Finite Group Databasebeta

← C22×#15688892 #26213720

Subgroups — C22×#15688892 #26213720

Order 12032.

2494 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C212yesmaximal
2C2×#15688892C2112yesmaximal, direct product factor
3C23×C4⋊C4×C47C211yesmaximal, characteristic
4C22×#10625329—28no—
5C22×#10625329—26no—
6C22×#10625329C4116yescomplemented
7C22×#10625329C22112yes—
8C23×C47⋊C8—22no—
9#15688892C22116yesdirect product factor
10C24×C4×C47C412yes—
11C24×C4×C47C2211yescharacteristic
12C22×C4⋊C4×C47—22no—
13C22×C4⋊C4×C47—26no—
14C22×C4⋊C4×C47C416yes—
15C22×C4⋊C4×C47C2216yes—
16C2×#10625329—248no—
17C2×#10625329—224no—
18C2×#10625329—212no—
19C2×#10625329C2×C4148yescomplemented
20C2×#10625329C2318yes—
21C22×C47⋊C8—216no—
22C22×C47⋊C8—212no—
23C25×C47C2×C411yescharacteristic
24C23×C4×C47—26no—
25C23×C4×C47—22no—
26C23×C4×C47—48no—
27C23×C4×C47C2×C4112yes—
28C23×C4×C47C2×C413yes—
29C23×C4×C47C2×C418yes—
30C23×C4×C47D414yes—
31C23×C4×C47D412yes—
32C23×C4×C47D416yes—
33C23×C4×C47Q814yes—
34C23×C4×C47C2313yes—
35C2×C4⋊C4×C47—212no—
36C2×C4⋊C4×C47—212no—
37C2×C4⋊C4×C47—212no—
38C2×C4⋊C4×C47—212no—
39C2×C4⋊C4×C47C2×C4112yes—
40C2×C4⋊C4×C47C2314yes—
41C24×C47—24no—
42C24×C47—24no—
43C24×C47C4214yes—
44C24×C47C22⋊C413yes—
45C24×C47C22⋊C411yescharacteristic
46C24×C47C4⋊C414yes—
47C24×C47C22×C413yes—
48C22×C4×C47—212no—
49C22×C4×C47—212no—
50C22×C4×C47—424no—
51C22×C4×C47—48no—
52C22×C4×C47—424no—
53C22×C4×C47C42112yes—
54C22×C4×C47C22⋊C416yes—
55C22×C4×C47C22⋊C416yes—
56C22×C4×C47C22⋊C418yes—
57C22×C4×C47C22⋊C4124yes—
58C22×C4×C47C4⋊C4112yes—
59C22×C4×C47C4⋊C4124yes—
60C22×C4×C47C4⋊C418yes—
61C22×C4×C47C22×C418yes—
62C22×C4×C47C22×C413yes—
63C22×C4×C47C22×C4112yes—
64C22×C4×C47C2×D416yes—
65C22×C4×C47C2×D416yes—
66C22×C4×C47C2×D416yes—
67C22×C4×C47C2×Q816yes—
68C22×C4×C47C2411yescharacteristic
69C4⋊C4×C47—224no—
70C4⋊C4×C47—28no—
71#10625329—248no—
72#10625329—28no—
73#10625329C22×C4116yescomplemented
74C2×C47⋊C8—248no—
75C2×C47⋊C8—28no—
76C23×C47—216no—
77C23×C47—216no—
78C23×C47—26no—
79C23×C47—26no—
80C23×C47—26no—
81C23×C47—26no—
82C23×C47#95112yes—
83C23×C47#9514yes—
84C23×C47#10014yes—
85C23×C47#10114yes—
86C23×C47C2×C4216yes—
87C23×C47C2×C22⋊C413yes—
88C23×C47C2×C22⋊C413yes—
89C23×C47C2×C4⋊C416yes—
90C23×C47C23×C411yescharacteristic
91C2×C4×C47—28no—
92C2×C4×C47—424no—
93C2×C4×C47—424no—
94C2×C4×C47—48no—
95C2×C4×C47#95124yes—
