Finite Group Databasebeta

← C3×#11925350 #72229738

Subgroups — C3×#11925350 #72229738

Order 3600.

410 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 70 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×C32×D52C211yesmaximal, characteristic
2C2×C3×C3⋊#1044C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#1043C211yesmaximal, characteristic, complemented
4#11925350C311yesmaximal, characteristic, direct product factor
5C3×#87831—31nomaximal
6C2×C32×C5⋊D5C2211yescharacteristic
7C2×C32×C5×D5C412yescomplemented
8C32×D52—21no—
9C32×D52—21no—
10C3×C3⋊#1044—21no—
11C3×C3⋊#1043—21no—
12C3×#6763—52nomaximal
13C2×C3×D52—21no—
14C2×C3×D52D311yescharacteristic
15C2×C3×D52C2×C311yescharacteristic
16C2×C3⋊#1044C2×C311yescharacteristic, complemented
17C2×C3⋊#1043C2×C311yescharacteristic, complemented
18C2×C3×#1044—31no—
19C2×C3×#1043—31no—
20C2×C32×C52C2×C411yescharacteristic
21C32×C5⋊D5D411yescharacteristic
22C32×C5⋊D5D411yescharacteristic
23C32×C5×D5—22no—
24#87831—31noHall
25C22×C32×D5—52no—
26C2×C3×ΓL(1,𝔽16)—52no—
27C2×C3×ΓL(1,𝔽16)—52no—
28C2×C3×ΓL(1,𝔽16)—102no—
29C2×C3×C5⋊D5—21no—
30C2×C3×C5⋊D5C2×D311yescharacteristic
31C2×C3×C5⋊D5C22×C311yescharacteristic
32C2×C3×C5×D5—22no—
33C2×C3×C5×D5Dic312yescomplemented
34C2×C3×C5×D5C3×C412yescomplemented
35C3×D52—22no—
36C3×D52—22no—
37C3×D52—21no—
38C3×D52—21no—
39C3×D52—21no—
40C3×D52—21no—
41C3⋊#1044—21no—
42C3⋊#1043—21no—
43C3×#1044—61no—
44C3×#1043—61no—
45#6763—52no—
46C3×C22⋊AGL(1,𝔽5)—152no—
47C32×C52C22⋊C411yescharacteristic, complemented, Hall
48C2×D52C3×D311yescharacteristic
49C2×#1044—31no—
50C2×#1043—31no—
51C22×C32×C5—52no—
52C2×C32×D5—52no—
53C2×C32×D5—102no—
54C2×C32×D5—102no—
55C2×C32×D5—102no—
56C2×C32×D5AGL(1,𝔽5)12yescomplemented
57C3×ΓL(1,𝔽16)—102no—
58C3×ΓL(1,𝔽16)—102no—
59C3×ΓL(1,𝔽16)—104no—
60C2×C3×C52—21no—
61C2×C3×C52C2×Dic311yescharacteristic
62C2×C3×C52C2×C3×C411yescharacteristic
63C3×C5⋊D5—21no—
64C3×C5⋊D5—21no—
65C3×C5⋊D5#6711yescharacteristic
66C3×C5⋊D5#6711yescharacteristic
67C3×C5⋊D5C3×D411yescharacteristic
68C3×C5⋊D5C3×D411yescharacteristic
69C3×C5×D5—24no—
70C3×C5×D5—22no—
71C3×C5×D5—22no—
72C3×#217—251noHall
73C22×C3×D5—52no—
74C22×C3×D5—52no—
75C22×C3×D5—102no—
76C2×ΓL(1,𝔽16)—52no—
77C2×ΓL(1,𝔽16)—52no—
78C2×ΓL(1,𝔽16)—102no—
79C2×C3×AGL(1,𝔽5)—152no—
80C2×C3×AGL(1,𝔽5)—152no—
