Finite Group Databasebeta

← C22×C4×C3⋊#157 #13181511

Subgroups — C22×C4×C3⋊#157 #13181511

Order 1728.

1576 conjugacy classes of subgroups; only the 421 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 421 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
1C23×C3⋊#157C212yesmaximal
2C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
3C2×C4×C3⋊#157C2112yesmaximal, direct product factor
4C22×C4×#157—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3⋊#157C4116yesdirect product factor
7C22×C3⋊#157C22112yes—
8C22×C33×C4C411yescharacteristic, complemented
9C22×C3×C3⋊Dic3C411yescharacteristic, complemented
10C2×C3×C4×C3⋊D3C416yescomplemented
11C2×C3×C4×C3⋊D3C2216yes—
12C4×C3⋊#157C22116yesdirect product factor
13C23×#157—32no—
14C22×C4×C3⋊D3D311yescharacteristic
15C22×C3×C4×D3—62no—
16C2×C4×#157—312no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3C2×C413yes—
19C22×C3×C3⋊D3C2×C418yes—
20C22×C3×C3⋊D3C2313yes—
21C2×C3⋊#157C2×C4148yesdirect product factor
22C2×C3⋊#157C2318yes—
23C2×C33×C4C2×C416yescomplemented
24C2×C3×C3⋊Dic3C2×C416yescomplemented
25C3×C4×C3⋊D3C2×C4112yescomplemented
26C3×C4×C3⋊D3C2314yes—
27C22×C4×Dic3—91nomaximal
28C23×C3⋊D3C2×D311yescharacteristic
29C23×C3×D3—62no—
30C22×#157—316no—
31C22×#157—312no—
32C22×C32×C4—22no—
33C22×C32×C4—42no—
34C22×C32×C4Dic311yescharacteristic, complemented
35C22×C3⋊Dic3Dic311yescharacteristic, complemented
36C2×C4×C3⋊D3Dic316yescomplemented
37C2×C4×C3⋊D3C2×D316yes—
38C22×C3×Dic3—62no—
39C2×C3×C4×D3—612no—
40C2×C3×C4×D3—612no—
41C4×#157—316no—
42C22×C33C4214yescomplemented
43C22×C33C22×C413yes—
44C2×C3×C3⋊D3C42112yescomplemented
45C2×C3×C3⋊D3C22×C413yes—
46C2×C3×C3⋊D3C22×C4112yes—
47C2×C3×C3⋊D3C2411yescharacteristic
48C3⋊#157C22×C4116yesdirect product factor
49C33×C4C22×C414yescomplemented
50C3×C3⋊Dic3C22×C414yescomplemented
51C23×C3×C4—91no—
52C23×Dic3—92no—
53C22×C4×D3—62no—
54C2×C4×Dic3—912no—
55C23×C32—22no—
56C23×C32—42no—
57C23×C32C2×Dic311yescharacteristic
58C22×C3⋊D3C4×D318yes—
59C22×C3⋊D3C2×Dic313yes—
60C22×C3⋊D3C22×D313yes—
61C22×C3×D3—616no—
62C22×C3×D3—66no—
63C22×C3×D3—66no—
64C2×#157—348no—
65C2×#157—38no—
66C2×C32×C4—212no—
67C2×C32×C4—412no—
68C2×C32×C4C2×Dic316yescomplemented
69C2×C3⋊Dic3C2×Dic316yescomplemented
70C4×C3⋊D3C2×Dic3112yescomplemented
71C4×C3⋊D3C22×D314yes—
72C2×C3×Dic3—612no—
73C3×C4×D3—624no—
74C3×C4×D3—68no—
75C22×C42—271noSylow
76C2×C33C2×C4216yescomplemented
77C2×C33C23×C411yescharacteristic
78C3×C3⋊D3C2×C4216yescomplemented
79C3×C3⋊D3C23×C412yes—
80C24×C3—91no—
81C23×D3—62no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4—91no—
85C22×C3×C4—96no—
86C22×C3×C4—96no—
87C22×C3×C4#15711yescharacteristic, complemented
88C22×Dic3—62no—
89C22×Dic3—916no—
90C22×Dic3—912no—
91C2×C4×D3—612no—
92C2×C4×D3—612no—
93C4×Dic3—916no—
94C22×C32—26no—
95C22×C32—28no—
96C22×C32—48no—
97C22×C32—46no—
98C22×C32C4×Dic314yescomplemented
99C22×C32C22×Dic313yes—
100C2×C3⋊D3C4×Dic3112yescomplemented
101C2×C3⋊D3C2×C4×D3112yes—
102C2×C3⋊D3C22×Dic313yes—
103C2×C3⋊D3C23×D311yescharacteristic
104C2×C3×D3—624no—
105C2×C3×D3—66no—
106C2×C3×D3—624no—
107C2×C3×D3—62no—
108#157—316no—
109C32×C4—28no—
110C32×C4—48no—
111C32×C4C22×Dic314yescomplemented
112C3⋊Dic3C22×Dic314yescomplemented
113C3×Dic3—68no—
114C23×C4—91no—
115C23×C4—272no—
116C2×C42—2712no—
117C33C22×C4211yescharacteristic, complemented, Sylow
118C23×C3—22no—
119C23×C3—42no—
120C23×C3—93no—
121C23×C3—93no—
122C23×C3—98no—
123C23×C3C2×#15711yescharacteristic
