Finite Group Databasebeta

← #28312301

Subgroups — #28312301

Order 13568.

396 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290581C211yesmaximal, characteristic
2C4×#13133278C211yesmaximal, characteristic
3#20289813C212yesmaximal
4#20289813C212yesmaximal
5#20279907C211yesmaximal, characteristic
6#16811626—21no—
7C2×#13133278C2212yes—
8C2×#13133278C2212yes—
9C4×#11719940—21no—
10#16811503—21no—
11#16811503—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810556—21no—
14#16810556—21no—
15#16810528C2211yescharacteristic
16#16810528C2211yescharacteristic
17#16810504—21no—
18#16810504—21no—
19#16810492C411yescharacteristic
20#16810488C411yescharacteristic
21C4×C4⋊Dic53C411yescharacteristic
22C2×#125×C53C411yescharacteristic
23C2×#11719940—22no—
24C2×#11719940—22no—
25#13133284—41no—
26C4⋊#88324—41no—
27#13133278—22no—
28#13133278—24no—
29#13133278Q814yes—
30#13133272—41no—
31C4×#88324—41no—
32C22×Dic106C2311yescharacteristic
33C4×Dic106—41no—
34C4×Dic106—41no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919D411yescharacteristic
40C2×#11719919D411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#125×C53—41no—
53#125×C53—41no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
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—41no—
71C2×Dic106—41no—
72C2×Dic106C2×Q812yes—
73C2×Dic106#4112yes—
74#11719940—24no—
75#11719940—42no—
76#11719940—42no—
77#11719940—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C42×C53—21no—
86C42×C53—21no—
87C4⋊Dic53—41no—
88C4⋊Dic53—42no—
89#11719919—41no—
90#11719919—41no—
91#11719919—42no—
92#11719919—42no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53C4×Q812yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53C22⋊Q812yes—
128C2×Dic53#12411yescharacteristic
129C2×Dic53C4⋊Q811yescharacteristic
130Dic106—42no—
131Dic106—42no—
132#88324—44no—
133#88324—82no—
134#13303—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53#39111yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53#41011yescharacteristic
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—22no—
143C4×C53—41no—
144C4×C53—81no—
145C4×C53—81no—
146Dic53—22no—
147Dic53—21no—
148Dic53—21no—
149Dic53—42no—
150Dic53—41no—
151Dic53—41no—
152Dic53—81no—
153#2565—531no—
154#1801—531no—
155C4×#101—531no—
156#1394—532no—
157#1394—532no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53#199611yescharacteristic
161C2×C53#213211yescharacteristic
162C2×C53#213311yescharacteristic
163C2×#125#615711yescharacteristic
164C2×C4×Q8—531no—
165#423—1061no—
166C2×#101—532no—
167C2×#101—532no—
168C4×M16—1061no—
169#398—1061no—
170#398—1061no—
171#391—531no—
172#391—531no—
173#387—1061no—
174#387—1061no—
175#384—531no—
176#382—531no—
177C4×C4⋊C4—531no—
178#343—1061no—
179#343—1061no—
180C53#1330311yescharacteristic, complemented, Sylow
181C22×Q8—531no—
182C2×M16—1062no—
183C2×M16—1062no—
184#125—41no—
185#125—41no—
186C4×Q8—2121no—
187C4×Q8—2121no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—531no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4C2×#615711yescharacteristic
197C2×C4⋊C4C2×#615711yescharacteristic
198C2×C42—531no—
199C2×C42—531no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42C2×#615711yescharacteristic
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205#101—534no—
206#101—1062no—
207#101—1064no—
208#97—2121no—
209C4×C8—2121no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C2×Q8—532no—
218C2×Q8—532no—
219C2×Q8—2121no—
220C2×Q8—2121no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—1061no—
228C22×C4—1061no—
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22⋊#615711yescharacteristic
231C22×C4C22×#615711yescharacteristic
232M16—1064no—
233M16—2122no—
234M16—2122no—
235M16—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C4⋊C4—2122no—
247C4⋊C4—2122no—
248C4⋊C4—2122no—
249C42—21no—
250C42—21no—
251C42—1061no—
252C42—1061no—
253C42—1061no—
254C42—1061no—
255C42—2121no—
256C42—2121no—
257C23C2×C22⋊#615711yescharacteristic
258Q8—2122no—
259Q8—2122no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—532no—
271C2×C4—531no—
272C2×C4—531no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1061no—
277C2×C4—1062no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4#1313327511yescharacteristic
288C2×C4#1313327511yescharacteristic
289C2×C4Q8×#615712yes—
290C8—4242no—
291C8—2124no—
292C22—21no—
293C22—21no—
294C22(C2×#6157)⋊Q811yescharacteristic
295C22C2×#1313327511yescharacteristic
296C22#1681160411yescharacteristic
297C4—22no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—1061no—
304C4—1062no—
305C4—1061no—
306C4—4241no—
307C4—2122no—
308C4—2121no—
309C4—2121no—
310C2—21no—
311C2—21no—
312C2#2029090411yescharacteristic
313C2#2029100611yescharacteristic
314C2#2029101011yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#125×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenterC2×#125#6157
hypocenterC53#13303
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
180. C53#13303semidirect

Maximal subgroups

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

subgroupindex
#202905812
C4×#131332782
#202898132
#202799072
#1330353