Finite Group Databasebeta

← #72225762

Subgroups — #72225762

Order 29400.

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

normal subgroups only

This group has 832 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
1#72225761C311yesmaximal, characteristic, complemented, Hall
2D5×#64423⋊C2—52nomaximal
3D52×AGL(1,𝔽7)—72nomaximal
4#28413321C212yesmaximal, complemented
5#28413316C212yesmaximal, complemented
6C5⋊D5×#64424C211yesmaximal, characteristic, complemented
7C5×#64424⋊D5C212yesmaximal, complemented
8C5⋊D5×#3861C2211yescharacteristic, complemented
9C5⋊(C5⋊#64424)C2211yescharacteristic, complemented
10C5×C5⋊#64423C2214yescomplemented
11C52×#64424C2211yescharacteristic, complemented
12D352C2×C312yescomplemented
13C7×(C5⋊D5)⋊D7C2×C312yescomplemented
14C5⋊D5×C7⋊D7C2×C311yescharacteristic, complemented
15C5×D35⋊D7C2×C312yescomplemented
16C52×#3861C2311yescharacteristic, complemented, Hall
17D5×#64423—54no—
18#64424⋊D5D512yescomplemented
19#64423⋊D5—54no—
20D5×#64424—52no—
21D5×#64424—102no—
22C5×#64423⋊C2—52no—
23C5⋊(C5⋊(C7⋊D7))C22×C311yescharacteristic, complemented
24C72×C5⋊D5C22×C311yescharacteristic, complemented
25C5×C7×D35C22×C314yescomplemented
26C52×C7⋊D7C22×C311yescharacteristic, complemented
27D52×ΓL(1,𝔽8)—72no—
28(C5⋊AGL(1,𝔽7))⋊D5—72no—
29D5×C5⋊AGL(1,𝔽7)—74no—
30AGL(1,𝔽7)×C5⋊D5—72no—
31AGL(1,𝔽7)×C5⋊D5—143no—
32C5×D5×AGL(1,𝔽7)—74no—
33D5×D72—52no—
34D5×#3861—52no—
35D5×#3861—102no—
36C5⋊#64423C2×D514yescomplemented
37C5⋊#64424—52no—
38C5⋊#64424—102no—
39C5×#64423—54no—
40C5×#64424—22no—
41C5×#64424C2×D512yescomplemented
42D52×D7—72no—
43C52×C72C23×C311yescharacteristic, complemented, Hall
44C2×#64423⋊C2—251noHall
45C5⋊(C5⋊AGL(1,𝔽7))—72no—
46C5⋊(C5⋊AGL(1,𝔽7))—143no—
47ΓL(1,𝔽8)×C5⋊D5—72no—
48ΓL(1,𝔽8)×C5⋊D5—143no—
49C5×C5⋊AGL(1,𝔽7)—74no—
50C5×D5×ΓL(1,𝔽8)—74no—
51C52×AGL(1,𝔽7)—72no—
52C52×AGL(1,𝔽7)—143no—
53D35⋊D7C3×D512yescomplemented
54D7×D35—54no—
55D5×C7⋊D7—52no—
56D5×C7⋊D7—102no—
57C7×D5×D7—54no—
58C5×D72—52no—
59C2×D5×AGL(1,𝔽7)—354no—
60C5×#3861—22no—
61C5×#3861C22×D512yescomplemented
62(C5⋊D5)⋊D7AGL(1,𝔽7)12yescomplemented
63D5×D35—74no—
64C7×D52—72no—
65D7×C5⋊D5—72no—
66D7×C5⋊D5—143no—
67C5×D5×D7—74no—
68C2×C3×D52—491noHall
69C2×#64424—251no—
70C2×#64423—252no—
71#64423⋊C2—52no—
72#64423⋊C2—252no—
73C52×ΓL(1,𝔽8)—72no—
74C52×ΓL(1,𝔽8)—143no—
75C5⋊(C7⋊D7)—52no—
76C5⋊(C7⋊D7)—102no—
77C7×D35C2×C3×D514yescomplemented
78C72×D5—52no—
79C72×D5—102no—
80C5×C7⋊D7—22no—
81C5×C7⋊D7C2×C3×D512yescomplemented
82C5×C7×D7—54no—
83C2×C5⋊AGL(1,𝔽7)—354no—
84C2×D5×ΓL(1,𝔽8)—354no—
85C2×C5×AGL(1,𝔽7)—354no—
86D5×AGL(1,𝔽7)—74no—
87D5×AGL(1,𝔽7)—354no—
88D5×AGL(1,𝔽7)—354no—
