Finite Group Databasebeta

← #72231071

Subgroups — #72231071

Order 2400.

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

normal subgroups only

This group has 428 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
1C2×#10597183C211yesmaximal, characteristic, complemented
2C2×C3×D5⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3#11925420C211yesmaximal, characteristic
4#11925355C211yesmaximal, characteristic, complemented
5#11925350C211yesmaximal, characteristic, complemented
6Dic3×#1044C211yesmaximal, characteristic, complemented
7Dic3⋊#1043C211yesmaximal, characteristic, complemented
8#11717365—31nomaximal, Hall
9C2×D15⋊D5C2211yescharacteristic
10C2×C3×D52C2211yescharacteristic
11C2×C3⋊#1044C2211yescharacteristic
12C2×C3⋊#1043C2211yescharacteristic
13C2×C3×#1044C2211yescharacteristic
14C2×C3×#1043C2211yescharacteristic
15#10597183—22no—
16C3×D5⋊AGL(1,𝔽5)—22no—
17#10597121C412yescomplemented
18Dic3×C5⋊D5C2211yescharacteristic
19C5×#1202C412yescomplemented
20AGL(1,𝔽5)×#67—52nomaximal
21C2×D5⋊AGL(1,𝔽5)—31no—
22C2×D5⋊AGL(1,𝔽5)D311yescharacteristic, complemented
23#87877—31no—
24C4×#1044—31no—
25C4⋊#1043—31no—
26#87831—31no—
27#87831—31no—
28C2×C5×D15C2×C412yescomplemented
29C2×C3×C5⋊D5C2311yescharacteristic
30C2×C3×C5×D5C2×C412yescomplemented
31D15⋊D5—21no—
32D15⋊D5—21no—
33C3×D52—21no—
34C3×D52—21no—
35C3⋊#1044—21no—
36C3⋊#1043—21no—
37C3×#1044D412yescomplemented
38C3×#1043—21no—
39C3⋊#1039C2×C411yescharacteristic, complemented
40C52×Dic3C2×C411yescharacteristic, complemented
41C22×C3×AGL(1,𝔽5)—52no—
42C2×D3×AGL(1,𝔽5)—52no—
43D5×#67—52no—
44#6763—52no—
45Dic3⋊AGL(1,𝔽5)—52no—
46Dic3⋊AGL(1,𝔽5)—102no—
47(C2×D3)⋊AGL(1,𝔽5)—52no—
48Dic3×AGL(1,𝔽5)—52no—
49C2×D52—31no—
50C2×D52C2×D311yescharacteristic
51C2×#1044—31no—
52C2×#1044C2×D311yescharacteristic
53C2×#1043—31no—
54C2×#1043C2×D311yescharacteristic
55D5⋊AGL(1,𝔽5)—22no—
56D5⋊AGL(1,𝔽5)—62no—
57C4×C5⋊D5—31no—
58C5×D20—32no—
59#6037—32no—
60D4×AGL(1,𝔽5)—152no—
61C2×C3×C52C22×C411yescharacteristic
62C5×D15—22no—
63C3×C5⋊D5C2×D411yescharacteristic
64C3×C5⋊D5#4111yescharacteristic
65C3×C5×D5—22no—
66C22×C3×D5—52no—
67C2×D3×D5—52no—
68C2×ΓL(1,𝔽16)—52no—
69C2×ΓL(1,𝔽16)—52no—
70C2×ΓL(1,𝔽16)—102no—
71C2×C3×AGL(1,𝔽5)—52no—
72C2×C3×AGL(1,𝔽5)—52no—
73C2×C3×AGL(1,𝔽5)—104no—
74C2×C3×AGL(1,𝔽5)—102no—
75D3×AGL(1,𝔽5)—104no—
76#1220—52no—
77C5×#67—52no—
78#1202AGL(1,𝔽5)12yescomplemented
