Finite Group Databasebeta

← #72231029

Subgroups — #72231029

Order 2400.

372 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 64 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
1D3×#6036C211yesmaximal, characteristic
2#11925361C211yesmaximal, characteristic, complemented
3#11925356C211yesmaximal, characteristic, complemented
4#11925349C211yesmaximal, characteristic, complemented
5(C2×D3)⋊#1044C211yesmaximal, characteristic
6(C2×D3)⋊#1043C211yesmaximal, characteristic
7C3×#87830C211yesmaximal, characteristic, complemented
8#11717367—31nomaximal, Hall
9C2×D3×C5⋊D5C2211yescharacteristic
10C2×C3⋊#1044C2211yescharacteristic
11C2×C3⋊#1043C2211yescharacteristic
12C2×C3×#1044C2211yescharacteristic
13C2×C3×#1043C2211yescharacteristic
14#10597126C2211yescharacteristic
15#10597120C412yescomplemented
16C5×D3×Dic5C412yescomplemented
17C3×#6036C2211yescharacteristic
18#91909—52nomaximal
19C2×#6036—31no—
20C22⋊#1044—31no—
21C22⋊#1043—31no—
22#87832—31no—
23#87832—31no—
24#87830—31no—
25#87830D311yescharacteristic, complemented
26C2×C3⋊(C5⋊D5)C2×C411yescharacteristic
27C2×C52×D3C2×C411yescharacteristic
28C2×C3×C5⋊D5C2311yescharacteristic
29D3×C5⋊D5—21no—
30D3×C5⋊D5—21no—
31C3⋊#1044—21no—
32C3⋊#1043—21no—
33C3×#1044—21no—
34C3×#1043—21no—
35C5×Dic15C2×C412yes—
36C3×C5×Dic5C2×C412yes—
37C4×D3×D5—52no—
38C4×ΓL(1,𝔽16)—52no—
39C3×C4×AGL(1,𝔽5)—52no—
40Dic3⋊AGL(1,𝔽5)—52no—
41Dic3⋊AGL(1,𝔽5)—52no—
42(C2×D3)⋊AGL(1,𝔽5)—52no—
43(C2×D3)⋊AGL(1,𝔽5)—52no—
44(C2×D3)⋊AGL(1,𝔽5)—102no—
45C22×C5⋊D5—31no—
46C2×#1044—31no—
47C2×#1044C2×D311yescharacteristic
48C2×#1043—31no—
49C2×#1043C2×D311yescharacteristic
50C2×C5×Dic5—32no—
51#6036—31no—
52#6036—32no—
53#6036C2×D311yescharacteristic
54#4130—152no—
55C2×C3×C52C22×C411yescharacteristic
56C3⋊(C5⋊D5)—21no—
57C52×D3—21no—
58C3×C5⋊D5#4111yescharacteristic
59C3×C5⋊D5#4111yescharacteristic
60C2×D3×D5—52no—
61C2×D3×D5—102no—
62C2×ΓL(1,𝔽16)—52no—
63C2×ΓL(1,𝔽16)—52no—
64C2×ΓL(1,𝔽16)—102no—
65C2×C3×AGL(1,𝔽5)—52no—
66C2×C3×AGL(1,𝔽5)—52no—
67C2×C3×AGL(1,𝔽5)—102no—
68C4×D15—52no—
69C4×C5×D3—52no—
70C3×C4×D5—52no—
71#1200—52no—
72D3×Dic5AGL(1,𝔽5)12yescomplemented
73D5×Dic3—52no—
74C22×C52—31no—
75C2×C5⋊D5—31no—
76C2×C5⋊D5—61no—
77C2×C5⋊D5—61no—
78C2×C5⋊D5C22×D311yescharacteristic
79#1044—21no—
80#1044—61no—
81#1043—21no—
82#1043—61no—
83C5×Dic5—32no—
84C5×Dic5C4×D312yes—
85#872—251noHall
86C2×C4×D5—152no—
87C22⋊AGL(1,𝔽5)—152no—
88C22⋊AGL(1,𝔽5)—152no—
89C22⋊AGL(1,𝔽5)—302no—
90C4⋊AGL(1,𝔽5)—152no—
91C4⋊AGL(1,𝔽5)—152no—
92C4×AGL(1,𝔽5)—52no—
93C4×AGL(1,𝔽5)—152no—
94C3×C52#11711yescharacteristic, complemented, Hall
95C2×D15—52no—
96C2×D15—102no—
97C2×C5×D3—22no—
98C2×C5×D3C2×AGL(1,𝔽5)12yes—
99C2×C3×D5—52no—
100C2×C3×D5—102no—
101D3×D5—102no—
102D3×D5—102no—
103D3×D5—202no—
104D3×D5—202no—
105ΓL(1,𝔽16)—102no—
106ΓL(1,𝔽16)—102no—
107ΓL(1,𝔽16)—202no—
108C3×AGL(1,𝔽5)—102no—
109C3×AGL(1,𝔽5)—102no—
110C3×AGL(1,𝔽5)—202no—
111C3×C4×C5—52no—
112Dic15C2×AGL(1,𝔽5)12yes—
113C3×Dic5C2×AGL(1,𝔽5)12yes—
114C5×Dic3—52no—
115C2×C52—61no—
116C2×C52C2×C4×D311yescharacteristic
