Finite Group Databasebeta

← #28308174

Subgroups — #28308174

Order 13568.

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

normal subgroups only

This group has 598 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#20289645C214yesmaximal, complemented
3C2×#16810530C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yes—
7C22×C4⋊Dic53C2211yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9#16810530—22no—
10#16810530C412yes—
11#16810530C2212yes—
12C2×#13133124—22no—
13C23×Q8×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—41no—
20C2×C4⋊Dic53—41no—
21C2×C4⋊Dic53C2×C411yescharacteristic
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53C2311yescharacteristic
24C2×C4×Dic53—21no—
25C2×C4×Dic53—42no—
26C2×C4×Dic53C2×C412yes—
27#13133124—24no—
28#13133124—24no—
29C22×Q8×C53—24no—
30C22×Q8×C53—21no—
31C22×Q8×C53—42no—
32C22×Q8×C53—41no—
33C22×Q8×C53—41no—
34C22×Q8×C53C2×C412yes—
35C23×C4×C53—21no—
36C23×C4×C53C2×C411yescharacteristic
37C24×C53C22⋊C411yescharacteristic
38C2×Q8×C53—24no—
39C2×Q8×C53—22no—
40C2×Q8×C53—48no—
41C2×Q8×C53—48no—
42C2×Q8×C53—42no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C2×Q8×C53—42no—
46C2×Q8×C53M1614yescomplemented
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—24no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—41no—
55C22×C4×C53—41no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—24no—
59C22×Dic53—22no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—42no—
63C22×Dic53—42no—
64C22×Dic53—44no—
65C22×Dic53C22⋊C412yes—
66C22×Dic53C2×D412yes—
67#11719940—84no—
68C2×#88324—44no—
69C4⋊Dic53—41no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72C4×Dic53—42no—
73C4×Dic53—42no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53#9911yescharacteristic
79C23×C53#9911yescharacteristic
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—48no—
82Q8×C53—44no—
83Q8×C53—44no—
84C2×C4×C53—24no—
85C2×C4×C53—24no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—24no—
91C2×C4×C53—44no—
92C2×C4×C53—44no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53#10012yes—
96C2×C4×C53C2×M1612yes—
97C2×Dic53—24no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—48no—
101C2×Dic53—42no—
102C2×Dic53—44no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—82no—
107#88324—84no—
108#11506—531nomaximal, Sylow
109C22×C53—24no—
110C22×C53—24no—
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#35412yes—
118C22×C53#35612yes—
119C22×C53#40711yescharacteristic
120C22×C53C2×#9911yescharacteristic
121C22×C53C2×#10011yescharacteristic
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—44no—
125Dic53—44no—
126Dic53—42no—
127Dic53—42no—
128C2×#391—531no—
129#1463—532no—
130#1319—534no—
131C2×C53—21no—
132C2×C53—22no—
133C2×C53—22no—
134C2×C53—21no—
135C2×C53#146211yescharacteristic
136C2×C53C2×#35411yescharacteristic
137C2×C53C2×#35611yescharacteristic
138C23×Q8#615711yescharacteristic
139C22×C4⋊C4—531no—
140C22×C42—531no—
141#407—1062no—
142#391—532no—
143#391—532no—
144#391—1062no—
145C2×#95—1062no—
146#324—534no—
147#324—1062no—
148C53#1150611yescharacteristic, complemented, Sylow
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8C2×#615712yes—
155C23×C4—21no—
156C23×C4—532no—
157C23×C4—1061no—
158C23×C4C2×#615711yescharacteristic
159C2×M16—2122no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—2121no—
164C2×C4⋊C4—2121no—
165C2×C42—532no—
166C2×C42—1061no—
167C2×C42—2122no—
168C22⋊C8—1064no—
169C22⋊C8—2122no—
170#95—1064no—
171#95—1064no—
172C24C22⋊#615711yescharacteristic
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—48no—
176C2×Q8—48no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8#1171994014yescomplemented
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—532no—
192C22×C4—532no—
193C22×C4—1062no—
194C22×C4—1064no—
195C22×C4—1062no—
196C22×C4—1062no—
197C22×C4—2124no—
198C22×C4—2122no—
199C22×C4—2122no—
200C22×C4C22⋊#615711yescharacteristic
201C22×C4C22×#615711yescharacteristic
202M16—4244no—
203C2×C8—2124no—
204C4⋊C4—2122no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—2122no—
208C42—2122no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23#1313327511yescharacteristic
214C23#1313327511yescharacteristic
215C23C2×C22⋊#615711yescharacteristic
216Q8—44no—
217Q8—48no—
218Q8—44no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—1064no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—4242no—
234C2×C4—2128no—
235C2×C4—2122no—
236C2×C4—2124no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4#1313327712yes—
241C2×C4C2×#1171994012yes—
242C8—4244no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1681152912yes—
252C22#1681153312yes—
253C22#1681159311yescharacteristic
254C22C2×#1313327511yescharacteristic
255C22C2×#1313327711yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—2124no—
260C4—2122no—
261C4—2122no—
262C2—21no—
263C2—22no—
264C2—22no—
265C2—21no—
266C2#2028996011yescharacteristic
267C2C2×#1681152911yescharacteristic
268C2C2×#1681153311yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC23×Q8×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC23×Q8#6157
hypocenterC53#11506
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #20289645C2semidirect
46. C2×Q8×C53M16semidirect
148. C53#11506semidirect
181. C2×Q8#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896452
C2×#168105302
#1150653