Finite Group Databasebeta

← #27279126

Subgroups — #27279126

Order 13568.

572 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16810562C211yesmaximal, characteristic
2#20274803C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2313C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8#16810562—22no—
9#16810562C414yescomplemented
10C2×#13133124—21no—
11C2×#13133124—21no—
12C2×#13133124C2211yescharacteristic
13#16810385—24no—
14#16810349—22no—
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#387—24no—
17C53×#381—21no—
18C53×#380C2212yes—
19C4×C22⋊C4×C53—21no—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C22×#88322—22no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53C2×C412yes—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—24no—
30#13133124—24no—
31#13133124—21no—
32#13133124C2×C412yes—
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C412yes—
39C2×C4⋊C4×C53D414yes—
40C2×C4⋊C4×C53D412yes—
41C2×C22⋊C4×C53—22no—
42C2×C42×C53—22no—
43#95×C53—22no—
44#95×C53—24no—
45#95×C53—22no—
46C24×C53C2×D411yescharacteristic
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—24no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53#4112yes—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—42no—
65C22×Dic53—42no—
66C22×Dic53—44no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—42no—
70C4⋊C4×C53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—44no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C42×C53—82no—
76C4⋊Dic53—24no—
77C4⋊Dic53—42no—
78C2×#88322—44no—
79C2×#88322—42no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53C2×C22⋊C411yescharacteristic
83C23×C53#12011yescharacteristic
84C23×C53#12311yescharacteristic
85C23×C53#13512yes—
86C23×C53#13712yes—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—44no—
100C2×C4×C53—44no—
101C2×C4×C53—44no—
102C2×C4×C53—81no—
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12112yes—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—44no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—84no—
111C2×Dic53—82no—
112C2×Dic53—82no—
113#88322—82no—
114#9022—531nomaximal, Sylow
115C22×C53—24no—
116C22×C53—24no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#38611yescharacteristic
122C22×C53#41611yescharacteristic
123C22×C53#41911yescharacteristic
124C22×C53#45212yes—
125C22×C53#45412yes—
126C22×C53#45812yes—
127C22×C53#48512yes—
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—42no—
131C4×C53—42no—
132C4×C53—84no—
133Dic53—82no—
134Dic53—82no—
135C2×#400—531no—
136#2313D5311yescharacteristic
137#1824—532no—
138C2×#324—531no—
139#1294—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—22no—
143C2×C53#161514yes—
144C2×C53#187311yescharacteristic
145C2×C53#188811yescharacteristic
146C2×C53#189011yescharacteristic
147C22×C4⋊C4—531no—
148C22×C4⋊C4C2×D5311yescharacteristic
149C2×C22⋊C8—532no—
150#400—534no—
151#400—1062no—
152#387—24no—
153#381—21no—
154#380C2×D5312yes—
155C4×C22⋊C4—21no—
156C2×#95—531no—
157C2×#95—1061no—
158C2×#95—1061no—
159#340—1064no—
160#324—1062no—
161C53#902211yescharacteristic, complemented, Sylow
162C23×C4—1061no—
163C23×C4—1061no—
164C23×C4C22⋊D5311yescharacteristic
165C23×C4C22⋊D5311yescharacteristic
166C23×C4C22×D5311yescharacteristic
167C22×C8—1062no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4C4×D5312yes—
175C2×C4⋊C4D21212yes—
176C2×C4⋊C4C22⋊D5314yes—
177C2×C22⋊C4—22no—
178C2×C42—22no—
179C22⋊C8—1062no—
180C22⋊C8—1062no—
181#95—22no—
182#95—24no—
183#95—22no—
184#95—532no—
185#95—1061no—
186#95—1064no—
187#95—1064no—
188C24C2×C22⋊D5311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4—2122no—
201C22×C4—2122no—
202C22×C4—2124no—
203C22×C4—2121no—
204C22×C4—2121no—
205C22×C4—2122no—
206C22×C4#1171992112yes—
207C22×C4#1171992112yes—
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4C2×D21211yescharacteristic
210C22×C4#1171994512yes—
211C22×C4C2×C22⋊D5312yes—
212C2×C8—2124no—
213C2×C8—2122no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—44no—
217C4⋊C4—1064no—
218C4⋊C4—2122no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—82no—
222C23—22no—
223C23—22no—
224C23C2×#1171992111yescharacteristic
225C23#1313318611yescharacteristic
226C23#1313320011yescharacteristic
227C23#1313323712yes—
228C23#1313324712yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—81no—
245C2×C4—4244no—
246C2×C4—4242no—
247C2×C4—4242no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2124no—
251C2×C4—2122no—
252C2×C4—2122no—
253C2×C4C4×C22⋊D5312yes—
254C2×C4#1313319112yes—
255C8—4242no—
256C22—24no—
257C22—24no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22#1681050211yescharacteristic
263C22#1681061711yescharacteristic
264C22#1681063111yescharacteristic
265C22#1681078512yes—
266C22#1681079612yes—
267C22#1681081312yes—
268C22#1681096412yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—84no—
274C4—4242no—
275C4—4242no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2#2027374614yes—
280C2#2027502711yescharacteristic
281C2#2027511811yescharacteristic
282C2#2027512811yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2313C2
socleC23×C53#120
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2313D53
hypocenterC53#9022
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
9. #16810562C4semidirect
161. C53#9022semidirect

Maximal subgroups

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

subgroupindex
C2×#168105622
#202748032
C2×#168103492
#202725932
C53×#23132
#902253