117C5⋊D5—61no—
118C5⋊D5#23511yescharacteristic
119C5⋊D5#23511yescharacteristic
120C2×C4×D3—251no—
121C3×C42—251no—
122(C2×D3)⋊C4—251no—
123(C2×D3)⋊C4—251no—
124Dic3⋊C4—251no—
125Dic3⋊C4—251no—
126C4×Dic3—251no—
127C22×D5—152no—
128C22×D5—302no—
129C2×AGL(1,𝔽5)—52no—
130C2×AGL(1,𝔽5)—52no—
131C2×AGL(1,𝔽5)—102no—
132C2×AGL(1,𝔽5)—152no—
133C2×AGL(1,𝔽5)—152no—
134C2×AGL(1,𝔽5)—302no—
135C2×C4×C5—152no—
136C2×Dic5—32no—
137C4×D5—52no—
138C4×D5—152no—
139C4×D5—152no—
140C4×D5—152no—
141#117—751noSylow
142C2×C3×C5—22no—
143C2×C3×C5C22×AGL(1,𝔽5)12yes—
144D15—102no—
145D15—202no—
146C3×D5—52no—
147C3×D5—52no—
148C3×D5—102no—
149C3×D5—102no—
150C5×D3—22no—
151C5×D3—42no—
152C52#87211yescharacteristic, complemented, Sylow
153C22×D3—251no—
154C2×C3×C4—251no—
155C2×C3×C4—251no—
156C2×C3×C4—251no—
157C2×Dic3—251no—
158C2×Dic3—251no—
159C2×Dic3—251no—
160C4×D3—52no—
161C4×D3—252no—
162C22×C5—32no—
163C22×C5—62no—
164C2×D5—52no—
165C2×D5—102no—
166C2×D5—152no—
167C2×D5—302no—
168C2×D5—302no—
169C2×D5—302no—
170C2×D5—602no—
171C2×D5—602no—
172AGL(1,𝔽5)—102no—
173AGL(1,𝔽5)—102no—
174AGL(1,𝔽5)—202no—
175AGL(1,𝔽5)—302no—
176AGL(1,𝔽5)—302no—
177AGL(1,𝔽5)—602no—
178C4×C5—52no—
179C4×C5—152no—
180Dic5—32no—
181Dic5D3×AGL(1,𝔽5)12yes—
182C22×C4—751no—
183C4⋊C4—751no—
184C4⋊C4—751no—
185C22⋊C4—751no—
186C22⋊C4—751no—
187C42—251no—
188C42—751no—
189C3×C5—22no—
190C3×C5#413012yescomplemented
191C22×C3—251no—
192C2×D3—251no—
193C2×D3—501no—
194C2×D3—501no—
195C2×D3D5⋊AGL(1,𝔽5)11yescharacteristic
196C3×C4—52no—
197C3×C4—501no—
198C3×C4—501no—
199Dic3—52no—
200Dic3—501no—
201Dic3—501no—
202C2×C5—22no—
203C2×C5—62no—
204C2×C5—122no—
205C2×C5C2×D3×AGL(1,𝔽5)12yes—
206D5—52no—
207D5—52no—
208D5—102no—
209D5—102no—
210D5—302no—
211D5—602no—
212C23—751no—
213C2×C4—152no—
214C2×C4—251no—
215C2×C4—251no—
216C2×C4—251no—
217C2×C4—752no—
218C2×C4—751no—
219C2×C4—751no—
220C2×C4—751no—
221C2×C3—251no—
222C2×C3—251no—
223C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
224D3—21no—
225D3—501no—
226C5—22no—
227C5#9190912yescomplemented
228C22—31no—
229C22—251no—
230C22—751no—
231C22—1501no—
232C22—1501no—
233C4—52no—
234C4—152no—
235C4—501no—
236C4—501no—
237C4—1501no—
238C4—1501no—
239C3#1171736711yescharacteristic, complemented, Sylow
240C2—61no—
241C2—251no—
242C2—251no—
243C2—1501no—
244C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radicalC2×D3×C5⋊D5C22
cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52#117
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #11925361C2semidirect
3. #11925356C2semidirect
4. #11925349C2semidirect
7. C3×#87830C2semidirect
15. #10597120C4semidirect
16. C5×D3×Dic5C4semidirect
25. #87830D3semidirect
72. D3×Dic5AGL(1,𝔽5)semidirect
94. C3×C52#117semidirect
152. C52#872semidirect
190. C3×C5#4130semidirect
227. C5#91909semidirect
239. C3#11717367semidirect

Maximal subgroups

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

subgroupindex
D3×#60362
#119253612
#119253562
#119253492
(C2×D3)⋊#10442
(C2×D3)⋊#10432
C3×#878302
#117173673
#919095