Finite Group Databasebeta

← #72231463

Subgroups — #72231463

Order 2400.

1040 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 156 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×D3×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×C4⋊(C3⋊(C5⋊D5))C211yesmaximal, characteristic, complemented
3C2×#10597116C211yesmaximal, characteristic, complemented
4C2×#10597115C211yesmaximal, characteristic, complemented
5C52×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C3×#87808C211yesmaximal, characteristic, complemented
7#11925120C211yesmaximal, characteristic, complemented
8#11717202—31nomaximal, Hall
9C22×C3⋊(C5⋊D5)C2211yescharacteristic
10C22×C3⋊(C5⋊D5)C2211yescharacteristic, complemented
11C22×C52×D3C2211yescharacteristic
12C22×C3×C5⋊D5C2211yescharacteristic
13C2×D3×C5⋊D5—22no—
14C2×D3×C5⋊D5—22no—
15C2×D3×C5⋊D5—22no—
16C2×C3×C4×C52C2211yescharacteristic, complemented
17C4⋊(C3⋊(C5⋊D5))—22no—
18C2×C52×Dic3C2211yescharacteristic, complemented
19C2×C3×#1039C2211yescharacteristic, complemented
20#10597116—22no—
21#10597115—22no—
22#91476—56nomaximal
23C23×C5⋊D5—31no—
24C2×C22⋊(C5⋊D5)—31no—
25C2×C4⋊(C5⋊D5)—31no—
26C2×C4⋊(C5⋊D5)—31no—
27C52×C22⋊C4—31no—
28#87808—31no—
29#87808D311yescharacteristic, complemented
30C22×C3×C52C2311yescharacteristic
31C2×C3⋊(C5⋊D5)—21no—
32C2×C3⋊(C5⋊D5)—21no—
33C2×C3⋊(C5⋊D5)—21no—
34C2×C3⋊(C5⋊D5)—21no—
35C2×C3⋊(C5⋊D5)—21no—
36C2×C3⋊(C5⋊D5)D412yescomplemented
37C2×C52×D3—21no—
38C2×C52×D3—21no—
39C2×C52×D3D412yescomplemented
40C2×C3×C5⋊D5—21no—
41C2×C3×C5⋊D5—21no—
42C2×C3×C5⋊D5D412yescomplemented
43D3×C5⋊D5—28no—
44C3×C4×C52—21no—
45C52×Dic3—21no—
46C3×#1039—21no—
47C22×D3×D5—56no—
48C2×D60—56no—
49C2×#1203—56no—
50C2×#1202—56no—
51C5×(C2×D3)⋊C4D516yescomplemented
52C3×#685—56no—
53#6667—56no—
54C23×C52—31no—
55C22×C5⋊D5—31no—
56C22×C5⋊D5—31no—
57C22×C5⋊D5—62no—
58C22×C5⋊D5—62no—
59C22×C5⋊D5—62no—
60C22×C5⋊D5C2×D311yescharacteristic
61C2×C4×C52—31no—
62C2×C4×C52C2×D311yescharacteristic, complemented
63C22⋊(C5⋊D5)—62no—
64C2×#1039C2×D311yescharacteristic, complemented
65C4⋊(C5⋊D5)—62no—
66C4⋊(C5⋊D5)—62no—
67#4028—156no—
68C2×C3×C52C2×D411yescharacteristic
69C2×C3×C52C2×D411yescharacteristic
70C2×C3×C52C2×D411yescharacteristic
71C3⋊(C5⋊D5)—22no—
72C3⋊(C5⋊D5)—41no—
73C52×D3—22no—
74C3×C5⋊D5—22no—
75C22×D15—56no—
76C22×D15—56no—
77C22×C5×D3C2×D516yes—
78C22×C3×D5—56no—
79C2×D3×D5—1012no—
80C2×D3×D5—1012no—
81C2×D3×D5—1012no—
82C2×C3×C4×C5C2×D516yescomplemented
83D60—1012no—
84C2×C5×Dic3C2×D516yescomplemented
85C2×C3×Dic5—56no—
86#1203—1012no—
