Finite Group Databasebeta

← #72229795

Subgroups — #72229795

Order 2400.

444 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 81 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, complemented
2C2×D5⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
3C3×#87886C211yesmaximal, characteristic
4#11925533C211yesmaximal, characteristic, complemented
5#11925351C211yesmaximal, characteristic, complemented
6#11925350C211yesmaximal, characteristic, complemented
7#11925349C211yesmaximal, characteristic, complemented
8#11717377—31nomaximal, Hall
9C22×C3×C5⋊D5C2211yescharacteristic
10C2×C3×D52C2211yescharacteristic
11C2×C3⋊#1044—22no—
12C2×C3⋊#1044C2211yescharacteristic
13C2×C3⋊#1044C2211yescharacteristic
14C2×C3⋊#1043C2211yescharacteristic
15C2×C3⋊#1043C2211yescharacteristic
16D5⋊ΓL(1,𝔽16)—22no—
17C3×C5×C22⋊D5C412yescomplemented
18C3×#6038C412yescomplemented
19C3×#6036C2211yescharacteristic
20D4×ΓL(1,𝔽16)—52nomaximal
21C22×#1044—31no—
22C2×D5⋊AGL(1,𝔽5)—31no—
23#87886D311yescharacteristic
24C22⋊#1043—31no—
25#87832—31no—
26#87831—31no—
27#87830—31no—
28C22×C3×C52C2×C411yescharacteristic, complemented
29C2×C3×C5⋊D5—21no—
30C2×C3×C5⋊D5—21no—
31C2×C3×C5⋊D5C2×C411yescharacteristic, complemented
32C2×C3×C5⋊D5C2311yescharacteristic
33C2×C3×C5×D5C2×C412yescomplemented
34C3×D52—21no—
35C3×D52—21no—
36C3⋊#1044—21no—
37C3⋊#1044D412yescomplemented
38C3⋊#1043—21no—
39C3⋊#1043—21no—
40C3×C5×Dic5C2×C412yescomplemented
41C22×ΓL(1,𝔽16)—52no—
42C22×ΓL(1,𝔽16)—52no—
43C3×D4×D5—52no—
44#6763—52no—
45#6763—52no—
46#6763—102no—
47C4⋊ΓL(1,𝔽16)—52no—
48C4×ΓL(1,𝔽16)—52no—
49C22×C5⋊D5C2×D311yescharacteristic
50C2×D52C2×D311yescharacteristic
51C2×#1044—31no—
52C2×#1044—31no—
53C2×#1044—62no—
54C2×#1043—31no—
55C2×#1043—31no—
56D5⋊AGL(1,𝔽5)—62no—
57C5×C22⋊D5Dic312yescomplemented
58#6038Dic312yescomplemented
59#6036C2×D311yescharacteristic
60D4×AGL(1,𝔽5)—152no—
61C2×C3×C52—21no—
62C2×C3×C52C22×C411yescharacteristic
63C3×C5⋊D5—21no—
64C3×C5⋊D5C2×D411yescharacteristic
65C3×C5⋊D5#4111yescharacteristic
66C3×C5×D5—22no—
67C22×C3×D5—52no—
68C22×C3×D5—52no—
69C22×C3×D5—102no—
70C2×ΓL(1,𝔽16)—52no—
71C2×ΓL(1,𝔽16)—52no—
72C2×ΓL(1,𝔽16)—52no—
73C2×ΓL(1,𝔽16)—52no—
74C2×ΓL(1,𝔽16)—104no—
75C2×ΓL(1,𝔽16)—104no—
76C2×ΓL(1,𝔽16)—102no—
77C2×ΓL(1,𝔽16)—102no—
78C3×C5×D4—52no—
79C3×C22⋊D5—52no—
80C3×C22⋊D5AGL(1,𝔽5)12yescomplemented
81C3×D20—52no—
82C3×C4×D5—52no—
83C22×C52C2×Dic311yescharacteristic, complemented
84C2×C5⋊D5—21no—
85C2×C5⋊D5—21no—
86C2×C5⋊D5C2×Dic311yescharacteristic, complemented
87C2×C5⋊D5C22×D311yescharacteristic
88C2×C5×D5C2×Dic312yescomplemented
