Finite Group Databasebeta

← #27279147

Subgroups — #27279147

Order 13568.

616 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277318C211yesmaximal, characteristic
2#20274801C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272594C212yesmaximal
5C2×C53×#398C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810509—24no—
8#16810486—21no—
9#16810483C2212yes—
10C4×C22⋊Dic53—21no—
11#16810385—24no—
12#16810349—22no—
13C22×C4⋊C4×C53—21no—
14C22×C4⋊C4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#398—22no—
17C53×#398C414yescomplemented
18C2×#95×C53—21no—
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21C2×C4⋊Dic53D414yes—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—22no—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133124—22no—
27#13133124—24no—
28#13133124—22no—
29C23×C4×C53—21no—
30C23×C4×C53—21no—
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C2×C4⋊C4×C53—24no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—44no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44#95×C53—24no—
45#95×C53—21no—
46#95×C53C2×C412yes—
47C24×C53C2×D411yescharacteristic
48C22×C4×C53—24no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22⋊C412yes—
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C22×Dic53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C4⋊C4×C53—48no—
76C4⋊C4×C53—42no—
77C22⋊Dic53—42no—
78C22⋊Dic53—42no—
79C4⋊Dic53—44no—
80C4×Dic53—82no—
81C2×#88322—44no—
82C2×#88322—42no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53#12011yescharacteristic
87C23×C53#12311yescharacteristic
88C23×C53#13512yes—
89C23×C53AGL(1,ℤ/8)12yes—
90C2×C4×C53—24no—
91C2×C4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—48no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—81no—
110C2×C4×C53—82no—
111C2×C4×C53—82no—
112C2×C4×C53C4×D412yes—
113C2×C4×C53#12112yes—
114C2×Dic53—44no—
115C2×Dic53—44no—
116C2×Dic53—44no—
117#88322—82no—
118#9026—531nomaximal, Sylow
119C22×C53—24no—
120C22×C53—24no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53#38611yescharacteristic
126C22×C53#41611yescharacteristic
127C22×C53#41811yescharacteristic
128C22×C53#44912yes—
129C22×C53#45612yes—
130C22×C53#45812yes—
131C22×C53#48212yes—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—44no—
135C4×C53—42no—
136C4×C53—42no—
137C4×C53—82no—
138Dic53—84no—
139C2×#398D5311yescharacteristic
140#2313—531no—
141#1823—532no—
142C2×#324—531no—
143#1294—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53—22no—
147C2×C53#160914yes—
148C2×C53#187511yescharacteristic
149C2×C53#188611yescharacteristic
150C2×C53#189011yescharacteristic
151C22×C4⋊C4—21no—
152C22×C4⋊C4C2×D5311yescharacteristic
153C22×C4⋊C4C2×D5311yescharacteristic
154C2×C22⋊C8—532no—
155#398—22no—
156#398Dic5314yescomplemented
157#387—1064no—
158#381—1061no—
159#380—532no—
160C4×C22⋊C4—1061no—
161C2×#95—21no—
162C2×#95C2×D5311yescharacteristic
163#340—1064no—
164#324—1062no—
165C53#902611yescharacteristic, complemented, Sylow
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22⋊D5311yescharacteristic
169C23×C4C22⋊D5311yescharacteristic
170C23×C4C22×D5311yescharacteristic
171C22×C8—1062no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—534no—
180C2×C4⋊C4C2×Dic5312yes—
181C2×C4⋊C4C2×Dic5312yes—
182C2×C4⋊C4C22⋊D5312yes—
183C2×C22⋊C4—1062no—
184C2×C42—1062no—
185C22⋊C8—1062no—
186C22⋊C8—1062no—
187#95—21no—
188#95—24no—
189#95—1062no—
190#95—1064no—
191#95—1062no—
192#95C2×Dic5312yes—
193C24C2×C22⋊D5311yescharacteristic
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1064no—
212C22×C4C22⋊Dic5312yes—
213C22×C4C22⋊Dic5312yes—
214C22×C4D4×D5312yes—
215C22×C4#1171994712yes—
216C22×C4C22×Dic5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218C2×C8—2124no—
219C2×C8—2122no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—2124no—
226C22⋊C4—2122no—
227C22⋊C4—2122no—
228C42—4242no—
229C23—22no—
230C23—22no—
231C23C2×C22⋊Dic5311yescharacteristic
232C23#1313318511yescharacteristic
233C23#1313319811yescharacteristic
234C23#1313323712yes—
235C23#1313324312yes—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—48no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—2124no—
259C2×C4—2124no—
260C2×C4—2124no—
261C2×C4D4×Dic5312yes—
262C2×C4#1313318912yes—
263C8—4242no—
264C22—24no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22#1681050311yescharacteristic
271C22#1681061911yescharacteristic
272C22#1681062811yescharacteristic
273C22#1681076612yes—
274C22#1681080512yes—
275C22#1681081212yes—
276C22#1681094412yes—
277C4—22no—
278C4—22no—
279C4—44no—
280C4—42no—
281C4—42no—
282C4—82no—
283C4—4244no—
284C2—21no—
285C2—21no—
286C2—22no—
287C2#2027371014yes—
288C2#2027503811yescharacteristic
289C2#2027510911yescharacteristic
290C2#2027512911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#398C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#398D53
hypocenterC53#9026
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
17. C53×#398C4semidirect
156. #398Dic53semidirect
165. C53#9026semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202748012
C2×#168103492
#202725942
C2×C53×#3982
#902653