Finite Group Databasebeta

← #10624391⋊(C5⋊D5) #49137614

Subgroups — #10624391⋊(C5⋊D5) #49137614

Order 36300.

256 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C5⋊(C5⋊((C11⋊D11)⋊C3))C211yesmaximal, characteristic, complemented
2C5⋊(C5⋊#10624391)C211yesmaximal, characteristic, complemented
3C52×#10624391C211yesmaximal, characteristic, complemented
4D11×C5⋊D55—31nomaximal, Hall
5C52×C112⋊C3C2211yescharacteristic, complemented, Hall
6#10624391⋊D5—51nomaximal
7#10624391⋊D5—52nomaximal
8#10624391⋊D5—51nomaximal
9#10624391⋊D5—51nomaximal
10#10624391⋊D5—51nomaximal
11C5⋊(C5⋊(C11⋊D11))D311yescharacteristic, complemented
12C11×C5⋊D55—31no—
13C52×C11×D11—31no—
14C5⋊((C11⋊D11)⋊C3)—52no—
15C5⋊((C11⋊D11)⋊C3)—51no—
16C5⋊((C11⋊D11)⋊C3)—51no—
17C5⋊((C11⋊D11)⋊C3)—51no—
18C5⋊((C11⋊D11)⋊C3)—51no—
19C5⋊#10624391—51no—
20C5⋊#10624391—51no—
21C5⋊#10624391—52no—
22C5⋊#10624391—51no—
23C5⋊#10624391—51no—
24C5×#10624391D511yescharacteristic, complemented
25C5×#10624391D511yescharacteristic, complemented
26C5×#10624391D512yescomplemented
27C5×#10624391D511yescharacteristic, complemented
28C5×#10624391D511yescharacteristic, complemented
29C52×C112C2×D311yescharacteristic, complemented, Hall
30D11×D55—152no—
31D11×D55—151no—
32D11×D55—151no—
33D11×D55—151no—
34D11×D55—151no—
35C5×C112⋊C3C2×D511yescharacteristic, complemented
36C5×C112⋊C3C2×D511yescharacteristic, complemented
37C5×C112⋊C3C2×D512yescomplemented
38C5×C112⋊C3C2×D511yescharacteristic, complemented
39C5×C112⋊C3C2×D511yescharacteristic, complemented
40#10624391⋊C2—251noHall
41C5⋊(C11⋊D11)—52no—
42C5⋊(C11⋊D11)—51no—
43C5⋊(C11⋊D11)—51no—
44C5⋊(C11⋊D11)—51no—
45C5⋊(C11⋊D11)—51no—
46C11×D55—151no—
47C11×D55—151no—
48C11×D55—151no—
49C11×D55—151no—
50C11×D55—152no—
51C5×C11×D11—31no—
52C5×C11×D11—31no—
53C5×C11×D11—31no—
54C5×C11×D11—31no—
55C5×C11×D11—32no—
56C2×C5⋊D55—331no—
57D11×C5⋊D5—331no—
58(C11⋊D11)⋊C3—251no—
59#10624391—251no—
60#10624391C5⋊D511yescharacteristic, complemented
61C5×C112D3×D511yescharacteristic, complemented
62C5×C112D3×D511yescharacteristic, complemented
63C5×C112D3×D511yescharacteristic, complemented
64C5×C112D3×D511yescharacteristic, complemented
65C5×C112D3×D512yescomplemented
66C2×C52×C11—331no—
67C5⋊D55—31no—
68C5⋊D55—661no—
69C5⋊D55—661no—
70C5⋊D55—661no—
71C11×C5⋊D5—331no—
72C52×D11—31no—
73D112—751noHall
74C112⋊C3C2×C5⋊D511yescharacteristic, complemented, Hall
75D3×C5⋊D5—1211nomaximal, Hall
76C52×C11—61no—
77C52×C11—61no—
78C52×C11—61no—
79C11⋊D11—251no—
80C11×D11—31no—
81C11×D11—751no—
82C2×D55—1652no—
83C2×D55—1651no—
84C2×D55—1651no—
85C2×D55—1651no—
86C2×D55—1651no—
87D5×D11—1652no—
88D5×D11—1651no—
89D5×D11—1651no—
90D5×D11—1651no—
91D5×D11—1651no—
92C3⋊(C5⋊D5)—2421no—
93C52×D3—2421no—
94C3×C5⋊D5—1211no—
95C112D3×C5⋊D511yescharacteristic, complemented, Sylow
96C2×C5×C11—331no—
97C2×C5×C11—331no—
98C2×C5×C11—331no—
99C2×C5×C11—331no—
100C2×C5×C11—332no—
101D55—151no—
102D55—151no—
103D55—151no—
104D55—151no—
105D55—152no—
106D55—3302no—
107D55—3301no—
108D55—3301no—
109D55—3301no—
110D55—3301no—
111D55—3302no—
112D55—3301no—
113D55—3301no—
114D55—3301no—
115D55—3302no—
116D55—3301no—
