Finite Group Databasebeta

← #72230638

Subgroups — #72230638

Order 2400.

412 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C4×D3×C5⋊D5C211yesmaximal, characteristic
2C4×C3⋊#1044C211yesmaximal, characteristic, complemented
3C3×C4×#1044C211yesmaximal, characteristic, complemented
4Dic3⋊#1044C212yesmaximal, complemented
5(C2×D3)⋊#1044C212yesmaximal
6#11717451—31nomaximal, Hall
7C2×D3×C5⋊D5C2211yescharacteristic
8C2×C3⋊#1044C2212yes—
9C2×C3×#1044C2212yes—
10C4×C3⋊(C5⋊D5)C411yescharacteristic, complemented
11C4×C52×D3C411yescharacteristic, complemented
12C3×C4×C5⋊D5C2211yescharacteristic
13#10597113C411yescharacteristic, complemented
14D3×#1039C411yescharacteristic, complemented
15Dic3×C5⋊D5C2211yescharacteristic
16#91909—52nomaximal
17C2×C4×C5⋊D5—31no—
18C22⋊#1044—32no—
19C4⋊#1044—32no—
20C4×#1044—31no—
21C4×#1044D311yescharacteristic, complemented
22C2×C3⋊(C5⋊D5)C2×C411yescharacteristic
23C2×C52×D3C2×C411yescharacteristic
24C2×C3×C5⋊D5C2311yescharacteristic
25D3×C5⋊D5—22no—
26C3⋊#1044—22no—
27C3×#1044—22no—
28C3×C4×C52C2×C411yescharacteristic
29C3⋊#1039C2×C411yescharacteristic
30C52×Dic3C2×C411yescharacteristic
31C3×#1039C2×C411yescharacteristic
32C4×D3×D5—52no—
33C4×D3×D5—102no—
34C4×ΓL(1,𝔽16)—52no—
35C3×C4×AGL(1,𝔽5)—52no—
36Dic3⋊AGL(1,𝔽5)—54no—
37(C2×D3)⋊AGL(1,𝔽5)—54no—
38C22×C5⋊D5—31no—
39C2×#1044—32no—
40C2×#1044C2×D312yes—
41C2×C4×C52—31no—
42C2×#1039—31no—
43C4×C5⋊D5—31no—
44C4×C5⋊D5—31no—
45C4×C5⋊D5—31no—
46C4×C5⋊D5C2×D311yescharacteristic
47#4130—152no—
48C2×C3×C52C22×C411yescharacteristic
49C3⋊(C5⋊D5)—21no—
50C52×D3—21no—
51C3×C5⋊D5#4112yes—
52C2×D3×D5—52no—
53C2×D3×D5—102no—
54C2×ΓL(1,𝔽16)—54no—
55C2×C3×AGL(1,𝔽5)—54no—
56C4×D15—52no—
57C4×D15—102no—
58C4×C5×D3—22no—
59C4×C5×D3AGL(1,𝔽5)12yescomplemented
60C3×C4×D5—52no—
61C3×C4×D5—102no—
62#1200—52no—
63#1200—102no—
64D3×Dic5—52no—
65D3×Dic5—102no—
66D5×Dic3—52no—
67D5×Dic3—102no—
68C22×C52—31no—
69C2×C5⋊D5—31no—
70C2×C5⋊D5—62no—
71C2×C5⋊D5C22×D311yescharacteristic
72#1044—22no—
73#1044—62no—
74C4×C52—31no—
75C4×C52C4×D311yescharacteristic
76#1039—31no—
77#1039C4×D311yescharacteristic
78#872—251noHall
79C2×C4×D5—152no—
80C2×C4×D5—302no—
81C22⋊AGL(1,𝔽5)—154no—
82C4⋊AGL(1,𝔽5)—154no—
83C4×AGL(1,𝔽5)—52no—
84C4×AGL(1,𝔽5)—152no—
85C3×C52#11711yescharacteristic, complemented, Hall
86C2×D15—52no—
87C2×D15—102no—
88C2×C5×D3—22no—
89C2×C5×D3C2×AGL(1,𝔽5)12yes—
90C2×C3×D5—52no—
91C2×C3×D5—102no—
92D3×D5—104no—
93D3×D5—104no—
94D3×D5—104no—
95ΓL(1,𝔽16)—104no—
96C3×AGL(1,𝔽5)—104no—
97C3×C4×C5—22no—
98C3×C4×C5C2×AGL(1,𝔽5)12yes—
99Dic15—52no—
100Dic15—102no—
101C3×Dic5—52no—
102C3×Dic5—102no—
103C5×Dic3—22no—
104C5×Dic3C2×AGL(1,𝔽5)12yes—
105C2×C52—61no—
106C2×C52C2×C4×D311yescharacteristic
107C5⋊D5—61no—
108C5⋊D5#23512yes—
109C2×C4×D3—251no—
110C3×C42—251no—
111(C2×D3)⋊C4—252no—
112Dic3⋊C4—252no—
113C4×Dic3—251no—
114C22×D5—152no—
115C22×D5—302no—
