Finite Group Databasebeta

← #27282051

Subgroups — #27282051

Order 13568.

544 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279023C211yesmaximal, characteristic
2#20274793C212yesmaximal, complemented
3#20273053C211yesmaximal, characteristic
4#20272620C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133198C412yes—
7C22×C4⋊Dic53C2211yescharacteristic
8#16810587C2212yes—
9#16810553—21no—
10#16810548—21no—
11#16810521—21no—
12#16810484C412yes—
13#16810483—21no—
14#16810483C2211yescharacteristic
15#16810391—24no—
16#16810349—21no—
17#16810347—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432—21no—
21C53×#407C2212yes—
22C2×C22⋊C8×C53—21no—
23C23×Dic53C2×C412yes—
24C2×#11719917—22no—
25#13133198—44no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53C2×C412yes—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—21no—
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42C2×M16×C53D414yes—
43C22×C8×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C4⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54#97×C53—22no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C22×C411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—41no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76M16×C53—44no—
77C2×C8×C53—24no—
78C2×C8×C53—82no—
79C4⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C42×C53—21no—
83C42×C53—21no—
84C22⋊Dic53—44no—
85C22⋊Dic53—42no—
86C4⋊Dic53—42no—
87C4⋊Dic53—41no—
88C4⋊Dic53—42no—
89C4⋊Dic53—41no—
90#11719919—44no—
91#11719917—44no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53#10012yes—
102C23×C53C2×C22⋊C411yescharacteristic
103C23×C53C4×D412yes—
104D4×C53—44no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—81no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12112yes—
120C2×C4×C53#12311yescharacteristic
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—84no—
128C8×C53—44no—
129#88322—82no—
130#9507—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—22no—
134C22×C53—22no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—81no—
143C22×C53#38611yescharacteristic
144C22×C53C2×#10011yescharacteristic
145C22×C53#41311yescharacteristic
146C4×C53—22no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53#45812yes—
152Dic53—82no—
153Dic53—82no—
154#2973D5311yescharacteristic
155#2486—531no—
156#1821—532no—
157#1465—531no—
158#1305—532no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—42no—
162C2×C53—42no—
163C2×C53#189011yescharacteristic
164C2×C53#190011yescharacteristic
165C2×C53#191511yescharacteristic
166C2×#123—532no—
167C2×C4×D4C2×D5311yescharacteristic
168C22×C4⋊C4—531no—
169#435—24no—
170#432—21no—
171#407—532no—
172#407C2×D5312yes—
173C2×C22⋊C8—21no—
174#398—1061no—
175#397—1061no—
176#389—1061no—
177#381—532no—
178#380—531no—
179#380—1061no—
180#343—1064no—
181#324—1061no—
182#323—1061no—
183C53#950711yescharacteristic, complemented, Sylow
184C22×D4C4×D5311yescharacteristic
185C23×C4—532no—
186C23×C4C22⋊D5311yescharacteristic
187C23×C4C22×D5311yescharacteristic
188C2×M16—1062no—
189C2×M16C22⋊D5314yes—
190C22×C8—22no—
191#123—2124no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1062no—
200C2×C4⋊C4D21211yescharacteristic
201C2×C22⋊C4—532no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4C4×D5311yescharacteristic
204C2×C22⋊C4D21211yescharacteristic
205C2×C42C22⋊D5311yescharacteristic
206C4⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—1062no—
211C22⋊C8—1062no—
212#97—22no—
213#95—1061no—
214#95—1062no—
215#95—1062no—
216C24#1171992111yescharacteristic
217C24C2×C4×D5311yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1064no—
226C22×C4—1062no—
227C22×C4—1062no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4#1171992111yescharacteristic
231C22×C4#1171992111yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4C2×D21211yescharacteristic
234C22×C4#1171994512yes—
235C22×C4C2×C22⋊D5312yes—
236C22×C4C2×C22⋊D5311yescharacteristic
237M16—44no—
238M16—2124no—
239C2×C8—24no—
240C2×C8—82no—
241C2×C8—2122no—
242C4⋊C4—22no—
243C4⋊C4—2124no—
244C4⋊C4—2122no—
245C4⋊C4—2121no—
246C4⋊C4—2122no—
247C4⋊C4—2121no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2124no—
251C22⋊C4—2122no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#1313313312yes—
263C23C2×#1171992111yescharacteristic
264C23C4×C22⋊D5312yes—
265D4—44no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—4244no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4#1313318611yescharacteristic
287C2×C4#1313319112yes—
288C2×C4#1313320011yescharacteristic
289C8—44no—
290C8—4242no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1681050211yescharacteristic
304C22C2×#1313313311yescharacteristic
305C22#1681060511yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—4242no—
312C4—4242no—
313C4#1681081312yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2027512811yescharacteristic
319C2#2027517811yescharacteristic
320C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#9507
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20274793C2semidirect
4. #20272620C2semidirect
183. C53#9507semidirect

Maximal subgroups

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

subgroupindex
#202790232
#202747932
#202730532
#202726202
C53×#29732
#950753