Finite Group Databasebeta

← #27294666

Subgroups — #27294666

Order 13568.

480 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133132C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272715C214yesmaximal, complemented
4C53×#2547C211yesmaximal, characteristic
5C2×#13133132—21no—
6#16810587—21no—
7C4×C22⋊Dic53—21no—
8#16810390C2212yes—
9#16810349—21no—
10#16810347C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×#124×C53—21no—
13C2×C4×D4×C53C2211yescharacteristic
14C53×#397C412yescomplemented
15C53×#397C2212yes—
16C53×#391—21no—
17C53×#380—21no—
18C53×#380—21no—
19C4×C4⋊C4×C53C411yescharacteristic, complemented
20C2×#11719917—41no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53—41no—
23#13133132—24no—
24C22⋊#88322—22no—
25C22⋊#88322—41no—
26#13133124—41no—
27C22×Q8×C53—21no—
28C22×D4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4⋊Q8×C53—41no—
32C4⋊Q8×C53—41no—
33#124×C53—44no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53—21no—
45C2×C42×C53C2×C411yescharacteristic
46#95×C53—22no—
47#95×C53—22no—
48#95×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C2×D411yescharacteristic
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53C2×Q8×C53—42no—
54C2×D4×C53—22no—
55C2×D4×C53—22no—
56C2×D4×C53—41no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—22no—
62C22×C4×C53—22no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—44no—
79C22⋊C4×C53—44no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—42no—
83C22⋊Dic53—24no—
84C22⋊Dic53—42no—
85C4×Dic53—82no—
86#11719917—82no—
87C2×#88322—42no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53AGL(1,ℤ/8)12yes—
99Q8×C53—84no—
100D4×C53—44no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
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—81no—
119C2×C4×C53—81no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×C4×C53#10412yes—
123C2×Dic53—44no—
124C2×Dic53—82no—
125C2×Dic53—82no—
126#88322—82no—
127#11777—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53C2×#9911yescharacteristic
141C22×C53#41811yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—82no—
149C4×C53—82no—
150C4×C53#35412yes—
151Dic53—84no—
152#2547D5311yescharacteristic
153C4×#99—531no—
154#1465—531no—
155#1344—534no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—81no—
160C2×C53#150711yescharacteristic
161C2×C53#212311yescharacteristic
162C2×C53C2×#35411yescharacteristic
163C2×C4⋊Q8Dic5311yescharacteristic, complemented
164C2×#124—21no—
165C2×C4×D4C2×D5311yescharacteristic
166C2×#99—1061no—
167#407—1061no—
168#397Dic5312yescomplemented
169#397C2×D5312yes—
170#391—21no—
171#380—21no—
172#380—21no—
173C4×C4⋊C4Dic5311yescharacteristic, complemented
174C4×C22⋊C4—1061no—
175#342—532no—
176#324—1061no—
177#323—532no—
178C53#1177711yescharacteristic, complemented, Sylow
179C22×Q8—21no—
180C22×D4C22⋊D5311yescharacteristic
181C23×C4C22⋊D5311yescharacteristic
182C23×C4C22⋊D5311yescharacteristic
183C2×M16—2121no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186#124—44no—
187C4×D4—41no—
188C4×D4—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic5312yes—
192C2×C4⋊C4C22⋊D5311yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1062no—
196C2×C22⋊C4C2×Dic5311yescharacteristic
197C2×C22⋊C4C22×D5311yescharacteristic
198C2×C42—21no—
199C2×C42—2121no—
200C2×C42C2×Dic5311yescharacteristic
201#99—1064no—
202C22⋊C8—1062no—
203C22⋊C8—2121no—
204#95—22no—
205#95—22no—
206#95—2121no—
207#95C2×Dic5312yes—
208C24C22⋊Dic5311yescharacteristic
209C24C2×C22⋊D5311yescharacteristic
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—2122no—
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22×Dic5311yescharacteristic
229C22×C4C2×C22⋊D5311yescharacteristic
230M16—4242no—
231C2×C8—2122no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—1064no—
242C22⋊C4—2122no—
243C42—21no—
244C42—21no—
245C42—42no—
246C42—4242no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic5311yescharacteristic
257C23#1313324312yes—
258Q8—84no—
259D4—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—4242no—
280C2×C4—4242no—
281C2×C4—2124no—
282C2×C4#1313313211yescharacteristic
283C2×C4#1313313211yescharacteristic
284C2×C4#1313314412yes—
285C8—4242no—
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22C2×#1313313211yescharacteristic
299C22#1681062811yescharacteristic
300C22C2×#1313314411yescharacteristic
301C4—22no—
302C4—21no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—82no—
307C4—82no—
308C4—4244no—
309C4#1681041812yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2027323111yescharacteristic
315C2#2027636911yescharacteristic
316C2C2×#1681041811yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2547C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2547D53
hypocenterC53#11777
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #20272715C2semidirect
11. C2×C4⋊Q8×C53C4semidirect
14. C53×#397C4semidirect
19. C4×C4⋊C4×C53C4semidirect
163. C2×C4⋊Q8Dic53semidirect
168. #397Dic53semidirect
173. C4×C4⋊C4Dic53semidirect
178. C53#11777semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730532
#202727152
C53×#25472
#1177753