Finite Group Databasebeta

← C22×#15689172 #27079971

Subgroups — C22×#15689172 #27079971

Order 12032.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625329C212yesmaximal
2C2×#15689172C2124yesmaximal, direct product factor
3C22×#13093960C214yesmaximal, complemented
4C23×M16×C47C211yesmaximal, characteristic
5C22×#10625329—22no—
6C22×#10625329—212no—
7C22×#10625329C2214yes—
8C22×#10625329C22124yes—
9C23×C47⋊C8—22no—
10#15689172C22164yesdirect product factor
11C2×#13093960C22148yescomplemented
12C24×C4×C47C2211yescharacteristic
13C22×M16×C47C414yes—
14C22×M16×C47C4112yes—
15C22×M16×C47C22112yes—
16C23×C8×C47C2212yes—
17C2×#10625329—224no—
18C2×#10625329—212no—
19C2×#10625329—248no—
20C2×#10625329C23124yes—
21C2×#10625329C23132yes—
22C22×C47⋊C8—224no—
23C22×C47⋊C8—24no—
24#13093960C23164yescomplemented
25C25×C47Q811yescharacteristic
26C23×C4×C47C2×C412yes—
27C23×C4×C47C2×C416yes—
28C23×C4×C47D412yes—
29C23×C4×C47D416yes—
30C23×C4×C47Q811yescharacteristic
31C23×C4×C47Q816yes—
32C23×C4×C47C2311yescharacteristic
33C23×C4×C47C2316yes—
34C2×M16×C47C2×C4148yes—
35C2×M16×C47C2×C4148yes—
36C2×M16×C47C23116yes—
37C22×C8×C47C2×C4112yes—
38C22×C8×C47C2×C414yes—
39C22×C8×C47C23112yes—
40C24×C47—44no—
41C24×C47C4⋊C412yes—
42C24×C47C4⋊C416yes—
43C24×C47C2×Q811yescharacteristic
44C24×C47C2×Q816yes—
45C22×C4×C47C4⋊C4112yes—
46C22×C4×C47C4⋊C4112yes—
47C22×C4×C47C4⋊C416yes—
48C22×C4×C47C4⋊C4124yes—
49C22×C4×C47C4⋊C412yes—
50C22×C4×C47C22×C413yes—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C22×C4112yes—
53C22×C4×C47C22×C4112yes—
54C22×C4×C47C2×D4112yes—
55C22×C4×C47C2×D413yes—
56C22×C4×C47C2×D4112yes—
57C22×C4×C47C2×D411yescharacteristic
58C22×C4×C47C2×Q813yes—
59C22×C4×C47C2×Q816yes—
60C22×C4×C47C2×Q8112yes—
61C22×C4×C47C2413yes—
62C22×C4×C47C2414yes—
63M16×C47C22×C4164yes—
64C2×C8×C47C22×C4124yes—
65C2×C8×C47C22×C4124yes—
66C2×C8×C47C2418yes—
67#10625329—232no—
68#10625329—224no—
69#10625329C24116yes—
70C2×C47⋊C8—224no—
71C2×C47⋊C8—232no—
72C23×C47—28no—
73C23×C47—24no—
74C23×C47—412no—
75C23×C47—412no—
76C23×C47C2×C4⋊C413yes—
77C23×C47C2×C4⋊C4112yes—
78C23×C47C2×C4⋊C4112yes—
79C23×C47C2×C4⋊C411yescharacteristic
80C23×C47C22×Q813yes—
81C23×C47C22×Q814yes—
82C2×C4×C47C2×C4⋊C416yes—
83C2×C4×C47C2×C4⋊C4124yes—
84C2×C4×C47C2×C4⋊C4124yes—
85C2×C4×C47C2×C4⋊C4112yes—
86C2×C4×C47C2×C4⋊C413yes—
87C2×C4×C47C2×C4⋊C4112yes—
88C2×C4×C47C2×C4⋊C413yes—
89C2×C4×C47C23×C413yes—
90C2×C4×C47C23×C413yes—
91C2×C4×C47C23×C418yes—
92C2×C4×C47C22×D413yes—
93C2×C4×C47C22×D418yes—
94C2×C4×C47C22×D413yes—
95C2×C4×C47C22×Q813yes—
96C2×C4×C47C22×Q814yes—
97C2×C4×C47C2511yescharacteristic
98C8×C47C23×C4116yes—
99C47⋊C8—216no—
100C22×#430—471nomaximal, Sylow
101C22×C47—224no—
102C22×C47—26no—
103C22×C47—26no—
104C22×C47—412no—
105C22×C47—44no—
106C22×C47#43014yescomplemented
107C22×C47C22×C4⋊C413yes—
108C22×C47C22×C4⋊C418yes—
109C22×C47C22×C4⋊C413yes—
110C22×C47C23×Q811yescharacteristic
111C4×C47C22×C4⋊C413yes—
112C4×C47C22×C4⋊C413yes—
113C4×C47C22×C4⋊C418yes—
114C4×C47C24×C411yescharacteristic
115C4×C47C23×D411yescharacteristic
116C23×M16—472no—
117C23×M16D4711yescharacteristic
118C2×#430—4724no—
119C22×#108—474no—
120C2×C47—28no—
121C2×C47—23no—
122C2×C47—21no—
123C2×C47C2×#43016yescomplemented
124C2×C47C23×C4⋊C411yescharacteristic
125C24×C4C2×D4711yescharacteristic
126C22×M16—474no—
127C22×M16—4724no—
