Finite Group Databasebeta

← #27287605

Subgroups — #27287605

Order 13568.

630 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 133 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#20274871C211yesmaximal, characteristic, complemented
2#20274715C212yesmaximal, complemented
3C2×D4⋊#88322C212yesmaximal, complemented
4#20272598C211yesmaximal, characteristic, complemented
5C53×#2537C211yesmaximal, characteristic
6C2×#13133148—22no—
7C2×C4⋊#88322C2212yes—
8C2×#13133138—22no—
9C2×C4×#88322C2211yescharacteristic
10#16810493C2211yescharacteristic
11#16810385—22no—
12D4⋊#88322—24no—
13C2×#121×C53—22no—
14C2×C4×D4×C53C2212yes—
15C53×#394—22no—
16C53×#384C2211yescharacteristic
17C53×#380—22no—
18C22×#88322—21no—
19C22×#88322—22no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53—21no—
22#13133148—44no—
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25#13133138—44no—
26C4×#88322—21no—
27C4×#88322D412yescomplemented
28C22×D4×C53—22no—
29C22×D4×C53D412yes—
30C23×C4×C53—22no—
31#121×C53—44no—
32#121×C53—44no—
33C4×D4×C53—22no—
34C4×D4×C53—22no—
35C4×D4×C53—22no—
36C4×D4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53D412yes—
40C2×C22⋊C4×C53—24no—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#95×C53—22no—
44C24×C53—22no—
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—22no—
48C2×D4×C53—22no—
49C2×D4×C53—22no—
50C2×D4×C53—48no—
51C2×D4×C53—82no—
52C2×D4×C53SD1614yes—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—41no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C22⋊C4×C53—48no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C42×C53C2×D411yescharacteristic
73C42×C53#4111yescharacteristic
74C42×C53#4112yes—
75C4⋊Dic53—42no—
76C4⋊Dic53—42no—
77C4⋊Dic53—42no—
78C2×#88322—22no—
79C2×#88322—22no—
80C2×#88322—22no—
81C2×#88322—42no—
82C2×#88322—42no—
83C2×#88322—42no—
84C23×C53—24no—
85C23×C53—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53#12011yescharacteristic
92D4×C53—24no—
93D4×C53—24no—
94D4×C53—42no—
95D4×C53—42no—
96D4×C53—88no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—82no—
105C2×C4×C53—82no—
106C2×C4×C53—82no—
107C2×C4×C53—82no—
108C2×C4×C53#12111yescharacteristic
109C2×C4×C53#12112yes—
110C2×C4×C53#12312yes—
111C2×C4×C53#12411yescharacteristic
112C2×C4×C53C2×SD1612yes—
113C2×C4×C53#13512yes—
114C2×C4×C53AGL(1,ℤ/8)12yes—
115C2×C4×C53AGL(1,ℤ/8)12yes—
116C2×Dic53—82no—
117C2×Dic53—81no—
118#88322—24no—
119#88322—82no—
120#10432—531nomaximal, Sylow
121C22×C53—44no—
122C22×C53—44no—
123C22×C53—44no—
124C22×C53—82no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53#39411yescharacteristic
129C22×C53#44912yes—
130C22×C53#45012yes—
131C22×C53#45311yescharacteristic
132C22×C53#45611yescharacteristic
133C4×C53—21no—
134C4×C53—21no—
135C4×C53—82no—
136C4×C53—82no—
137C4×C53#46512yes—
138C4×C53#46912yes—
139C4×C53#48112yes—
140C4×C53#48212yes—
141Dic53—161no—
142#2537D5311yescharacteristic
143#1844—531no—
144#1802—532no—
145C2×D4⋊C8—532no—
146#1296—531no—
147C2×C53—84no—
148C2×C53#162411yescharacteristic
149C2×C53#162911yescharacteristic
150C2×C53#165412yes—
151C2×C53#203412yes—
152C2×C53#203911yescharacteristic
153C2×#121—22no—
154C2×C4×D4C2×D5312yes—
155C2×#106—1062no—
156C2×C4⋊C8—532no—
157C2×#102—1062no—
158C2×C4×C8—531no—
159#394—22no—
160#384—531no—
161#384C2×D5311yescharacteristic
162#380—22no—
163#340—1062no—
164D4⋊C8—1064no—
165C53#1043211yescharacteristic, complemented, Sylow
166C22×D4—22no—
167C22×D4C22⋊D5312yes—
168C23×C4—22no—
169C22×C8—1061no—
170C22×C8—1062no—
171#121—44no—
172#121—44no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—1062no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4C22⋊D5312yes—
181C2×C4⋊C4C22⋊D5312yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C42C22×D5311yescharacteristic
185#106—2124no—
186C4⋊C8—1062no—
187C4⋊C8—1062no—
188#102—2124no—
189C4×C8—532no—
190C4×C8—1061no—
191#95—22no—
192C24—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—48no—
199C2×D4—82no—
200C2×D4#1171992314yes—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—2121no—
210C22×C4C2×C22⋊D5311yescharacteristic
211C22×C4C2×C22⋊D5312yes—
212C2×C8—1062no—
213C2×C8—1062no—
214C2×C8—1062no—
215C2×C8—2122no—
216C2×C8—2122no—
217C2×C8—2122no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—2122no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C22⋊C4—48no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42D4×D5311yescharacteristic
230C42#1171994711yescharacteristic
231C42#1171994712yes—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23#1313318611yescharacteristic
240D4—24no—
241D4—24no—
242D4—42no—
243D4—42no—
244D4—88no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—4242no—
257C2×C4—4241no—
258C2×C4#1313318811yescharacteristic
259C2×C4#1313318912yes—
260C2×C4#1313319812yes—
261C2×C4#1313320411yescharacteristic
262C2×C4C2×#1171992312yes—
263C2×C4#1313323712yes—
264C2×C4#1313324212yes—
265C2×C4#1313324312yes—
266C8—1064no—
267C8—4242no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22#1681053711yescharacteristic
276C22#1681076812yes—
277C22#1681077412yes—
278C22#1681079111yescharacteristic
279C22#1681080611yescharacteristic
280C4—21no—
281C4—21no—
282C4—82no—
283C4—82no—
284C4—8481no—
285C4#1681085012yes—
286C4#1681087412yes—
287C4#1681093912yes—
288C4#1681094412yes—
289C2—84no—
290C2#2027379811yescharacteristic
291C2#2027383111yescharacteristic
292C2#2027399912yes—
293C2#2027590112yes—
294C2#2027593611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2537C2
socleC23×C53#120
hypercenter#2537D53
hypocenterC53#10432
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #20274871C2semidirect
2. #20274715C2semidirect
3. C2×D4⋊#88322C2semidirect
4. #20272598C2semidirect
27. C4×#88322D4semidirect
165. C53#10432semidirect

Maximal subgroups

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

subgroupindex
#202748712
#202747152
C2×D4⋊#883222
#202725982
C53×#25372
#1043253