Finite Group Databasebeta

← #27293740

Subgroups — #27293740

Order 13568.

424 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279888C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272658C214yesmaximal, complemented
4C53×#1465C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×C22⋊#88322—21no—
7#16810535—21no—
8#16810535—21no—
9#16810526C412yes—
10#16810488C2212yes—
11#16810483—21no—
12#16810483—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349C2212yes—
15#16810347—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324C2212yes—
19C53×#323—21no—
20C22×Dic106—21no—
21C22×#88322—41no—
22#13133217—41no—
23#13133217—41no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C4×Dic53C2×C412yes—
30C22⋊#88322—22no—
31C22⋊#88322—41no—
32#13133124—21no—
33#13133124—22no—
34#13133124—22no—
35C22×D4×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C2×M16×C53—41no—
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C22⋊C8×C53—22no—
46C22⋊C8×C53—41no—
47C24×C53C22⋊C411yescharacteristic
48C24×C53C22⋊C411yescharacteristic
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C2×Dic106—42no—
66C2×Dic106—82no—
67M16×C53—82no—
68C2×C8×C53—42no—
69C4⋊C4×C53—41no—
70C22⋊C4×C53—41no—
71C22⋊C4×C53—41no—
72C42×C53—21no—
73C42×C53—21no—
74C22⋊Dic53—82no—
75C22⋊Dic53—82no—
76C4⋊Dic53—42no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80C4×Dic53—24no—
81C2×#88322—42no—
82C2×#88322—82no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—41no—
89C23×C53—41no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53#10112yes—
93C23×C53C2×C22⋊C411yescharacteristic
94D4×C53—44no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—81no—
107C2×C4×C53—81no—
108C2×C4×C53#9911yescharacteristic
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#13112yes—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117Dic106—84no—
118C8×C53—82no—
119#88322—82no—
120#11569—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—22no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53#40811yescharacteristic
134C22×C53C2×#9911yescharacteristic
135C22×C53C2×#10111yescharacteristic
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53#35412yes—
142Dic53—82no—
143Dic53—82no—
144#2563—531no—
145#1465D5311yescharacteristic
146#1459—531no—
147#1324—534no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—42no—
151C2×C53—81no—
152C2×C53#147211yescharacteristic
153C2×C53C2×#35411yescharacteristic
154C2×C53#213411yescharacteristic
155C2×C4⋊Q8—531no—
156C2×C4×D4C2×D5311yescharacteristic
157#407—21no—
158C2×C22⋊C8—1061no—
159#393—1061no—
160#393—1061no—
161#390—532no—
162#382—532no—
163#380—1061no—
164#380—1061no—
165C4×C4⋊C4—531no—
166#324—532no—
167#324C2×D5312yes—
168#323—21no—
169#323—1061no—
170C53#1156911yescharacteristic, complemented, Sylow
171C22×Q8—1061no—
172C22×D4C4×D5311yescharacteristic
173C23×C4D21211yescharacteristic
174C23×C4C22⋊D5311yescharacteristic
175C2×M16—41no—
176C22×C8—2121no—
177C4⋊Q8—2121no—
178C4⋊Q8—2121no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C22×D5311yescharacteristic
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1062no—
187C2×C22⋊C4D21211yescharacteristic
188C2×C22⋊C4C22⋊D5311yescharacteristic
189C2×C42—532no—
190C2×C42C4×D5311yescharacteristic
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—1062no—
194C22⋊C8—2121no—
195#95—1062no—
196#95—1062no—
197#95—1061no—
198C24#1171992111yescharacteristic
199C24#1171992111yescharacteristic
200C2×Q8—4242no—
201C2×Q8—2122no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4#1171992111yescharacteristic
214C22×C4#1171992111yescharacteristic
215C22×C4C2×C4×D5311yescharacteristic
216C22×C4C2×D21211yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218M16—82no—
219C2×C8—42no—
220C2×C8—4242no—
221C2×C8—2122no—
222C4⋊C4—41no—
223C4⋊C4—2122no—
224C4⋊C4—2122no—
225C4⋊C4—2121no—
226C4⋊C4—2121no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—4242no—
230C22⋊C4—4242no—
231C42—21no—
232C42—21no—
233C42—1064no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1313313512yes—
244C23C2×#1171992111yescharacteristic
245Q8—4244no—
246D4—44no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4#1313313111yescharacteristic
267C2×C4#1313313111yescharacteristic
268C2×C4#1313322512yes—
269C8—82no—
270C8—4242no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22#1681059111yescharacteristic
284C22C2×#1313313111yescharacteristic
285C22C2×#1313313511yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—4242no—
292C4—4242no—
293C4#1681041712yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2027308011yescharacteristic
299C2C2×#1681041711yescharacteristic
300C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11569
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20272658C2semidirect
170. C53#11569semidirect

Maximal subgroups

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

subgroupindex
#202798882
#202730332
#202726582
C53×#14652
#1156953