Finite Group Databasebeta

← C2×C3×D3×#170 #13088649

Subgroups — C2×C3×D3×#170 #13088649

Order 1440.

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

normal subgroups only

This group has 326 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×D3×Dic5C211yesmaximal, characteristic
2C2×C3×C3⋊#170C211yesmaximal, characteristic, complemented
3C2×C32×#170C211yesmaximal, characteristic, complemented
4C3×D3×#170C214yesmaximal, direct product factor
5C22×C3×#170—31nomaximal
6C2×D3×#170C311yesmaximal, characteristic, direct product factor
7C22×C3×C5×D3C411yescharacteristic
8C2×C3×Dic15C411yescharacteristic
9C2×C32×Dic5C2211yescharacteristic
10C3×D3×Dic5C412yes—
11C3×D3×Dic5C2212yes—
12C3×C3⋊#170C2212yescomplemented
13C32×#170C2212yescomplemented
14C2×C3×D3×C8—51nomaximal, Hall
15C22×C3×Dic5—31no—
16C2×D3×Dic5C2×C311yescharacteristic
17C2×C3⋊#170C2×C311yescharacteristic, complemented
18C2×C3×#170—21no—
19C2×C3×#170—31no—
20C2×C3×#170—34no—
21C2×C3×#170D311yescharacteristic, direct product factor
22C2×C3×#170C2×C311yescharacteristic, complemented
23D3×#170C2×C314yesdirect product factor
24C22×C32×C5C2×C411yescharacteristic
25C2×C3×C5×D3C814yescomplemented
26C2×C3×C5×D3C2×C412yes—
27C3×Dic15C2×C412yes—
28C32×Dic5C2×C411yescharacteristic
29C32×Dic5C2311yescharacteristic
30C22×#170—31noHall
31C2×C3×C4×D3—51no—
32C2×C32×C8—51no—
33C2×C3×C3⋊C8—51no—
34C3×D3×C8—54no—
35C23×C3×C5—31no—
36C22×C5×D3C3×C411yescharacteristic
37C2×Dic15C3×C411yescharacteristic
38C2×C3×Dic5—21no—
39C2×C3×Dic5—31no—
40C2×C3×Dic5—32no—
41C2×C3×Dic5—32no—
42C2×C3×Dic5C2×D311yescharacteristic
43C2×C3×Dic5C22×C311yescharacteristic
44D3×Dic5C3×C412yes—
45D3×Dic5C22×C312yes—
46C3⋊#170C22×C312yescomplemented
47C3×#170—22no—
48C3×#170—32no—
49C3×#170C2×D312yesdirect product factor
50C3×#170C22×C312yescomplemented
51C22×C3×C8—151no—
52C2×D3×C8—51no—
53C2×C32×C5C2×C812yescomplemented
54C2×C32×C5C22×C411yescharacteristic
55C3×C5×D3C2×C814yescomplemented
56C22×Dic5—31no—
57C2×#170—31no—
58C2×#170—34no—
59C2×#170C3×D311yescharacteristic, direct product factor
60C22×C3×D3AGL(1,𝔽5)11yescharacteristic
61C2×C32×C4—51no—
62C2×C3×Dic3—51no—
63C3×C4×D3—52no—
64C3×C4×D3—52no—
65C32×C8—52no—
66C3×C3⋊C8—52no—
67C22×C3×C5—21no—
68C22×C3×C5—34no—
69C22×C3×C5—32no—
70C22×C3×C5C4×D311yescharacteristic
71C22×C3×C5C2×C3×C411yescharacteristic
72C2×C5×D3C3×C814yescomplemented
73C2×C5×D3C2×C3×C412yes—
74Dic15C2×C3×C412yes—
75C3×Dic5—21no—
76C3×Dic5—21no—
77C3×Dic5—32no—
78C3×Dic5C4×D311yescharacteristic
79C3×Dic5C2×C3×C411yescharacteristic
80C3×Dic5C22×D311yescharacteristic
81C3×Dic5C23×C311yescharacteristic
82C22×C3×C4—151no—
83C2×C4×D3—51no—
84C2×C3×C8—51no—
85C2×C3×C8—51no—
86C2×C3×C8—101no—
87C2×C3×C8—154no—
88C2×C3×C8—151no—
89C2×C3⋊C8—51no—
90D3×C8—54no—
91C32×C5C22×C811yescharacteristic, complemented, Hall
92C23×C5—31no—
93C2×Dic5—31no—
94C2×Dic5—32no—
95C2×Dic5—32no—
96C2×Dic5C2×C3×D311yescharacteristic
97#170—32no—
98#170C2×C3×D312yesdirect product factor
99C22×C32C2×AGL(1,𝔽5)11yescharacteristic
100C2×C3×D3#17014yesdirect product factor
101C2×C3×D3C2×AGL(1,𝔽5)12yes—
102C32×C4—51no—
103C32×C4—51no—
104C3×Dic3—52no—
105C22×C8—151noSylow
106C2×C3×C5—22no—
107C2×C3×C5—21no—
108C2×C3×C5—34no—
109C2×C3×C5D3×C812yescomplemented
110C2×C3×C5C2×C3×C812yescomplemented
111C2×C3×C5C2×C4×D311yescharacteristic
112C2×C3×C5C22×C3×C411yescharacteristic
113C5×D3C2×C3×C814yescomplemented
114C23×C3—31no—
