Finite Group Databasebeta

← C3×M16×#157 #13173244

Subgroups — C3×M16×#157 #13173244

Order 1728.

460 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 140 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×C3×C4×#157C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4C3×#63767C212yesmaximal, complemented
5C3×C8×#157C212yesmaximal, complemented
6M16×#157C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C412yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C3×#3792C412yescomplemented
10C3×C4×#157C412yes—
11C3×C4×#157C2211yescharacteristic
12C3×C4×#157C2211yescharacteristic
13C3×#3788C412yescomplemented
14C3×#3787C2212yes—
15C33×M16C411yescharacteristic, complemented
16C3×#3748C411yescharacteristic, complemented
17C3×#3743C412yescomplemented
18C3×C8×C3⋊D3C2212yes—
19C2×C4×#157C2×C311yescharacteristic
20#64283C2×C311yescharacteristic
21M16×C3⋊D3C2×C311yescharacteristic
22C3×D3×M16—62no—
23#63767C2×C312yescomplemented
24C8×#157C2×C312yescomplemented
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C2×C3×#157—24no—
27C2×C3×#157C2×C412yes—
28C2×C3×#157C2×C412yes—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C3×#622C2×C414yescomplemented
36C33×C8C2×C412yescomplemented
37C3×C3⋊(C3⋊C8)C2×C412yescomplemented
38C3×C4×M16—91nomaximal
39C22×#157C3×C412yes—
40C2×C4×C3⋊D3C22×C311yescharacteristic
41C2×C3×C4×D3—62no—
42#3792C3×C412yescomplemented
43C4×#157C3×C412yes—
44C4×#157C22×C311yescharacteristic
45C4×#157C22×C311yescharacteristic
46#3788C3×C412yescomplemented
47#3787C22×C312yes—
48C32×M16—22no—
49C32×M16—42no—
50C32×M16C3×C411yescharacteristic, complemented
51#3748C3×C411yescharacteristic, complemented
52#3743C3×C412yescomplemented
53C8×C3⋊D3C22×C312yes—
54C3×#208—62no—
55C3×ΓL(1,𝔽25)—64no—
56C3×D3×C8—64no—
57C22×C33C4211yescharacteristic
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3C4211yescharacteristic
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C3×#157—22no—
62C3×#157M1614yesdirect product factor
63C33×C4C4211yescharacteristic
64C33×C4C22×C411yescharacteristic
65C3×C3⋊Dic3C4211yescharacteristic
66C3×C3⋊Dic3C22×C411yescharacteristic
67C2×C3×M16—91no—
68C2×C3×M16—91no—
69C2×C3×C42—91no—
70D3×M16—62no—
71C3×#97—92no—
72C3×C4×C8—92no—
73C22×C3⋊D3C2×C3×C411yescharacteristic
74C22×C3×D3—62no—
75C2×#157—24no—
76C2×#157C2×C3×C412yes—
77C2×#157C2×C3×C412yes—
78C2×C32×C4—22no—
79C2×C32×C4—42no—
80C2×C32×C4C2×C3×C411yescharacteristic
81C2×C3⋊Dic3C2×C3×C411yescharacteristic
82C4×C3⋊D3C2×C3×C411yescharacteristic
83C4×C3⋊D3C2×C3×C411yescharacteristic
84C4×C3⋊D3C2×C3×C411yescharacteristic
85C4×C3⋊D3C23×C311yescharacteristic
86C2×C3×Dic3—62no—
87C3×C4×D3—62no—
88C3×C4×D3—62no—
89C3×C4×D3—62no—
90C3×C4×D3—62no—
91#622C2×C3×C414yescomplemented
92C32×C8—24no—
93C32×C8—44no—
94C32×C8C2×C3×C412yescomplemented
95C3⋊(C3⋊C8)C2×C3×C412yescomplemented
96C3×C3⋊C8—64no—
97C4×M16—91noSylow
98C2×C33—21no—
99C2×C33C2×C4211yescharacteristic
100C3×C3⋊D3—21no—
101C3×C3⋊D3C2×M1612yes—
102C22×C3×C4—91no—
103C22×C3×C4—92no—
104C2×C4×D3—62no—
105C3×M16—22no—
106C3×M16—42no—
107C3×M16—92no—
108C3×M16—92no—
109C3×M16—91no—
110C3×M16—92no—
111C3×M16#15711yescharacteristic, direct product factor
112C2×C3×C8—92no—
113C2×C3×C8—92no—
114C3×C42—91no—
115C3×C42—91no—
116C3×C42—92no—
117#208—62no—
118ΓL(1,𝔽25)—64no—
119D3×C8—64no—
120C22×C32—22no—
121C22×C32—42no—
122C22×C32C3×C4211yescharacteristic
123C2×C3⋊D3—22no—
124C2×C3⋊D3C3×C4211yescharacteristic
125C2×C3⋊D3C22×C3×C411yescharacteristic
126C2×C3×D3—62no—
127C2×C3×D3—62no—
128C2×C3×D3—124no—
129#157—22no—
130#157C3×M1614yesdirect product factor
131C32×C4—22no—
132C32×C4—22no—
