Finite Group Databasebeta

← #27300079

Subgroups — #27300079

Order 13568.

544 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16810418C212yesmaximal
2#20274703C212yesmaximal
3#20272840C212yesmaximal
4C4×#124×C53C211yesmaximal, characteristic
5C2×#13133132C2214yes—
6#16810483—22no—
7#16810418—24no—
8#16810390—22no—
9C2×#124×C53C2211yescharacteristic
10C2×C4×Q8×C53C411yescharacteristic, complemented
11C2×C4×D4×C53C2211yescharacteristic
12C53×#391C411yescharacteristic, complemented
13C53×#390C2211yescharacteristic
14C53×#388—22no—
15C53×#382C411yescharacteristic, complemented
16C4×C22⋊C4×C53—22no—
17C43×C53C411yescharacteristic, complemented
18C2×C4⋊Dic53—42no—
19C2×C22⋊Dic53—24no—
20#13133132—24no—
21#13133132—24no—
22#13133124—42no—
23C22×Q8×C53C2×C411yescharacteristic
24C22×D4×C53C2311yescharacteristic
25C23×C4×C53D412yes—
26#124×C53—22no—
27#124×C53D412yes—
28#124×C53Q812yes—
29C4×Q8×C53—41no—
30C4×Q8×C53—41no—
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53C2×C411yescharacteristic
35C2×C4⋊C4×C53C2×C411yescharacteristic
36C2×C22⋊C4×C53—22no—
37C2×C22⋊C4×C53—22no—
38C2×C22⋊C4×C53D412yes—
39C2×C42×C53—21no—
40C2×C42×C53—21no—
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47C24×C53C2×D412yes—
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C4⋊C412yescomplemented
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C2×D411yescharacteristic
55C2×D4×C53C2×Q811yescharacteristic
56C2×D4×C53#4112yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—22no—
62C22×C4×C53—21no—
63C22×C4×C53—44no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×Dic53—44no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—24no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—44no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—42no—
80C42×C53—42no—
81C42×C53—41no—
82C42×C53—41no—
83C42×C53—41no—
84C42×C53C4⋊C412yescomplemented
85C22⋊Dic53—44no—
86C22⋊Dic53—44no—
87C4⋊Dic53—84no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53C2×C22⋊C411yescharacteristic
93C23×C53#12112yes—
94C23×C53C22⋊Q812yes—
95Q8×C53—42no—
96Q8×C53—42no—
97D4×C53—41no—
98D4×C53—41no—
99D4×C53—42no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—44no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—82no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53C2×C4⋊C411yescharacteristic
123C2×C4×C53#11711yescharacteristic
124C2×Dic53—84no—
125C2×Dic53—84no—
126C2×Dic53—84no—
127#12798—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—22no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—82no—
137C22×C53#38011yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53#41011yescharacteristic
140C4×C53—21no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—42no—
144C4×C53—42no—
145C4×C53—42no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—81no—
149C4×C53#35412yescomplemented
150Dic53—164no—
151C4×#124D5311yescharacteristic
152C2×#354—532no—
153#1799—532no—
154#1393—532no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53—42no—
159C2×C53#179911yescharacteristic
160C2×C53C2×#35411yescharacteristic
161C2×C53#212611yescharacteristic
162C2×#124C2×D5311yescharacteristic
163C2×C4×Q8Dic5311yescharacteristic, complemented
164C2×C4×D4C2×D5311yescharacteristic
165C2×#99—534no—
166#391Dic5311yescharacteristic, complemented
167#390C2×D5311yescharacteristic
168#388—22no—
169#382Dic5311yescharacteristic, complemented
170#380—1062no—
171C4×C22⋊C4—22no—
172C43Dic5311yescharacteristic, complemented
173#354—1064no—
174#342—1062no—
175C53#1279811yescharacteristic, complemented, Sylow
176C22×Q8C2×Dic5311yescharacteristic
177C22×D4C22×D5311yescharacteristic
178C23×C4C22⋊D5312yes—
179#124—22no—
180#124Dic10612yes—
181#124D21212yes—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—2122no—
188C2×C4⋊C4C2×Dic5311yescharacteristic
189C2×C4⋊C4C2×Dic5311yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—1064no—
193C2×C22⋊C4C22⋊D5312yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×Dic5311yescharacteristic
198C2×C42C2×Dic5311yescharacteristic
199C2×C42C2×Dic5311yescharacteristic
200#99—1064no—
201#99—1064no—
202#95—22no—
203#95—22no—
204#95—2122no—
205C24C2×C22⋊D5312yes—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊Dic5312yescomplemented
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C2×Dic10611yescharacteristic
213C2×D4C2×D21211yescharacteristic
214C2×D4#1171994512yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—2124no—
223C22×C4C22⋊Dic5312yes—
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22×Dic5311yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—4244no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—2124no—
236C22⋊C4—2124no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42C4⋊Dic5312yescomplemented
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23C2×C22⋊Dic5311yescharacteristic
251C23#1313319112yes—
252C23#1313319612yes—
253Q8—42no—
254Q8—42no—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—4244no—
279C2×C4—4244no—
280C2×C4—4244no—
281C2×C4#1313313211yescharacteristic
282C2×C4#1313313211yescharacteristic
283C2×C4C2×C4⋊Dic5311yescharacteristic
284C2×C4#1313317611yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22#1681048311yescharacteristic
295C22C2×#1313313211yescharacteristic
296C22#1681059711yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4—8484no—
307C4#1681041812yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027470311yescharacteristic
313C2C2×#1681041811yescharacteristic
314C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#124×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#124D53
hypocenterC53#12798
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
10. C2×C4×Q8×C53C4semidirect
12. C53×#391C4semidirect
15. C53×#382C4semidirect
17. C43×C53C4semidirect
51. C2×Q8×C53C4⋊C4semidirect
84. C42×C53C4⋊C4semidirect
149. C4×C53#354semidirect
163. C2×C4×Q8Dic53semidirect
166. #391Dic53semidirect
169. #382Dic53semidirect
172. C43Dic53semidirect
175. C53#12798semidirect
209. C2×Q8C4⋊Dic53semidirect
245. C42C4⋊Dic53semidirect
307. C4#16810418semidirect

Maximal subgroups

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

subgroupindex
C2×#168104182
#202747032
#202728402
C4×#124×C532
#1279853