Finite Group Databasebeta

← C2×#20272661 #27293246

Subgroups — C2×#20272661 #27293246

Order 13568.

598 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16810349C212yesmaximal
2#20272661C214yesmaximal, direct product factor
3C2×C53×#384C211yesmaximal, characteristic
4C2×C22⋊#88322—22no—
5#16810349—22no—
6#16810349C2214yes—
7C22×C4⋊C4×C53—22no—
8C22×C4⋊C4×C53C411yescharacteristic
9C22×C4⋊C4×C53C2211yescharacteristic
10C22×C42×C53C411yescharacteristic
11C53×#384—22no—
12C53×#384C412yes—
13C53×#384C2212yes—
14C22×#88322—42no—
15C22⋊#88322—24no—
16C22⋊#88322—42no—
17C23×C4×C53—21no—
18C23×C4×C53—21no—
19C23×C4×C53C2×C411yescharacteristic
20C23×C4×C53D412yes—
21C2×C4⋊C4×C53—24no—
22C2×C4⋊C4×C53—21no—
23C2×C4⋊C4×C53—24no—
24C2×C4⋊C4×C53—41no—
25C2×C4⋊C4×C53—41no—
26C2×C4⋊C4×C53—48no—
27C2×C4⋊C4×C53—41no—
28C2×C4⋊C4×C53—41no—
29C2×C4⋊C4×C53C2×C412yes—
30C2×C4⋊C4×C53C2×C411yescharacteristic
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53C2311yescharacteristic
33C2×C42×C53—21no—
34C2×C42×C53—22no—
35C2×C42×C53C814yescomplemented
36C2×C42×C53C2×C412yes—
37C24×C53C22⋊C411yescharacteristic
38C22×C4×C53—24no—
39C22×C4×C53—24no—
40C22×C4×C53—24no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53—44no—
49C22×C4×C53—44no—
50C22×C4×C53—41no—
51C22×C4×C53—41no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C2×C812yes—
55C22×C4×C53M1612yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D412yes—
58C4⋊C4×C53—41no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—48no—
62C4⋊C4×C53—48no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C42×C53—24no—
66C42×C53—22no—
67C42×C53C2×C814yescomplemented
68C2×#88322—44no—
69C2×#88322—84no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C22⋊C812yes—
73C23×C53#9911yescharacteristic
74C23×C53#9911yescharacteristic
75C23×C53#10112yes—
76C23×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53—216no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—44no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53—82no—
97C2×C4×C53—82no—
98C2×C4×C53—82no—
99C2×C4×C53C22⋊C812yes—
100C2×C4×C53C22×C811yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102#88322—84no—
103C2×#1325—531nomaximal, Sylow
104C22×C53—24no—
105C22×C53—24no—
106C22×C53—21no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53#32412yes—
111C22×C53#32412yes—
112C22×C53#35312yes—
113C22×C53#35612yes—
114C22×C53C2×C22⋊C811yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C4×C53—28no—
118C4×C53—22no—
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—42no—
122C4×C53—42no—
123C4×C53—82no—
124C2×#384D5311yescharacteristic
125C2×#324—532no—
126#1325—534no—
127C2×C53—21no—
128C2×C53—22no—
129C2×C53—21no—
130C2×C53#132514yescomplemented
131C2×C53C2×#32411yescharacteristic
132C2×C53C2×#35311yescharacteristic
133C2×C53C2×#35611yescharacteristic
134C22×C4⋊C4—22no—
135C22×C4⋊C4Dic5311yescharacteristic
136C22×C4⋊C4C2×D5311yescharacteristic
137C22×C42Dic5311yescharacteristic
138C2×C22⋊C8—1062no—
139#384—22no—
140#384Dic5312yes—
141#384C2×D5312yes—
142#324—534no—
143#324—1062no—
144C53C2×#132511yescharacteristic, complemented, Sylow
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×Dic5311yescharacteristic
148C23×C4C22⋊D5312yes—
149C22×C8—2122no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C2×Dic5312yes—
159C2×C4⋊C4C2×Dic5311yescharacteristic
160C2×C4⋊C4C22⋊D5312yes—
161C2×C4⋊C4C22×D5311yescharacteristic
162C2×C42—22no—
163C2×C42—21no—
164C2×C42#8832214yescomplemented
165C2×C42C2×Dic5312yes—
166C22⋊C8—1064no—
167C22⋊C8—2122no—
168C24C22⋊Dic5311yescharacteristic
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4C2×#8832212yes—
184C22×C4#1171991712yes—
185C22×C4C22⋊Dic5311yescharacteristic
186C22×C4C22⋊Dic5312yes—
187C22×C4C22×Dic5311yescharacteristic
188C22×C4C2×C22⋊D5312yes—
189C2×C8—4244no—
190C2×C8—2124no—
191C4⋊C4—41no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—48no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C42—24no—
199C42—22no—
200C42C2×#8832214yescomplemented
201C23—22no—
202C23—22no—
203C23C22⋊#8832212yes—
204C23#1313313211yescharacteristic
205C23#1313313211yescharacteristic
206C23#1313313612yes—
207C23C2×C22⋊Dic5311yescharacteristic
208C2×C4—24no—
209C2×C4—216no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4C22⋊#8832212yes—
231C2×C4C22×#8832211yescharacteristic
232C2×C4C2×#1171991711yescharacteristic
233C8—4244no—
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#1681034912yes—
241C22#1681034912yes—
242C22#1681041612yes—
243C22#1681042212yes—
244C22C2×C22⋊#8832211yescharacteristic
245C22C2×#1313313211yescharacteristic
246C22C2×#1313313611yescharacteristic
247C4—28no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C2—21no—
255C2—21no—
256C2—22no—
257C2#2027266114yesdirect product factor
258C2C2×#1681034911yescharacteristic
259C2C2×#1681041611yescharacteristic
260C2C2×#1681042211yescharacteristic
subgroupG/H
centerC23#13133132
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC2×C53×#384C2
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenterC2×#384D53
hypocenterC53C2×#1325
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. #20272661C2direct (N × H)
35. C2×C42×C53C8semidirect
67. C42×C53C2×C8semidirect
130. C2×C53#1325semidirect
144. C53C2×#1325semidirect
164. C2×C42#88322semidirect
200. C42C2×#88322semidirect
257. C2#20272661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168103492
#202726612
C2×C53×#3842
C2×#132553