87#1202—1012no—
88C22×C52—32no—
89C22×C52—61no—
90C22×C52—61no—
91C22×C52C22×D311yescharacteristic
92C2×C5⋊D5—21no—
93C2×C5⋊D5—21no—
94C2×C5⋊D5—32no—
95C2×C5⋊D5—68no—
96C2×C5⋊D5—61no—
97C2×C5⋊D5—61no—
98C2×C5⋊D5—61no—
99C2×C5⋊D5—61no—
100C2×C5⋊D5—61no—
101C2×C5⋊D5D1212yescomplemented
102C4×C52—21no—
103C4×C52—61no—
104#1039—21no—
105#882—251noHall
106C23×D5—156no—
107C2×C22⋊D5—156no—
108C2×D20—156no—
109C2×D20—156no—
110C5×C22⋊C4—36no—
111#685—56no—
112#685—156no—
113C3×C52#12011yescharacteristic, complemented, Hall
114C22×C3×C5C22×D516yes—
115C2×D15—512no—
116C2×D15—106no—
117C2×D15—106no—
118C2×D15—106no—
119C2×D15—106no—
120C2×D15—106no—
121C2×C5×D3—26no—
122C2×C5×D3—26no—
123C2×C5×D3D20112yescomplemented
124C2×C3×D5—512no—
125C2×C3×D5—106no—
126C2×C3×D5—106no—
127D3×D5—1048no—
128C3×C4×C5—26no—
129C3×Dic5—106no—
130C5×Dic3—26no—
131C2×C52—62no—
132C2×C52C2×D1211yescharacteristic
133C2×C52D3×D411yescharacteristic
134C2×C52D3×D411yescharacteristic
135C5⋊D5—22no—
136C5⋊D5—62no—
137C5⋊D5—121no—
138C23×D3—251no—
139C2×#67—251no—
140C2×D12—251no—
141C2×D12—251no—
142C3×C22⋊C4—251no—
143(C2×D3)⋊C4—251no—
144(C2×D3)⋊C4C5⋊D511yescharacteristic, complemented
145C23×C5—36no—
146C22×D5—56no—
147C22×D5—156no—
148C22×D5—156no—
149C22×D5—3012no—
150C22×D5—3012no—
151C22×D5—3012no—
152C2×C4×C5—36no—
153C2×C4×C5D3×D516yescomplemented
154C22⋊D5—3012no—
155C2×Dic5—56no—
156D20—3012no—
157D20—3012no—
158#120—751noSylow
159C2×C3×C5C2×D2016yes—
160C2×C3×C5D4×D516yes—
161C2×C3×C5D4×D516yes—
162D15—1012no—
163D15—206no—
164C3×D5—1012no—
165C5×D3—212no—
166C52#88211yescharacteristic, complemented, Sylow
167C23×C3—251no—
168C22×D3—251no—
169C22×D3—251no—
170C22×D3—502no—
171C22×D3—502no—
172C22×D3—502no—
173C22×D3C2×C5⋊D511yescharacteristic
174C2×C3×C4—251no—
175C2×C3×C4C2×C5⋊D511yescharacteristic, complemented
176#67—502no—
177C2×Dic3C2×C5⋊D511yescharacteristic, complemented
178D12—502no—
179D12—502no—
180C22×C5—312no—
181C22×C5—66no—
182C22×C5—66no—
183C22×C5C2×D3×D516yes—
184C2×D5—512no—
185C2×D5—106no—
186C2×D5—106no—
187C2×D5—1512no—
188C2×D5—3048no—
189C2×D5—306no—
190C2×D5—306no—
191C2×D5—306no—
192C2×D5—306no—
193C2×D5—306no—
194C4×C5—26no—
195C4×C5—66no—
196Dic5—106no—
197C24—751no—
198C2×D4—751no—
199C2×D4—751no—
200C2×D4—751no—
201C22⋊C4—31no—
202C22⋊C4—251no—
203C22⋊C4—751no—
204C3×C5#402816yescomplemented
205C22×C3—252no—
206C22×C3—501no—
207C22×C3—501no—
208C22×C3C22×C5⋊D511yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—252no—
212C2×D3—508no—
213C2×D3—501no—