89D5×AGL(1,𝔽7)—354no—
90D5×AGL(1,𝔽7)—7012no—
91D5×AGL(1,𝔽7)—704no—
92D5×AGL(1,𝔽7)—706no—
93C2×D72—251noHall
94C5⋊D35—72no—
95C5⋊D35—143no—
96C7×C5⋊D5—23no—
97C7×C5⋊D5C2×AGL(1,𝔽7)12yescomplemented
98C5×D35C2×AGL(1,𝔽7)14yescomplemented
99C5×C7×D5—74no—
100C52×D7—72no—
101C52×D7—143no—
102C2×C3×C5⋊D5—491no—
103C2×C3×C5×D5—492no—
104C3×D52—492no—
105C3×D52—492no—
106C2×#3861—251no—
107#64424—251no—
108#64424D5211yescharacteristic, complemented
109#64423—54no—
110C2×D5×D7—354no—
111C5×C72—22no—
112C5×C72C22×C3×D512yescomplemented
113C2×C5×ΓL(1,𝔽8)—354no—
114C5⋊AGL(1,𝔽7)—74no—
115C5⋊AGL(1,𝔽7)—354no—
116C5⋊AGL(1,𝔽7)—7012no—
117C5⋊AGL(1,𝔽7)—704no—
118C5⋊AGL(1,𝔽7)—706no—
119D5×ΓL(1,𝔽8)—74no—
120D5×ΓL(1,𝔽8)—354no—
121D5×ΓL(1,𝔽8)—7012no—
122D5×ΓL(1,𝔽8)—704no—
123D5×ΓL(1,𝔽8)—706no—
124C5×AGL(1,𝔽7)—74no—
125C5×AGL(1,𝔽7)—146no—
126C5×AGL(1,𝔽7)—144no—
127C5×AGL(1,𝔽7)—1412no—
128C5×AGL(1,𝔽7)—354no—
129C2×D52—491noHall
130C2×C7⋊D7—251no—
131C2×C7×D7—252no—
132D72—52no—
133D72—252no—
134C52×C7—23no—
135C52×C7C22×AGL(1,𝔽7)12yescomplemented
136C22×AGL(1,𝔽7)—1752no—
137C2×C3×C52—491no—
138C3×C5⋊D5—491no—
139C3×C5⋊D5—491no—
140C3×C5×D5—494no—
141#3861C2×D5211yescharacteristic, complemented, Hall
142C2×D35—354no—
143C2×C7×D5—354no—
144C2×C5×D7—354no—
145D5×D7—54no—
146D5×D7—74no—
147D5×D7—354no—
148D5×D7—354no—
149D5×D7—7012no—
150D5×D7—704no—
151D5×D7—706no—
152C22×C3×D5—2452no—
153C5×ΓL(1,𝔽8)—74no—
154C5×ΓL(1,𝔽8)—146no—
155C5×ΓL(1,𝔽8)—144no—
156C5×ΓL(1,𝔽8)—1412no—
157C2×C5⋊D5—491no—
158C2×C5×D5—492no—
159D52—72no—
160D52—492no—
161C2×C72—251no—
162C7⋊D7—251no—
163C7⋊D7C3×D5211yescharacteristic, complemented
164C7×D7—54no—
165C22×ΓL(1,𝔽8)—1752no—
166C2×AGL(1,𝔽7)—354no—
167C2×AGL(1,𝔽7)—3503no—
168C2×AGL(1,𝔽7)—1752no—
169C2×AGL(1,𝔽7)—1752no—
170C2×AGL(1,𝔽7)—1754no—
171C3×C52—491noHall
172C2×C5×C7—354no—
173D35—354no—
174D35—7012no—
175D35—704no—
176D35—706no—
177D35D5×AGL(1,𝔽7)14yescomplemented
178C5×D7—54no—
179C5×D7—74no—
180C5×D7—146no—
181C5×D7—144no—
182C5×D7—1412no—
183C7×D5—54no—
184C7×D5—74no—
185C7×D5—106no—
186C7×D5—104no—
187C7×D5—1012no—
188C22×C3×C5—2452no—
189C2×C3×D5—492no—
190C2×C3×D5—4902no—
191C2×C3×D5—2454no—
192C2×C3×D5—2454no—
193C2×C3×D5—2452no—
194C22×D7—1752no—
195C2×C52—491no—
196C5⋊D5—491no—
197C5⋊D5#64423⋊C211yescharacteristic, complemented
198C5×D5—74no—
199C72C2×C3×D5211yescharacteristic, complemented, Sylow
200C2×ΓL(1,𝔽8)—354no—
201C2×ΓL(1,𝔽8)—3503no—
202C2×ΓL(1,𝔽8)—1752no—
203AGL(1,𝔽7)—72no—
204AGL(1,𝔽7)—143no—
205AGL(1,𝔽7)—354no—
206AGL(1,𝔽7)—3503no—
207AGL(1,𝔽7)—1752no—
208C22×D5—2452no—
209C5×C7—212no—
210C5×C7—24no—
211C5×C7—26no—
212C5×C7C2×D5×AGL(1,𝔽7)14yescomplemented
