Finite Group Databasebeta

← #28303355

Subgroups — #28303355

Order 13568.

526 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 100 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#20290441C212yesmaximal
2#20289591C212yesmaximal, complemented
3#20289591C212yesmaximal, complemented
4#20277118C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6C2×#13133272C2214yes—
7#16811489—22no—
8C2×#13133176C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10#16810492C411yescharacteristic
11#16810492C411yescharacteristic
12#16810488C2211yescharacteristic
13#16810488C2211yescharacteristic
14#16810481—22no—
15C4×#11719919—22no—
16C2×#117×C53C411yescharacteristic
17C22×#88324—24no—
18C22⋊#88324—28no—
19#13133272—28no—
20C23×Dic53D412yes—
21#13133176—42no—
22C2×C4⋊Dic53C2×C411yescharacteristic
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×C22⋊Dic53—21no—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D412yes—
31C2×C4×Dic53C2311yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C23×C4×C53C2×C411yescharacteristic
35#117×C53—42no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C22⋊C4×C53—21no—
39C2×C42×C53—21no—
40C24×C53C22⋊C411yescharacteristic
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—24no—
49C22×Dic53—24no—
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53—42no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D412yes—
55C2×#88324—28no—
56C2×#88324—44no—
57C2×#88324—44no—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—41no—
60C22⋊C4×C53—42no—
61C42×C53—42no—
62C22⋊Dic53—42no—
63C4⋊Dic53—41no—
64C4⋊Dic53—41no—
65#11719919—44no—
66#11719919—44no—
67C4×Dic53—22no—
68C4×Dic53—24no—
69C4×Dic53—42no—
70C4×Dic53#4114yes—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53#10112yes—
76C23×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53—28no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—41no—
83C2×C4×C53—41no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53#10012yes—
87C2×C4×C53#11712yes—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—48no—
93C2×Dic53—41no—
94C2×Dic53—41no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53#10414yes—
98C2×Dic53#12314yes—
99#88324—48no—
100#8721—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—24no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—41no—
106C22×C53#38111yescharacteristic
107C22×C53C2×#10011yescharacteristic
108C22×C53C2×#10111yescharacteristic
109C22×C53C2×#10412yes—
110C22×C53#42112yes—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—81no—
114C4×C53—81no—
115Dic53—24no—
116Dic53—24no—
117Dic53—81no—
118Dic53—81no—
119#2285—531no—
120#1749—532no—
121#1295—532no—
122#1295—532no—
123C2×C53—42no—
124C2×C53#152511yescharacteristic
125C2×C53#153112yes—
126C2×C53#153611yescharacteristic
127C2×C53#182111yescharacteristic
128C2×C53#182612yes—
129C2×#117—531no—
130C2×#117#615711yescharacteristic
131C22×C42—531no—
132C2×C22⋊C8—1062no—
133C2×#97—534no—
134#384—531no—
135#384—531no—
136#382—531no—
137#382—531no—
138#380—1062no—
139C4×C4⋊C4—1062no—
140#336—1062no—
141C53#872111yescharacteristic, complemented, Sylow
142C23×C4—532no—
143C23×C4C2×#615711yescharacteristic
144C22×C8—1064no—
145#117—42no—
146#117—2122no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C4⋊C4—1062no—
150C2×C4⋊C4—1062no—
151C2×C4⋊C4C2×#615711yescharacteristic
152C2×C4⋊C4C2×#615711yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—1061no—
155C2×C42—21no—
156C2×C42—531no—
157C2×C42—532no—
158C2×C42—531no—
159C2×C42—1064no—
160C2×C42—1061no—
161C22⋊C8—1068no—
162#97—1068no—
163#95—1062no—
164#95—1062no—
165C24C22⋊#615711yescharacteristic
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—1064no—
174C22×C4—1064no—
175C22×C4—1061no—
176C22×C4—1061no—
177C22×C4—2122no—
178C22×C4C22⋊#615711yescharacteristic
179C22×C4C22×#615711yescharacteristic
180C2×C8—1068no—
181C2×C8—2124no—
182C2×C8—2124no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—2121no—
186C4⋊C4—2121no—
187C4⋊C4—2124no—
188C4⋊C4—2124no—
189C22⋊C4—42no—
190C22⋊C4—2122no—
191C42—42no—
192C42—534no—
193C42—1064no—
194C42—1062no—
195C42—2122no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23#1313327812yes—
201C23C2×C22⋊#615711yescharacteristic
202C2×C4—28no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—534no—
212C2×C4—534no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—2128no—
218C2×C4—2121no—
219C2×C4—2121no—
220C2×C4—2122no—
221C2×C4—2122no—
222C2×C4#1313327712yes—
223C2×C4#1313329612yes—
224C8—2128no—
225C22—22no—
226C22—24no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22#1681156611yescharacteristic
231C22C2×#1313327711yescharacteristic
232C22C2×#1313327811yescharacteristic
233C22D53⋊#10412yes—
234C22#1681162012yes—
235C4—22no—
236C4—22no—
237C4—81no—
238C4—81no—
239C4—1064no—
240C4—1064no—
241C4—4241no—
242C4—4241no—
243C2—42no—
244C2#2029016911yescharacteristic
245C2#2029018712yes—
246C2#2029020311yescharacteristic
247C2#2029061411yescharacteristic
248C2#2029062312yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8721
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289591C2semidirect
3. #20289591C2semidirect
141. C53#8721semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895912
#202771182
#872153