Finite Group Databasebeta

← C3×#11925360 #72231327

Subgroups — C3×#11925360 #72231327

Order 3600.

378 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 62 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×D15⋊D5C211yesmaximal, characteristic
2C2×C3×C3⋊#1043C211yesmaximal, characteristic, complemented
3C2×C32×#1044C211yesmaximal, characteristic, complemented
4#11925360C311yesmaximal, characteristic, direct product factor
5C3×#87831—31nomaximal
6C2×C3×C5×D15C412yescomplemented
7C2×C32×C5⋊D5C2211yescharacteristic
8C3×D15⋊D5—21no—
9C3×D15⋊D5—21no—
10C3×C3⋊#1043—21no—
11C32×#1044—21no—
12C3×(C2×D3)⋊AGL(1,𝔽5)—52nomaximal
13C2×D15⋊D5C2×C311yescharacteristic
14C2×C3×D52—31no—
15C2×C3⋊#1043C2×C311yescharacteristic, complemented
16C2×C3×#1044—21no—
17C2×C3×#1044D311yescharacteristic, complemented
18C2×C3×#1044C2×C311yescharacteristic, complemented
19C2×C3×#1043—31no—
20C2×C32×C52C2×C411yescharacteristic
21C3×C5×D15—22no—
22C32×C5⋊D5D411yescharacteristic
23C32×C5⋊D5D411yescharacteristic
24#87831—31noHall
25C2×C3×D3×D5—52no—
26C2×C3×ΓL(1,𝔽16)—52no—
27C2×C3×ΓL(1,𝔽16)—102no—
28C2×C32×AGL(1,𝔽5)—52no—
29C2×C5×D15C3×C412yescomplemented
30C2×C3×C5⋊D5—21no—
31C2×C3×C5⋊D5C2×D311yescharacteristic
32C2×C3×C5⋊D5C22×C311yescharacteristic
33C2×C3×C5×D5—32no—
34D15⋊D5—21no—
35D15⋊D5—21no—
36C3×D52—61no—
37C3×D52—61no—
38C3⋊#1043—21no—
39C3×#1044—21no—
40C3×#1044—21no—
41C3×#1044—21no—
42C3×#1044—21no—
43C3×#1043—61no—
44C3×C22⋊AGL(1,𝔽5)—152no—
45(C2×D3)⋊AGL(1,𝔽5)—52no—
46C32×C52C22⋊C411yescharacteristic, complemented, Hall
47C2×D52—31no—
48C2×#1044C3×D311yescharacteristic, complemented
49C2×#1043—31no—
50C2×C3×D15AGL(1,𝔽5)12yescomplemented
51C2×C3×C5×D3—52no—
52C2×C32×D5—52no—
53C2×C32×D5—102no—
54C3×D3×D5—102no—
55C3×D3×D5—102no—
56C3×ΓL(1,𝔽16)—102no—
57C3×ΓL(1,𝔽16)—102no—
58C3×ΓL(1,𝔽16)—102no—
59C32×AGL(1,𝔽5)—102no—
60C2×C3×C52—21no—
61C2×C3×C52C4×D311yescharacteristic
62C2×C3×C52C2×C3×C411yescharacteristic
63C5×D15—22no—
64C3×C5⋊D5—21no—
65C3×C5⋊D5—21no—
66C3×C5⋊D5D1211yescharacteristic
67C3×C5⋊D5#6711yescharacteristic
68C3×C5⋊D5C3×D411yescharacteristic
69C3×C5⋊D5C3×D411yescharacteristic
70C3×C5×D5—62no—
71C3×(C2×D3)⋊C4—251noHall
72C22×C3×D5—152no—
73C2×D3×D5—52no—
74C2×ΓL(1,𝔽16)—52no—
75C2×ΓL(1,𝔽16)—102no—
76C2×C3×AGL(1,𝔽5)—52no—
