Finite Group Databasebeta

← #72231049

Subgroups — #72231049

Order 2400.

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

normal subgroups only

This group has 356 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
1D52×Dic3C211yesmaximal, characteristic
2#11925359C211yesmaximal, characteristic, complemented
3#11925354C211yesmaximal, characteristic, complemented
4#11925350C211yesmaximal, characteristic
5Dic3⋊#1044C211yesmaximal, characteristic, complemented
6Dic3⋊#1043C211yesmaximal, characteristic, complemented
7C3×#87831C211yesmaximal, characteristic
8#11717354—31nomaximal, Hall
9C2×C3×D52C2211yescharacteristic
10C2×C3⋊#1044C2211yescharacteristic
11C2×C3⋊#1043C2211yescharacteristic
12C2×C3×#1044C2211yescharacteristic
13C2×C3×#1043C2211yescharacteristic
14#10597126C2211yescharacteristic
15D5×Dic15C412yescomplemented
16Dic3×C5⋊D5C2211yescharacteristic
17C5×D5×Dic3C412yescomplemented
18#91923—52nomaximal
19C4×D52—31no—
20C4⋊#1044—31no—
21C4⋊#1043—31no—
22#87831—31no—
23#87831D311yescharacteristic
24#87830—31no—
25#87830—31no—
26C2×C3×C5⋊D5C2311yescharacteristic
27C2×C3×C5×D5C2×C412yes—
28C3×D52—21no—
29C3×D52—21no—
30C3⋊#1044—21no—
31C3⋊#1043—21no—
32C3×#1044—21no—
33C3×#1043—21no—
34C3⋊#1039C2×C411yescharacteristic
35C5×Dic15C2×C412yes—
36C52×Dic3C2×C411yescharacteristic
37C2×D5×Dic3—52no—
38#6763—52no—
39C3×C22⋊AGL(1,𝔽5)—52no—
40Dic3⋊AGL(1,𝔽5)—52no—
41Dic3⋊AGL(1,𝔽5)—52no—
42Dic3⋊AGL(1,𝔽5)—102no—
43Dic3×AGL(1,𝔽5)—52no—
44Dic3×AGL(1,𝔽5)—52no—
45C2×D52C2×D311yescharacteristic
46C2×#1044—31no—
47C2×#1044C2×D311yescharacteristic
48C2×#1043—31no—
49C2×#1043C2×D311yescharacteristic
50C4×C5⋊D5—31no—
51C4×C5×D5—32no—
52#6036—31no—
53D5×Dic5—32no—
54#4130—152no—
55C2×C3×C52C22×C411yescharacteristic
56C3×C5⋊D5#4111yescharacteristic
57C3×C5⋊D5#4111yescharacteristic
58C3×C5×D5—22no—
59C22×C3×D5—52no—
60C2×ΓL(1,𝔽16)—52no—
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×Dic15—52no—
67C2×C5×Dic3—52no—
68D5×Dic3—52no—
69D5×Dic3—52no—
70D5×Dic3—52no—
71D5×Dic3—102no—
72D5×Dic3AGL(1,𝔽5)12yescomplemented
73C2×C5⋊D5C22×D311yescharacteristic
74C2×C5×D5C4×D312yes—
75D52—21no—
76D52—21no—
77#1044—21no—
78#1044—61no—
79#1043—21no—
80#1043—61no—
81C4×C52—31no—
82#1039—31no—
83C5×Dic5—32no—
84#877—251noHall
85C2×C4×D5—152no—
86C22⋊AGL(1,𝔽5)—52no—
87C22⋊AGL(1,𝔽5)—152no—
88C4⋊AGL(1,𝔽5)—152no—
89C4⋊AGL(1,𝔽5)—152no—
90C4⋊AGL(1,𝔽5)—302no—
91C4×AGL(1,𝔽5)—152no—
92C4×AGL(1,𝔽5)—152no—
93C3×C52#11711yescharacteristic, complemented, Hall
94C22×C3×C5—52no—
95C2×C3×D5—52no—
96C2×C3×D5—102no—
97C2×C3×D5—102no—
98C2×C3×D5—102no—
99C2×C3×D5C2×AGL(1,𝔽5)12yes—
100ΓL(1,𝔽16)—102no—
101ΓL(1,𝔽16)—102no—
102ΓL(1,𝔽16)—202no—
103C3×AGL(1,𝔽5)—102no—
104C3×AGL(1,𝔽5)—102no—
105C3×AGL(1,𝔽5)—202no—
106Dic15—52no—
107Dic15—102no—
108Dic15C2×AGL(1,𝔽5)12yes—
109C5×Dic3—22no—
110C5×Dic3—52no—
111C5×Dic3C2×AGL(1,𝔽5)12yes—
