4—21no—
294C4—42no—
295C4—41no—
296C4—41no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—4241no—
303C4—4241no—
304C2—21no—
305C2—21no—
306C2#2027367811yescharacteristic
307C2#2027672811yescharacteristic
308C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2769C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2769D53
hypocenterC53#12241
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2769

Complements

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

NH ≅ G/Ntype
181. C53#12241semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202737562
#202737222
#202727422
#202727212
#202726592
C53×#27692
#1224153