Finite Group Databasebeta

← #72260010

Subgroups — #72260010

Order 2400.

412 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C22×C3⋊#1044C211yesmaximal, characteristic
2C22×C3×#1043C211yesmaximal, characteristic
3C2×#10597126C211yesmaximal, characteristic
4#11716776—31nomaximal, Hall
5C22×C3×C5⋊D5C2211yescharacteristic
6C2×C3⋊#1044—21no—
7C2×C3⋊#1044—21no—
8C2×C3⋊#1044C412yescomplemented
9C2×C3×#1043—21no—
10C2×C3×#1043—21no—
11C2×C3×#1043C412yescomplemented
12C2×C5×Dic15C412yescomplemented
13#10597126—21no—
14#10597126—21no—
15(C2×AGL(1,𝔽5))⋊Dic3—52nomaximal
16C22×#1044—31no—
17C22×#1043D311yescharacteristic
18C2×#6036—31no—
19C22×C3×C52C2×C411yescharacteristic
20C2×C3×C5⋊D5C2×C411yescharacteristic
21C2×C3×C5⋊D5C2×C411yescharacteristic
22C2×C3×C5⋊D5D411yescharacteristic
23C2×C3×C5⋊D5D411yescharacteristic
24C2×C3×C5⋊D5D411yescharacteristic
25C2×C3×C5⋊D5Q811yescharacteristic
26C3⋊#1044—22no—
27C3×#1043—22no—
28C5×Dic15—22no—
29C22×ΓL(1,𝔽16)—52no—
30C22×C3×AGL(1,𝔽5)—52no—
31C22×C3×AGL(1,𝔽5)—102no—
32C2×D5×Dic3—52no—
33C22×C5⋊D5C2×D311yescharacteristic
34C2×#1044—32no—
35C2×#1044—61no—
36C2×#1044—61no—
37C2×#1043—21no—
38C2×#1043—21no—
39C2×#1043Dic312yescomplemented
40C2×C5×Dic5—32no—
41#6036—61no—
42#6036—61no—
43(C2×AGL(1,𝔽5))⋊C4—152no—
44C2×C3×C52C4211yescharacteristic
45C2×C3×C52C22⋊C411yescharacteristic
46C2×C3×C52C4⋊C411yescharacteristic
47C3×C5⋊D5C22⋊C411yescharacteristic
48C3×C5⋊D5C22⋊C411yescharacteristic
49C3×C5⋊D5C4⋊C411yescharacteristic
50C3×C5⋊D5C4⋊C411yescharacteristic
51C22×C3×D5—52no—
52C22×C3×D5—102no—
53C2×ΓL(1,𝔽16)—54no—
54C2×ΓL(1,𝔽16)—102no—
55C2×ΓL(1,𝔽16)—102no—
56C2×C3×AGL(1,𝔽5)—54no—
57C2×C3×AGL(1,𝔽5)—102no—
58C2×C3×AGL(1,𝔽5)—102no—
59C2×C3×AGL(1,𝔽5)—104no—
60C2×C3×AGL(1,𝔽5)—104no—
61C2×C3×AGL(1,𝔽5)—104no—
62C2×Dic15AGL(1,𝔽5)12yescomplemented
63C2×C5×Dic3—52no—
64D5×Dic3—102no—
65D5×Dic3—102no—
66C22×C52C4×D311yescharacteristic
67C2×C5⋊D5Dic611yescharacteristic
68C2×C5⋊D5C4×D311yescharacteristic
69C2×C5⋊D5D1211yescharacteristic
70C2×C5⋊D5C2×Dic311yescharacteristic
71C2×C5⋊D5#6711yescharacteristic
72C2×C5⋊D5#6711yescharacteristic
73#1044—62no—
74#1043—22no—
75C5×Dic5—62no—
76C3⋊#95—251noHall
77C22×AGL(1,𝔽5)—52no—
78C22×AGL(1,𝔽5)—102no—
79C22×AGL(1,𝔽5)—152no—
80C2×C4×D5—152no—
81C3×C52#9511yescharacteristic, complemented, Hall
82C22×C3×C5—22no—
83C22×C3×C5C2×AGL(1,𝔽5)12yes—
84C2×C3×D5—52no—
85C2×C3×D5—52no—
86C2×C3×D5—52no—
87C2×C3×D5—52no—
88C2×C3×D5—52no—
89C2×C3×D5—52no—
90C2×C3×D5—102no—
91C2×C3×D5—102no—
92C2×C3×D5—102no—
93C2×C3×D5—102no—
94C2×C3×D5—102no—
95C2×C3×D5—102no—
96ΓL(1,𝔽16)—104no—
97C3×AGL(1,𝔽5)—104no—
98C3×AGL(1,𝔽5)—108no—
99Dic15—22no—
100C5×Dic3—102no—
101C2×C52C4×Dic311yescharacteristic
102C2×C52Dic3⋊C411yescharacteristic
103C2×C52(C2×D3)⋊C411yescharacteristic
104C5⋊D5Dic3⋊C411yescharacteristic
105C5⋊D5C4⋊Dic311yescharacteristic
106C5⋊D5(C2×D3)⋊C411yescharacteristic
107C5⋊D5#21711yescharacteristic
108C22×C3×C4—251no—
109C22×Dic3—251no—
110C22×Dic3—251no—
111C22×D5—52no—
112C22×D5—102no—
113C2×AGL(1,𝔽5)—54no—
