Finite Group Databasebeta

← C3×#11925361 #72231323

Subgroups — C3×#11925361 #72231323

Order 3600.

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

normal subgroups only

This group has 314 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×C3⋊#1043C211yesmaximal, characteristic
2C2×C32×#1044C211yesmaximal, characteristic
3C3×#10597126C211yesmaximal, characteristic
4#11925361C311yesmaximal, characteristic, direct product factor
5C3×#87832—31nomaximal
6C2×C32×C5⋊D5C2211yescharacteristic
7C3×C3⋊#1043—21no—
8C32×#1044—21no—
9C3×C5×Dic15C412yescomplemented
10C3×Dic3⋊AGL(1,𝔽5)—52nomaximal
11C2×C3⋊#1043C2×C311yescharacteristic
12C2×C3×#1044—21no—
13C2×C3×#1044D311yescharacteristic
14C2×C3×#1044C2×C311yescharacteristic
15C2×C3×#1043—31no—
16#10597126C2×C311yescharacteristic
17C3×#6036—31no—
18C2×C32×C52C2×C411yescharacteristic
19C32×C5⋊D5D411yescharacteristic
20C32×C5⋊D5Q811yescharacteristic
21#87832—31noHall
22C2×C3×ΓL(1,𝔽16)—52no—
23C2×C3×ΓL(1,𝔽16)—102no—
24C2×C32×AGL(1,𝔽5)—52no—
25C3×D5×Dic3—52no—
26C2×C3×C5⋊D5—21no—
27C2×C3×C5⋊D5C2×D311yescharacteristic
28C2×C3×C5⋊D5C22×C311yescharacteristic
29C3⋊#1043—21no—
30C3×#1044—21no—
31C3×#1044—21no—
32C3×#1044—21no—
33C3×#1044—21no—
34C3×#1043—61no—
35C5×Dic15C3×C412yescomplemented
36C3×C5×Dic5—32no—
37C3×C4⋊AGL(1,𝔽5)—152no—
38Dic3⋊AGL(1,𝔽5)—52no—
39C32×C52C4⋊C411yescharacteristic, complemented, Hall
40C2×#1044C3×D311yescharacteristic
41C2×#1043—31no—
42#6036—31no—
43C2×C32×D5—52no—
44C2×C32×D5—102no—
45C3×ΓL(1,𝔽16)—102no—
46C3×ΓL(1,𝔽16)—102no—
47C3×ΓL(1,𝔽16)—102no—
48C32×AGL(1,𝔽5)—102no—
49C3×Dic15AGL(1,𝔽5)12yescomplemented
50C3×C5×Dic3—52no—
51C2×C3×C52—21no—
52C2×C3×C52C4×D311yescharacteristic
53C2×C3×C52C2×C3×C411yescharacteristic
54C3×C5⋊D5—21no—
55C3×C5⋊D5—21no—
56C3×C5⋊D5Dic611yescharacteristic
57C3×C5⋊D5#6711yescharacteristic
58C3×C5⋊D5C3×D411yescharacteristic
59C3×C5⋊D5C3×Q811yescharacteristic
60C3×Dic3⋊C4—251noHall
61C2×ΓL(1,𝔽16)—52no—
62C2×ΓL(1,𝔽16)—102no—
63C2×C3×AGL(1,𝔽5)—52no—
64C2×C3×AGL(1,𝔽5)—52no—
65C2×C3×AGL(1,𝔽5)—102no—
66C2×C3×AGL(1,𝔽5)—152no—
67C2×C3×AGL(1,𝔽5)—302no—
68C3×C4×D5—152no—
69D5×Dic3—52no—
70C2×C5⋊D5C2×C3×D311yescharacteristic
71#1044—21no—
72#1043—61no—
73C5×Dic5—32no—
74C2×C32×C5—22no—
75C2×C32×C5C2×AGL(1,𝔽5)12yes—
76C32×D5—52no—
77C32×D5—52no—
78C32×D5—102no—
79C32×D5—102no—
80C4⋊AGL(1,𝔽5)—152no—
81C3×C52—21no—
82C3×C52Dic3⋊C411yescharacteristic, complemented
83C3×C52C3×C4⋊C411yescharacteristic, complemented
84C2×C32×C4—251no—
85C2×C3×Dic3—251no—
86C2×C3×Dic3—251no—
87C2×C3×D5—52no—
88C2×C3×D5—52no—
89C2×C3×D5—102no—
90C2×C3×D5—102no—
91C2×C3×D5—102no—
92C2×C3×D5—202no—
93ΓL(1,𝔽16)—102no—
94ΓL(1,𝔽16)—102no—
95ΓL(1,𝔽16)—102no—
96C3×AGL(1,𝔽5)—102no—
97C3×AGL(1,𝔽5)—102no—
98C3×AGL(1,𝔽5)—102no—
99C3×AGL(1,𝔽5)—102no—
100C3×AGL(1,𝔽5)—302no—
101C3×AGL(1,𝔽5)—302no—
102C3×AGL(1,𝔽5)—302no—
103C3×C4×C5—152no—
104Dic15C3×AGL(1,𝔽5)12yescomplemented
105C3×Dic5—32no—
106C5×Dic3—52no—
107C2×C52C3×C4×D311yescharacteristic