133C32×C4—42no—
134C32×C4—42no—
135C32×C4C3×C4211yescharacteristic
136C32×C4C22×C3×C411yescharacteristic
137C3⋊Dic3C3×C4211yescharacteristic
138C3⋊Dic3C22×C3×C411yescharacteristic
139C3×Dic3—62no—
140C3×Dic3—62no—
141C2×M16—91no—
142C2×M16—91no—
143C2×C42—91no—
144#97—92no—
145C4×C8—92no—
146C33C4×M1611yescharacteristic, complemented, Sylow
147C23×C3—91no—
148C22×D3—62no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4—92no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4—92no—
158C2×C3×C4—184no—
159C2×C3×C4C2×#15711yescharacteristic
160C2×Dic3—62no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—62no—
164C4×D3—62no—
165C3×C8—24no—
166C3×C8—44no—
167C3×C8—94no—
168C3×C8—92no—
169C3×C8C2×#15712yescomplemented
170C3⋊C8—64no—
171C2×C32—22no—
172C2×C32—21no—
173C2×C32—42no—
174C2×C32—42no—
175C2×C32—82no—
176C2×C32C2×C3×C4211yescharacteristic
177C3⋊D3—21no—
178C3⋊D3C2×C3×M1612yes—
179C3×D3—64no—
180C3×D3—122no—
181C22×C4—91no—
182C22×C4—92no—
183M16—92no—
184M16—92no—
185M16—91no—
186M16—92no—
187M16C3×#15711yescharacteristic, direct product factor
188C2×C8—92no—
189C2×C8—92no—
190C42—91no—
191C42—91no—
192C42—92no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—91no—
196C22×C3—91no—
197C22×C3—182no—
198C22×C3C4×#15711yescharacteristic
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—124no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—42no—
205C3×C4—42no—
206C3×C4—94no—
207C3×C4—91no—
208C3×C4—91no—
209C3×C4—182no—
210C3×C4C4×#15711yescharacteristic
211C3×C4C22×#15711yescharacteristic
212Dic3—62no—
213Dic3—62no—
214C32—22no—
215C32—42no—
216C32C3×C4×M1611yescharacteristic, complemented
217C23—91no—
218C2×C4—92no—
219C2×C4—91no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—92no—
225C2×C4—184no—
226C2×C4C2×C3×#15711yescharacteristic
227C8—94no—
228C8—92no—
229C8C2×C3×#15712yescomplemented
230C2×C3—21no—
231C2×C3—22no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—82no—
235C2×C3—92no—
236C2×C3—181no—
237C2×C3C2×C4×#15711yescharacteristic
238D3—64no—
239D3—122no—
240C22—91no—
241C22—91no—
242C22—182no—
243C22C3×C4×#15711yescharacteristic
244C4—94no—
245C4—91no—
246C4—91no—
247C4—182no—
248C4C3×C4×#15711yescharacteristic
249C4C22×C3×#15711yescharacteristic
250C3—22no—
251C3—42no—
252C3M16×#15711yescharacteristic, direct product factor
253C2—21no—
254C2—92no—
255C2—181no—
256C2C2×C3×C4×#15711yescharacteristic
subgroupG/H
centerC3×C4C22×#157
commutatorC2×C32C2×C3×C42
FrattiniC4C22×C3×#157
FittingC33×M16C4
socleC2×C33C2×C42
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC4×C3⋊D3C23×C3
hypercenterC3×M16#157
hypocenterC32C3×C4×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. C3×#63767C2semidirect
5. C3×C8×#157C2semidirect
6. M16×#157C3direct (N × H)
9. C3×#3792C4semidirect
13. C3×#3788C4semidirect
15. C33×M16C4semidirect
16. C3×#3748C4semidirect
17. C3×#3743C4semidirect
23. #63767C2×C3semidirect
24. C8×#157C2×C3semidirect
35. C3×#622C2×C4semidirect
36. C33×C8C2×C4semidirect
37. C3×C3⋊(C3⋊C8)C2×C4semidirect
42. #3792C3×C4semidirect
46. #3788C3×C4semidirect
50. C32×M16C3×C4semidirect
51. #3748C3×C4semidirect
52. #3743C3×C4semidirect
62. C3×#157M16direct (N × H)
91. #622C2×C3×C4semidirect
94. C32×C8C2×C3×C4semidirect
95. C3⋊(C3⋊C8)C2×C3×C4semidirect
111. C3×M16#157direct (N × H)
130. #157C3×M16direct (N × H)
146. C33C4×M16semidirect
169. C3×C8C2×#157semidirect
187. M16C3×#157direct (N × H)
216. C32C3×C4×M16semidirect
229. C8C2×C3×#157semidirect
252. C3M16×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C3×#642832
C3×M16×C3⋊D32
C3×#637672
C3×C8×#1572
M16×#1573
C3×C4×M169