114C2×AGL(1,𝔽5)—104no—
115C2×AGL(1,𝔽5)—102no—
116C2×AGL(1,𝔽5)—102no—
117C2×AGL(1,𝔽5)—104no—
118C2×AGL(1,𝔽5)—104no—
119C2×AGL(1,𝔽5)—154no—
120C2×AGL(1,𝔽5)—302no—
121C2×AGL(1,𝔽5)—302no—
122C2×C4×C5—152no—
123C2×Dic5—32no—
124C4×D5—302no—
125C4×D5—302no—
126#95—751noSylow
127C2×C3×C5—22no—
128C2×C3×C5—22no—
129C2×C3×C5—22no—
130C2×C3×C5C4×AGL(1,𝔽5)12yes—
131C2×C3×C5C4⋊AGL(1,𝔽5)12yes—
132C2×C3×C5C22⋊AGL(1,𝔽5)12yes—
133C3×D5—52no—
134C3×D5—52no—
135C3×D5—52no—
136C3×D5—52no—
137C3×D5—102no—
138C3×D5—102no—
139C3×D5—102no—
140C3×D5—102no—
141C52C3⋊#9511yescharacteristic, complemented, Sylow
142C23×C3—251no—
143C2×C3×C4—252no—
144C2×C3×C4—501no—
145C2×C3×C4—501no—
146C2×Dic3—52no—
147C2×Dic3—252no—
148C2×Dic3—501no—
149C2×Dic3—501no—
150C2×Dic3—501no—
151C2×Dic3—501no—
152C22×C5—22no—
153C22×C5D3×AGL(1,𝔽5)12yes—
154C2×D5—52no—
155C2×D5—52no—
156C2×D5—52no—
157C2×D5—52no—
158C2×D5—52no—
159C2×D5—52no—
160C2×D5—102no—
161C2×D5—102no—
162C2×D5—102no—
163C2×D5—102no—
164C2×D5—102no—
165C2×D5—102no—
166AGL(1,𝔽5)—104no—
167AGL(1,𝔽5)—108no—
168AGL(1,𝔽5)—304no—
169C4×C5—302no—
170Dic5—62no—
171C22×C4—251no—
172C22×C4—751no—
173C22×C4—751no—
174C3×C5—22no—
175C3×C5(C2×AGL(1,𝔽5))⋊C412yescomplemented
176C22×C3—251no—
177C22×C3—251no—
178C22×C3—251no—
179C22×C3—251no—
180C22×C3—251no—
181C22×C3—251no—
182C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic
183C3×C4—502no—
184Dic3—102no—
185Dic3—502no—
186C2×C5—22no—
187C2×C5—22no—
188C2×C5—22no—
189C2×C5Dic3×AGL(1,𝔽5)12yes—
190C2×C5(C2×D3)⋊AGL(1,𝔽5)12yes—
191C2×C5Dic3⋊AGL(1,𝔽5)12yes—
192D5—52no—
193D5—52no—
194D5—52no—
195D5—52no—
196D5—102no—
197D5—102no—
198D5—102no—
199D5—102no—
200C23—251no—
201C2×C4—152no—
202C2×C4—252no—
203C2×C4—501no—
204C2×C4—501no—
205C2×C4—752no—
206C2×C4—1501no—
207C2×C4—1501no—
208C2×C4—1501no—
209C2×C4—1501no—
210C2×C3—251no—
211C2×C3—251no—
212C2×C3—251no—
213C2×C3—251no—
214C2×C3#8783011yescharacteristic
215C2×C3#8783111yescharacteristic
216C2×C3#8783211yescharacteristic
217C5—22no—
218C5(C2×AGL(1,𝔽5))⋊Dic312yescomplemented
219C22—251no—
220C22—251no—
221C22—251no—
222C22—251no—
223C22—251no—
224C22—251no—
225C22#1059718211yescharacteristic
226C4—302no—
227C4—502no—
228C4—1502no—
229C3#1171677611yescharacteristic, complemented, Sylow
230C2—251no—
231C2—251no—
232C2—251no—
233C2—251no—
234C2#1192535411yescharacteristic
235C2#1192535511yescharacteristic
236C2#1192535611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#10597182
commutatorC2×C3×C52C42
Fitting, socleC22×C3×C52C2×C4
prime radical, cosocleC22×C3×C5⋊D5C22
hypocenterC3×C52#95
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C2×C3⋊#1044C4semidirect
11. C2×C3×#1043C4semidirect
12. C2×C5×Dic15C4semidirect
39. C2×#1043Dic3semidirect
62. C2×Dic15AGL(1,𝔽5)semidirect
81. C3×C52#95semidirect
141. C52C3⋊#95semidirect
175. C3×C5(C2×AGL(1,𝔽5))⋊C4semidirect
218. C5(C2×AGL(1,𝔽5))⋊Dic3semidirect
229. C3#11716776semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#10442
C22×C3×#10432
C2×#105971262
#117167763
(C2×AGL(1,𝔽5))⋊Dic35