Finite Group Databasebeta

← #72230748

Subgroups — #72230748

Order 2400.

444 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 67 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×#10597182C211yesmaximal, characteristic, complemented
3#11925664C211yesmaximal, characteristic
4C3×C22⋊#1043C211yesmaximal, characteristic, complemented
5#11925356C211yesmaximal, characteristic, complemented
6#11925355C211yesmaximal, characteristic, complemented
7#11925354C211yesmaximal, characteristic, complemented
8#11717377—31nomaximal, Hall
9C22×C3×C5⋊D5C2211yescharacteristic
10C2×D15⋊D5C2211yescharacteristic
11C2×C3⋊#1044—22no—
12C2×C3⋊#1044C2211yescharacteristic
13C2×C3⋊#1044C2211yescharacteristic
14C2×C3×#1043C2211yescharacteristic
15C2×C3×#1043C2211yescharacteristic
16#10597182—22no—
17C5×#1220C412yescomplemented
18#10597127C412yescomplemented
19#10597126C2211yescharacteristic
20#67⋊AGL(1,𝔽5)—52nomaximal
21C22×#1044—31no—
22C2×D5⋊AGL(1,𝔽5)—31no—
23#87886—31no—
24C22⋊#1043D311yescharacteristic, complemented
25#87832—31no—
26#87831—31no—
27#87830—31no—
28C22×C3×C52C2×C411yescharacteristic, complemented
29C2×C5×D15C2×C412yescomplemented
30C2×C3×C5⋊D5—21no—
31C2×C3×C5⋊D5—21no—
32C2×C3×C5⋊D5C2×C411yescharacteristic, complemented
33C2×C3×C5⋊D5C2311yescharacteristic
34D15⋊D5—21no—
35D15⋊D5—21no—
36C3⋊#1044—21no—
37C3⋊#1044D412yescomplemented
38C3×#1043—21no—
39C3×#1043—21no—
40C5×Dic15C2×C412yescomplemented
41C22×ΓL(1,𝔽16)—52no—
42C2×D3×AGL(1,𝔽5)—52no—
43D5×#67—52no—
44C3×C22⋊AGL(1,𝔽5)—52no—
45C3×C22⋊AGL(1,𝔽5)—102no—
46Dic3⋊AGL(1,𝔽5)—52no—
47(C2×D3)⋊AGL(1,𝔽5)—52no—
48Dic3×AGL(1,𝔽5)—52no—
49C22×C5⋊D5C2×D311yescharacteristic
50C2×D52—31no—
51C2×#1044—31no—
52C2×#1044—31no—
53C2×#1044—62no—
54C2×#1043C2×D311yescharacteristic
55C2×#1043C2×D311yescharacteristic
56D5⋊AGL(1,𝔽5)—62no—
57C5×C22⋊D5—32no—
58#6038—32no—
59#6036—31no—
60D4×AGL(1,𝔽5)—152no—
61C2×C3×C52—21no—
62C2×C3×C52C22×C411yescharacteristic
63C5×D15—22no—
64C3×C5⋊D5—21no—
65C3×C5⋊D5C2×D411yescharacteristic
66C3×C5⋊D5#4111yescharacteristic
67C22×C3×D5—52no—
68C22×C3×D5—102no—
69C2×D3×D5—52no—
70C2×ΓL(1,𝔽16)—52no—
71C2×ΓL(1,𝔽16)—52no—
72C2×ΓL(1,𝔽16)—104no—
73C2×C3×AGL(1,𝔽5)—52no—
74C2×C3×AGL(1,𝔽5)—52no—
75C2×C3×AGL(1,𝔽5)—102no—
76C2×C3×AGL(1,𝔽5)—102no—
77D3×AGL(1,𝔽5)—104no—
78#1220AGL(1,𝔽5)12yescomplemented
79C5×#67—52no—
80#1202—52no—
81#1201—52no—
82D5×Dic3—52no—
83C22×C52C4×D311yescharacteristic, complemented
84C2×C5⋊D5—21no—