96C2×C4×C47#95124yes—
97C2×C4×C47C2×C4216yes—
98C2×C4×C47C2×C22⋊C4112yes—
99C2×C4×C47C2×C22⋊C4112yes—
100C2×C4×C47C2×C22⋊C416yes—
101C2×C4×C47C2×C4⋊C4112yes—
102C2×C4×C47C2×C4⋊C4112yes—
103C2×C4×C47C2×C4⋊C416yes—
104C2×C4×C47C23×C412yes—
105C2×C4×C47C22×D412yes—
106C2×C4×C47C22×D411yescharacteristic
107C2×C4×C47C22×Q811yescharacteristic
108C47⋊C8—216no—
109C22×#343—471nomaximal, Sylow
110C22×C47—23no—
111C22×C47—21no—
112C22×C47—23no—
113C22×C47—21no—
114C22×C47—224no—
115C22×C47—212no—
116C22×C47—212no—
117C22×C47#343116yescomplemented
118C22×C47C2×#9516yes—
119C22×C47C2×#9516yes—
120C22×C47C2×#10016yes—
121C22×C47C2×#10116yes—
122C22×C47C22×C4211yescharacteristic
123C22×C47C22×C22⋊C411yescharacteristic
124C22×C47C22×C4⋊C411yescharacteristic
125C4×C47—48no—
126C4×C47C2×#95112yes—
127C4×C47C22×C22⋊C412yes—
128C4×C47C22×C4⋊C412yes—
129C23×M16—472no—
130C23×C4⋊C4D4711yescharacteristic
131C2×#343—4712no—
132C2×C47—21no—
133C2×C47—26no—
134C2×C47—21no—
135C2×C47C2×#343112yescomplemented
136C2×C47C22×#9511yescharacteristic
137C2×C47C22×#10011yescharacteristic
138C2×C47C22×#10111yescharacteristic
139C24×C4Dic4712yes—
140C24×C4C2×D4711yescharacteristic
141C22×M16—4716no—
142C22×M16—4712no—
143C22×M16—948no—
144C22×M16—946no—
145C23×C8—942no—
146C22×C4⋊C4—26no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4Dic4716yes—
149C22×C4⋊C4C2×D4716yes—
150#343—4716no—
151C47C22×#34311yescharacteristic, complemented, Sylow
152C25C2×Dic4711yescharacteristic
153C23×C4—26no—
154C23×C4—22no—
155C23×C4—48no—
156C23×C4Dic9414yes—
157C23×C4C4×D4718yes—
158C23×C4D18814yes—
159C23×C4C2×Dic47112yes—
160C23×C4C2×Dic4713yes—
161C23×C4C22⋊D4716yes—
162C23×C4C22⋊D4712yes—
163C23×C4C22×D4713yes—
164C2×M16—4748no—
165C2×M16—478no—
166C2×M16—9448no—
167C2×M16—9424no—
168C2×M16—9412no—
169C22×C8—9416no—
170C22×C8—9412no—
171C2×C4⋊C4—212no—
172C2×C4⋊C4—212no—
173C2×C4⋊C4—212no—
174C2×C4⋊C4—212no—
175C2×C4⋊C4C2×Dic47112yes—
176C2×C4⋊C4C22×D4714yes—
177C24—24no—
178C24—24no—
179C24C4×Dic4714yes—
180C24C4⋊Dic4714yes—
181C24C22⋊Dic4713yes—
182C24C22⋊Dic4711yescharacteristic
183C24C22×Dic4713yes—
184C22×C4—212no—
185C22×C4—212no—
186C22×C4—424no—
187C22×C4—48no—
188C22×C4—424no—
189C22×C4C4×Dic47112yes—
190C22×C4Dic47⋊C4124yes—
191C22×C4Dic47⋊C418yes—
192C22×C4C4⋊Dic47112yes—
193C22×C4(C2×D47)⋊C418yes—
194C22×C4(C2×D47)⋊C4124yes—
195C22×C4C22⋊Dic4716yes—