108C5⋊D5C3×Dic611yescharacteristic
109C5⋊D5C3×#6711yescharacteristic
110C3×C4⋊C4—751no—
111Dic3⋊C4—251no—
112C32×C5—22no—
113C32×C5C4⋊AGL(1,𝔽5)12yescomplemented
114C2×AGL(1,𝔽5)—52no—
115C2×AGL(1,𝔽5)—152no—
116C2×AGL(1,𝔽5)—302no—
117C4×D5—152no—
118C22×C32—251no—
119C32×C4—501no—
120C3×Dic3—52no—
121C3×Dic3—501no—
122C2×C3×C5—22no—
123C2×C3×C5—22no—
124C2×C3×C5—22no—
125C2×C3×C5—42no—
126C2×C3×C5D3×AGL(1,𝔽5)12yes—
127C2×C3×C5C2×C3×AGL(1,𝔽5)12yes—
128C3×D5—52no—
129C3×D5—52no—
130C3×D5—52no—
131C3×D5—52no—
132C3×D5—102no—
133C3×D5—102no—
134C3×D5—102no—
135C3×D5—102no—
136C3×D5—102no—
137C3×D5—102no—
138C3×D5—202no—
139C3×D5—202no—
140C52C3×Dic3⋊C411yescharacteristic, complemented, Sylow
141C2×C3×C4—251no—
142C2×C3×C4—251no—
143C2×C3×C4—501no—
144C2×C3×C4—751no—
145C2×C3×C4—751no—
146C2×Dic3—251no—
147C2×Dic3—251no—
148C2×D5—52no—
149C2×D5—102no—
150AGL(1,𝔽5)—102no—
151AGL(1,𝔽5)—302no—
152AGL(1,𝔽5)—302no—
153AGL(1,𝔽5)—302no—
154C4×C5—152no—
155Dic5—32no—
156C2×C32—251no—
157C2×C32—251no—
158C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic
159C4⋊C4—751noSylow
160C3×C5—22no—
161C3×C5—22no—
162C3×C5—22no—
163C3×C5—42no—
164C3×C5Dic3⋊AGL(1,𝔽5)12yescomplemented
165C3×C5C3×C4⋊AGL(1,𝔽5)12yescomplemented
166C22×C3—251no—
167C22×C3—251no—
168C22×C3—501no—
169C3×C4—152no—
170C3×C4—501no—
171C3×C4—501no—
172C3×C4—501no—
173C3×C4—501no—
174C3×C4—1501no—
175Dic3—52no—
176Dic3—501no—
177C2×C5—22no—
178C2×C5C3×D3×AGL(1,𝔽5)12yes—
179D5—52no—
180D5—52no—
181D5—102no—
182D5—102no—
183C32#8783211yescharacteristic, complemented, Sylow
184C2×C4—251no—
185C2×C4—751no—
186C2×C4—751no—
187C2×C3—21no—
188C2×C3—251no—
189C2×C3—251no—
190C2×C3—251no—
191C2×C3—251no—
192C2×C3—501no—
193C2×C3—501no—
194C2×C3C3×D5⋊AGL(1,𝔽5)11yescharacteristic
195C2×C3#1059718311yescharacteristic
196C5—22no—
197C5C3×Dic3⋊AGL(1,𝔽5)12yescomplemented
198C22—251no—
199C4—152no—
200C4—501no—
201C4—1501no—
202C3—21no—
203C3C3×#8783211yescharacteristic, complemented
204C3#1192536111yescharacteristic, direct product factor
205C2—251no—
206C2—251no—
207C2C3×#1059718311yescharacteristic
subgroupG/H
center, hypercenterC2×C3#10597183
commutatorC2×C3×C52C2×C3×C4
FrattiniC2C3×#10597183
Fitting, socleC2×C32×C52C2×C4
prime radicalC2×C32×C5⋊D5C22
cosocleC2×C3×C5⋊D5C22×C3
hypocenterC3×C52C3×C4⋊C4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
4. #11925361C3direct (N × H)
9. C3×C5×Dic15C4semidirect
35. C5×Dic15C3×C4semidirect
39. C32×C52C4⋊C4semidirect
49. C3×Dic15AGL(1,𝔽5)semidirect
82. C3×C52Dic3⋊C4semidirect
83. C3×C52C3×C4⋊C4semidirect
104. Dic15C3×AGL(1,𝔽5)semidirect
113. C32×C5C4⋊AGL(1,𝔽5)semidirect
140. C52C3×Dic3⋊C4semidirect
164. C3×C5Dic3⋊AGL(1,𝔽5)semidirect
165. C3×C5C3×C4⋊AGL(1,𝔽5)semidirect
183. C32#87832semidirect
197. C5C3×Dic3⋊AGL(1,𝔽5)semidirect
203. C3C3×#87832semidirect
204. C3#11925361direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#10432
C2×C32×#10442
C3×#105971262
#119253613
C3×#878323
C3×Dic3⋊AGL(1,𝔽5)5