85C2×C5⋊D5—21no—
86C2×C5⋊D5C4×D311yescharacteristic, complemented
87C2×C5⋊D5C22×D311yescharacteristic
88C2×C5×D5—32no—
89D52—61no—
90D52—61no—
91#1044—32no—
92#1044—61no—
93#1043—21no—
94#1043—21no—
95C5×Dic5—32no—
96#67⋊C4—251noHall
97C22×AGL(1,𝔽5)—152no—
98C22×AGL(1,𝔽5)—152no—
99D4×D5—152no—
100C22⋊AGL(1,𝔽5)—52no—
101C22⋊AGL(1,𝔽5)—102no—
102C22⋊AGL(1,𝔽5)—152no—
103C4⋊AGL(1,𝔽5)—152no—
104C4×AGL(1,𝔽5)—152no—
105C3×C52C4×D411yescharacteristic, complemented, Hall
106C22×C3×C5—22no—
107C22×C3×C5C2×AGL(1,𝔽5)12yescomplemented
108C2×D15C2×AGL(1,𝔽5)12yescomplemented
109C2×C5×D3—52no—
110C2×C3×D5—52no—
111C2×C3×D5—52no—
112C2×C3×D5—102no—
113C2×C3×D5—102no—
114C2×C3×D5—102no—
115C2×C3×D5—102no—
116C2×C3×D5—202no—
117C2×C3×D5—202no—
118D3×D5—102no—
119D3×D5—102no—
120ΓL(1,𝔽16)—54no—
121ΓL(1,𝔽16)—102no—
122C3×AGL(1,𝔽5)—102no—
123C3×AGL(1,𝔽5)—102no—
124C3×AGL(1,𝔽5)—202no—
125C3×AGL(1,𝔽5)—202no—
126Dic15C2×AGL(1,𝔽5)12yescomplemented
127C5×Dic3—52no—
128C2×C52—21no—
129C2×C52C2×C4×D311yescharacteristic
130C5⋊D5—21no—
131C5⋊D5D3×D411yescharacteristic
132C5⋊D5#23711yescharacteristic
133C5×D5—62no—
134C2×#67—251no—
135C22×Dic3—251no—
136C2×C4×D3—251no—
137C3×C22⋊C4—251no—
138(C2×D3)⋊C4—251no—
139Dic3⋊C4—251no—
140C4×Dic3—251no—
141C22×D5—52no—
142C22×D5—102no—
143C22×D5—152no—
144C2×AGL(1,𝔽5)—52no—
145C2×AGL(1,𝔽5)—52no—
146C2×AGL(1,𝔽5)—102no—
147C2×AGL(1,𝔽5)—102no—
148C2×AGL(1,𝔽5)—152no—
149C2×AGL(1,𝔽5)—152no—
150C2×AGL(1,𝔽5)—304no—
151C2×AGL(1,𝔽5)—304no—
152C5×D4—152no—
153C22⋊D5—32no—
154C22⋊D5—152no—
155D20—152no—
156C4×D5—152no—
157C4×D4—751noSylow
158C2×C3×C5—22no—
159C2×C3×C5—22no—
160C2×C3×C5—42no—
161C2×C3×C5C22×AGL(1,𝔽5)12yes—
162D15—22no—
163C3×D5—52no—
164C3×D5—52no—
165C3×D5—102no—
166C3×D5—102no—
167C3×D5—102no—
168C3×D5—202no—
169C5×D3—102no—
170C52#67⋊C411yescharacteristic, complemented, Sylow
171C23×C3—251no—
172C22×D3—251no—
173C2×C3×C4—251no—
174C2×C3×C4—251no—
175#67—52no—
176#67—252no—
177C2×Dic3—251no—
178C2×Dic3—251no—
179C2×Dic3—251no—
180C2×Dic3—502no—
181C4×D3—502no—
182C22×C5—22no—
183C22×C5—152no—
184C22×C5D3×AGL(1,𝔽5)12yescomplemented
185C2×D5—32no—
186C2×D5—52no—
187C2×D5—52no—
188C2×D5—102no—
189C2×D5—102no—
190C2×D5—102no—
191C2×D5—102no—
192C2×D5—202no—
193C2×D5—202no—
194C2×D5—302no—
195C2×D5—302no—
196AGL(1,𝔽5)—102no—
197AGL(1,𝔽5)—102no—
198AGL(1,𝔽5)—154no—
199AGL(1,𝔽5)—202no—
200AGL(1,𝔽5)—202no—
201AGL(1,𝔽5)—302no—
202C4×C5—152no—
203Dic5—32no—
204C2×D4—751no—