115C22×D3C3×AGL(1,𝔽5)11yescharacteristic
116C2×C3×C4—51no—
117C2×C3×C4—51no—
118C2×C3×C4—101no—
119C2×C3×C4—152no—
120C2×C3×C4—152no—
121C2×C3×C4—151no—
122C2×Dic3—51no—
123C4×D3—52no—
124C4×D3—52no—
125C3×C8—52no—
126C3×C8—52no—
127C3×C8—102no—
128C3×C8—152no—
129C3⋊C8—52no—
130C22×C5—34no—
131C22×C5—32no—
132C22×C5C3×C4×D311yescharacteristic
133Dic5—32no—
134Dic5C3×C4×D311yescharacteristic
135Dic5C22×C3×D311yescharacteristic
136C2×C32C2×#17012yescomplemented
137C2×C32C22×AGL(1,𝔽5)11yescharacteristic
138C3×D3C2×#17014yesdirect product factor
139C22×C4—151no—
140C2×C8—51no—
141C2×C8—154no—
142C2×C8—151no—
143C3×C5—21no—
144C3×C5C2×D3×C811yescharacteristic, complemented
145C3×C5C22×C3×C811yescharacteristic, complemented
146C22×C3—21no—
147C22×C3—34no—
148C22×C3—32no—
149C22×C3D3×AGL(1,𝔽5)11yescharacteristic
150C22×C3C2×C3×AGL(1,𝔽5)11yescharacteristic
151C2×D3C3×#17014yesdirect product factor
152C2×D3C2×C3×AGL(1,𝔽5)12yes—
153C3×C4—51no—
154C3×C4—51no—
155C3×C4—51no—
156C3×C4—51no—
157C3×C4—101no—
158C3×C4—101no—
159C3×C4—152no—
160Dic3—52no—
161C2×C5—34no—
162C2×C5C3×D3×C812yescomplemented
163C2×C5C2×C3×C4×D311yescharacteristic
164C32C22×#17011yescharacteristic, complemented, Sylow
165C23—31no—
166C2×C4—51no—
167C2×C4—152no—
168C2×C4—152no—
169C2×C4—151no—
170C8—52no—
171C8—152no—
172C2×C3—21no—
173C2×C3—22no—
174C2×C3—34no—
175C2×C3D3×#17012yesdirect product factor
176C2×C3C2×C3×#17012yescomplemented
177C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
178C2×C3C22×C3×AGL(1,𝔽5)11yescharacteristic
179D3C2×C3×#17014yesdirect product factor
180C5C2×C3×D3×C811yescharacteristic, complemented, Sylow
181C22—32no—
182C22—34no—
183C22C3×D3×AGL(1,𝔽5)11yescharacteristic
184C4—51no—
185C4—51no—
186C4—152no—
187C3—21no—
188C3C2×D3×#17011yescharacteristic, direct product factor
189C3C22×C3×#17011yescharacteristic, complemented
190C2—34no—
191C2C3×D3×#17012yesdirect product factor
192C2C2×C3×D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×AGL(1,𝔽5)
commutator, hypocenterC3×C5C22×C3×C8
FrattiniC2C2×C3×D3×AGL(1,𝔽5)
Fitting, socleC22×C32×C5C2×C4
prime radicalC22×C3×C5×D3C4
cosocleC3×Dic5C23×C3
LCSG ▷ C3×C5
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#170C2semidirect
3. C2×C32×#170C2semidirect
4. C3×D3×#170C2direct (N × H)
6. C2×D3×#170C3direct (N × H)
12. C3×C3⋊#170C22semidirect
13. C32×#170C22semidirect
17. C2×C3⋊#170C2×C3semidirect
21. C2×C3×#170D3direct (N × H)
22. C2×C3×#170C2×C3semidirect
23. D3×#170C2×C3direct (N × H)
25. C2×C3×C5×D3C8semidirect
46. C3⋊#170C22×C3semidirect
49. C3×#170C2×D3direct (N × H)
50. C3×#170C22×C3semidirect
53. C2×C32×C5C2×C8semidirect
55. C3×C5×D3C2×C8semidirect
59. C2×#170C3×D3direct (N × H)
72. C2×C5×D3C3×C8semidirect
91. C32×C5C22×C8semidirect
98. #170C2×C3×D3direct (N × H)
100. C2×C3×D3#170direct (N × H)
109. C2×C3×C5D3×C8semidirect
110. C2×C3×C5C2×C3×C8semidirect
113. C5×D3C2×C3×C8semidirect
136. C2×C32C2×#170semidirect
138. C3×D3C2×#170direct (N × H)
144. C3×C5C2×D3×C8semidirect
145. C3×C5C22×C3×C8semidirect
151. C2×D3C3×#170direct (N × H)
162. C2×C5C3×D3×C8semidirect
164. C32C22×#170semidirect
175. C2×C3D3×#170direct (N × H)
176. C2×C3C2×C3×#170semidirect
179. D3C2×C3×#170direct (N × H)
180. C5C2×C3×D3×C8semidirect
188. C3C2×D3×#170direct (N × H)
189. C3C22×C3×#170semidirect
191. C2C3×D3×#170direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic52
C2×C3×C3⋊#1702
C2×C32×#1702
C3×D3×#1702
C22×C3×#1703
C2×D3×#1703
C2×C3×D3×C85