81C2×C3×AGL(1,𝔽5)—302no—
82C2×C5⋊D5C2×C3×D311yescharacteristic
83C2×C5×D5C3×Dic312yescomplemented
84D52—21no—
85D52—21no—
86#1044—61no—
87#1043—61no—
88C2×C32×C5—22no—
89C2×C32×C5—102no—
90C2×C32×C5C2×AGL(1,𝔽5)12yes—
91C32×D5—22no—
92C32×D5—52no—
93C32×D5—52no—
94C32×D5—102no—
95C32×D5—102no—
96C22⋊AGL(1,𝔽5)—152no—
97C3×C52—21no—
98C3×C52#21711yescharacteristic, complemented
99C3×C52C3×C22⋊C411yescharacteristic, complemented
100C23×C32—251no—
101C2×C3×Dic3—251no—
102C2×C3×Dic3—251no—
103C22×C3×C5—52no—
104C22×C3×C5—52no—
105C22×C3×C5—102no—
106C2×C3×D5—22no—
107C2×C3×D5—52no—
108C2×C3×D5—52no—
109C2×C3×D5—102no—
110C2×C3×D5—102no—
111C2×C3×D5—102no—
112C2×C3×D5—102no—
113C2×C3×D5—102no—
114C2×C3×D5—102no—
115C2×C3×D5—102no—
116C2×C3×D5—104no—
117C2×C3×D5—104no—
118C2×C3×D5—202no—
119C2×C3×D5C3×AGL(1,𝔽5)12yescomplemented
120C2×C3×D5ΓL(1,𝔽16)12yescomplemented
121ΓL(1,𝔽16)—102no—
122ΓL(1,𝔽16)—102no—
123ΓL(1,𝔽16)—104no—
124C3×AGL(1,𝔽5)—302no—
125C3×AGL(1,𝔽5)—302no—
126C3×AGL(1,𝔽5)—304no—
127C2×C52C2×C3×Dic311yescharacteristic
128C5⋊D5C3×#6711yescharacteristic
129C5⋊D5C3×#6711yescharacteristic
130C5×D5—22no—
131C3×C22⋊C4—751no—
132#217—251no—
133C32×C5—22no—
134C32×C5C22⋊AGL(1,𝔽5)12yescomplemented
135C22×D5—52no—
136C2×AGL(1,𝔽5)—152no—
137C2×AGL(1,𝔽5)—152no—
138C2×AGL(1,𝔽5)—302no—
139C22×C32—52no—
140C22×C32—251no—
141C22×C32—501no—
142C22×C32—501no—
143C3×Dic3—501no—
144C3×Dic3—501no—
145C2×C3×C5—22no—
146C2×C3×C5—22no—
147C2×C3×C5—22no—
148C2×C3×C5—42no—
149C2×C3×C5—104no—
150C2×C3×C5—102no—
151C2×C3×C5—102no—
152C2×C3×C5C2×C3×AGL(1,𝔽5)12yes—
153C2×C3×C5C2×ΓL(1,𝔽16)12yes—
154C3×D5—24no—
155C3×D5—22no—
156C3×D5—22no—
157C3×D5—52no—
158C3×D5—52no—
159C3×D5—52no—
160C3×D5—52no—
161C3×D5—102no—
162C3×D5—102no—
163C3×D5—102no—
164C3×D5—102no—
165C3×D5—102no—
166C3×D5—102no—
167C3×D5—202no—
168C3×D5—202no—
169C52C3×#21711yescharacteristic, complemented, Sylow
170C23×C3—251no—
171C23×C3—251no—
172C23×C3—501no—
173C2×C3×C4—751no—
174C2×C3×C4—751no—
175C2×Dic3—251no—
176C2×Dic3—251no—
177C22×C5—52no—
178C2×D5—52no—
179C2×D5—102no—
180C2×D5—102no—
181C2×D5—102no—
182C2×D5C3×ΓL(1,𝔽16)12yescomplemented
183AGL(1,𝔽5)—302no—
184AGL(1,𝔽5)—302no—
185AGL(1,𝔽5)—304no—
186C2×C32—102no—
187C2×C32—251no—
188C2×C32—251no—
189C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic
190C22⋊C4—751noSylow
191C3×C5—22no—
192C3×C5—22no—
193C3×C5—22no—
194C3×C5—42no—
195C3×C5C3×C22⋊AGL(1,𝔽5)12yescomplemented
196C3×C5#676312yescomplemented
197C22×C3—52no—
198C22×C3—52no—
199C22×C3—102no—
200C22×C3—251no—
201C22×C3—251no—
202C22×C3—501no—
203C22×C3—501no—
204C22×C3—501no—
205C22×C3—501no—
206C22×C3—501no—
207C22×C3—502no—
208C22×C3—502no—
209C3×C4—1501no—
210C3×C4—1501no—
211Dic3—501no—
212Dic3—501no—
213C2×C5—22no—
214C2×C5—102no—
215C2×C5C2×C3×ΓL(1,𝔽16)12yes—
216D5—22no—
217D5—52no—
218D5—52no—
219D5—102no—
220D5—102no—
221C32#8783111yescharacteristic, complemented, Sylow
222C23—251no—
223C2×C4—751no—
224C2×C4—751no—
225C2×C3—21no—
226C2×C3—104no—
227C2×C3—102no—
228C2×C3—102no—
229C2×C3—251no—
230C2×C3—251no—
231C2×C3—251no—
232C2×C3—251no—
233C2×C3—501no—
234C2×C3—501no—
235C2×C3C3×D5⋊AGL(1,𝔽5)11yescharacteristic
236C2×C3D5⋊ΓL(1,𝔽16)11yescharacteristic
237C5—22no—
238C5C3×#676312yescomplemented
239C22—52no—
240C22—251no—
241C22—501no—
242C22—501no—
243C4—1501no—
244C4—1501no—
245C3—21no—
246C3C3×#8783111yescharacteristic, complemented
247C3#1192535011yescharacteristic, direct product factor
248C2—102no—
249C2—251no—
250C2—251no—
251C2C3×D5⋊ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, hypercenterC2×C3D5⋊ΓL(1,𝔽16)
commutatorC2×C3×C52C2×C3×C4
FrattiniC2C3×D5⋊ΓL(1,𝔽16)
Fitting, socleC2×C32×C52C2×C4
prime radicalC2×C32×D52C2
cosocleC2×C3×C5⋊D5C22×C3
hypocenterC3×C52C3×C22⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2×C3

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. #11925350C3direct (N × H)
7. C2×C32×C5×D5C4semidirect
16. C2×C3⋊#1044C2×C3semidirect
17. C2×C3⋊#1043C2×C3semidirect
33. C2×C3×C5×D5Dic3semidirect
34. C2×C3×C5×D5C3×C4semidirect
47. C32×C52C22⋊C4semidirect
56. C2×C32×D5AGL(1,𝔽5)semidirect
83. C2×C5×D5C3×Dic3semidirect
98. C3×C52#217semidirect
99. C3×C52C3×C22⋊C4semidirect
119. C2×C3×D5C3×AGL(1,𝔽5)semidirect
120. C2×C3×D5ΓL(1,𝔽16)semidirect
134. C32×C5C22⋊AGL(1,𝔽5)semidirect
169. C52C3×#217semidirect
182. C2×D5C3×ΓL(1,𝔽16)semidirect
195. C3×C5C3×C22⋊AGL(1,𝔽5)semidirect
196. C3×C5#6763semidirect
221. C32#87831semidirect
238. C5C3×#6763semidirect
246. C3C3×#87831semidirect
247. C3#11925350direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D522
C2×C3×C3⋊#10442
C2×C3×C3⋊#10432
#119253503
C3×#878313
C3×#67635