Finite Group Databasebeta

← #72231366

Subgroups — #72231366

Order 3600.

314 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 54 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⋊#1044C211yesmaximal, characteristic
2C2×C3×C3⋊#1043C211yesmaximal, characteristic
3#14268268C211yesmaximal, characteristic
4#11925361—31nomaximal
5#11925356—31nomaximal
6C2×C32×C5⋊D5C2211yescharacteristic
7C3×C3⋊#1044—21no—
8C3×C3⋊#1043—21no—
9C5×C3⋊Dic15C412yescomplemented
10#10623979—52nomaximal
11C2×C3⋊#1044D311yescharacteristic
12C2×C3⋊#1043D311yescharacteristic
13C2×C3×#1044—31no—
14C2×C3×#1043—31no—
15#10597126—31no—
16#10597126—31no—
17#10597126—61no—
18C2×C32×C52C2×C411yescharacteristic
19C32×C5⋊D5D411yescharacteristic
20C32×C5⋊D5Q811yescharacteristic
21#87832—91noHall
22C2×C3×ΓL(1,𝔽16)—52no—
23C2×C3×ΓL(1,𝔽16)—52no—
24C2×C3×ΓL(1,𝔽16)—102no—
25D5×C3⋊Dic3—52no—
26C2×C3×C5⋊D5—21no—
27C2×C3×C5⋊D5C2×D311yescharacteristic
28C2×C3×C5⋊D5C2×D311yescharacteristic
29C3⋊#1044—21no—
30C3⋊#1043—21no—
31C3×#1044—61no—
32C3×#1043—61no—
33C5×Dic15—32no—
34C5×Dic15—32no—
35C5×Dic15—62no—
36Dic3⋊AGL(1,𝔽5)—152no—
37Dic3⋊AGL(1,𝔽5)—152no—
38C32×C52C4⋊C411yescharacteristic, complemented, Hall
39C2×#1044—31no—
40C2×#1043—31no—
41#6036—91no—
42C2×C32×D5—52no—
43C2×C32×D5—102no—
44C3×ΓL(1,𝔽16)—102no—
45C3×ΓL(1,𝔽16)—102no—
46C3×ΓL(1,𝔽16)—102no—
47C3×ΓL(1,𝔽16)—102no—
48C3⋊Dic15AGL(1,𝔽5)12yescomplemented
49C5×C3⋊Dic3—52no—
50C2×C3×C52—21no—
51C2×C3×C52C4×D311yescharacteristic
52C2×C3×C52C4×D311yescharacteristic
53C3×C5⋊D5—21no—
54C3×C5⋊D5—21no—
55C3×C5⋊D5Dic611yescharacteristic
56C3×C5⋊D5Dic611yescharacteristic
57C3×C5⋊D5#6711yescharacteristic
58C3×C5⋊D5#6711yescharacteristic
59#3724—251noHall
60C2×ΓL(1,𝔽16)—52no—
61C2×ΓL(1,𝔽16)—52no—
62C2×ΓL(1,𝔽16)—102no—
63C2×C3×AGL(1,𝔽5)—152no—
64C2×C3×AGL(1,𝔽5)—152no—
65C2×C3×AGL(1,𝔽5)—302no—
66D5×Dic3—152no—
67D5×Dic3—152no—
68D5×Dic3—302no—
69C2×C5⋊D5D3211yescharacteristic
70#1044—61no—
71#1043—61no—
72C5×Dic5—92no—
73C2×C32×C5—22no—
74C2×C32×C5C2×AGL(1,𝔽5)12yes—
75C32×D5—52no—
76C32×D5—52no—
77C32×D5—102no—
78C32×D5—102no—
79C4⋊AGL(1,𝔽5)—452no—
80C3×C52—21no—
81C3×C52Dic3⋊C411yescharacteristic, complemented
82C3×C52Dic3⋊C411yescharacteristic, complemented
83C2×C3⋊Dic3—251no—
84C2×C3×Dic3—251no—
85C2×C3×Dic3—251no—
86C2×C3×D5—52no—
87C2×C3×D5—52no—
88C2×C3×D5—102no—
89C2×C3×D5—102no—
90C2×C3×D5—102no—
91C2×C3×D5—202no—
92ΓL(1,𝔽16)—102no—
93ΓL(1,𝔽16)—102no—
94ΓL(1,𝔽16)—102no—
95ΓL(1,𝔽16)—102no—
96C3×AGL(1,𝔽5)—302no—
97C3×AGL(1,𝔽5)—302no—
98C3×AGL(1,𝔽5)—302no—
99C3×AGL(1,𝔽5)—302no—
100Dic15—32no—
101Dic15—32no—
102Dic15—62no—
103C5×Dic3—152no—
104C5×Dic3—152no—
105C5×Dic3—302no—
106C2×C52#62411yescharacteristic
107C5⋊D5#62511yescharacteristic
