Finite Group Databasebeta

← C2×#14268438 #72231331

Subgroups — C2×#14268438 #72231331

Order 3600.

570 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 78 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×#11744719C211yesmaximal, characteristic
2C2×C3×C3⋊#1044C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#1043C211yesmaximal, characteristic, complemented
4#14268438C214yesmaximal, direct product factor
5C2×#10597183—31nomaximal
6C2×#10597182—31nomaximal
7C2×C5×C3⋊D15C412yescomplemented
8C2×C32×C5⋊D5C2211yescharacteristic
9#11744719C412yescomplemented
10#11744719C2212yes—
11C3×C3⋊#1044C2212yescomplemented
12C3×C3⋊#1043C2212yescomplemented
13C2×ΓL(1,𝔽16)⋊D3—52nomaximal
14C2×D15⋊D5—31no—
15C2×D15⋊D5—31no—
16C2×D15⋊D5—61no—
17C2×C3⋊#1044D311yescharacteristic, complemented
18C2×C3⋊#1043D311yescharacteristic, complemented
19C2×C3×#1044—31no—
20C2×C3×#1043—31no—
21#10597183—34no—
22#10597182—34no—
23C2×C32×C52C2×C411yescharacteristic, complemented
24C5×C3⋊D15C2×C414yescomplemented
25C32×C5⋊D5C2×C411yescharacteristic, complemented
26C32×C5⋊D5C2311yescharacteristic
27C2×D5⋊AGL(1,𝔽5)—91noHall
28C2×D5×C3⋊D3—52no—
29C2×C3×ΓL(1,𝔽16)—52no—
30C2×C3×ΓL(1,𝔽16)—52no—
31C2×C3×ΓL(1,𝔽16)—102no—
32ΓL(1,𝔽16)⋊D3—58no—
33C2×C5×D15—32no—
34C2×C5×D15—32no—
35C2×C5×D15—62no—
36C2×C3×C5⋊D5—21no—
37C2×C3×C5⋊D5C2×D311yescharacteristic
38C2×C3×C5⋊D5C2×D311yescharacteristic
39D15⋊D5—32no—
40D15⋊D5—32no—
41D15⋊D5—32no—
42D15⋊D5—32no—
43D15⋊D5—62no—
44D15⋊D5—62no—
45C3⋊#1044C2×D312yescomplemented
46C3⋊#1043C2×D312yescomplemented
47C3×#1044—32no—
48C3×#1043—32no—
49C2×D3×AGL(1,𝔽5)—152no—
50C2×D3×AGL(1,𝔽5)—152no—
51C32×C52C22×C411yescharacteristic, complemented, Hall
52C2×D52—91no—
53C2×#1044—31no—
54C2×#1043—31no—
55D5⋊AGL(1,𝔽5)—94no—
56C2×C3⋊D15AGL(1,𝔽5)12yescomplemented
57C2×C5×C3⋊D3—52no—
58C2×C32×D5—52no—
59C2×C32×D5—102no—
60D5×C3⋊D3—54no—
61D5×C3⋊D3—54no—
62C3×ΓL(1,𝔽16)—54no—
63C3×ΓL(1,𝔽16)—54no—
64C3×ΓL(1,𝔽16)—104no—
65C2×C3×C52—21no—
66C2×C3×C52C4×D311yescharacteristic, complemented
67C2×C3×C52C4×D311yescharacteristic, complemented
68C5×D15—34no—
69C5×D15—34no—
70C5×D15—64no—
71C3×C5⋊D5—21no—
72C3×C5⋊D5—21no—
73C3×C5⋊D5C4×D311yescharacteristic, complemented
74C3×C5⋊D5C4×D311yescharacteristic, complemented
75C3×C5⋊D5C22×D311yescharacteristic
76C3×C5⋊D5C22×D311yescharacteristic
77C2×#624—251noHall
78C2×D3×D5—152no—
79C2×D3×D5—152no—
80C2×D3×D5—302no—
81C2×ΓL(1,𝔽16)—52no—
82C2×ΓL(1,𝔽16)—52no—
83C2×ΓL(1,𝔽16)—102no—
84C2×C3×AGL(1,𝔽5)—152no—
85C2×C3×AGL(1,𝔽5)—152no—
86C2×C3×AGL(1,𝔽5)—302no—
87D3×AGL(1,𝔽5)—158no—
88D3×AGL(1,𝔽5)—158no—
