Finite Group Databasebeta

← #27293270

Subgroups — #27293270

Order 13568.

562 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16810347C212yesmaximal
2#20272663C214yesmaximal, complemented
3C53×#2315C211yesmaximal, characteristic
4C2×C22⋊#88322—22no—
5#16810347—22no—
6#16810347C2214yes—
7C22×C22⋊C4×C53C411yescharacteristic
8C22×C22⋊C4×C53C411yescharacteristic
9C22×C22⋊C4×C53C2211yescharacteristic
10C53×#385—22no—
11C53×#385C412yes—
12C53×#385C2212yes—
13C53×#380—22no—
14C22×#88322—42no—
15C22⋊#88322—24no—
16C22⋊#88322—42no—
17C25×C53C2×C411yescharacteristic
18C23×C4×C53—21no—
19C23×C4×C53C2×C412yes—
20C23×C4×C53D412yes—
21C2×C4⋊C4×C53—22no—
22C2×C4⋊C4×C53—22no—
23C2×C22⋊C4×C53—21no—
24C2×C22⋊C4×C53—21no—
25C2×C22⋊C4×C53—21no—
26C2×C22⋊C4×C53—22no—
27C2×C22⋊C4×C53—42no—
28C2×C22⋊C4×C53—42no—
29C2×C22⋊C4×C53C814yescomplemented
30C2×C22⋊C4×C53C2×C412yes—
31C2×C22⋊C4×C53C2×C411yescharacteristic
32C2×C22⋊C4×C53D412yes—
33C2×C22⋊C4×C53C2311yescharacteristic
34#95×C53—22no—
35#95×C53—22no—
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—22no—
46C22×C4×C53—24no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53—42no—
50C22×C4×C53—82no—
51C22×C4×C53—82no—
52C22×C4×C53—82no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C2×D412yes—
55C4⋊C4×C53—82no—
56C4⋊C4×C53—82no—
57C22⋊C4×C53—24no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—41no—
60C22⋊C4×C53—42no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—42no—
63C22⋊C4×C53—42no—
64C22⋊C4×C53C2×C814yescomplemented
65C2×#88322—44no—
66C2×#88322—84no—
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—24no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—44no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—82no—
100C2×C4×C53—84no—
101C2×C4×C53—84no—
102#88322—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—44no—
126C4×C53—42no—
127C4×C53—42no—
128C4×C53—82no—
129#2315D5311yescharacteristic
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⋊C4Dic5311yescharacteristic
141C22×C22⋊C4Dic5311yescharacteristic
142C22×C22⋊C4C2×D5311yescharacteristic
143C2×C22⋊C8—1062no—
144#385—22no—
145#385Dic5312yes—
146#385C2×D5312yes—
147#380—22no—
148#323—534no—
149#323—1062no—
150C53#1145311yescharacteristic, complemented, Sylow
151C25C2×Dic5311yescharacteristic
152C23×C4—21no—
153C23×C4C2×Dic5312yes—
154C23×C4C22⋊D5312yes—
155C22×C8—2122no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4#8832214yescomplemented
165C2×C22⋊C4C2×Dic5312yes—
166C2×C22⋊C4C2×Dic5311yescharacteristic
167C2×C22⋊C4C22⋊D5312yes—
168C2×C22⋊C4C22×D5311yescharacteristic
169C22⋊C8—1064no—
170C22⋊C8—2122no—
171#95—22no—
172#95—22no—
173C24—22no—
174C24—22no—
175C24—82no—
176C24C2×#8832212yes—
177C24#1171991712yes—
178C24C22⋊Dic5311yescharacteristic
179C24C22⋊Dic5311yescharacteristic
180C24C22×Dic5311yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C22⋊Dic5312yes—
191C22×C4C2×C22⋊D5312yes—
192C2×C8—4244no—
193C2×C8—2124no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—24no—
197C22⋊C4—22no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4C2×#8832214yescomplemented
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—
218C23C22⋊#8832212yes—
219C23C22⋊#8832212yes—
220C23#1313313212yes—
221C23#1313313612yes—
222C23C2×C22⋊Dic5311yescharacteristic
223C23C22×#8832211yescharacteristic
224C23C2×#1171991711yescharacteristic
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—84no—
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#1681034914yes—
258C22C2×C22⋊#8832211yescharacteristic
259C22C2×#1313313211yescharacteristic
260C22C2×#1313313611yescharacteristic
261C4—44no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—42no—
269C2—41no—
270C2—41no—
271C2C2×#1681034911yescharacteristic
272C2#2027636112yes—
subgroupG/H
centerC22C2×#13133132
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2315C2
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#2315D53
hypocenterC53#11453
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #20272663C2semidirect
29. C2×C22⋊C4×C53C8semidirect
64. C22⋊C4×C53C2×C8semidirect
150. C53#11453semidirect
164. C2×C22⋊C4#88322semidirect
203. C22⋊C4C2×#88322semidirect

Maximal subgroups

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

subgroupindex
C2×#168103472
#202726632
C53×#23152
#1145353