Finite Group Databasebeta

← #27293739

Subgroups — #27293739

Order 13568.

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

normal subgroups only

This group has 424 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
1#20273053C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272659C214yesmaximal, complemented
4C53×#2563C211yesmaximal, characteristic
5#16810587—21no—
6C2×C22⋊#88322—21no—
7#16810349C2212yes—
8#16810349C2212yes—
9#16810347—21no—
10#16810347—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C4×D4×C53C2211yescharacteristic
13C53×#393—21no—
14C53×#393—21no—
15C53×#390C412yes—
16C53×#382C2212yes—
17C53×#380—21no—
18C53×#380—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—41no—
21C22×#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24C22⋊#88322—41no—
25C22⋊#88322—41no—
26C22×Q8×C53—21no—
27C22×D4×C53C2×C411yescharacteristic
28C23×C4×C53D411yescharacteristic
29C23×C4×C53D411yescharacteristic
30C4⋊Q8×C53—41no—
31C4⋊Q8×C53—41no—
32C4×D4×C53—41no—
33C4×D4×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C412yes—
37C2×C4⋊C4×C53C2311yescharacteristic
38C2×C22⋊C4×C53—22no—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53D411yescharacteristic
42C2×C42×C53C2×C412yes—
43C2×C42×C53C2×C411yescharacteristic
44#95×C53—21no—
45#95×C53—22no—
46#95×C53—22no—
47C24×C53C22⋊C411yescharacteristic
48C24×C53C22⋊C411yescharacteristic
49C2×Q8×C53—42no—
50C2×Q8×C53—82no—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53—82no—
75C22⋊C4×C53—82no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—24no—
79#11719917—82no—
80C2×#88322—42no—
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
94Q8×C53—84no—
95D4×C53—44no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—81no—
114C2×C4×C53—81no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#13112yes—
118#88322—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—82no—
142C4×C53—82no—
143C4×C53#35412yes—
144#2563D5311yescharacteristic
145#1465—531no—
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⋊Q8Dic5311yescharacteristic
156C2×C4×D4C2×D5311yescharacteristic
157#407—1061no—
158C2×C22⋊C8—1061no—
159#393—21no—
160#393—21no—
161#390Dic5312yes—
162#382C2×D5312yes—
163#380—21no—
164#380—21no—
165C4×C4⋊C4Dic5311yescharacteristic
166#324—532no—
167#324—532no—
168#323—1061no—
169#323—1061no—
170C53#1156911yescharacteristic, complemented, Sylow
171C22×Q8—21no—
172C22×D4C2×Dic5311yescharacteristic
173C23×C4C22⋊D5311yescharacteristic
174C23×C4C22⋊D5311yescharacteristic
175C2×M16—2121no—
176C22×C8—2121no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×Dic5312yes—
184C2×C4⋊C4C22×D5311yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊D5311yescharacteristic
188C2×C22⋊C4C22⋊D5311yescharacteristic
189C2×C42C2×Dic5312yes—
190C2×C42C2×Dic5311yescharacteristic
191C22⋊C8—1062no—
192C22⋊C8—1062no—
193C22⋊C8—2121no—
194C22⋊C8—2121no—
195#95—22no—
196#95—22no—
197#95—21no—
198C24C22⋊Dic5311yescharacteristic
199C24C22⋊Dic5311yescharacteristic
200C2×Q8—42no—
201C2×Q8—82no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C22⋊Dic5311yescharacteristic
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22×Dic5311yescharacteristic
216C22×C4C2×C22⋊D5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218M16—4242no—
219C2×C8—4242no—
220C2×C8—2122no—
221C2×C8—2122no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#1313313612yes—
244C23C2×C22⋊Dic5311yescharacteristic
245Q8—84no—
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—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#1313313211yescharacteristic
267C2×C4#1313313211yescharacteristic
268C2×C4#1313322412yes—
269C8—4242no—
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#1681059011yescharacteristic
284C22C2×#1313313211yescharacteristic
285C22C2×#1313313611yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4#1681041812yes—
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2027307911yescharacteristic
299C2C2×#1681041811yescharacteristic
300C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2563C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2563D53
hypocenterC53#11569
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #20272659C2semidirect
170. C53#11569semidirect

Maximal subgroups

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

subgroupindex
#202730532
#202730332
#202726592
C53×#25632
#1156953