89C2×C5⋊D5D3211yescharacteristic
90C2×C5×D5—92no—
91D52—92no—
92D52—92no—
93#1044—32no—
94#1043—32no—
95C2×C32×C5—22no—
96C2×C32×C5C2×AGL(1,𝔽5)12yescomplemented
97C3⋊D15C2×AGL(1,𝔽5)14yescomplemented
98C5×C3⋊D3—54no—
99C32×D5—52no—
100C32×D5—52no—
101C32×D5—102no—
102C32×D5—102no—
103C22×AGL(1,𝔽5)—452no—
104C3×C52—21no—
105C3×C52C2×C4×D311yescharacteristic, complemented
106C3×C52C2×C4×D311yescharacteristic, complemented
107C22×C3⋊D3—251no—
108C2×C3×Dic3—251no—
109C2×C3×Dic3—251no—
110#624—254no—
111C2×D15—32no—
112C2×D15—32no—
113C2×D15—62no—
114C2×C5×D3—152no—
115C2×C5×D3—152no—
116C2×C5×D3—302no—
117C2×C3×D5—52no—
118C2×C3×D5—52no—
119C2×C3×D5—102no—
120C2×C3×D5—102no—
121C2×C3×D5—102no—
122C2×C3×D5—202no—
123D3×D5—154no—
124D3×D5—154no—
125D3×D5—154no—
126D3×D5—154no—
127D3×D5—304no—
128D3×D5—304no—
129ΓL(1,𝔽16)—54no—
130ΓL(1,𝔽16)—54no—
131ΓL(1,𝔽16)—104no—
132C3×AGL(1,𝔽5)—154no—
133C3×AGL(1,𝔽5)—154no—
134C3×AGL(1,𝔽5)—304no—
135C2×C52#62411yescharacteristic, complemented
136C5⋊D5#62411yescharacteristic, complemented
137C5⋊D5C2×D3211yescharacteristic
138C5×D5—94no—
139C2×C4×D3—751no—
140C2×C4×D3—751no—
141C32×C5—22no—
142C32×C5C22×AGL(1,𝔽5)12yescomplemented
143C22×D5—452no—
144C2×AGL(1,𝔽5)—152no—
145C2×AGL(1,𝔽5)—152no—
146C2×AGL(1,𝔽5)—302no—
147C2×AGL(1,𝔽5)—458no—
148C22×C32—251no—
149C2×C3⋊D3—52no—
150C2×C3⋊D3—252no—
151C2×C3⋊D3—252no—
152C3×Dic3—252no—
153C3×Dic3—252no—
154C2×C3×C5—22no—
155C2×C3×C5—22no—
156C2×C3×C5—22no—
157C2×C3×C5—42no—
158C2×C3×C5D3×AGL(1,𝔽5)12yescomplemented
159C2×C3×C5D3×AGL(1,𝔽5)12yescomplemented
160D15—34no—
161D15—34no—
162D15—64no—
163C3×D5—52no—
164C3×D5—52no—
165C3×D5—52no—
166C3×D5—52no—
167C3×D5—102no—
168C3×D5—102no—
169C3×D5—102no—
170C3×D5—102no—
171C3×D5—102no—
172C3×D5—102no—
173C3×D5—202no—
174C3×D5—202no—
175C5×D3—154no—
176C5×D3—154no—
177C5×D3—304no—
178C52C2×#62411yescharacteristic, complemented, Sylow
179C22×D3—751no—
180C22×D3—751no—
181C22×D3—1501no—
182C2×C3×C4—751no—
183C2×C3×C4—751no—
184C2×Dic3—251no—
185C2×Dic3—251no—
186C4×D3—754no—
187C4×D3—754no—
188C22×C5—452no—
189C2×D5—52no—
190C2×D5—92no—
191C2×D5—102no—
192C2×D5—454no—
193C2×D5—454no—
194AGL(1,𝔽5)—154no—
195AGL(1,𝔽5)—154no—
196AGL(1,𝔽5)—304no—
197C2×C32—251no—
198C2×C32—251no—
199C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
200C3⋊D3—54no—
201C22×C4—2251noSylow
202C3×C5—22no—
203C3×C5—22no—
204C3×C5—22no—
205C3×C5—42no—
206C3×C5C2×D3×AGL(1,𝔽5)12yescomplemented
207C3×C5C2×D3×AGL(1,𝔽5)12yescomplemented
208C22×C3—251no—
209C22×C3—251no—
210C22×C3—501no—
211C2×D3—152no—
212C2×D3—152no—
213C2×D3—302no—
214C2×D3—752no—
215C2×D3—752no—