79#1201—52no—
80D5×Dic3—52no—
81D5×Dic3—102no—
82C2×C5⋊D5C22×D311yescharacteristic
83C2×C5×D5—32no—
84C2×C5×D5C4×D312yescomplemented
85D52—21no—
86D52—21no—
87D52—61no—
88D52—61no—
89#1044—61no—
90#1044#6712yescomplemented
91#1043—21no—
92#1043—61no—
93C4×C52—31no—
94#1039—31no—
95C4×#67—251noHall
96C22×AGL(1,𝔽5)—52no—
97C22×AGL(1,𝔽5)—152no—
98D4×D5—152no—
99C22⋊AGL(1,𝔽5)—152no—
100C22⋊AGL(1,𝔽5)—152no—
101C4⋊AGL(1,𝔽5)—152no—
102C4⋊AGL(1,𝔽5)—302no—
103C4×AGL(1,𝔽5)—152no—
104C3×C52C4×D411yescharacteristic, complemented, Hall
105C22×C3×C5—52no—
106C2×D15C2×AGL(1,𝔽5)12yescomplemented
107C2×C5×D3—52no—
108C2×C3×D5—52no—
109C2×C3×D5—102no—
110C2×C3×D5—102no—
111C2×C3×D5—102no—
112C2×C3×D5C2×AGL(1,𝔽5)12yescomplemented
113D3×D5—102no—
114D3×D5—102no—
115ΓL(1,𝔽16)—102no—
116ΓL(1,𝔽16)—102no—
117ΓL(1,𝔽16)—202no—
118C3×AGL(1,𝔽5)—54no—
119C3×AGL(1,𝔽5)—102no—
120C3×AGL(1,𝔽5)—202no—
121Dic15—52no—
122Dic15—102no—
123C5×Dic3—22no—
124C5×Dic3C2×AGL(1,𝔽5)12yescomplemented
125C2×C52C2×C4×D311yescharacteristic
126C5⋊D5#23511yescharacteristic
127C5⋊D5C2×#6711yescharacteristic
128C5×D5—22no—
129C5×D5—62no—
130C22×C3×C4—251no—
131C2×#67—251no—
132C2×C4×D3—251no—
133#217—251no—
134(C2×D3)⋊C4—251no—
135Dic3⋊C4—251no—
136C4×Dic3—251no—
137C22×D5—52no—
138C22×D5—152no—
139C2×AGL(1,𝔽5)—52no—
140C2×AGL(1,𝔽5)—52no—
141C2×AGL(1,𝔽5)—102no—
142C2×AGL(1,𝔽5)—104no—
143C2×AGL(1,𝔽5)—152no—
144C2×AGL(1,𝔽5)—152no—
145C2×AGL(1,𝔽5)—302no—
146C2×AGL(1,𝔽5)—304no—
147C5×D4—152no—
148C22⋊D5—152no—
149C22⋊D5—152no—
150D20—32no—
151C4×D5—152no—
152C4×D5—302no—
153C4×D4—751noSylow
154C2×C3×C5—22no—
155C2×C3×C5—102no—
156C2×C3×C5C22×AGL(1,𝔽5)12yes—
157D15—22no—
158C3×D5—22no—
159C3×D5—52no—
160C3×D5—52no—
161C3×D5—102no—
162C3×D5—102no—
163C5×D3—102no—
164C52C4×#6711yescharacteristic, complemented, Sylow
165C23×C3—251no—
166C22×D3—251no—
167C2×C3×C4—251no—
168C2×C3×C4—251no—
169C2×C3×C4—502no—
170#67—52no—
171#67—252no—
172C2×Dic3—251no—
173C2×Dic3—251no—
174C2×Dic3—251no—
175C4×D3—502no—
176C22×C5—52no—
177C22×C5—152no—
178C2×D5—32no—
179C2×D5—52no—
180C2×D5—102no—
181C2×D5—102no—
182C2×D5—102no—
183C2×D5—302no—
184C2×D5—302no—
185C2×D5D3×AGL(1,𝔽5)12yescomplemented
186AGL(1,𝔽5)—54no—
187AGL(1,𝔽5)—102no—
188AGL(1,𝔽5)—202no—
189AGL(1,𝔽5)—302no—
190AGL(1,𝔽5)—302no—
191AGL(1,𝔽5)—602no—
192C4×C5—32no—
193C4×C5—62no—
194Dic5—152no—
195Dic5—302no—
196C2×D4—751no—
197C22×C4—251no—
