Finite Group Databasebeta

← #28308217

Subgroups — #28308217

Order 13568.

542 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289960C212yesmaximal
2#20289659C214yesmaximal, complemented
3#20277329C211yesmaximal, characteristic
4#16811593—22no—
5#16811466—22no—
6#16811466C2214yes—
7C22×C22⋊Dic53C411yescharacteristic
8C22×C22⋊Dic53C2211yescharacteristic
9#16810498—22no—
10#16810498C412yes—
11#16810498C2212yes—
12#16810483—22no—
13C22×C22⋊C4×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53C2×C412yes—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—22no—
20C2×C4⋊Dic53—22no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—41no—
24C2×C22⋊Dic53—41no—
25C2×C22⋊Dic53—41no—
26C2×C22⋊Dic53—41no—
27C2×C22⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53D412yes—
29C2×C22⋊Dic53C2311yescharacteristic
30#13133124—22no—
31#13133124—22no—
32C25×C53C2×C411yescharacteristic
33C23×C4×C53—21no—
34C2×C22⋊C4×C53—24no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53C2×C412yes—
37C24×C53—21no—
38C24×C53—21no—
39C24×C53—22no—
40C24×C53—22no—
41C24×C53—82no—
42C24×C53C22⋊C411yescharacteristic
43C24×C53C22⋊C411yescharacteristic
44C24×C53C22×C411yescharacteristic
45C22×C4×C53—22no—
46C22×C4×C53—24no—
47C22×C4×C53—82no—
48C22×Dic53—22no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—42no—
52C22×Dic53—82no—
53C22×Dic53—82no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56#11719940—84no—
57C2×#88324—44no—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—22no—
60C22⋊C4×C53M1614yescomplemented
61C22⋊Dic53—41no—
62C22⋊Dic53—42no—
63C22⋊Dic53—41no—
64C22⋊Dic53—42no—
65C22⋊Dic53—41no—
66C22⋊Dic53—41no—
67C4⋊Dic53—82no—
68C4⋊Dic53—82no—
69C23×C53—24no—
70C23×C53—24no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53—82no—
81C23×C53—82no—
82C23×C53—84no—
83C23×C53—84no—
84C23×C53#9911yescharacteristic
85C23×C53#9911yescharacteristic
86C23×C53#10112yes—
87C23×C53C2×C22⋊C411yescharacteristic
88C23×C53C2×M1612yes—
89C2×C4×C53—24no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—84no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—44no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—82no—
102C2×Dic53—84no—
103#88324—84no—
104#11519—531nomaximal, Sylow
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—84no—
121C22×C53—84no—
122C22×C53#40711yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C4×C53—44no—
126Dic53—42no—
127Dic53—42no—
128Dic53—82no—
129#2315—531no—
130#1462—532no—
131#1326—534no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—41no—
135C2×C53—41no—
136C2×C53—42no—
137C2×C53—42no—
138C2×C53#146311yescharacteristic
139C2×C53#212112yes—
140C22×C22⋊C4—531no—
141C22×C22⋊C4—531no—
142C22×C22⋊C4#615711yescharacteristic
143#407—1062no—
144#385—532no—
145#385—532no—
146#385—1062no—
147#380—1062no—
148#323—534no—
149#323—1062no—
150C53#1151911yescharacteristic, complemented, Sylow
151C25C2×#615711yescharacteristic
152C23×C4—21no—
153C23×C4—532no—
154C23×C4—532no—
155C2×M16—2122no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1062no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—531no—
162C2×C22⋊C4—531no—
163C2×C22⋊C4—1061no—
164C2×C22⋊C4—1061no—
165C2×C22⋊C4—2121no—
166C2×C22⋊C4—2121no—
167C2×C22⋊C4—2121no—
168C2×C22⋊C4—2121no—
169C2×C22⋊C4C2×#615712yes—
170C22⋊C8—1064no—
171C22⋊C8—2122no—
172#95—1062no—
173#95—1062no—
174C24—21no—
175C24—21no—
176C24—22no—
177C24—22no—
178C24—82no—
179C24C22⋊#615711yescharacteristic
180C24C22⋊#615711yescharacteristic
181C24C22×#615711yescharacteristic
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—82no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4—4242no—
191C22×C4—4242no—
192C22×C4—2122no—
193M16—4244no—
194C2×C8—2124no—
195C4⋊C4—4242no—
196C4⋊C4—4242no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—2121no—
200C22⋊C4—2122no—
201C22⋊C4—2121no—
202C22⋊C4—2122no—
203C22⋊C4—2121no—
204C22⋊C4—2121no—
205C22⋊C4#1171994014yescomplemented
206C23—24no—
207C23—24no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23#1313327611yescharacteristic
222C23#1313327611yescharacteristic
223C23#1313327812yes—
224C23C2×C22⋊#615711yescharacteristic
225C23C2×#1171994012yes—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—4244no—
233C2×C4—4242no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4—2124no—
238C2×C4—2122no—
239C2×C4—2122no—
240C8—4244no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—22no—
247C22—22no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—84no—
257C22—84no—
258C22#1681159311yescharacteristic
259C22C2×#1313327611yescharacteristic
260C22C2×#1313327811yescharacteristic
261C4—44no—
262C4—4242no—
263C4—2122no—
264C4—2122no—
265C2—21no—
266C2—21no—
267C2—41no—
268C2—41no—
269C2—42no—
270C2—42no—
271C2#2028996311yescharacteristic
272C2#2029097212yes—
subgroupG/H
centerC22C2×#13133276
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#11519
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289659C2semidirect
60. C22⋊C4×C53M16semidirect
150. C53#11519semidirect
205. C22⋊C4#11719940semidirect

Maximal subgroups

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

subgroupindex
#202899602
#202896592
#202773292
#1151953