108C5⋊D5#62711yescharacteristic
109Dic3⋊C4—751no—
110Dic3⋊C4—751no—
111C32×C5—22no—
112C32×C5C4⋊AGL(1,𝔽5)12yescomplemented
113C2×AGL(1,𝔽5)—152no—
114C2×AGL(1,𝔽5)—152no—
115C2×AGL(1,𝔽5)—302no—
116C4×D5—452no—
117C22×C32—251no—
118C3⋊Dic3—52no—
119C3×Dic3—501no—
120C3×Dic3—501no—
121C2×C3×C5—22no—
122C2×C3×C5—22no—
123C2×C3×C5—22no—
124C2×C3×C5—42no—
125C2×C3×C5D3×AGL(1,𝔽5)12yes—
126C2×C3×C5D3×AGL(1,𝔽5)12yes—
127C3×D5—52no—
128C3×D5—52no—
129C3×D5—52no—
130C3×D5—52no—
131C3×D5—102no—
132C3×D5—102no—
133C3×D5—102no—
134C3×D5—102no—
135C3×D5—102no—
136C3×D5—102no—
137C3×D5—202no—
138C3×D5—202no—
139C52#372411yescharacteristic, complemented, Sylow
140C2×C3×C4—751no—
141C2×C3×C4—751no—
142C2×Dic3—251no—
143C2×Dic3—251no—
144C2×Dic3—751no—
145C2×Dic3—751no—
146C2×Dic3—1501no—
147C2×D5—52no—
148C2×D5—102no—
149AGL(1,𝔽5)—302no—
150AGL(1,𝔽5)—302no—
151AGL(1,𝔽5)—302no—
152AGL(1,𝔽5)—302no—
153C4×C5—452no—
154Dic5—92no—
155C2×C32—251no—
156C2×C32—251no—
157C2×C32D5⋊AGL(1,𝔽5)11yescharacteristic
158C4⋊C4—2251noSylow
159C3×C5—22no—
160C3×C5—22no—
161C3×C5—22no—
162C3×C5—42no—
163C3×C5Dic3⋊AGL(1,𝔽5)12yescomplemented
164C3×C5Dic3⋊AGL(1,𝔽5)12yescomplemented
165C22×C3—251no—
166C22×C3—251no—
167C22×C3—501no—
168C3×C4—1501no—
169C3×C4—1501no—
170Dic3—152no—
171Dic3—152no—
172Dic3—302no—
173Dic3—501no—
174Dic3—501no—
175C2×C5—22no—
176C2×C5ΓL(1,𝔽16)⋊D312yes—
177D5—52no—
178D5—52no—
179D5—102no—
180D5—102no—
181C32#8783211yescharacteristic, complemented, Sylow
182C2×C4—751no—
183C2×C4—751no—
184C2×C4—2251no—
185C2×C3—21no—
186C2×C3—251no—
187C2×C3—251no—
188C2×C3—251no—
189C2×C3—251no—
190C2×C3—501no—
191C2×C3—501no—
192C2×C3#1059718211yescharacteristic
193C2×C3#1059718311yescharacteristic
194C5—22no—
195C5#1062397912yescomplemented
196C22—251no—
197C4—452no—
198C4—1501no—
199C4—1501no—
200C3—21no—
201C3#1192535611yescharacteristic, complemented
202C3#1192536111yescharacteristic, complemented
203C2—251no—
204C2—251no—
205C2#1426843811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#14268438
commutator, Fitting, socleC2×C32×C52C2×C4
prime radical, cosocleC2×C32×C5⋊D5C22
hypocenterC32×C52C4⋊C4
LCSG ▷ C2×C32×C52 ▷ C32×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
9. C5×C3⋊Dic15C4semidirect
38. C32×C52C4⋊C4semidirect
48. C3⋊Dic15AGL(1,𝔽5)semidirect
81. C3×C52Dic3⋊C4semidirect
82. C3×C52Dic3⋊C4semidirect
112. C32×C5C4⋊AGL(1,𝔽5)semidirect
139. C52#3724semidirect
163. C3×C5Dic3⋊AGL(1,𝔽5)semidirect
164. C3×C5Dic3⋊AGL(1,𝔽5)semidirect
181. C32#87832semidirect
195. C5#10623979semidirect
201. C3#11925356semidirect
202. C3#11925361semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊#10442
C2×C3×C3⋊#10432
#142682682
#119253613
#119253563
#106239795