89D52—21no—
90D52—21no—
91#1044—32no—
92#1044—61no—
93#1043—61no—
94#1043—61no—
95C5×Dic5C2×Dic312yescomplemented
96D4×Dic3—251noHall
97C22×AGL(1,𝔽5)—152no—
98C22×AGL(1,𝔽5)—152no—
99D4×D5—52no—
100C22⋊AGL(1,𝔽5)—152no—
101C22⋊AGL(1,𝔽5)—152no—
102C22⋊AGL(1,𝔽5)—302no—
103C4⋊AGL(1,𝔽5)—152no—
104C4×AGL(1,𝔽5)—152no—
105C3×C52C4×D411yescharacteristic, complemented, Hall
106C22×C3×C5—22no—
107C22×C3×C5—52no—
108C22×C3×C5C2×AGL(1,𝔽5)12yescomplemented
109C2×C3×D5—52no—
110C2×C3×D5—52no—
111C2×C3×D5—102no—
112C2×C3×D5—102no—
113C2×C3×D5—102no—
114C2×C3×D5—102no—
115C2×C3×D5—102no—
116C2×C3×D5—102no—
117C2×C3×D5—202no—
118C2×C3×D5—202no—
119C2×C3×D5C2×AGL(1,𝔽5)12yescomplemented
120ΓL(1,𝔽16)—54no—
121ΓL(1,𝔽16)—102no—
122ΓL(1,𝔽16)—102no—
123ΓL(1,𝔽16)—102no—
124ΓL(1,𝔽16)—202no—
125ΓL(1,𝔽16)—202no—
126C3×C4×C5—52no—
127C3×Dic5C2×AGL(1,𝔽5)12yescomplemented
128C2×C52—21no—
129C2×C52C22×Dic311yescharacteristic
130C5⋊D5—21no—
131C5⋊D5D3×D411yescharacteristic
132C5⋊D5#23711yescharacteristic
133C5×D5—22no—
134C2×C3×D4—251no—
135C22×Dic3—251no—
136C22×Dic3—251no—
137#217—251no—
138#217—251no—
139C4⋊Dic3—251no—
140C4×Dic3—251no—
141C22×D5—52no—
142C22×D5—52no—
143C22×D5—102no—
144C2×AGL(1,𝔽5)—152no—
145C2×AGL(1,𝔽5)—152no—
146C2×AGL(1,𝔽5)—152no—
147C2×AGL(1,𝔽5)—152no—
148C2×AGL(1,𝔽5)—302no—
149C2×AGL(1,𝔽5)—302no—
150C2×AGL(1,𝔽5)—304no—
151C2×AGL(1,𝔽5)—304no—
152C5×D4—52no—
153C22⋊D5—52no—
154C22⋊D5ΓL(1,𝔽16)12yescomplemented
155D20—52no—
156C4×D5—52no—
157C4×D4—751noSylow
158C2×C3×C5—22no—
159C2×C3×C5—22no—
160C2×C3×C5—42no—
161C2×C3×C5—102no—
162C2×C3×C5C22×AGL(1,𝔽5)12yes—
163C3×D5—22no—
164C3×D5—52no—
165C3×D5—52no—
166C3×D5—102no—
167C3×D5—102no—
168C3×D5—102no—
169C3×D5—202no—
170C52D4×Dic311yescharacteristic, complemented, Sylow
171C23×C3—251no—
172C23×C3—251no—
173C3×D4—52no—
174C3×D4—252no—
175C2×C3×C4—251no—
176C2×Dic3—251no—
177C2×Dic3—251no—
178C2×Dic3—251no—
179C2×Dic3—251no—
180C2×Dic3—502no—
181C2×Dic3—502no—
182C22×C5—22no—
183C22×C5—52no—
184C22×C5C2×ΓL(1,𝔽16)12yescomplemented
185C2×D5—52no—
186C2×D5—52no—
187C2×D5—102no—
188C2×D5—102no—
189C2×D5—102no—
190C2×D5—102no—
191C2×D5—102no—
192C2×D5—102no—
193C2×D5—202no—
194C2×D5—202no—
195C2×D5C2×ΓL(1,𝔽16)12yescomplemented
196AGL(1,𝔽5)—154no—
197AGL(1,𝔽5)—302no—
198AGL(1,𝔽5)—302no—
199AGL(1,𝔽5)—302no—
200AGL(1,𝔽5)—602no—
201AGL(1,𝔽5)—602no—
202C4×C5—52no—
203Dic5C2×ΓL(1,𝔽16)12yescomplemented
204C2×D4—251no—
205C22×C4—751no—
206C22×C4—751no—
207C4⋊C4—751no—
208C22⋊C4—751no—
209C22⋊C4—751no—
210C42—751no—
211C3×C5—22no—
212C3×C5D4×AGL(1,𝔽5)12yescomplemented
