Finite Group Databasebeta

← C2×C3×#10597126 #72230608

Subgroups — C2×C3×#10597126 #72230608

Order 3600.

688 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C32×C5⋊D5C211yesmaximal, characteristic
2C2×C3×C5×Dic15C212yesmaximal, complemented
3C3×#10597126C214yesmaximal, direct product factor
4C2×#10597126C311yesmaximal, characteristic, direct product factor
5C2×C3×#6036—31nomaximal
6C22×C32×C52C2211yescharacteristic
7C2×C32×C5⋊D5C414yescomplemented
8C2×C32×C5⋊D5C2212yes—
9C3×C5×Dic15C2214yescomplemented
10C2×C3×D5×Dic3—52nomaximal
11C22×C3×C5⋊D5—21no—
12C22×C3×C5⋊D5D311yescharacteristic
13C22×C3×C5⋊D5C2×C311yescharacteristic
14C2×C5×Dic15C2×C312yescomplemented
15C2×C3×C5×Dic5—32no—
16#10597126C2×C314yesdirect product factor
17C3×#6036—34no—
18C2×C32×C52C2×C412yescomplemented
19C2×C32×C52C2311yescharacteristic
20C32×C5⋊D5C2×C414yescomplemented
21C2×#6036—31noHall
22C22×C32×D5—52no—
23C22×C32×D5—102no—
24C2×C3×Dic15D512yescomplemented
25C2×C3×C5×Dic3—52no—
26C3×D5×Dic3—58no—
27C22×C3×C52—21no—
28C22×C3×C52C2×D311yescharacteristic
29C22×C3×C52C22×C311yescharacteristic
30C2×C3×C5⋊D5—24no—
31C2×C3×C5⋊D5—22no—
32C2×C3×C5⋊D5Dic314yescomplemented
33C2×C3×C5⋊D5C3×C414yescomplemented
34C2×C3×C5⋊D5C2×D312yes—
35C2×C3×C5⋊D5C22×C312yes—
36C5×Dic15C22×C314yescomplemented
37C3×C5×Dic5—34no—
38C2×C3×C4×D5—152no—
39C2×D5×Dic3—52no—
40C32×C52C22×C411yescharacteristic, complemented, Hall
41C22×C5⋊D5C3×D311yescharacteristic
42C2×C5×Dic5—32no—
43#6036—34no—
44C22×C32×C5—22no—
45C22×C32×C5C2×D512yes—
46C2×C32×D5—58no—
47C2×C32×D5—54no—
48C2×C32×D5—108no—
49C2×C32×D5—104no—
50C3×Dic15C2×D514yescomplemented
51C3×C5×Dic3—54no—
52C2×C3×C52—22no—
53C2×C3×C52—21no—
54C2×C3×C52C2×Dic312yescomplemented
55C2×C3×C52C2×C3×C412yescomplemented
56C2×C3×C52C22×D311yescharacteristic
57C2×C3×C52C23×C311yescharacteristic
58C3×C5⋊D5—24no—
59C3×C5⋊D5C2×Dic314yescomplemented
60C3×C5⋊D5C2×C3×C414yescomplemented
61C22×C3×Dic3—251noHall
62C22×C3×D5—52no—
63C22×C3×D5—52no—
64C22×C3×D5—102no—
65C22×C3×D5—102no—
66C22×C3×D5—102no—
67C22×C3×D5—104no—
68C2×C3×C4×C5—152no—
69C2×Dic15C3×D512yescomplemented
70C2×C5×Dic3—52no—
71C2×C3×Dic5—32no—
72C3×C4×D5—158no—
73D5×Dic3—58no—
74C22×C52C2×C3×D311yescharacteristic
75C2×C5⋊D5C3×Dic314yescomplemented
76C2×C5⋊D5C2×C3×D312yes—
77C5×Dic5—34no—
78C2×C32×C5—24no—
79C2×C32×C5—22no—
80C2×C32×C5C4×D514yescomplemented
81C2×C32×C5C22×D512yes—
82C32×D5—58no—
83C32×D5—108no—
84C2×C4×D5—152no—
85C3×C52—21no—
86C3×C52C22×Dic311yescharacteristic, complemented
87C3×C52C22×C3×C411yescharacteristic, complemented
88C23×C32—251no—
89C2×C3×Dic3—52no—
90C2×C3×Dic3—254no—
91C22×C3×C5—24no—
92C22×C3×C5—22no—
93C22×C3×C5—22no—
94C22×C3×C5—22no—
95C22×C3×C5D3×D512yes—
96C22×C3×C5C2×C3×D512yes—
97C2×C3×D5—58no—
98C2×C3×D5—58no—
99C2×C3×D5—54no—
100C2×C3×D5—54no—
101C2×C3×D5—108no—
102C2×C3×D5—108no—
103C2×C3×D5—104no—
104C2×C3×D5—104no—
105C2×C3×D5—108no—
106C2×C3×D5—104no—
107C2×C3×D5—1016no—
108C2×C3×D5—108no—
109C3×C4×C5—154no—
110Dic15C2×C3×D514yescomplemented
111C3×Dic5—34no—
112C5×Dic3—54no—
113C2×C52C2×C3×Dic312yescomplemented
114C2×C52C22×C3×D311yescharacteristic
115C5⋊D5C2×C3×Dic314yescomplemented
116C22×C3×C4—751no—
117C22×Dic3—251no—
118C32×C5—22no—
119C32×C5C2×C4×D512yescomplemented
120C22×D5—52no—
121C22×D5—102no—
122C2×C4×C5—152no—
123C2×Dic5—32no—
124C4×D5—158no—
125C22×C32—254no—
126C22×C32—252no—
127C22×C32D5211yescharacteristic
128C3×Dic3—54no—
129C2×C3×C5—28no—
130C2×C3×C5—24no—
131C2×C3×C5—24no—
132C2×C3×C5—22no—