214C2×D3—501no—
215C2×D3—501no—
216C2×D3—501no—
217C2×D3—501no—
218C2×D3C4⋊(C5⋊D5)12yescomplemented
219C3×C4—21no—
220C3×C4—501no—
221Dic3—21no—
222C2×C5—612no—
223C2×C5D5×D1216yes—
224C2×C5D3×D2016yes—
225C2×C5#679016yes—
226D5—1012no—
227D5—3012no—
228D5—606no—
229C23—31no—
230C23—251no—
231C23—751no—
232C23—751no—
233C23—1502no—
234C23—1502no—
235C23—1502no—
236D4—1502no—
237D4—1502no—
238D4—1502no—
239C2×C4—31no—
240C2×C4—251no—
241C2×C4D3×C5⋊D511yescharacteristic, complemented
242C2×C3—502no—
243C2×C3C2×C4⋊(C5⋊D5)11yescharacteristic
244C2×C3D4×C5⋊D511yescharacteristic
245C2×C3D4×C5⋊D511yescharacteristic
246D3—22no—
247D3—502no—
248D3—1001no—
249C5#9147616yescomplemented
250C22—32no—
251C22—61no—
252C22—61no—
253C22—252no—
254C22—501no—
255C22—501no—
256C22—752no—
257C22—1508no—
258C22—1501no—
259C22—1501no—
260C22—1501no—
261C22—1501no—
262C22—1501no—
263C22C2×D3×C5⋊D511yescharacteristic
264C4—21no—
265C4—61no—
266C4—501no—
267C3#1171720211yescharacteristic, complemented, Sylow
268C2—62no—
269C2—3001no—
270C2—502no—
271C2—1502no—
272C2D12×C5⋊D511yescharacteristic
273C2D3×C4⋊(C5⋊D5)11yescharacteristic
274C2#1192561911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C5⋊D5
commutator, socle, cosocleC22×C3×C52C23
FittingC2×C3×C4×C52C22
hypercenterC2×C4D3×C5⋊D5
hypocenterC3×C52#120
LCSG ▷ C22×C3×C52 ▷ C3×C52
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×D3×C5⋊D5C2semidirect
2. C2×C4⋊(C3⋊(C5⋊D5))C2semidirect
3. C2×#10597116C2semidirect
4. C2×#10597115C2semidirect
5. C52×(C2×D3)⋊C4C2semidirect
6. C3×#87808C2semidirect
7. #11925120C2semidirect
10. C22×C3⋊(C5⋊D5)C22semidirect
16. C2×C3×C4×C52C22semidirect
18. C2×C52×Dic3C22semidirect
19. C2×C3×#1039C22semidirect
29. #87808D3semidirect
36. C2×C3⋊(C5⋊D5)D4semidirect
39. C2×C52×D3D4semidirect
42. C2×C3×C5⋊D5D4semidirect
51. C5×(C2×D3)⋊C4D5semidirect
62. C2×C4×C52C2×D3semidirect
64. C2×#1039C2×D3semidirect
82. C2×C3×C4×C5C2×D5semidirect
84. C2×C5×Dic3C2×D5semidirect
101. C2×C5⋊D5D12semidirect
113. C3×C52#120semidirect
123. C2×C5×D3D20semidirect
144. (C2×D3)⋊C4C5⋊D5semidirect
153. C2×C4×C5D3×D5semidirect
166. C52#882semidirect
175. C2×C3×C4C2×C5⋊D5semidirect
177. C2×Dic3C2×C5⋊D5semidirect
204. C3×C5#4028semidirect
218. C2×D3C4⋊(C5⋊D5)semidirect
241. C2×C4D3×C5⋊D5semidirect
249. C5#91476semidirect
267. C3#11717202semidirect

Maximal subgroups

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

subgroupindex
C22×D3×C5⋊D52
C2×C4⋊(C3⋊(C5⋊D5))2
C2×#105971162
C2×#105971152
C52×(C2×D3)⋊C42
C3×#878082
#119251202
#117172023
#914765