205C22×C4—751no—
206C22×C4—751no—
207C4⋊C4—751no—
208C22⋊C4—251no—
209C22⋊C4—751no—
210C42—751no—
211C3×C5—22no—
212C3×C5D4×AGL(1,𝔽5)12yescomplemented
213C22×C3—251no—
214C22×C3—251no—
215C22×C3—501no—
216C22×C3—501no—
217C22×C3D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
218C2×D3—52no—
219C2×D3—501no—
220C2×D3—501no—
221C3×C4—501no—
222C3×C4—501no—
223Dic3—52no—
224Dic3—252no—
225Dic3—501no—
226C2×C5—22no—
227C2×C5—22no—
228C2×C5—42no—
229C2×C5—302no—
230C2×C5C2×D3×AGL(1,𝔽5)12yes—
231D5—52no—
232D5—52no—
233D5—62no—
234D5—102no—
235D5—102no—
236D5—102no—
237D5—202no—
238C23—251no—
239C23—751no—
240D4—152no—
241D4—752no—
242C2×C4—251no—
243C2×C4—251no—
244C2×C4—751no—
245C2×C4—751no—
246C2×C4—751no—
247C2×C4—1502no—
248C2×C4—1502no—
249C2×C3—21no—
250C2×C3—251no—
251C2×C3—251no—
252C2×C3—501no—
253C2×C3C2×D5⋊AGL(1,𝔽5)11yescharacteristic
254D3—102no—
255C5—22no—
256C5#67⋊AGL(1,𝔽5)12yescomplemented
257C22—152no—
258C22—251no—
259C22—251no—
260C22—501no—
261C22—501no—
262C22—1501no—
263C22—1501no—
264C22#1059718211yescharacteristic, complemented
265C4—152no—
266C4—501no—
267C4—501no—
268C4—752no—
269C4—1501no—
270C3#1171737711yescharacteristic, complemented, Sylow
271C2—21no—
272C2—251no—
273C2—251no—
274C2—302no—
275C2—501no—
276C2C2×#1059718211yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10597182
commutator, socleC2×C3×C52C22×C4
FittingC22×C3×C52C2×C4
prime radical#11925664C2
cosocleC2×C3×C5⋊D5C23
hypercenterC22#10597182
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×#10597182C2semidirect
4. C3×C22⋊#1043C2semidirect
5. #11925356C2semidirect
6. #11925355C2semidirect
7. #11925354C2semidirect
17. C5×#1220C4semidirect
18. #10597127C4semidirect
24. C22⋊#1043D3semidirect
28. C22×C3×C52C2×C4semidirect
29. C2×C5×D15C2×C4semidirect
32. C2×C3×C5⋊D5C2×C4semidirect
37. C3⋊#1044D4semidirect
40. C5×Dic15C2×C4semidirect
78. #1220AGL(1,𝔽5)semidirect
83. C22×C52C4×D3semidirect
86. C2×C5⋊D5C4×D3semidirect
105. C3×C52C4×D4semidirect
107. C22×C3×C5C2×AGL(1,𝔽5)semidirect
108. C2×D15C2×AGL(1,𝔽5)semidirect
126. Dic15C2×AGL(1,𝔽5)semidirect
170. C52#67⋊C4semidirect
184. C22×C5D3×AGL(1,𝔽5)semidirect
212. C3×C5D4×AGL(1,𝔽5)semidirect
217. C22×C3D5⋊AGL(1,𝔽5)semidirect
256. C5#67⋊AGL(1,𝔽5)semidirect
264. C22#10597182semidirect
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×#105971822
#119256642
C3×C22⋊#10432
#119253562
#119253552
#119253542
#117173773
#67⋊AGL(1,𝔽5)5