133C2×C3×C5—24no—
134C2×C3×C5—22no—
135C2×C3×C5—22no—
136C2×C3×C5—24no—
137C2×C3×C5D5×Dic314yescomplemented
138C2×C3×C5C3×C4×D514yescomplemented
139C2×C3×C5C2×D3×D512yes—
140C2×C3×C5C22×C3×D512yes—
141C3×D5—58no—
142C3×D5—58no—
143C3×D5—108no—
144C3×D5—108no—
145C3×D5—108no—
146C3×D5—1016no—
147C52C22×C3×Dic311yescharacteristic, complemented, Sylow
148C23×C3—251no—
149C23×C3—251no—
150C23×C3—501no—
151C2×C3×C4—152no—
152C2×C3×C4—754no—
153C2×Dic3—52no—
154C2×Dic3—254no—
155C22×C5—22no—
156C22×C5C3×D3×D512yes—
157C2×D5—58no—
158C2×D5—54no—
159C2×D5—108no—
160C2×D5—104no—
161C4×C5—154no—
162Dic5—34no—
163C2×C32—254no—
164C2×C32#603612yescomplemented
165C2×C32C2×D5211yescharacteristic
166C22×C4—751noSylow
167C3×C5—24no—
168C3×C5—22no—
169C3×C5—22no—
170C3×C5—22no—
171C3×C5C2×D5×Dic312yescomplemented
172C3×C5C2×C3×C4×D512yescomplemented
173C22×C3—21no—
174C22×C3—254no—
175C22×C3—252no—
176C22×C3—252no—
177C22×C3—254no—
178C22×C3—504no—
179C22×C3—502no—
180C22×C3C3×D5211yescharacteristic
181C22×C3D15⋊D511yescharacteristic
182C3×C4—154no—
183Dic3—54no—
184C2×C5—24no—
185C2×C5—22no—
186C2×C5C3×D5×Dic314yescomplemented
187C2×C5C2×C3×D3×D512yes—
188D5—58no—
189D5—108no—
190C32C2×#603611yescharacteristic, complemented, Sylow
191C23—251no—
192C2×C4—152no—
193C2×C4—754no—
194C2×C3—22no—
195C2×C3—21no—
196C2×C3—254no—
197C2×C3—254no—
198C2×C3—504no—
199C2×C3C3×#603612yescomplemented
200C2×C3#1059712612yesdirect product factor
201C2×C3C2×C3×D5211yescharacteristic
202C2×C3C2×D15⋊D511yescharacteristic
203C5—22no—
204C5C2×C3×D5×Dic312yescomplemented
205C22—254no—
206C22—252no—
207C22C3×D15⋊D511yescharacteristic
208C4—154no—
209C3—21no—
210C3C2×C3×#603611yescharacteristic, complemented
211C3C2×#1059712611yescharacteristic, direct product factor
212C2—254no—
213C2C3×#1059712612yesdirect product factor
214C2C2×C3×D15⋊D511yescharacteristic
subgroupG/H
center, hypercenterC22×C3D15⋊D5
commutator, hypocenterC3×C52C22×C3×C4
FrattiniC2C2×C3×D15⋊D5
Fitting, socleC22×C32×C52C22
prime radicalC22×C32×C5⋊D5C2
cosocleC2×C3×C52C23×C3
LCSG ▷ C3×C52
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×Dic15C2semidirect
3. C3×#10597126C2direct (N × H)
4. C2×#10597126C3direct (N × H)
7. C2×C32×C5⋊D5C4semidirect
9. C3×C5×Dic15C22semidirect
14. C2×C5×Dic15C2×C3semidirect
16. #10597126C2×C3direct (N × H)
18. C2×C32×C52C2×C4semidirect
20. C32×C5⋊D5C2×C4semidirect
24. C2×C3×Dic15D5semidirect
32. C2×C3×C5⋊D5Dic3semidirect
33. C2×C3×C5⋊D5C3×C4semidirect
36. C5×Dic15C22×C3semidirect
40. C32×C52C22×C4semidirect
50. C3×Dic15C2×D5semidirect
54. C2×C3×C52C2×Dic3semidirect
55. C2×C3×C52C2×C3×C4semidirect
59. C3×C5⋊D5C2×Dic3semidirect
60. C3×C5⋊D5C2×C3×C4semidirect
69. C2×Dic15C3×D5semidirect
75. C2×C5⋊D5C3×Dic3semidirect
80. C2×C32×C5C4×D5semidirect
86. C3×C52C22×Dic3semidirect
87. C3×C52C22×C3×C4semidirect
110. Dic15C2×C3×D5semidirect
113. C2×C52C2×C3×Dic3semidirect
115. C5⋊D5C2×C3×Dic3semidirect
119. C32×C5C2×C4×D5semidirect
137. C2×C3×C5D5×Dic3semidirect
138. C2×C3×C5C3×C4×D5semidirect
147. C52C22×C3×Dic3semidirect
164. C2×C32#6036semidirect
171. C3×C5C2×D5×Dic3semidirect
172. C3×C5C2×C3×C4×D5semidirect
186. C2×C5C3×D5×Dic3semidirect
190. C32C2×#6036semidirect
199. C2×C3C3×#6036semidirect
200. C2×C3#10597126direct (N × H)
204. C5C2×C3×D5×Dic3semidirect
210. C3C2×C3×#6036semidirect
211. C3C2×#10597126direct (N × H)
213. C2C3×#10597126direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C5⋊D52
C2×C3×C5×Dic152
C3×#105971262
C2×#105971263
C2×C3×#60363
C2×C3×D5×Dic35