213C2×C3×C5—492no—
214C2×C3×C5—982no—
215C2×C3×C5—2454no—
216C3×D5—494no—
217C3×D5—4902no—
218C3×D5—4902no—
219C3×D5—2452no—
220C3×D5—2452no—
221C22×C7—1752no—
222C2×D7—252no—
223C2×D7—354no—
224C2×D7—3503no—
225C2×D7—1754no—
226C2×D7—1752no—
227C52C2×#64423⋊C211yescharacteristic, complemented, Sylow
228C23×C3—12251noHall
229ΓL(1,𝔽8)—72no—
230ΓL(1,𝔽8)—143no—
231C22×C5—2452no—
232C2×D5—354no—
233C2×D5—492no—
234C2×D5—4902no—
235C2×D5—2452no—
236C2×D5—2454no—
237C3×C5—492no—
238C3×C5—982no—
239C2×C7—252no—
240C2×C7—354no—
241C2×C7—503no—
242D7—54no—
243D7—72no—
244D7—143no—
245D7—3503no—
246D7—1752no—
247C22×C3—12252no—
248C22×C3—12251no—
249C22×C3—12252no—
250C22×C3—2452no—
251C2×C5—354no—
252C2×C5—492no—
253C2×C5—982no—
254D5—52no—
255D5—74no—
256D5—102no—
257D5—4902no—
258D5—2452no—
259C23—12251noSylow
260C7—23no—
261C7D52×AGL(1,𝔽7)12yescomplemented
262C2×C3—491no—
263C2×C3—12251no—
264C2×C3—12251no—
265C2×C3—2454no—
266C5—22no—
267C5D5×#64423⋊C212yescomplemented
268C22—1752no—
269C22—12251no—
270C22—12252no—
271C22—2452no—
272C3—491noSylow
273C2—251no—
274C2—354no—
275C2—491no—
276C2—12251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C72C23×C3
LCSG ▷ C52×C72
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72225761C3semidirect
4. #28413321C2semidirect
5. #28413316C2semidirect
6. C5⋊D5×#64424C2semidirect
7. C5×#64424⋊D5C2semidirect
8. C5⋊D5×#3861C22semidirect
9. C5⋊(C5⋊#64424)C22semidirect
10. C5×C5⋊#64423C22semidirect
11. C52×#64424C22semidirect
12. D352C2×C3semidirect
13. C7×(C5⋊D5)⋊D7C2×C3semidirect
14. C5⋊D5×C7⋊D7C2×C3semidirect
15. C5×D35⋊D7C2×C3semidirect
16. C52×#3861C23semidirect
18. #64424⋊D5D5semidirect
23. C5⋊(C5⋊(C7⋊D7))C22×C3semidirect
24. C72×C5⋊D5C22×C3semidirect
25. C5×C7×D35C22×C3semidirect
26. C52×C7⋊D7C22×C3semidirect
36. C5⋊#64423C2×D5semidirect
41. C5×#64424C2×D5semidirect
43. C52×C72C23×C3semidirect
53. D35⋊D7C3×D5semidirect
61. C5×#3861C22×D5semidirect
62. (C5⋊D5)⋊D7AGL(1,𝔽7)semidirect
77. C7×D35C2×C3×D5semidirect
81. C5×C7⋊D7C2×C3×D5semidirect
97. C7×C5⋊D5C2×AGL(1,𝔽7)semidirect
98. C5×D35C2×AGL(1,𝔽7)semidirect
108. #64424D52semidirect
112. C5×C72C22×C3×D5semidirect
135. C52×C7C22×AGL(1,𝔽7)semidirect
141. #3861C2×D52semidirect
163. C7⋊D7C3×D52semidirect
177. D35D5×AGL(1,𝔽7)semidirect
197. C5⋊D5#64423⋊C2semidirect
199. C72C2×C3×D52semidirect
212. C5×C7C2×D5×AGL(1,𝔽7)semidirect
227. C52C2×#64423⋊C2semidirect
261. C7D52×AGL(1,𝔽7)semidirect
267. C5D5×#64423⋊C2semidirect

Maximal subgroups

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

subgroupindex
#722257613
D5×#64423⋊C25
D52×AGL(1,𝔽7)7
#284133212
#284133162
C5⋊D5×#644242
C5×#64424⋊D52