77C2×C3×AGL(1,𝔽5)—52no—
78C2×C3×AGL(1,𝔽5)—102no—
79C2×C3×AGL(1,𝔽5)—152no—
80C2×C3×AGL(1,𝔽5)—302no—
81C2×C5⋊D5C2×C3×D311yescharacteristic
82C2×C5×D5—32no—
83D52—61no—
84D52—61no—
85#1044—21no—
86#1043—61no—
87C2×C32×C5—22no—
88C2×C32×C5C2×AGL(1,𝔽5)12yes—
89C3×D15—22no—
90C3×C5×D3—102no—
91C32×D5—52no—
92C32×D5—52no—
93C32×D5—102no—
94C32×D5—102no—
95C22⋊AGL(1,𝔽5)—152no—
96C3×C52—21no—
97C3×C52(C2×D3)⋊C411yescharacteristic, complemented
98C3×C52C3×C22⋊C411yescharacteristic, complemented
99C22×C3×D3—251no—
100C2×C32×C4—251no—
101C2×C3×Dic3—251no—
102C22×C3×C5—152no—
103C2×D15C3×AGL(1,𝔽5)12yescomplemented
104C2×C5×D3—52no—
105C2×C3×D5—32no—
106C2×C3×D5—52no—
107C2×C3×D5—52no—
108C2×C3×D5—102no—
109C2×C3×D5—102no—
110C2×C3×D5—102no—
111C2×C3×D5—202no—
112C2×C3×D5—302no—
113C2×C3×D5—302no—
114D3×D5—102no—
115D3×D5—102no—
116ΓL(1,𝔽16)—102no—
117ΓL(1,𝔽16)—102no—
118ΓL(1,𝔽16)—102no—
119C3×AGL(1,𝔽5)—102no—
120C3×AGL(1,𝔽5)—102no—
121C3×AGL(1,𝔽5)—102no—
122C3×AGL(1,𝔽5)—102no—
123C3×AGL(1,𝔽5)—302no—
124C3×AGL(1,𝔽5)—302no—
125C3×AGL(1,𝔽5)—302no—
126C2×C52C3×C4×D311yescharacteristic
127C5⋊D5C3×D1211yescharacteristic
128C5⋊D5C3×#6711yescharacteristic
129C5×D5—62no—
130C3×C22⋊C4—751no—
131(C2×D3)⋊C4—251no—
132C32×C5—22no—
133C32×C5C22⋊AGL(1,𝔽5)12yescomplemented
134C22×D5—152no—
135C2×AGL(1,𝔽5)—52no—
136C2×AGL(1,𝔽5)—152no—
137C2×AGL(1,𝔽5)—302no—
138C22×C32—251no—
139C2×C3×D3—52no—
140C2×C3×D3—501no—
141C2×C3×D3—501no—
142C32×C4—501no—
143C3×Dic3—501no—
144C2×C3×C5—22no—
145C2×C3×C5—22no—
146C2×C3×C5—22no—
147C2×C3×C5—42no—
148C2×C3×C5—302no—
149C2×C3×C5D3×AGL(1,𝔽5)12yes—
150C2×C3×C5C2×C3×AGL(1,𝔽5)12yes—
151D15—22no—
152C3×D5—52no—
153C3×D5—52no—
154C3×D5—52no—
155C3×D5—52no—
156C3×D5—62no—
157C3×D5—102no—
158C3×D5—102no—
159C3×D5—102no—
160C3×D5—102no—
161C3×D5—102no—
162C3×D5—102no—
163C3×D5—202no—
164C3×D5—202no—
165C5×D3—102no—
166C52C3×(C2×D3)⋊C411yescharacteristic, complemented, Sylow
167C23×C3—751no—
168C22×D3—251no—
169C2×C3×C4—251no—
170C2×C3×C4—251no—
171C2×C3×C4—501no—
172C2×C3×C4—751no—
173C2×Dic3—251no—
174C22×C5—152no—
175C2×D5—32no—
176C2×D5—52no—
177C2×D5—102no—
178C2×D5—302no—
179C2×D5—302no—
180AGL(1,𝔽5)—102no—
181AGL(1,𝔽5)—302no—
182AGL(1,𝔽5)—302no—
183AGL(1,𝔽5)—302no—
184C2×C32—251no—
185C2×C32—251no—