128C22×M16—9412no—
129C22×M16—942no—
130C22×M16Dic4714yes—
131C22×M16Dic47112yes—
132C22×M16C2×D47112yes—
133C23×C8—942no—
134C23×C8C2×D4712yes—
135#430—4764no—
136C2×#108—4748no—
137C47C22×#43011yescharacteristic, complemented, Sylow
138C25Dic9411yescharacteristic
139C23×C4Dic9411yescharacteristic
140C23×C4Dic9416yes—
141C23×C4D18816yes—
142C23×C4D18812yes—
143C23×C4C2×Dic4712yes—
144C23×C4C2×Dic4716yes—
145C23×C4C22×D4716yes—
146C23×C4C22×D4711yescharacteristic
147C2×M16—4724no—
148C2×M16—4732no—
149C2×M16—9424no—
150C2×M16—9412no—
151C2×M16—9448no—
152C2×M16C2×Dic47148yes—
153C2×M16C2×Dic47148yes—
154C2×M16C22×D47116yes—
155C22×C8—9424no—
156C22×C8—944no—
157C22×C8C2×Dic4714yes—
158C22×C8C2×Dic47112yes—
159C22×C8C22×D47112yes—
160#108—4764no—
161C24—44no—
162C24C4⋊Dic4712yes—
163C24C4⋊Dic4716yes—
164C24C2×Dic9411yescharacteristic
165C24C2×Dic9416yes—
166C22×C4C4⋊Dic4716yes—
167C22×C4C4⋊Dic47112yes—
168C22×C4C4⋊Dic47124yes—
169C22×C4C4⋊Dic4712yes—
170C22×C4C4⋊Dic47112yes—
171C22×C4C2×Dic9416yes—
172C22×C4C2×Dic94112yes—
173C22×C4C2×Dic9413yes—
174C22×C4C2×D188112yes—
175C22×C4C2×D18813yes—
176C22×C4C2×D188112yes—
177C22×C4C2×D18811yescharacteristic
178C22×C4C22×Dic4711yescharacteristic
179C22×C4C22×Dic4713yes—
180C22×C4C22×Dic47112yes—
181C22×C4C22×Dic47112yes—
182C22×C4C23×D4713yes—
183C22×C4C23×D4714yes—
184M16—4716no—
185M16—9432no—
186M16—9424no—
187M16C22×Dic47164yes—
188C2×C8—9424no—
189C2×C8—9432no—
190C2×C8C22×Dic47124yes—
191C2×C8C22×Dic47124yes—
192C2×C8C23×D4718yes—
193C23—28no—
194C23—24no—
195C23—412no—
196C23—412no—
197C23C2×C4⋊Dic47112yes—
198C23C2×C4⋊Dic4713yes—
199C23C2×C4⋊Dic47112yes—
200C23C2×C4⋊Dic4711yescharacteristic
201C23C22×Dic9413yes—
202C23C22×Dic9414yes—
203C2×C4C2×C4⋊Dic47124yes—
204C2×C4C2×C4⋊Dic4716yes—
205C2×C4C2×C4⋊Dic47124yes—
206C2×C4C2×C4⋊Dic47112yes—
207C2×C4C2×C4⋊Dic47112yes—
208C2×C4C2×C4⋊Dic4713yes—
209C2×C4C2×C4⋊Dic4713yes—
210C2×C4C23×Dic4713yes—
211C2×C4C23×Dic4713yes—
212C2×C4C23×Dic4718yes—
213C2×C4C22×D18813yes—
214C2×C4C22×D18818yes—
215C2×C4C22×D18813yes—
216C2×C4C22×Dic9413yes—
217C2×C4C22×Dic9414yes—
218C2×C4C24×D4711yescharacteristic
219C8—9416no—
220C8C23×Dic47116yes—
221C22—224no—
222C22—26no—
223C22—26no—
224C22—412no—
225C22—44no—
226C22#1568917214yesdirect product factor
227C22C22×C4⋊Dic4713yes—
228C22C22×C4⋊Dic4718yes—
229C22C22×C4⋊Dic4713yes—
230C22C23×Dic9411yescharacteristic
231C4C22×C4⋊Dic4713yes—
232C4C22×C4⋊Dic4718yes—
233C4C22×C4⋊Dic4713yes—
234C4C24×Dic4711yescharacteristic
235C4C23×D18811yescharacteristic
236C2—28no—
237C2—23no—
238C2—21no—
239C2C2×#1568917216yesdirect product factor
240C2C23×C4⋊Dic4711yescharacteristic
subgroupG/H
centerC22×C4C2×D188
commutatorC4×C47C24×C4
FrattiniC2×C4C24×D47
FittingC23×M16×C47C2
socleC23×C47C2×C4⋊C4
prime radicalC25×C47Q8
cosocleC2×C4×C47C25
hypercenterC23×M16D47
hypocenterC47C22×#430
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C23×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15689172C2direct (N × H)
3. C22×#13093960C2semidirect
10. #15689172C22direct (N × H)
11. C2×#13093960C22semidirect
24. #13093960C23semidirect
106. C22×C47#430semidirect
123. C2×C47C2×#430semidirect
137. C47C22×#430semidirect
226. C22#15689172direct (N × H)
239. C2C2×#15689172direct (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×#156891722
C22×#130939602
C23×M16×C472
C22×#43047