Finite Group Databasebeta

← #27294281

Subgroups — #27294281

Order 13568.

468 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 115 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#20274448C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272715C214yesmaximal
4C53×#2621C211yesmaximal, characteristic
5C2×#13133176—21no—
6#16810676—21no—
7#16810390—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×C22⋊Q8×C53—21no—
13C2×#117×C53C2211yescharacteristic
14C53×#398—21no—
15C53×#398C412yescomplemented
16C53×#397—21no—
17C53×#397C2212yes—
18C53×#391—21no—
19C53×#382C411yescharacteristic, complemented
20#13133176—24no—
21#13133176—42no—
22C2×C4⋊Dic53—41no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—41no—
25C4⋊#88322—41no—
26C22⋊#88322—22no—
27C4×#88322—41no—
28C22×Q8×C53—21no—
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32C22⋊Q8×C53—42no—
33C22⋊Q8×C53—42no—
34#117×C53—41no—
35#117×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—21no—
48#95×C53—22no—
49#95×C53—22no—
50#95×C53C2×C412yes—
51C24×C53C2×D411yescharacteristic
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53—42no—
55C2×Q8×C53—42no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53D812yes—
66C22×C4×C53SD1612yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—44no—
75C4⋊C4×C53—42no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—42no—
79C22⋊C4×C53—42no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—24no—
85C22⋊Dic53—42no—
86C4⋊Dic53—44no—
87C4×Dic53—44no—
88C2×#88322—42no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C23×C53C2×D811yescharacteristic
95C23×C53C2×SD1611yescharacteristic
96Q8×C53—82no—
97Q8×C53—82no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53#9911yescharacteristic
120C2×C4×C53#10212yes—
121C2×C4×C53#10212yes—
122C2×C4×C53#10412yes—
123C2×Dic53—44no—
124C2×Dic53—82no—
125C2×Dic53—82no—
126#88322—82no—
127#11694—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53C2×#9911yescharacteristic
138C22×C53C2×#10211yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—82no—
146C4×C53—82no—
147C4×C53#32712yes—
148C4×C53#32712yes—
149Dic53—84no—
150#2621D5311yescharacteristic
151#1740—531no—
152#1458—531no—
153#1344—534no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—41no—
157C2×C53—41no—
158C2×C53C2×#32711yescharacteristic
159C2×C53#212311yescharacteristic
160C2×C53#212911yescharacteristic
161C2×C4⋊Q8Dic5311yescharacteristic, complemented
162C2×C22⋊Q8—21no—
163C2×#117—1061no—
164C2×#117C2×D5311yescharacteristic
165#431—1061no—
166#398—21no—
167#398Dic5312yescomplemented
168#397—21no—
169#397C2×D5312yes—
170#391—21no—
171#382Dic5311yescharacteristic, complemented
172#342—532no—
173#342—1061no—
174#324—1061no—
175#323—532no—
176C53#1169411yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C23×C4C22⋊D5311yescharacteristic
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—41no—
184#117—41no—
185#117—1064no—
186#117—2122no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—2121no—
191C2×C4⋊C4C2×Dic5312yes—
192C2×C4⋊C4C2×Dic5311yescharacteristic
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4C22⋊D5311yescharacteristic
197C2×C22⋊C4C22×D5311yescharacteristic
198C2×C42—2121no—
199C2×C42C2×Dic5311yescharacteristic
200C2×C42C22⋊D5311yescharacteristic
201C4⋊C8—2121no—
202C22⋊C8—1062no—
203C4×C8—2121no—
204#95—22no—
205#95—21no—
206#95—22no—
207#95C2×Dic5312yes—
208C24C2×C22⋊D5311yescharacteristic
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—2122no—
219C22×C4#1171992212yes—
220C22×C4#1171992312yes—
221C22×C4C22⋊Dic5311yescharacteristic
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22×Dic5311yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227C2×C8—2122no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—2124no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—1064no—
240C22⋊C4—2122no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—2124no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊Dic5311yescharacteristic
251C23C2×#1171992211yescharacteristic
252C23C2×#1171992311yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—4242no—
276C2×C4—4242no—
277C2×C4—2124no—
278C2×C4#1313313211yescharacteristic
279C2×C4#1313313211yescharacteristic
280C2×C4#1313313812yes—
281C2×C4#1313313812yes—
282C2×C4#1313314412yes—
283C8—4242no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#1313313211yescharacteristic
294C22C2×#1313313811yescharacteristic
295C22C2×#1313314411yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—4244no—
304C4#1681035712yes—
305C4#1681035712yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#1681035711yescharacteristic
311C2#2027636911yescharacteristic
312C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2621C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2621D53
hypocenterC53#11694
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C53C4semidirect
15. C53×#398C4semidirect
19. C53×#382C4semidirect
161. C2×C4⋊Q8Dic53semidirect
167. #398Dic53semidirect
171. #382Dic53semidirect
176. C53#11694semidirect

Maximal subgroups

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

subgroupindex
#202744482
#202730292
#202727152
C53×#26212
#1169453