Finite Group Databasebeta

← #28299824

Subgroups — #28299824

Order 13568.

396 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C4×C4⋊#88324C211yesmaximal, characteristic
2#20290432C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5#20283101C211yesmaximal, characteristic
6C2×C4⋊#88324—21no—
7C2×#13133272—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810522C2211yescharacteristic
13#16810521C411yescharacteristic
14#16810519C2211yescharacteristic
15#16810509—21no—
16#16810508—21no—
17#16810505—21no—
18#16810504—21no—
19#16810488C411yescharacteristic
20#16810488C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25C4⋊#88324—24no—
26#13133272—24no—
27C4×#88324—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53C2311yescharacteristic
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×#88324—24no—
63C2×#88324—24no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C42×C53—22no—
72C4⋊Dic53—22no—
73C4⋊Dic53—42no—
74C4⋊Dic53—41no—
75#11719919—42no—
76#11719919—41no—
77#11719919—41no—
78C4×Dic53—22no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—41no—
85C23×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53#10412yes—
99C2×C4×C53#10412yes—
100C2×C4×C53#13112yes—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53#10112yes—
117C2×Dic53#13512yes—
118C2×Dic53AGL(1,ℤ/8)11yescharacteristic
119C2×Dic53#13711yescharacteristic
120#88324—44no—
121#88324—44no—
122#7677—531nomaximal, Sylow
123C22×C53#40811yescharacteristic
124C22×C53#41011yescharacteristic
125C22×C53C2×#10111yescharacteristic
126C22×C53#41611yescharacteristic
127C22×C53#41711yescharacteristic
128C22×C53C2×#10411yescharacteristic
129C22×C53C2×#10411yescharacteristic
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—81no—
134C4×C53—81no—
135Dic53—22no—
136Dic53—22no—
137Dic53—21no—
138Dic53—21no—
139Dic53—41no—
140Dic53—81no—
141Dic53—81no—
142#2864—531no—
143C4×C4⋊C8—531no—
144#1745—531no—
145#1271—532no—
146#1271—532no—
147C2×C53#147011yescharacteristic
148C2×C53#148311yescharacteristic
149C2×C53#149411yescharacteristic
150C2×C53#150011yescharacteristic
151C2×C53#150911yescharacteristic
152C2×C53#152611yescharacteristic
153C2×C53#155211yescharacteristic
154C2×C4⋊C8—1061no—
155C2×#97—1061no—
156C2×C4×C8—1061no—
157#389—531no—
158#389—531no—
159#389—531no—
160#389#615711yescharacteristic
161#387—1061no—
162#387—1061no—
163#387—1061no—
164#387—1061no—
165#382—531no—
166#382—531no—
167C43—531no—
168#336—532no—
169#336—532no—
170#336—1061no—
171C53#767711yescharacteristic, complemented, Sylow
172C22×C8—1062no—
173C22×C8—1062no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4C2×#615711yescharacteristic
181C2×C4⋊C4C2×#615711yescharacteristic
182C2×C42—531no—
183C2×C42—531no—
184C2×C42—531no—
185C2×C42—531no—
186C2×C42—531no—
187C2×C42—531no—
188C2×C42C2×#615711yescharacteristic
189C4⋊C8—1064no—
190#97—1064no—
191C4×C8—2122no—
192#95—21no—
193#95—1061no—
194#95—1061no—
195#95—1061no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4C22⋊#615711yescharacteristic
209C22×C4C22⋊#615711yescharacteristic
210C22×C4C22×#615711yescharacteristic
211C2×C8—1064no—
212C2×C8—1064no—
213C2×C8—2122no—
214C2×C8—2122no—
215C2×C8—2122no—
216C2×C8—2122no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—1062no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C42—22no—
227C42—1061no—
228C42—1062no—
229C42—1062no—
230C42—1061no—
231C42—1062no—
232C42—1062no—
233C42—2121no—
234C23C2×C22⋊#615711yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—532no—
248C2×C4—532no—
249C2×C4—531no—
250C2×C4—531no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4#1313328112yes—
267C2×C4#1313328212yes—
268C2×C4#1313330412yes—
269C8—2124no—
270C8—2124no—
271C22#1681159611yescharacteristic
272C22#1681160211yescharacteristic
273C22D53⋊#10111yescharacteristic
274C22#1681161311yescharacteristic
275C22#1681161411yescharacteristic
276C22C2×#1313328111yescharacteristic
277C22C2×#1313328211yescharacteristic
278C4—22no—
279C4—22no—
280C4—22no—
281C4—81no—
282C4—81no—
283C4—1062no—
284C4—1061no—
285C4—1061no—
286C4—1062no—
287C4—4241no—
288C4—4241no—
289C4—2121no—
290C2#2028998711yescharacteristic
291C2#2029003211yescharacteristic
292C2#2029006611yescharacteristic
293C2#2029008811yescharacteristic
294C2#2029011711yescharacteristic
295C2#2029017311yescharacteristic
296C2#2029023711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7677
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
171. C53#7677semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊#883242
#202904322
#202895342
#202895332
#202831012
#767753