Finite Group Databasebeta

← #27287089

Subgroups — #27287089

Order 13568.

500 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 109 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#20274799C212yesmaximal
2#20272982C211yesmaximal, characteristic
3#20272593C212yesmaximal
4#20272558C211yesmaximal, characteristic
5C53×#2646C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8C2×#13133124—22no—
9C2×#13133124C2211yescharacteristic
10#16810385—22no—
11#16810385—22no—
12C53×#400—22no—
13C53×#398—22no—
14C53×#385—22no—
15C53×#380C2212yes—
16C2×#95×C53C2211yescharacteristic
17C23×Dic53—21no—
18C23×Dic53—21no—
19C22×#88322—22no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53Q812yes—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133124—22no—
27#13133124—44no—
28C23×C4×C53D412yes—
29C23×C4×C53C2311yescharacteristic
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D412yes—
34C2×C22⋊C4×C53—22no—
35#95×C53—22no—
36#95×C53—22no—
37#95×C53—22no—
38#95×C53—22no—
39C24×C53C2×D411yescharacteristic
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—42no—
46C22×C4×C53—42no—
47C22×C4×C53—42no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×Q811yescharacteristic
50C22×C4×C53#4112yes—
51C22×C4×C53#4111yescharacteristic
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—41no—
55C22×Dic53—42no—
56C22×Dic53—41no—
57C22×Dic53—41no—
58C22×Dic53—41no—
59C22×Dic53—42no—
60C22×Dic53—42no—
61C22×Dic53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—82no—
65C4⋊C4×C53—82no—
66C22⋊C4×C53—42no—
67C22⋊C4×C53—42no—
68C4⋊Dic53—24no—
69C4⋊Dic53—42no—
70C2×#88322—42no—
71C2×#88322—42no—
72C2×#88322—42no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—22no—
76C23×C53#12011yescharacteristic
77C23×C53C22⋊Q811yescharacteristic
78C23×C53#12411yescharacteristic
79C23×C53#13512yes—
80C23×C53AGL(1,ℤ/8)12yes—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—44no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—82no—
97C2×C4×C53C22⋊Q812yes—
98C2×C4×C53#12412yes—
99C2×Dic53—44no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—44no—
103C2×Dic53—82no—
104C2×Dic53—81no—
105C2×Dic53—82no—
106C2×Dic53—81no—
107C2×Dic53—81no—
108C2×Dic53—81no—
109#88322—82no—
110#10346—531nomaximal, Sylow
111C22×C53—22no—
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53#39311yescharacteristic
120C22×C53#44912yes—
121C22×C53#45511yescharacteristic
122C22×C53#45611yescharacteristic
123C22×C53#45812yes—
124C22×C53#47812yes—
125C22×C53#48811yescharacteristic
126C22×C53#49011yescharacteristic
127C4×C53—22no—
128C4×C53—22no—
129C4×C53—82no—
130C4×C53—82no—
131C4×C53—82no—
132Dic53—82no—
133Dic53—82no—
134#2646D5311yescharacteristic
135#1823—532no—
136#1445—531no—
137#1294—532no—
138#1280—531no—
139C2×C53—21no—
140C2×C53—22no—
141C2×C53—21no—
142C2×C53#160912yes—
143C2×C53#161212yes—
144C2×C53#202511yescharacteristic
145C2×C53#202811yescharacteristic
146C2×C53#203011yescharacteristic
147C22×C4⋊C4—531no—
148C2×C22⋊C8—532no—
149#400—22no—
150#398—22no—
151#385—22no—
152#380C2×D5312yes—
153C2×#95—531no—
154C2×#95—1062no—
155C2×#95C2×D5311yescharacteristic
156#340—1062no—
157#340—1062no—
158C53#1034611yescharacteristic, complemented, Sylow
159C23×C4—1061no—
160C23×C4—1061no—
161C23×C4C22⋊D5312yes—
162C23×C4C22×D5311yescharacteristic
163C22×C8—1062no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4—1062no—
169C2×C4⋊C4—1061no—
170C2×C4⋊C4C22⋊D5312yes—
171C2×C4⋊C4C22⋊D5312yes—
172C2×C22⋊C4—22no—
173C22⋊C8—1062no—
174C22⋊C8—1062no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—1062no—
180#95—2124no—
181C24C2×C22⋊D5311yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4—2121no—
193C22×C4—2122no—
194C22×C4—2121no—
195C22×C4—2121no—
196C22×C4—2121no—
197C22×C4—2122no—
198C22×C4—2122no—
199C22×C4—2122no—
200C22×C4Q8×D5311yescharacteristic
201C22×C4#1171994912yes—
202C22×C4#1171994911yescharacteristic
203C22×C4C2×C22⋊D5312yes—
204C2×C8—2122no—
205C2×C8—2122no—
206C2×C8—2122no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—1064no—
212C4⋊C4—2122no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23#1313318611yescharacteristic
219C23#1313319311yescharacteristic
220C23#1313320611yescharacteristic
221C23#1313323712yes—
222C23#1313324212yes—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4—4242no—
240C2×C4—4241no—
241C2×C4—4242no—
242C2×C4—4241no—
243C2×C4—4241no—
244C2×C4—4241no—
245C2×C4—2124no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2124no—
249C2×C4#1313319312yes—
250C2×C4#1313320612yes—
251C8—4242no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22#1681053411yescharacteristic
261C22#1681076812yes—
262C22#1681080111yescharacteristic
263C22#1681080611yescharacteristic
264C22#1681081312yes—
265C22#1681092012yes—
266C22#1681098011yescharacteristic
267C22#1681098911yescharacteristic
268C4—22no—
269C4—22no—
270C4—82no—
271C4—82no—
272C4—82no—
273C4—4242no—
274C4—4242no—
275C2—21no—
276C2—22no—
277C2—21no—
278C2#2027371212yes—
279C2#2027372912yes—
280C2#2027585811yescharacteristic
281C2#2027587311yescharacteristic
282C2#2027588111yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2646C2
socleC23×C53#120
prime radicalC24×C53C2×D4
hypercenter#2646D53
hypocenterC53#10346
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2646

Complements

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

NH ≅ G/Ntype
158. C53#10346semidirect

Maximal subgroups

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

subgroupindex
#202747992
#202729822
#202725932
#202725582
C53×#26462
#1034653