Finite Group Databasebeta

← #27294926

Subgroups — #27294926

Order 13568.

340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280591C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272741C212yesmaximal
4#20272720C212yesmaximal
5C53×#1801C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic
7#16810680—21no—
8#16810562C411yescharacteristic
9#16810562C2211yescharacteristic
10#16810561—21no—
11#16810553C411yescharacteristic
12#16810553C2211yescharacteristic
13#16810542—21no—
14#16810507—21no—
15#16810507—21no—
16#16810491C411yescharacteristic
17#16810349—21no—
18#16810349C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#423—21no—
21C2×#101×C53—21no—
22C53×#343C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C4⋊#88322—41no—
32C22⋊#88322—22no—
33#13133128—41no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×Q8×C53D411yescharacteristic
45C2×M16×C53—22no—
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C4⋊C8×C53—41no—
56#101×C53—42no—
57C2×Q8×C53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72M16×C53—42no—
73M16×C53—42no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88#11719919—41no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92C2×#88322—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C53#13711yescharacteristic
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—42no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C8×C53—82no—
127#88322—82no—
128#11835—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41711yescharacteristic
133C22×C53#42111yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143Dic53—81no—
144Dic53—81no—
145Dic53—81no—
146Dic53—81no—
147#2634—531no—
148#1801D5311yescharacteristic
149#1467—531no—
150#1353—532no—
151#1346—532no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53#151311yescharacteristic
155C2×C53#212311yescharacteristic
156C2×C53#213311yescharacteristic
157C2×#125—531no—
158C2×C4×Q8C2×D5311yescharacteristic
159#432—1061no—
160#423—21no—
161C2×#101—21no—
162#400—531no—
163#400—531no—
164#400—1061no—
165#398—531no—
166#398—531no—
167#395—1061no—
168#387—1061no—
169#387—1061no—
170#383—531no—
171#343C2×D5312yes—
172#324—532no—
173#324—1061no—
174C53#1183511yescharacteristic, complemented, Sylow
175C22×Q8C22⋊D5311yescharacteristic
176C2×M16—22no—
177#125—2121no—
178#125—2121no—
179C4×Q8—41no—
180C4×Q8—41no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4C4×D5311yescharacteristic
188C2×C4⋊C4D21211yescharacteristic
189C2×C4⋊C4C22×D5311yescharacteristic
190C2×C42C4×D5311yescharacteristic
191C2×C42D21211yescharacteristic
192C2×C42C22⋊D5311yescharacteristic
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8—2121no—
196#101—42no—
197C22⋊C8—1062no—
198#97—2121no—
199#95—531no—
200#95—531no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4#1171992111yescharacteristic
218C22×C4#1171992111yescharacteristic
219C22×C4#1171992111yescharacteristic
220C22×C4#1171992111yescharacteristic
221C22×C4C2×C4×D5311yescharacteristic
222C22×C4C2×D21211yescharacteristic
223C22×C4C2×C22⋊D5311yescharacteristic
224M16—42no—
225M16—42no—
226C2×C8—42no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—2121no—
232C4⋊C4—2121no—
233C4⋊C4—2122no—
234C4⋊C4—2122no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2121no—
238C4⋊C4—2121no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C23C2×#1171992111yescharacteristic
246Q8—42no—
247Q8—42no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2122no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4#1313313111yescharacteristic
275C2×C4#1313313111yescharacteristic
276C2×C4#1313324311yescharacteristic
277C2×C4#1313324811yescharacteristic
278C8—82no—
279C8—4242no—
280C22—21no—
281C22—21no—
282C22C2×#1313313111yescharacteristic
283C22#1681062211yescharacteristic
284C22#1681064111yescharacteristic
285C4—21no—
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—4241no—
295C4—4241no—
296C4—4241no—
297C4—4241no—
298C2—21no—
299C2—21no—
300C2#2027325611yescharacteristic
301C2#2027637011yescharacteristic
302C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1801C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1801D53
hypocenterC53#11835
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
174. C53#11835semidirect

Maximal subgroups

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

subgroupindex
#202805912
#202730592
#202727412
#202727202
C53×#18012
#1183553