Finite Group Databasebeta

← #27281771

Subgroups — #27281771

Order 13568.

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

normal subgroups only

This group has 518 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#20279707C211yesmaximal, characteristic
2#20273440C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272572C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7C2×#13133204C412yes—
8#16810680C2212yes—
9#16810548—21no—
10#16810548—21no—
11#16810528C412yes—
12#16810483—21no—
13#16810483—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810371—24no—
16#16810347—21no—
17#16810347—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432C2212yes—
21C53×#407—21no—
22C2×C22⋊C8×C53—21no—
23C22×Dic106C2×C412yes—
24#13133217—22no—
25#13133217—22no—
26#13133204—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C4×Dic53C2×C412yes—
33C4⋊#88322—22no—
34C22⋊#88322—22no—
35#13133128—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—21no—
39C22×D4×C53C2×C411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C2×M16×C53—22no—
43C22×C8×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53C2311yescharacteristic
50C4⋊C8×C53D414yes—
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—21no—
63C22×C4×C53—21no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C2×Dic106—24no—
74C2×Dic106—42no—
75C2×Dic106—42no—
76M16×C53—82no—
77C2×C8×C53—24no—
78C2×C8×C53—82no—
79C4⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C42×C53C2×D412yes—
83C42×C53#4112yes—
84C22⋊Dic53—44no—
85C22⋊Dic53—44no—
86C4⋊Dic53—42no—
87C4⋊Dic53—41no—
88C4⋊Dic53—42no—
89C4⋊Dic53—41no—
90C4×Dic53—42no—
91C2×#88322—42no—
92C23×C53—21no—
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
103D4×C53—44no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—81no—
115C2×C4×C53—81no—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53#12112yes—
119C2×C4×C53#12311yescharacteristic
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126Dic106—84no—
127C8×C53—44no—
128#88322—82no—
129#9451—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—22no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53#38611yescharacteristic
143C22×C53C2×#10011yescharacteristic
144C22×C53#41311yescharacteristic
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53#45512yes—
151Dic53—82no—
152Dic53—82no—
153#2973D5311yescharacteristic
154#2547—531no—
155#1558—532no—
156#1466—531no—
157#1285—532no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53—42no—
161C2×C53—81no—
162C2×C53#188511yescharacteristic
163C2×C53#190011yescharacteristic
164C2×C53#190911yescharacteristic
165C2×C4⋊Q8—531no—
166C2×#124—532no—
167C2×C4×D4C2×D5311yescharacteristic
168#435—24no—
169#432—532no—
170#432C2×D5312yes—
171#407—21no—
172C2×C22⋊C8—21no—
173#397—1061no—
174#397—1061no—
175#391—532no—
176#380—1061no—
177#380—1061no—
178C4×C4⋊C4—531no—
179#333—1064no—
180#323—1061no—
181#323—1061no—
182C53#945111yescharacteristic, complemented, Sylow
183C22×Q8—532no—
184C22×D4C4×D5311yescharacteristic
185C23×C4C22⋊D5311yescharacteristic
186C23×C4C22⋊D5311yescharacteristic
187C2×M16—22no—
188C22×C8—22no—
189C4⋊Q8—1062no—
190C4⋊Q8—1062no—
191#124—2124no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1061no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C4×D5311yescharacteristic
198C2×C22⋊C4—1062no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4D21211yescharacteristic
201C2×C22⋊C4D21211yescharacteristic
202C2×C42—532no—
203C2×C42C22×D5311yescharacteristic
204C4⋊C8—1062no—
205C4⋊C8C22⋊D5314yes—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1062no—
210#97—22no—
211#97—1062no—
212#95—1061no—
213#95—1062no—
214#95—1062no—
215C24#1171992111yescharacteristic
216C24#1171992111yescharacteristic
217C2×Q8—1064no—
218C2×Q8—2122no—
219C2×Q8—2122no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4#1171992111yescharacteristic
232C22×C4#1171992111yescharacteristic
233C22×C4C2×C4×D5311yescharacteristic
234C22×C4C2×D21211yescharacteristic
235C22×C4C2×C22⋊D5311yescharacteristic
236M16—82no—
237C2×C8—24no—
238C2×C8—82no—
239C2×C8—2122no—
240C4⋊C4—22no—
241C4⋊C4—2122no—
242C4⋊C4—2121no—
243C4⋊C4—2122no—
244C4⋊C4—2121no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—2124no—
248C22⋊C4—2124no—
249C42—2122no—
250C42#1171994512yes—
251C42C2×C22⋊D5312yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1313313312yes—
262C23C2×#1171992111yescharacteristic
263Q8—4244no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4C4×C22⋊D5312yes—
284C2×C4#1313318611yescharacteristic
285C2×C4#1313319112yes—
286C2×C4#1313320011yescharacteristic
287C8—44no—
288C8—4242no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1681050211yescharacteristic
302C22C2×#1313313311yescharacteristic
303C22#1681060511yescharacteristic
304C4—22no—
305C4—21no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—4242no—
310C4—4242no—
311C4#1681080112yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2027510311yescharacteristic
317C2#2027517811yescharacteristic
318C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9451
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20273440C2semidirect
4. #20272572C2semidirect
182. C53#9451semidirect

Maximal subgroups

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

subgroupindex
#202797072
#202734402
#202730562
#202725722
C53×#29732
#945153