117D55—1651no—
118D55—1651no—
119D55—1651no—
120D55—1651no—
121C5×D11—31no—
122C5×D11—31no—
123C5×D11—31no—
124C5×D11—31no—
125C5×D11—32no—
126D5×C11—1651no—
127D5×C11—1651no—
128D5×C11—1651no—
129D5×C11—1651no—
130D5×C11—1652no—
131C2×C5⋊D5—3631noHall
132C3×C52—4841noHall
133D3×D5—6051no—
134D3×D5—6052no—
135D3×D5—6051no—
136D3×D5—6051no—
137D3×D5—6051no—
138C5×C11—31no—
139C5×C11—31no—
140C5×C11—31no—
141C5×C11—31no—
142C5×C11—61no—
143C5×C11—61no—
144C5×C11—61no—
145C5×C11—62no—
146C5×C11—61no—
147C5×C11—62no—
148C5×C11—61no—
149C5×C11—61no—
150C5×C11—61no—
151C5×C11—61no—
152C5×C11—62no—
153C2×C52—331no—
154C5⋊D5—331no—
155C5⋊D5—1211no—
156C2×D11—8251no—
157C2×D11—8251no—
158D15—6051no—
159D15—12101no—
160D15—12102no—
161D15—12101no—
162D15—12101no—
163C3×D5—6052no—
164C3×D5—6051no—
165C3×D5—6051no—
166C3×D5—6051no—
167C3×D5—6051no—
168C5×D3—1211no—
169C5×D3—2421no—
170C5×D3—2422no—
171C5×D3—2421no—
172C5×D3—2421no—
173C52#10624391⋊C211yescharacteristic, complemented, Sylow
174C2×C11—331no—
175C2×C11—8251no—
176D11—31no—
177D11—751no—
178D11—16501no—
179D11—16501no—
180D11—16501no—
181C2×D5—18152no—
182C2×D5—18151no—
183C2×D5—18151no—
184C2×D5—18151no—
185C2×D5—18151no—
186C3×C5—4841no—
187C3×C5—4841no—
188C3×C5—4842no—
189C3×C5—2421no—
190C3×C5—2421no—
191C2×D3—30251noHall
192C11—61no—
193C11—61no—
194C11—61no—
195C2×C5—331no—
196C2×C5—331no—
197C2×C5—331no—
198C2×C5—331no—
199C2×C5—332no—
200D5—6052no—
201D5—6051no—
202D5—6051no—
203D5—6051no—
204D5—6051no—
205D5—1651no—
206D5—1651no—
207D5—1651no—
208D5—1651no—
209D5—1652no—
210C2×C3—30251no—
211D3—60501no—
212D3—2421no—
213C5#10624391⋊D511yescharacteristic, complemented
214C5#10624391⋊D511yescharacteristic, complemented
215C5#10624391⋊D511yescharacteristic, complemented
216C5#10624391⋊D511yescharacteristic, complemented
217C5#10624391⋊D512yescomplemented
218C22—90751noSylow
219C3—4841noSylow
220C2—331no—
221C2—8251no—
222C2—30251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC52×C112⋊C3C22
Fitting, socleC52×C112C2×D3
LCSG ▷ C52×C112⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C5⋊((C11⋊D11)⋊C3))C2semidirect
2. C5⋊(C5⋊#10624391)C2semidirect
3. C52×#10624391C2semidirect
5. C52×C112⋊C3C22semidirect
11. C5⋊(C5⋊(C11⋊D11))D3semidirect
24. C5×#10624391D5semidirect
25. C5×#10624391D5semidirect
26. C5×#10624391D5semidirect
27. C5×#10624391D5semidirect
28. C5×#10624391D5semidirect
29. C52×C112C2×D3semidirect
35. C5×C112⋊C3C2×D5semidirect
36. C5×C112⋊C3C2×D5semidirect
37. C5×C112⋊C3C2×D5semidirect
38. C5×C112⋊C3C2×D5semidirect
39. C5×C112⋊C3C2×D5semidirect
60. #10624391C5⋊D5semidirect
61. C5×C112D3×D5semidirect
62. C5×C112D3×D5semidirect
63. C5×C112D3×D5semidirect
64. C5×C112D3×D5semidirect
65. C5×C112D3×D5semidirect
74. C112⋊C3C2×C5⋊D5semidirect
95. C112D3×C5⋊D5semidirect
173. C52#10624391⋊C2semidirect
213. C5#10624391⋊D5semidirect
214. C5#10624391⋊D5semidirect
215. C5#10624391⋊D5semidirect
216. C5#10624391⋊D5semidirect
217. C5#10624391⋊D5semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C5⋊((C11⋊D11)⋊C3))2
C5⋊(C5⋊#10624391)2
C52×#106243912
D11×C5⋊D553
#10624391⋊D55
D3×C5⋊D5121