Finite Group Databasebeta

← #28307987

Subgroups — #28307987

Order 13568.

562 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×#16811466C212yesmaximal
2#20289660C214yesmaximal, complemented
3#20277327C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811466—22no—
6#16811466C2214yes—
7C22×C22⋊Dic53C411yescharacteristic
8C22×C22⋊Dic53C2211yescharacteristic
9#16810496C412yes—
10#16810496C2212yes—
11#16810495—22no—
12#16810481—22no—
13C22×C22⋊C4×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×#11719919—22no—
20C2×#11719919—22no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—42no—
24C2×C22⋊Dic53C814yescomplemented
25C2×C22⋊Dic53C2×C412yes—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53D412yes—
28C2×C22⋊Dic53C2311yescharacteristic
29#13133124—22no—
30#13133124—22no—
31C25×C53C2×C411yescharacteristic
32C23×C4×C53C2×C412yes—
33C2×C22⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—42no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—82no—
39C24×C53C22⋊C411yescharacteristic
40C24×C53C22⋊C411yescharacteristic
41C24×C53C2×C812yes—
42C24×C53M1612yes—
43C24×C53C22×C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—42no—
46C22×C4×C53—82no—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—24no—
50C22×Dic53—22no—
51C22×Dic53—82no—
52C22×Dic53—82no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D412yes—
55C2×#88324—44no—
56C2×#88324—84no—
57C22⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C22⋊Dic53—24no—
60C22⋊Dic53—22no—
61C22⋊Dic53—41no—
62C22⋊Dic53—42no—
63C22⋊Dic53—41no—
64C22⋊Dic53C2×C814yescomplemented
65#11719919—82no—
66#11719919—82no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—24no—
72C23×C53—24no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53—42no—
76C23×C53—82no—
77C23×C53—82no—
78C23×C53—82no—
79C23×C53—82no—
80C23×C53—84no—
81C23×C53C22⋊C812yes—
82C23×C53C22⋊C812yes—
83C23×C53#9912yes—
84C23×C53#10112yes—
85C23×C53C2×C22⋊C411yescharacteristic
86C23×C53C22×C811yescharacteristic
87C23×C53C2×M1611yescharacteristic
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—84no—
92C2×Dic53—24no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—44no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—82no—
101C2×Dic53—84no—
102#88324—84no—
103#11453—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—42no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—41no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—42no—
119C22×C53—84no—
120C22×C53—84no—
121C22×C53#32414yes—
122C22×C53C2×C22⋊C811yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C4×C53—82no—
126Dic53—44no—
127Dic53—42no—
128Dic53—42no—
129#2315—531no—
130C2×#323—532no—
131#1326—534no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—24no—
135C2×C53—42no—
136C2×C53—41no—
137C2×C53—41no—
138C2×C53C2×#32411yescharacteristic
139C2×C53#212112yes—
140C22×C22⋊C4—531no—
141C22×C22⋊C4—531no—
142C22×C22⋊C4#615711yescharacteristic
143C2×C22⋊C8—1062no—
144#385—532no—
145#385—532no—
146#385—1062no—
147#380—1062no—
148#323—534no—
149#323—1062no—
150C53#1145311yescharacteristic, complemented, Sylow
151C25C2×#615711yescharacteristic
152C23×C4—532no—
153C23×C4—1061no—
154C23×C4C2×#615712yes—
155C22×C8—2122no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1062no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—534no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—531no—
165C2×C22⋊C4—531no—
166C2×C22⋊C4—1062no—
167C2×C22⋊C4—1061no—
168C2×C22⋊C4—2122no—
169C22⋊C8—1064no—
170C22⋊C8—2122no—
171#95—1062no—
172#95—1062no—
173C24—22no—
174C24—22no—
175C24—82no—
176C24D53⋊C812yes—
177C24#1171993612yes—
178C24C22⋊#615711yescharacteristic
179C24C22⋊#615711yescharacteristic
180C24C22×#615711yescharacteristic
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—82no—
184C22×C4—532no—
185C22×C4—532no—
186C22×C4—1064no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4—4242no—
191C22×C4—4242no—
192C2×C8—4244no—
193C2×C8—2124no—
194C4⋊C4—4242no—
195C4⋊C4—4242no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—534no—
199C22⋊C4—1064no—
200C22⋊C4—1062no—
201C22⋊C4—2121no—
202C22⋊C4—2122no—
203C22⋊C4—2121no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23#1313327412yes—
219C23#1313327412yes—
220C23#1313327512yes—
221C23#1313327812yes—
222C23C2×C22⋊#615711yescharacteristic
223C23C2×D53⋊C811yescharacteristic
224C23C2×#1171993611yescharacteristic
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—1064no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—4242no—
233C2×C4—4244no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2124no—
237C2×C4—2122no—
238C2×C4—2122no—
239C8—4244no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22#1681146914yes—
258C22C2×#1313327411yescharacteristic
259C22C2×#1313327511yescharacteristic
260C22C2×#1313327811yescharacteristic
261C4—82no—
262C4—2124no—
263C4—2122no—
264C4—2122no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—42no—
269C2—41no—
270C2—41no—
271C2C2×#1681146911yescharacteristic
272C2#2029097012yes—
subgroupG/H
centerC22C2×#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#11453
LCSG ▷ C23×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. #20289660C2semidirect
24. C2×C22⋊Dic53C8semidirect
64. C22⋊Dic53C2×C8semidirect
150. C53#11453semidirect

Maximal subgroups

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

subgroupindex
C2×#168114662
#202896602
#202773272
#1145353