Finite Group Databasebeta

← #27284789

Subgroups — #27284789

Order 13568.

698 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 87 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#20285539C211yesmaximal, characteristic
2C2×#16810779C212yesmaximal
3#20273080C212yesmaximal
4#20272602C211yesmaximal, characteristic
5C53×#1784C211yesmaximal, characteristic
6C22×#11719925—22no—
7C2×#13133196C2212yes—
8#16810954—22no—
9#16810881—22no—
10#16810781—22no—
11#16810779—24no—
12#16810779—24no—
13C2×#13133141—22no—
14C2×C22⋊#88322C2212yes—
15#16810348—24no—
16C23×Q8×C53C2211yescharacteristic
17C2×#101×C53—22no—
18C53×#407C2211yescharacteristic
19C53×#407C2211yescharacteristic
20C22×Dic106D412yes—
21C2×Dic212—22no—
22C2×#11719925—28no—
23C2×#11719925—48no—
24C22×#88322—22no—
25#13133196—22no—
26#13133196—22no—
27#13133196—22no—
28#13133196—22no—
29C2×C4⋊Dic53D412yes—
30C2×#11719919—22no—
31C2×C22⋊Dic53—22no—
32#13133149—22no—
33#13133142—24no—
34#13133141—28no—
35C22⋊#88322—22no—
36C22⋊#88322D414yes—
37C22×Q8×C53—21no—
38C22×Q8×C53—22no—
39C22×Q8×C53—22no—
40C22×Q8×C53—44no—
41C22×Q8×C53D412yes—
42C23×C4×C53—21no—
43C23×C4×C53C2311yescharacteristic
44C2×M16×C53—21no—
45C2×M16×C53D412yes—
46#101×C53—44no—
47C22⋊C8×C53—22no—
48C22⋊C8×C53—22no—
49C24×C53#4111yescharacteristic
50C2×Q8×C53—24no—
51C2×Q8×C53—28no—
52C2×Q8×C53—24no—
53C2×Q8×C53—44no—
54C2×Q8×C53—44no—
55C2×Q8×C53—42no—
56C2×Q8×C53—48no—
57C2×Q8×C53—41no—
58C2×Q8×C53—41no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—44no—
62C22×C4×C53—42no—
63C22×C4×C53C2×D412yes—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×Dic53—22no—
68C22×Dic53—22no—
69C2×Dic106—22no—
70C2×Dic106—22no—
71C2×Dic106—44no—
72M16×C53—42no—
73M16×C53—42no—
74C2×C8×C53—22no—
75C2×C8×C53—41no—
76C22⋊Dic53—42no—
77C22⋊Dic53—42no—
78#11719925—416no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81#11719919—42no—
82#11719919—42no—
83C2×#88322—24no—
84C2×#88322—44no—
85Dic212—82no—
86C23×C53—24no—
87C23×C53—41no—
88C23×C53#12112yes—
89C23×C53#12111yescharacteristic
90Q8×C53—28no—
91Q8×C53—44no—
92Q8×C53—44no—
93Q8×C53—44no—
94C2×C4×C53—24no—
95C2×C4×C53—22no—
96C2×C4×C53—24no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—48no—
101C2×C4×C53—42no—
102C2×C4×C53#12011yescharacteristic
103C2×C4×C53#12012yes—
104C2×C4×C53#12711yescharacteristic
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—44no—
108C2×Dic53—82no—
109Dic106—44no—
110Dic106—44no—
111C8×C53—42no—
112C8×C53—81no—
113#88322—44no—
114#9991—531nomaximal, Sylow
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—22no—
118C22×C53—21no—
119C22×C53—44no—
120C22×C53#39211yescharacteristic
121C22×C53#47314yes—
122C4×C53—22no—
123C4×C53—21no—
124C4×C53—44no—
125C4×C53B(4,𝔽2)12yes—
126Dic53—82no—
127Dic53—82no—
128#3195—531no—
129C2×#451—532no—
130#1784D5311yescharacteristic
131#1472—532no—
132#1298—531no—
133C2×C53—24no—
134C2×C53—41no—
135C2×C53#202012yes—
136C2×C53#202111yescharacteristic
137C23×Q8C2×D5311yescharacteristic
138C22×Q16—1062no—
139C2×C22⋊Q8—532no—
140#483—1062no—
141#470—1062no—
142#451—1062no—
143#451—1064no—
144#451—1064no—
145C2×#103—1062no—
146C2×#101—22no—
147#407C2×D5311yescharacteristic
148#407C2×D5311yescharacteristic
149C2×C22⋊C8—532no—
150#324—1064no—
151C53#999111yescharacteristic, complemented, Sylow
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—532no—
157C22×Q8D21212yes—
158C23×C4—21no—
159C23×C4C22×D5311yescharacteristic
160C2×Q16—1062no—
161C2×Q16—1068no—
162C2×Q16—2128no—
163C2×M16—21no—
164C2×M16C22⋊D5312yes—
165C22×C8—1062no—
166C22⋊Q8—1062no—
167C22⋊Q8—1062no—
168C22⋊Q8—1062no—
169C22⋊Q8—1062no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1062no—
172C2×C22⋊C4—1062no—
173#106—1062no—
174#103—1064no—
175#103—1068no—
176#101—44no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—534no—
180C22⋊C8—1062no—
181C24#1171994511yescharacteristic
182C2×Q8—24no—
183C2×Q8—28no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—48no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1062no—
192C2×Q8—1062no—
193C2×Q8—2124no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4C2×D21211yescharacteristic
201C22×C4D4×D5312yes—
202C22×C4D4×D5312yes—
203C22×C4C2×C22⋊D5311yescharacteristic
204Q16—4242no—
205Q16—21216no—
206M16—42no—
207M16—42no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—1064no—
211C2×C8—2124no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—2122no—
215C4⋊C4—2122no—
216C22⋊C4—2122no—
217C22⋊C4—2122no—
218C23—24no—
219C23—41no—
220C23#1313319012yes—
221C23#1313319111yescharacteristic
222Q8—28no—
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—2124no—
227Q8—2124no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—4242no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2124no—
240C2×C4#1313318511yescharacteristic
241C2×C4#1313318712yes—
242C2×C4#1313321311yescharacteristic
243C8—42no—
244C8—81no—
245C8—2124no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—44no—
251C22#1681053211yescharacteristic
252C22#1681089714yes—
253C4—21no—
254C4—22no—
255C4—44no—
256C4—4242no—
257C4—4242no—
258C4#1681080912yes—
259C2—24no—
260C2—41no—
261C2#2027583512yes—
262C2#2027584111yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1784C2
socleC22×C53#392
prime radicalC24×C53#41
hypercenter#1784D53
hypocenterC53#9991
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
151. C53#9991semidirect

Maximal subgroups

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

subgroupindex
#202855392
C2×#168107792
#202730802
#202726022
C53×#17842
#999153