Finite Group Databasebeta

← #28309172

Subgroups — #28309172

Order 13568.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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×#13133278C211yesmaximal, characteristic
2#20289977C211yesmaximal, characteristic
3#20289716C212yesmaximal
4#20289714C212yesmaximal
5#20280081C211yesmaximal, characteristic
6#16811644—21no—
7C2×#13133278—21no—
8C4×#11719940—21no—
9#16811503C2212yes—
10#16811469C2212yes—
11#16811468—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810561C411yescharacteristic
14#16810561C2211yescharacteristic
15#16810542C411yescharacteristic
16#16810542C2211yescharacteristic
17#16810529—21no—
18#16810521—21no—
19#16810504—21no—
20#16810504—21no—
21#16810489C411yescharacteristic
22C2×C4×Q8×C53C411yescharacteristic
23C2×#11719940—22no—
24C4⋊#88324—41no—
25#13133278—24no—
26C22⋊#88324—22no—
27#13133272—41no—
28#13133272—41no—
29C4×#88324—41no—
30C22×Dic106D411yescharacteristic
31C4×Dic106—41no—
32C4×Dic106—41no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919C2×C411yescharacteristic
38C2×#11719919C2311yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124C2×C411yescharacteristic
49#13133124C2×C411yescharacteristic
50C22×Q8×C53C2×C411yescharacteristic
51C4×Q8×C53—41no—
52C4×Q8×C53—41no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2×C411yescharacteristic
57C2×Q8×C53—42no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×Dic106—22no—
71C2×Dic106—22no—
72C2×Dic106—41no—
73#11719940—24no—
74#11719940—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—42no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85#11719919—41no—
86#11719919—41no—
87#11719919—42no—
88#11719919—42no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—42no—
95Q8×C53—42no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C2×C4×C53#10412yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53#13712yes—
125Dic106—44no—
126#88324—44no—
127#88324—82no—
128#11792—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53C2×#9911yescharacteristic
132C22×C53#41911yescharacteristic
133C22×C53C2×#10411yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—41no—
139C4×C53—81no—
140Dic53—22no—
141Dic53—41no—
142Dic53—41no—
143Dic53—41no—
144Dic53—81no—
145Dic53—81no—
146#2581—531no—
147C4×#101—531no—
148#1467—531no—
149#1354—532no—
150#1353—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#150811yescharacteristic
154C2×C53#212011yescharacteristic
155C2×C53#213311yescharacteristic
156C2×C4×Q8—531no—
157C2×C4×Q8#615711yescharacteristic
158#432—1061no—
159C2×#101—1061no—
160C4×M16—1061no—
161#400—531no—
162#400—531no—
163#395—531no—
164#395—531no—
165#391—1061no—
166#389—1061no—
167#387—1061no—
168#387—1061no—
169#382—531no—
170#343—532no—
171#324—532no—
172#324—1061no—
173C53#1179211yescharacteristic, complemented, Sylow
174C22×Q8—531no—
175C22×Q8C2×#615711yescharacteristic
176C2×M16—1062no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—2121no—
180C4×Q8—2121no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—1061no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C42—21no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42C2×#615711yescharacteristic
193C4⋊C8—2121no—
194#101—1064no—
195C22⋊C8—1062no—
196#97—2121no—
197#97—2121no—
198C4×C8—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—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—1062no—
211C2×Q8—1062no—
212C2×Q8—2121no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22⋊#615711yescharacteristic
223C22×C4C22×#615711yescharacteristic
224M16—1064no—
225M16—2122no—
226C2×C8—2122no—
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—2121no—
232C4⋊C4—2121no—
233C4⋊C4—2121no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C4⋊C4—2122no—
237C42—21no—
238C42—21no—
239C42—42no—
240C42—1061no—
241C42—1061no—
242C42—1061no—
243C42—1061no—
244C23C2×C22⋊#615711yescharacteristic
245Q8—42no—
246Q8—42no—
247Q8—2124no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—532no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4#1313327511yescharacteristic
275C2×C4#1313327511yescharacteristic
276C2×C4#1313328212yes—
277C8—4242no—
278C8—2124no—
279C22—21no—
280C22—21no—
281C22C2×#1313327511yescharacteristic
282C22#1681161611yescharacteristic
283C22C2×#1313328211yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—81no—
290C4—1062no—
291C4—4241no—
292C4—4241no—
293C4—2121no—
294C4—2121no—
295C4—2121no—
296C2—21no—
297C2—21no—
298C2#2029011511yescharacteristic
299C2#2029096611yescharacteristic
300C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C4×Q8#6157
hypocenterC53#11792
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
173. C53#11792semidirect

Maximal subgroups

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

subgroupindex
C4×#131332782
#202899772
#202897162
#202897142
#202800812
#1179253