198C22×C4—751no—
199C4⋊C4—751no—
200C22⋊C4—751no—
201C22⋊C4—751no—
202C42—751no—
203C3×C5—22no—
204C3×C5D4×AGL(1,𝔽5)12yescomplemented
205C22×C3—52no—
206C22×C3—251no—
207C22×C3—501no—
208C22×C3—501no—
209C2×D3—52no—
210C2×D3—501no—
211C2×D3—501no—
212C3×C4—252no—
213C3×C4—501no—
214Dic3—251no—
215Dic3—501no—
216Dic3—501no—
217Dic3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
218C2×C5—22no—
219C2×C5—102no—
220C2×C5—302no—
221C2×C5C2×D3×AGL(1,𝔽5)12yes—
222D5—22no—
223D5—52no—
224D5—52no—
225D5—62no—
226D5—102no—
227D5—102no—
228C23—251no—
229C23—751no—
230D4—152no—
231D4—752no—
232C2×C4—251no—
233C2×C4—251no—
234C2×C4—502no—
235C2×C4—751no—
236C2×C4—751no—
237C2×C4—751no—
238C2×C4—1502no—
239C2×C3—102no—
240C2×C3—251no—
241C2×C3—251no—
242C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
243D3—102no—
244C5—22no—
245C5AGL(1,𝔽5)×#6712yescomplemented
246C22—52no—
247C22—152no—
248C22—251no—
249C22—501no—
250C22—501no—
251C22—1501no—
252C22—1501no—
253C4—31no—
254C4—252no—
255C4—501no—
256C4—751no—
257C4—1501no—
258C4—1501no—
259C3#1171736511yescharacteristic, complemented, Sylow
260C2—102no—
261C2—251no—
262C2—251no—
263C2—302no—
264C2D3×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×D5⋊AGL(1,𝔽5)
commutator, Fitting, socleC2×C3×C52C22×C4
prime radical#11925420C2
cosocleC2×C3×C5⋊D5C23
hypocenterC3×C52C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#10597183C2semidirect
2. C2×C3×D5⋊AGL(1,𝔽5)C2semidirect
4. #11925355C2semidirect
5. #11925350C2semidirect
6. Dic3×#1044C2semidirect
7. Dic3⋊#1043C2semidirect
17. #10597121C4semidirect
19. C5×#1202C4semidirect
22. C2×D5⋊AGL(1,𝔽5)D3semidirect
28. C2×C5×D15C2×C4semidirect
30. C2×C3×C5×D5C2×C4semidirect
37. C3×#1044D4semidirect
39. C3⋊#1039C2×C4semidirect
40. C52×Dic3C2×C4semidirect
78. #1202AGL(1,𝔽5)semidirect
84. C2×C5×D5C4×D3semidirect
90. #1044#67semidirect
104. C3×C52C4×D4semidirect
106. C2×D15C2×AGL(1,𝔽5)semidirect
112. C2×C3×D5C2×AGL(1,𝔽5)semidirect
124. C5×Dic3C2×AGL(1,𝔽5)semidirect
164. C52C4×#67semidirect
185. C2×D5D3×AGL(1,𝔽5)semidirect
204. C3×C5D4×AGL(1,𝔽5)semidirect
217. Dic3D5⋊AGL(1,𝔽5)semidirect
245. C5AGL(1,𝔽5)×#67semidirect
259. C3#11717365semidirect

Maximal subgroups

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

subgroupindex
C2×#105971832
C2×C3×D5⋊AGL(1,𝔽5)2
#119254202
#119253552
#119253502
Dic3×#10442
Dic3⋊#10432
#117173653
AGL(1,𝔽5)×#675