112C2×C52C2×C4×D311yescharacteristic
113C5⋊D5#23711yescharacteristic
114C5⋊D5#23711yescharacteristic
115C5×D5—22no—
116C22×Dic3—251no—
117C3×C22⋊C4—251no—
118#217—251no—
119Dic3⋊C4—251no—
120Dic3⋊C4—251no—
121C4×Dic3—251no—
122C4×Dic3—251no—
123C22×D5—52no—
124C2×AGL(1,𝔽5)—52no—
125C2×AGL(1,𝔽5)—52no—
126C2×AGL(1,𝔽5)—102no—
127C2×AGL(1,𝔽5)—152no—
128C2×AGL(1,𝔽5)—152no—
129C2×AGL(1,𝔽5)—302no—
130C2×C4×C5—152no—
131C2×Dic5—152no—
132C4×D5—32no—
133C4×D5—152no—
134C4×D5—152no—
135C4×D5—152no—
136C4×D5—302no—
137#117—751noSylow
138C2×C3×C5—22no—
139C2×C3×C5—102no—
140C2×C3×C5C22×AGL(1,𝔽5)12yes—
141C3×D5—22no—
142C3×D5—52no—
143C3×D5—52no—
144C3×D5—102no—
145C3×D5—102no—
146C52#87711yescharacteristic, complemented, Sylow
147C23×C3—251no—
148C2×C3×C4—251no—
149C2×C3×C4—251no—
150C2×Dic3—52no—
151C2×Dic3—251no—
152C2×Dic3—252no—
153C2×Dic3—251no—
154C2×Dic3—251no—
155C2×Dic3—251no—
156C22×C5—52no—
157C2×D5—52no—
158C2×D5—102no—
159C2×D5—102no—
160C2×D5—102no—
161C2×D5D3×AGL(1,𝔽5)12yes—
162AGL(1,𝔽5)—102no—
163AGL(1,𝔽5)—102no—
164AGL(1,𝔽5)—202no—
165AGL(1,𝔽5)—302no—
166AGL(1,𝔽5)—302no—
167AGL(1,𝔽5)—602no—
168C4×C5—32no—
169C4×C5—62no—
170C4×C5—152no—
171Dic5—32no—
172Dic5—152no—
173Dic5—302no—
174C22×C4—751no—
175C4⋊C4—751no—
176C4⋊C4—751no—
177C22⋊C4—251no—
178C22⋊C4—751no—
179C42—751no—
180C42—751no—
181C3×C5—22no—
182C3×C5#413012yescomplemented
183C22×C3—52no—
184C22×C3—251no—
185C22×C3—501no—
186C22×C3—501no—
187C3×C4—501no—
188C3×C4—501no—
189Dic3—52no—
190Dic3—251no—
191Dic3—501no—
192Dic3—501no—
193Dic3D5⋊AGL(1,𝔽5)11yescharacteristic
194C2×C5—22no—
195C2×C5—102no—
196C2×C5C2×D3×AGL(1,𝔽5)12yes—
197D5—22no—
198D5—52no—
199D5—52no—
200D5—102no—
201D5—102no—
202C23—251no—
203C2×C4—152no—
204C2×C4—251no—
205C2×C4—251no—
206C2×C4—751no—
207C2×C4—751no—
208C2×C4—752no—
209C2×C4—751no—
210C2×C4—751no—
211C2×C3—102no—
212C2×C3—251no—
213C2×C3—251no—
214C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
215C5—22no—
216C5#9192312yescomplemented
217C22—52no—
218C22—251no—
219C22—501no—
220C22—501no—
221C4—31no—
222C4—152no—
223C4—501no—
224C4—501no—
225C4—751no—
226C4—1501no—
227C4—1501no—
228C3#1171735411yescharacteristic, complemented, Sylow
229C2—102no—
230C2—251no—
231C2—251no—
232C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radicalC2×C3×D52C22
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. #11925359C2semidirect
3. #11925354C2semidirect
5. Dic3⋊#1044C2semidirect
6. Dic3⋊#1043C2semidirect
15. D5×Dic15C4semidirect
17. C5×D5×Dic3C4semidirect
72. D5×Dic3AGL(1,𝔽5)semidirect
93. C3×C52#117semidirect
146. C52#877semidirect
182. C3×C5#4130semidirect
216. C5#91923semidirect
228. C3#11717354semidirect

Maximal subgroups

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

subgroupindex
D52×Dic32
#119253592
#119253542
#119253502
Dic3⋊#10442
Dic3⋊#10432
C3×#878312
#117173543
#919235