Finite Group Databasebeta

← #28309155

Subgroups — #28309155

Order 13568.

460 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13133276C211yesmaximal, characteristic
2#20289972C211yesmaximal, characteristic
3#20289710C214yesmaximal, complemented
4#20279889C211yesmaximal, characteristic
5C2×#13133276—21no—
6#16811594—21no—
7C4×C22⋊#6157—21no—
8#16811501C2212yes—
9#16811468—21no—
10#16811467C2212yes—
11C2×C4×D212C2211yescharacteristic
12#16810534C412yescomplemented
13#16810534C2212yes—
14#16810524—21no—
15#16810488—21no—
16#16810480—21no—
17#16810480—21no—
18C4×C4⋊Dic53C411yescharacteristic, complemented
19C2×C4⋊Q8×C53C411yescharacteristic, complemented
20C2×#11719936—41no—
21C2×C22⋊#6157—22no—
22C2×C4×#6157—41no—
23#13133276—24no—
24#13133274—41no—
25C22⋊#88324—22no—
26(C2×#6157)⋊C4—41no—
27C22×D212D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C4×D212—41no—
31C4×D212—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×#11719921C2×C411yescharacteristic
37C2×#11719921C2311yescharacteristic
38C2×C4×Dic53—21no—
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C22×Q8×C53C2×C412yes—
43C4⋊Q8×C53—41no—
44C4⋊Q8×C53—41no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C23×D53C2×D411yescharacteristic
50C22×#6157—42no—
51C2×Q8×C53—42no—
52C2×Q8×C53—82no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×D212—22no—
61C2×D212—22no—
62C2×D212—41no—
63C2×C4×D53—21no—
64C2×C4×D53—21no—
65C2×C4×D53—21no—
66C2×C4×D53—21no—
67C2×C4×D53—42no—
68C2×C4×D53—42no—
69C22⋊#6157—24no—
70C22⋊#6157—42no—
71C2×#88324—42no—
72C4×#6157—82no—
73#11719936—82no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C42×C53—21no—
78C42×C53—21no—
79#11719921—41no—
80#11719921—41no—
81#11719921—82no—
82#11719921—82no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85C4×Dic53—42no—
86C23×C53C2×C22⋊C411yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53#13712yes—
97C2×#6157—44no—
98C2×#6157—82no—
99C2×#6157—82no—
100Q8×C53—84no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—41no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#10412yes—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119D212—44no—
120C4×D53—42no—
121C4×D53—42no—
122C4×D53—81no—
123C4×D53—81no—
124#88324—82no—
125#11787—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53C2×#9911yescharacteristic
129C22×C53#41911yescharacteristic
130C22×C53C2×#10411yescharacteristic
131C2×D53—22no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—81no—
140C2×D53—81no—
141#6157—84no—
142C4×C53—21no—
143C4×C53—22no—
144C4×C53—41no—
145C4×C53—82no—
146C4×C53#35412yes—
147Dic53—41no—
148Dic53—41no—
149Dic53—82no—
150#2563—531no—
151C4×#99—531no—
152#1465—531no—
153#1351—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53#150811yescharacteristic
157C2×C53#212211yescharacteristic
158C2×C53C2×#35411yescharacteristic
159D53—42no—
160D53—81no—
161C2×C4⋊Q8#615711yescharacteristic, complemented
162C2×C4×D4—531no—
163C2×#99—1061no—
164#407—1061no—
165#393—532no—
166#393—532no—
167#390—1061no—
168#382—1061no—
169#380—1061no—
170#380—1061no—
171C4×C4⋊C4—531no—
172C4×C22⋊C4—1061no—
173#342—532no—
174#324—1061no—
175#323—532no—
176C53#1178711yescharacteristic, complemented, Sylow
177C22×Q8C2×#615712yes—
178C22×D4—531no—
179C23×C4—531no—
180C23×C4—531no—
181C2×M16—2121no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4×D4—2121no—
185C4×D4—2121no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—1062no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1062no—
195C2×C42—1061no—
196C2×C42—2121no—
197C2×C42C2×#615711yescharacteristic
198#99—1064no—
199C22⋊C8—1062no—
200C22⋊C8—2121no—
201#95—532no—
202#95—1062no—
203#95—1062no—
204#95—2121no—
205C24—531no—
206C24—531no—
207C2×Q8—42no—
208C2×Q8—82no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—2121no—
212C22×C4—22no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—1061no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—1062no—
220C22×C4—2122no—
221C22×C4—2122no—
222C22×C4—2122no—
223C22×C4C22⋊#615711yescharacteristic
224C22×C4C22⋊#615711yescharacteristic
225C22×C4C22×#615711yescharacteristic
226M16—4242no—
227C2×C8—2122no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—2121no—
232C4⋊C4—2122no—
233C22⋊C4—1064no—
234C22⋊C4—4242no—
235C22⋊C4—4242no—
236C22⋊C4—2121no—
237C22⋊C4—2121no—
238C22⋊C4—2122no—
239C42—21no—
240C42—21no—
241C42—4242no—
242C42—2122no—
243C23—532no—
244C23—1061no—
245C23—1061no—
246C23—1061no—
247C23—1061no—
248C23—1061no—
249C23—2121no—
250C23—2121no—
251C23—2121no—
252C23—2121no—
253C23C2×C22⋊#615711yescharacteristic
254Q8—84no—
255D4—2124no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—4241no—
267C2×C4—4241no—
268C2×C4—4242no—
269C2×C4—4242no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2124no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4#1313327511yescharacteristic
279C2×C4#1313327511yescharacteristic
280C2×C4#1313328212yes—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22—1062no—
285C22—4241no—
286C22—4241no—
287C22—2122no—
288C22—2122no—
289C22—2121no—
290C22—2121no—
291C22—2121no—
292C22—2121no—
293C22—2121no—
294C22C2×#1313327511yescharacteristic
295C22#1681161611yescharacteristic
296C22C2×#1313328211yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—82no—
301C4—4242no—
302C4—4244no—
303C4—2121no—
304C4—2121no—
305C4#1681152912yes—
306C2—21no—
307C2—21no—
308C2—4241no—
309C2—2122no—
310C2#2029011511yescharacteristic
311C2#2029097411yescharacteristic
312C2C2×#1681152911yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212D4
cosocleC2×#11719921C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11787
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
3. #20289710C2semidirect
12. #16810534C4semidirect
18. C4×C4⋊Dic53C4semidirect
19. C2×C4⋊Q8×C53C4semidirect
161. C2×C4⋊Q8#6157semidirect
176. C53#11787semidirect

Maximal subgroups

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

subgroupindex
C4×#131332762
#202899722
#202897102
#202798892
#1178753