196C22×C4C22⋊Dic4716yes—
197C22×C4C2×Dic9416yes—
198C22×C4C2×C4×D47112yes—
199C22×C4C2×D18816yes—
200C22×C4C22×Dic4718yes—
201C22×C4C22×Dic4713yes—
202C22×C4C2×C22⋊D4716yes—
203C22×C4C2×C22⋊D4716yes—
204C22×C4C23×D4711yescharacteristic
205M16—4716no—
206M16—9448no—
207M16—948no—
208C2×C8—9448no—
209C2×C8—948no—
210C4⋊C4—224no—
211C4⋊C4—28no—
212C23—26no—
213C23—26no—
214C23—26no—
215C23—26no—
216C23—216no—
217C23—216no—
218C23C47⋊#95112yes—
219C23C47⋊#9514yes—
220C23#1309394114yes—
221C23#1309394314yes—
222C23C2×C4×Dic4716yes—
223C23C2×C22⋊Dic4713yes—
224C23C2×C22⋊Dic4713yes—
225C23C2×C4⋊Dic4716yes—
226C23C23×Dic4711yescharacteristic
227C2×C4—28no—
228C2×C4—424no—
229C2×C4—424no—
230C2×C4—48no—
231C2×C4C47⋊#95124yes—
232C2×C4C47⋊#95124yes—
233C2×C4C2×C4×Dic4716yes—
234C2×C4C2×(C2×D47)⋊C4112yes—
235C2×C4C2×(C2×D47)⋊C4112yes—
236C2×C4C2×C22⋊Dic4716yes—
237C2×C4C2×Dic47⋊C4112yes—
238C2×C4C2×Dic47⋊C4112yes—
239C2×C4C2×C4⋊Dic4716yes—
240C2×C4C22×C4×D4712yes—
241C2×C4C22×C22⋊D4712yes—
242C2×C4C22×D18811yescharacteristic
243C2×C4C22×Dic9411yescharacteristic
244C8—9416no—
245C22—224no—
246C22—212no—
247C22—212no—
248C22—23no—
249C22—21no—
250C22—23no—
251C22—21no—
252C22#15688892116yesdirect product factor
253C22C2×C47⋊#9516yes—
254C22C2×C47⋊#9516yes—
255C22C2×#1309394116yes—
256C22C2×#1309394316yes—
257C22C22×C4×Dic4711yescharacteristic
258C22C22×C22⋊Dic4711yescharacteristic
259C22C22×C4⋊Dic4711yescharacteristic
260C4—48no—
261C4C2×C47⋊#95112yes—
262C4C22×(C2×D47)⋊C412yes—
263C4C22×Dic47⋊C412yes—
264C2—26no—
265C2—21no—
266C2—21no—
267C2C2×#15688892112yesdirect product factor
268C2C22×C47⋊#9511yescharacteristic
269C2C22×#1309394111yescharacteristic
270C2C22×#1309394311yescharacteristic
subgroupG/H
centerC24C22⋊Dic47
commutatorC22×C47C22×C42
FrattiniC22×C4C23×D47
FittingC23×C4⋊C4×C47C2
socleC24×C47C22⋊C4
prime radicalC25×C47C2×C4
cosocleC22×C4×C47C24
hypercenterC23×C4⋊C4D47
hypocenterC47C22×#343
LCSG ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15688892C2direct (N × H)
6. C22×#10625329C4semidirect
9. #15688892C22direct (N × H)
19. C2×#10625329C2×C4semidirect
73. #10625329C22×C4semidirect
117. C22×C47#343semidirect
135. C2×C47C2×#343semidirect
151. C47C22×#343semidirect
252. C22#15688892direct (N × H)
267. C2C2×#15688892direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#106253292
C2×#156888922
C23×C4⋊C4×C472
C22×#34347