116C2×AGL(1,𝔽5)—54no—
117C2×AGL(1,𝔽5)—154no—
118C2×C4×C5—32no—
119C2×C4×C5—62no—
120C2×Dic5—152no—
121C2×Dic5—302no—
122C4×D5—52no—
123C4×D5—102no—
124C4×D5—152no—
125C4×D5—152no—
126C4×D5—152no—
127C4×D5—302no—
128C4×D5—302no—
129C4×D5—302no—
130#117—751noSylow
131C2×C3×C5—22no—
132C2×C3×C5C22×AGL(1,𝔽5)12yes—
133D15—102no—
134D15—102no—
135D15—102no—
136C3×D5—54no—
137C3×D5—104no—
138C5×D3—22no—
139C5×D3—22no—
140C5×D3—22no—
141C52#87211yescharacteristic, complemented, Sylow
142C22×D3—251no—
143C2×C3×C4—251no—
144C2×C3×C4—252no—
145C2×Dic3—251no—
146C2×Dic3—252no—
147C4×D3—251no—
148C4×D3—251no—
149C4×D3—251no—
150C4×D3#104411yescharacteristic, complemented
151C22×C5—32no—
152C22×C5—62no—
153C2×D5—52no—
154C2×D5—102no—
155C2×D5—152no—
156C2×D5—302no—
157C2×D5—304no—
158C2×D5—304no—
159C2×D5—304no—
160AGL(1,𝔽5)—104no—
161AGL(1,𝔽5)—304no—
162C4×C5—22no—
163C4×C5—32no—
164C4×C5—62no—
165C4×C5D3×AGL(1,𝔽5)12yes—
166Dic5—52no—
167Dic5—102no—
168Dic5—152no—
169Dic5—302no—
170C22×C4—751no—
171C4⋊C4—752no—
172C22⋊C4—752no—
173C42—251no—
174C42—751no—
175C3×C5—22no—
176C3×C5#413012yescomplemented
177C22×C3—251no—
178C2×D3—251no—
179C2×D3—502no—
180C2×D3C2×#104411yescharacteristic
181C3×C4—251no—
182C3×C4—502no—
183C3×C4C2×#104411yescharacteristic
184Dic3—251no—
185Dic3—502no—
186Dic3C2×#104411yescharacteristic
187C2×C5—22no—
188C2×C5—62no—
189C2×C5—62no—
190C2×C5—62no—
191C2×C5C2×D3×AGL(1,𝔽5)12yes—
192D5—54no—
193D5—104no—
194D5—302no—
195D5—302no—
196D5—302no—
197C23—751no—
198C2×C4—31no—
199C2×C4—251no—
200C2×C4—252no—
201C2×C4—751no—
202C2×C4—751no—
203C2×C4—751no—
204C2×C4—751no—
205C2×C4—752no—
206C2×C3—252no—
207C2×C3C22×#104411yescharacteristic
208D3—21no—
209D3—501no—
210C5—22no—
211C5#9190912yescomplemented
212C22—31no—
213C22—251no—
214C22—751no—
215C22—1502no—
216C4—31no—
217C4—251no—
218C4—502no—
219C4—751no—
220C4—1502no—
221C4D3×#104411yescharacteristic
222C3#1171745111yescharacteristic, complemented, Sylow
223C2—61no—
224C2—252no—
225C2—1501no—
226C2C2×D3×#104411yescharacteristic
subgroupG/H
center, hypercenterC4D3×#1044
commutator, socleC2×C3×C52C22×C4
FrattiniC2C2×D3×#1044
FittingC3×C4×C52C2×C4
prime radicalC2×D3×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52#117
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C4×C3⋊#1044C2semidirect
3. C3×C4×#1044C2semidirect
4. Dic3⋊#1044C2semidirect
10. C4×C3⋊(C5⋊D5)C4semidirect
11. C4×C52×D3C4semidirect
13. #10597113C4semidirect
14. D3×#1039C4semidirect
21. C4×#1044D3semidirect
59. C4×C5×D3AGL(1,𝔽5)semidirect
85. C3×C52#117semidirect
141. C52#872semidirect
150. C4×D3#1044semidirect
176. C3×C5#4130semidirect
211. C5#91909semidirect
222. C3#11717451semidirect

Maximal subgroups

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

subgroupindex
C4×D3×C5⋊D52
C4×C3⋊#10442
C3×C4×#10442
Dic3⋊#10442
(C2×D3)⋊#10442
#117174513
#919095