186C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic
187C3×D3—102no—
188C22⋊C4—751noSylow
189C3×C5—22no—
190C3×C5—22no—
191C3×C5—22no—
192C3×C5—42no—
193C3×C5(C2×D3)⋊AGL(1,𝔽5)12yescomplemented
194C3×C5C3×C22⋊AGL(1,𝔽5)12yescomplemented
195C22×C3—152no—
196C22×C3—251no—
197C22×C3—251no—
198C22×C3—501no—
199C22×C3—1501no—
200C22×C3—1501no—
201C2×D3—52no—
202C2×D3—501no—
203C2×D3—501no—
204C3×C4—501no—
205C3×C4—501no—
206C3×C4—501no—
207C3×C4—501no—
208C3×C4—1501no—
209Dic3—501no—
210C2×C5—22no—
211C2×C5—302no—
212C2×C5C3×D3×AGL(1,𝔽5)12yes—
213D5—52no—
214D5—52no—
215D5—62no—
216D5—102no—
217D5—102no—
218C32#8783111yescharacteristic, complemented, Sylow
219C23—751no—
220C2×C4—251no—
221C2×C4—751no—
222C2×C3—21no—
223C2×C3—251no—
224C2×C3—251no—
225C2×C3—251no—
226C2×C3—251no—
227C2×C3—302no—
228C2×C3—501no—
229C2×C3—501no—
230C2×C3C3×D5⋊AGL(1,𝔽5)11yescharacteristic
231C2×C3#1059718311yescharacteristic
232D3—102no—
233C5—22no—
234C5C3×(C2×D3)⋊AGL(1,𝔽5)12yescomplemented
235C22—152no—
236C22—251no—
237C22—1501no—
238C22—1501no—
239C4—501no—
240C4—1501no—
241C3—21no—
242C3C3×#8783111yescharacteristic, complemented
243C3#1192536011yescharacteristic, direct product factor
244C2—251no—
245C2—251no—
246C2—302no—
247C2C3×#1059718311yescharacteristic
subgroupG/H
center, hypercenterC2×C3#10597183
commutatorC2×C3×C52C2×C3×C4
FrattiniC2C3×#10597183
Fitting, socleC2×C32×C52C2×C4
prime radicalC2×C3×D15⋊D5C2
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⋊#1043C2semidirect
3. C2×C32×#1044C2semidirect
4. #11925360C3direct (N × H)
6. C2×C3×C5×D15C4semidirect
15. C2×C3⋊#1043C2×C3semidirect
17. C2×C3×#1044D3semidirect
18. C2×C3×#1044C2×C3semidirect
29. C2×C5×D15C3×C4semidirect
46. C32×C52C22⋊C4semidirect
48. C2×#1044C3×D3semidirect
50. C2×C3×D15AGL(1,𝔽5)semidirect
97. C3×C52(C2×D3)⋊C4semidirect
98. C3×C52C3×C22⋊C4semidirect
103. C2×D15C3×AGL(1,𝔽5)semidirect
133. C32×C5C22⋊AGL(1,𝔽5)semidirect
166. C52C3×(C2×D3)⋊C4semidirect
193. C3×C5(C2×D3)⋊AGL(1,𝔽5)semidirect
194. C3×C5C3×C22⋊AGL(1,𝔽5)semidirect
218. C32#87831semidirect
234. C5C3×(C2×D3)⋊AGL(1,𝔽5)semidirect
242. C3C3×#87831semidirect
243. C3#11925360direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D15⋊D52
C2×C3×C3⋊#10432
C2×C32×#10442
#119253603
C3×#878313
C3×(C2×D3)⋊AGL(1,𝔽5)5