Finite Group Databasebeta

← #27294705

Subgroups — #27294705

Order 13568.

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

normal subgroups only

This group has 462 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#20273056C211yesmaximal, characteristic
3#20272739C214yesmaximal, complemented
4C53×#2487C211yesmaximal, characteristic
5#16810680—21no—
6C2×#13133132—21no—
7C4×C22⋊Dic53—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C2×C4×D4×C53C2211yescharacteristic
12C22×C4⋊C4×C53C411yescharacteristic, complemented
13C53×#400C412yescomplemented
14C53×#399C2212yes—
15C53×#389—21no—
16C53×#386—21no—
17C53×#385—21no—
18C53×#380—21no—
19C53×#380C411yescharacteristic, complemented
20C2×C22⋊Dic53—22no—
21C2×C4×Dic53—41no—
22C4⋊#88322—41no—
23#13133132—24no—
24C22⋊#88322—22no—
25#13133128—41no—
26#13133124—41no—
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4×D4×C53—41no—
32C4×D4×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—41no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C24×C53C2×D411yescharacteristic
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—41no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—82no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—82no—
75C22⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—82no—
79C42×C53—21no—
80C42×C53—21no—
81C22⋊Dic53—24no—
82C22⋊Dic53—42no—
83C4×Dic53—82no—
84C2×#88322—42no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53#10412yes—
94C23×C53C2×C22⋊C411yescharacteristic
95C23×C53#13712yes—
96D4×C53—44no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—81no—
117C2×C4×C53—84no—
118C2×C4×C53#9911yescharacteristic
119C2×C4×C53#9911yescharacteristic
120C2×Dic53—44no—
121C2×Dic53—82no—
122C2×Dic53—82no—
123#88322—82no—
124#11786—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—22no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—81no—
137C22×C53C2×#9911yescharacteristic
138C22×C53#41911yescharacteristic
139C22×C53C2×#10411yescharacteristic
140C4×C53—21no—
141C4×C53—41no—
142C4×C53—41no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—82no—
146C4×C53—82no—
147C4×C53#35212yes—
148Dic53—84no—
149#2487D5311yescharacteristic
150C4×#99—531no—
151#1466—531no—
152#1352—534no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53—42no—
156C2×C53—42no—
157C2×C53#150811yescharacteristic
158C2×C53C2×#35211yescharacteristic
159C2×C53#212811yescharacteristic
160C2×C4×D4C2×D5311yescharacteristic
161C22×C4⋊C4Dic5311yescharacteristic, complemented
162#432—1061no—
163C2×#99—1061no—
164#400Dic5312yescomplemented
165#399C2×D5312yes—
166#389—21no—
167#386—21no—
168#385—21no—
169#380—21no—
170#380Dic5311yescharacteristic, complemented
171C4×C22⋊C4—1061no—
172#342—532no—
173#323—532no—
174#323—1061no—
175C53#1178611yescharacteristic, complemented, Sylow
176C22×D4C22⋊D5311yescharacteristic
177C23×C4—21no—
178C23×C4C2×Dic5311yescharacteristic
179C23×C4C22⋊D5311yescharacteristic
180C4×D4—41no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×Dic5312yes—
188C2×C4⋊C4C2×Dic5311yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—1062no—
192C2×C22⋊C4C22⋊D5311yescharacteristic
193C2×C22⋊C4C22×D5311yescharacteristic
194C2×C42—2121no—
195C2×C42C22⋊D5311yescharacteristic
196C4⋊C8—2121no—
197#99—1064no—
198C22⋊C8—1062no—
199#97—2121no—
200#95—22no—
201#95—22no—
202#95—2121no—
203#95C2×Dic5312yes—
204C24C22⋊Dic5311yescharacteristic
205C24C2×C22⋊D5311yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—82no—
219C22×C4—2122no—
220C22×C4C22⋊Dic5311yescharacteristic
221C22×C4C22⋊Dic5311yescharacteristic
222C22×C4C22⋊Dic5311yescharacteristic
223C22×C4C22×Dic5311yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225C2×C8—2122no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C22⋊C4—1064no—
237C22⋊C4—2122no—
238C42—21no—
239C42—21no—
240C42—4242no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#1313314412yes—
250C23C2×C22⋊Dic5311yescharacteristic
251C23#1313324812yes—
252D4—44no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—84no—
274C2×C4—4242no—
275C2×C4—4242no—
276C2×C4—2124no—
277C2×C4#1313313211yescharacteristic
278C2×C4#1313313211yescharacteristic
279C8—4242no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#1313313211yescharacteristic
293C22#1681063211yescharacteristic
294C22C2×#1313314411yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—4244no—
303C4#1681041312yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#2027323511yescharacteristic
309C2C2×#1681041311yescharacteristic
310C2#2027638711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2487C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2487D53
hypocenterC53#11786
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
3. #20272739C2semidirect
12. C22×C4⋊C4×C53C4semidirect
13. C53×#400C4semidirect
19. C53×#380C4semidirect
161. C22×C4⋊C4Dic53semidirect
164. #400Dic53semidirect
170. #380Dic53semidirect
175. C53#11786semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730562
#202727392
C53×#24872
#1178653