Finite Group Databasebeta

← #28308194

Subgroups — #28308194

Order 13568.

578 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 88 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#20289963C212yesmaximal
2#20289657C214yesmaximal, complemented
3C2×#16810493C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yes—
7C22×C4⋊Dic53—22no—
8C22×C4⋊Dic53C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810493—22no—
11#16810493C412yes—
12#16810493C2212yes—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—24no—
21C2×C4⋊Dic53—24no—
22C2×C4⋊Dic53—41no—
23C2×C4⋊Dic53—41no—
24C2×C4⋊Dic53—48no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C412yes—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C2×C42×C53—24no—
34C2×C42×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—24no—
38C22×C4×C53—24no—
39C22×C4×C53—24no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—44no—
51C22×Dic53—44no—
52C22×Dic53—41no—
53C22×Dic53—41no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56#11719940—84no—
57C2×#88324—44no—
58C42×C53—24no—
59C42×C53—22no—
60C42×C53M1614yescomplemented
61C4⋊Dic53—41no—
62C4⋊Dic53—42no—
63C4⋊Dic53—41no—
64C4⋊Dic53—48no—
65C4⋊Dic53—48no—
66C4⋊Dic53—42no—
67C4⋊Dic53—42no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53#9911yescharacteristic
73C23×C53#9911yescharacteristic
74C23×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53—216no—
76C2×C4×C53—24no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—24no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53#10112yes—
86C2×C4×C53C2×M1612yes—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—44no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—44no—
93C2×Dic53—42no—
94C2×Dic53—82no—
95C2×Dic53—82no—
96C2×Dic53—82no—
97#88324—84no—
98#11512—531nomaximal, Sylow
99C22×C53—24no—
100C22×C53—24no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#35312yes—
108C22×C53#35612yes—
109C22×C53#40711yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53C2×#10111yescharacteristic
112C4×C53—28no—
113C4×C53—22no—
114C4×C53—22no—
115Dic53—42no—
116Dic53—42no—
117Dic53—82no—
118C2×#384—531no—
119#1463—532no—
120#1325—534no—
121C2×C53—21no—
122C2×C53—22no—
123C2×C53—22no—
124C2×C53—21no—
125C2×C53#146311yescharacteristic
126C2×C53C2×#35311yescharacteristic
127C2×C53C2×#35611yescharacteristic
128C22×C4⋊C4—531no—
129C22×C4⋊C4—531no—
130C22×C4⋊C4—1062no—
131C22×C42#615711yescharacteristic
132#407—1062no—
133#384—532no—
134#384—532no—
135#384—1062no—
136#324—534no—
137#324—1062no—
138C53#1151211yescharacteristic, complemented, Sylow
139C23×C4—21no—
140C23×C4—532no—
141C23×C4—1061no—
142C23×C4C2×#615711yescharacteristic
143C2×M16—2122no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—531no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—532no—
148C2×C4⋊C4—1064no—
149C2×C4⋊C4—1064no—
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—2121no—
152C2×C4⋊C4—2121no—
153C2×C4⋊C4—2121no—
154C2×C4⋊C4—2121no—
155C2×C4⋊C4—2128no—
156C2×C42—21no—
157C2×C42—24no—
158C2×C42C2×#615712yes—
159C22⋊C8—1064no—
160C22⋊C8—2122no—
161C24C22⋊#615711yescharacteristic
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—532no—
171C22×C4—532no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4—1062no—
175C22×C4—2124no—
176C22×C4—2121no—
177C22×C4—2121no—
178C22×C4—2124no—
179C22×C4C22⋊#615711yescharacteristic
180C22×C4C22×#615711yescharacteristic
181M16—4244no—
182C2×C8—2124no—
183C4⋊C4—2122no—
184C4⋊C4—2121no—
185C4⋊C4—2121no—
186C4⋊C4—2122no—
187C4⋊C4—2128no—
188C4⋊C4—2128no—
189C4⋊C4—2122no—
190C42—24no—
191C42—22no—
192C42#1171994014yescomplemented
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23#1313327611yescharacteristic
198C23#1313327611yescharacteristic
199C23C2×C22⋊#615711yescharacteristic
200C2×C4—24no—
201C2×C4—216no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—4242no—
213C2×C4—4242no—
214C2×C4—4242no—
215C2×C4—2124no—
216C2×C4—2122no—
217C2×C4—2124no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4#1313327812yes—
221C2×C4C2×#1171994012yes—
222C8—4244no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#1681152812yes—
232C22#1681153412yes—
233C22#1681159311yescharacteristic
234C22C2×#1313327611yescharacteristic
235C22C2×#1313327811yescharacteristic
236C4—28no—
237C4—22no—
238C4—22no—
239C4—4242no—
240C4—2122no—
241C4—2122no—
242C2—21no—
243C2—22no—
244C2—22no—
245C2—21no—
246C2#2028996311yescharacteristic
247C2C2×#1681152811yescharacteristic
248C2C2×#1681153411yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11512
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289657C2semidirect
60. C42×C53M16semidirect
138. C53#11512semidirect
192. C42#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896572
C2×#168104932
#1151253