124C22×D3—616no—
125C22×D3—66no—
126C22×D3—66no—
127C2×C3×C4—212no—
128C2×C3×C4—412no—
129C2×C3×C4—96no—
130C2×C3×C4—94no—
131C2×C3×C4—912no—
132C2×C3×C4C2×#15716yescomplemented
133C2×Dic3—612no—
134C2×Dic3—948no—
135C2×Dic3—98no—
136C4×D3—624no—
137C4×D3—68no—
138C2×C32—212no—
139C2×C32—22no—
140C2×C32—412no—
141C2×C32—42no—
142C2×C32C2×C4×Dic316yescomplemented
143C2×C32C23×Dic311yescharacteristic
144C3⋊D3C2×C4×Dic316yescomplemented
145C3⋊D3C22×C4×D312yes—
146C3×D3—612no—
147C3×D3—64no—
148C24—91no—
149C22×C4—91no—
150C22×C4—96no—
151C22×C4—96no—
152C22×C4—2712no—
153C22×C4—2716no—
154C22×C4C3⋊#15711yescharacteristic, direct product factor
155C42—2716no—
156C22×C3—26no—
157C22×C3—28no—
158C22×C3—46no—
159C22×C3—48no—
160C22×C3—912no—
161C22×C3—912no—
162C22×C3—91no—
163C22×C3—93no—
164C22×C3C4×#15714yescomplemented
165C22×C3C22×#15713yes—
166C2×D3—66no—
167C2×D3—62no—
168C2×D3—624no—
169C2×D3—624no—
170C3×C4—28no—
171C3×C4—48no—
172C3×C4—94no—
173C3×C4C22×#15714yescomplemented
174Dic3—68no—
175Dic3—916no—
176C32—22no—
177C32—42no—
178C32C22×C4×Dic311yescharacteristic, complemented
179C23—93no—
180C23—93no—
181C23—98no—
182C23C2×C3⋊#15711yescharacteristic
183C2×C4—96no—
184C2×C4—94no—
185C2×C4—912no—
186C2×C4—278no—
187C2×C4—2748no—
188C2×C4C2×C3⋊#15716yesdirect product factor
189C2×C3—22no—
190C2×C3—212no—
191C2×C3—42no—
192C2×C3—412no—
193C2×C3—92no—
194C2×C3—96no—
195C2×C3C2×C4×#15716yescomplemented
196C2×C3C23×#15711yescharacteristic
197D3—612no—
198D3—64no—
199C22—91no—
200C22—93no—
201C22—912no—
202C22—912no—
203C22C4×C3⋊#15714yesdirect product factor
204C22C22×C3⋊#15713yes—
205C4—94no—
206C4—2716no—
207C4C22×C3⋊#15714yesdirect product factor
208C3—22no—
209C3—42no—
210C3C22×C4×#15711yescharacteristic, complemented
211C2—92no—
212C2—96no—
213C2C2×C4×C3⋊#15716yesdirect product factor
214C2C23×C3⋊#15711yescharacteristic
subgroupG/H
center, hypercenterC22×C4C3⋊#157
commutator, hypocenterC33C22×C42
FrattiniC2C23×C3⋊#157
FittingC22×C33×C4C4
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
LCSG ▷ C33
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4×C3⋊#157C2direct (N × H)
6. C22×C3⋊#157C4direct (N × H)
8. C22×C33×C4C4semidirect
9. C22×C3×C3⋊Dic3C4semidirect
10. C2×C3×C4×C3⋊D3C4semidirect
12. C4×C3⋊#157C22direct (N × H)
21. C2×C3⋊#157C2×C4direct (N × H)
23. C2×C33×C4C2×C4semidirect
24. C2×C3×C3⋊Dic3C2×C4semidirect
25. C3×C4×C3⋊D3C2×C4semidirect
34. C22×C32×C4Dic3semidirect
35. C22×C3⋊Dic3Dic3semidirect
36. C2×C4×C3⋊D3Dic3semidirect
42. C22×C33C42semidirect
44. C2×C3×C3⋊D3C42semidirect
48. C3⋊#157C22×C4direct (N × H)
49. C33×C4C22×C4semidirect
50. C3×C3⋊Dic3C22×C4semidirect
68. C2×C32×C4C2×Dic3semidirect
69. C2×C3⋊Dic3C2×Dic3semidirect
70. C4×C3⋊D3C2×Dic3semidirect
76. C2×C33C2×C42semidirect
78. C3×C3⋊D3C2×C42semidirect
87. C22×C3×C4#157semidirect
98. C22×C32C4×Dic3semidirect
100. C2×C3⋊D3C4×Dic3semidirect
111. C32×C4C22×Dic3semidirect
112. C3⋊Dic3C22×Dic3semidirect
117. C33C22×C42semidirect
132. C2×C3×C4C2×#157semidirect
142. C2×C32C2×C4×Dic3semidirect
144. C3⋊D3C2×C4×Dic3semidirect
154. C22×C4C3⋊#157direct (N × H)
164. C22×C3C4×#157semidirect
173. C3×C4C22×#157semidirect
178. C32C22×C4×Dic3semidirect
188. C2×C4C2×C3⋊#157direct (N × H)
195. C2×C3C2×C4×#157semidirect
203. C22C4×C3⋊#157direct (N × H)
207. C4C22×C3⋊#157direct (N × H)
210. C3C22×C4×#157semidirect
213. C2C2×C4×C3⋊#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C22×C3×C4×C3⋊D32
C2×C4×C3⋊#1572
C22×C4×#1573
C22×C4×Dic39