216C2×D3—752no—
217C2×D3—752no—
218C2×D3—1502no—
219C2×D3—1502no—
220C3×C4—752no—
221C3×C4—752no—
222Dic3—252no—
223Dic3—252no—
224C2×C5—22no—
225C2×C5—454no—
226C2×C5ΓL(1,𝔽16)⋊D312yescomplemented
227D5—52no—
228D5—52no—
229D5—94no—
230D5—102no—
231D5—102no—
232C32C2×D5⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
233C23—2251no—
234C2×C4—751no—
235C2×C4—751no—
236C2×C4—2254no—
237C2×C3—21no—
238C2×C3—251no—
239C2×C3—251no—
240C2×C3—251no—
241C2×C3—251no—
242C2×C3—501no—
243C2×C3—501no—
244C2×C3#1059718211yescharacteristic, complemented
245C2×C3#1059718311yescharacteristic, complemented
246D3—154no—
247D3—154no—
248D3—304no—
249C5—22no—
250C5C2×ΓL(1,𝔽16)⋊D312yescomplemented
251C22—251no—
252C22—452no—
253C22—2252no—
254C22—2252no—
255C4—752no—
256C4—752no—
257C3—21no—
258C3C2×#1059718211yescharacteristic, complemented
259C3C2×#1059718311yescharacteristic, complemented
260C2—251no—
261C2—251no—
262C2—454no—
263C2#1426843811yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#14268438
commutator, hypocenterC32×C52C22×C4
FrattiniC1G
Fitting, socleC2×C32×C52C2×C4
prime radicalC2×#11744719C2
cosocleC32×C5⋊D5C23
LCSG ▷ C32×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#1044C2semidirect
3. C2×C3×C3⋊#1043C2semidirect
4. #14268438C2direct (N × H)
7. C2×C5×C3⋊D15C4semidirect
9. #11744719C4semidirect
11. C3×C3⋊#1044C22semidirect
12. C3×C3⋊#1043C22semidirect
17. C2×C3⋊#1044D3semidirect
18. C2×C3⋊#1043D3semidirect
23. C2×C32×C52C2×C4semidirect
24. C5×C3⋊D15C2×C4semidirect
25. C32×C5⋊D5C2×C4semidirect
45. C3⋊#1044C2×D3semidirect
46. C3⋊#1043C2×D3semidirect
51. C32×C52C22×C4semidirect
56. C2×C3⋊D15AGL(1,𝔽5)semidirect
66. C2×C3×C52C4×D3semidirect
67. C2×C3×C52C4×D3semidirect
73. C3×C5⋊D5C4×D3semidirect
74. C3×C5⋊D5C4×D3semidirect
96. C2×C32×C5C2×AGL(1,𝔽5)semidirect
97. C3⋊D15C2×AGL(1,𝔽5)semidirect
105. C3×C52C2×C4×D3semidirect
106. C3×C52C2×C4×D3semidirect
135. C2×C52#624semidirect
136. C5⋊D5#624semidirect
142. C32×C5C22×AGL(1,𝔽5)semidirect
158. C2×C3×C5D3×AGL(1,𝔽5)semidirect
159. C2×C3×C5D3×AGL(1,𝔽5)semidirect
178. C52C2×#624semidirect
199. C2×C32D5⋊AGL(1,𝔽5)semidirect
206. C3×C5C2×D3×AGL(1,𝔽5)semidirect
207. C3×C5C2×D3×AGL(1,𝔽5)semidirect
226. C2×C5ΓL(1,𝔽16)⋊D3semidirect
232. C32C2×D5⋊AGL(1,𝔽5)semidirect
244. C2×C3#10597182semidirect
245. C2×C3#10597183semidirect
250. C5C2×ΓL(1,𝔽16)⋊D3semidirect
258. C3C2×#10597182semidirect
259. C3C2×#10597183semidirect
263. C2#14268438direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117447192
C2×C3×C3⋊#10442
C2×C3×C3⋊#10432
#142684382
C2×#105971833
C2×#105971823
C2×ΓL(1,𝔽16)⋊D35