213C22×C3—52no—
214C22×C3—251no—
215C22×C3—251no—
216C22×C3—501no—
217C22×C3—501no—
218C22×C3—501no—
219C22×C3—501no—
220C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
221C3×C4—52no—
222Dic3—252no—
223Dic3—501no—
224Dic3—501no—
225Dic3—501no—
226C2×C5—22no—
227C2×C5—22no—
228C2×C5—42no—
229C2×C5—102no—
230C2×C5C22×ΓL(1,𝔽16)12yes—
231D5—22no—
232D5—52no—
233D5—52no—
234D5—102no—
235D5—102no—
236D5—102no—
237D5—202no—
238C23—251no—
239C23—251no—
240D4—52no—
241D4—252no—
242C2×C4—251no—
243C2×C4—751no—
244C2×C4—751no—
245C2×C4—751no—
246C2×C4—751no—
247C2×C4—1502no—
248C2×C4—1502no—
249C2×C3—21no—
250C2×C3—102no—
251C2×C3—251no—
252C2×C3—251no—
253C2×C3—501no—
254C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
255C5—22no—
256C5D4×ΓL(1,𝔽16)12yescomplemented
257C22—52no—
258C22—251no—
259C22—251no—
260C22—501no—
261C22—501no—
262C22—501no—
263C22—501no—
264C22D5⋊ΓL(1,𝔽16)11yescharacteristic, complemented
265C4—52no—
266C4—752no—
267C4—1501no—
268C4—1501no—
269C4—1501no—
270C3#1171737711yescharacteristic, complemented, Sylow
271C2—21no—
272C2—102no—
273C2—251no—
274C2—251no—
275C2—501no—
276C2C2×D5⋊ΓL(1,𝔽16)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×D5⋊ΓL(1,𝔽16)
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radicalC3×#87886C2
cosocleC2×C3×C5⋊D5C23
hypercenterC22D5⋊ΓL(1,𝔽16)
hypocenterC3×C52C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊#1044C2semidirect
2. C2×D5⋊ΓL(1,𝔽16)C2semidirect
4. #11925533C2semidirect
5. #11925351C2semidirect
6. #11925350C2semidirect
7. #11925349C2semidirect
17. C3×C5×C22⋊D5C4semidirect
18. C3×#6038C4semidirect
28. C22×C3×C52C2×C4semidirect
31. C2×C3×C5⋊D5C2×C4semidirect
33. C2×C3×C5×D5C2×C4semidirect
37. C3⋊#1044D4semidirect
40. C3×C5×Dic5C2×C4semidirect
57. C5×C22⋊D5Dic3semidirect
58. #6038Dic3semidirect
80. C3×C22⋊D5AGL(1,𝔽5)semidirect
83. C22×C52C2×Dic3semidirect
86. C2×C5⋊D5C2×Dic3semidirect
88. C2×C5×D5C2×Dic3semidirect
95. C5×Dic5C2×Dic3semidirect
105. C3×C52C4×D4semidirect
108. C22×C3×C5C2×AGL(1,𝔽5)semidirect
119. C2×C3×D5C2×AGL(1,𝔽5)semidirect
127. C3×Dic5C2×AGL(1,𝔽5)semidirect
154. C22⋊D5ΓL(1,𝔽16)semidirect
170. C52D4×Dic3semidirect
184. C22×C5C2×ΓL(1,𝔽16)semidirect
195. C2×D5C2×ΓL(1,𝔽16)semidirect
203. Dic5C2×ΓL(1,𝔽16)semidirect
212. C3×C5D4×AGL(1,𝔽5)semidirect
220. C22×C3D5⋊AGL(1,𝔽5)semidirect
256. C5D4×ΓL(1,𝔽16)semidirect
264. C22D5⋊ΓL(1,𝔽16)semidirect
270. C3#11717377semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#10442
C2×D5⋊ΓL(1,𝔽16)2
C3×#878862
#119255332
#119253512
#119253502
#119253492
#117173773
D4×ΓL(1,𝔽16)5