Finite Group Databasebeta

← C3⋊(C5⋊(C5⋊#64418)) #36348042

Subgroups — C3⋊(C5⋊(C5⋊#64418)) #36348042

Order 22050.

304 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C3×C52×#3862C211yesmaximal, characteristic, complemented, Hall
2C7×C3⋊(C5⋊D35)—31nomaximal
3C5⋊(C5⋊#64418)—31nomaximal
4C5⋊(C5⋊#64418)—31nomaximal
5C5⋊(C5⋊#64418)—31nomaximal
6C3⋊(C5⋊#64418)—51nomaximal
7C3⋊(C5⋊#64418)—52nomaximal
8C3⋊(C5⋊#64418)—51nomaximal
9C3⋊(C5⋊#64418)—51nomaximal
10C3⋊(C5⋊#64418)—51nomaximal
11C52×#3862D311yescharacteristic, complemented
12C52×#3862D311yescharacteristic, complemented
13C52×#3862D311yescharacteristic, complemented
14C3×C52×C72D311yescharacteristic, complemented
15C7×C5⋊D35—91noHall
16C3×C5×#3862D511yescharacteristic, complemented
17C3×C5×#3862D512yescomplemented
18C3×C5×#3862D511yescharacteristic, complemented
19C3×C5×#3862D511yescharacteristic, complemented
20C3×C5×#3862D511yescharacteristic, complemented
21C3×C52×ΓL(1,𝔽8)—141no—
22C7×D105—152no—
23C7×D105—151no—
24C7×D105—151no—
25C7×D105—151no—
26C7×D105—151no—
27C5⋊#64418—151no—
28C5⋊#64418—151no—
29C5⋊#64418—151no—
30C5⋊#64418—152no—
31C5⋊#64418—151no—
32C5⋊#64418—151no—
33C5⋊#64418—151no—
34C5⋊#64418—151no—
35C5⋊#64418—151no—
36C5⋊#64418—151no—
37C5⋊#64418—152no—
38C5⋊#64418—152no—
39C5⋊#64418—151no—
40C5⋊#64418—151no—
41C5⋊#64418—151no—
42C52×C72C3⋊D311yescharacteristic, complemented, Hall
43C3⋊(C5⋊D35)—31no—
44C7×C3⋊(C5⋊D5)—211no—
45C3⋊#64418—251noHall
46C3×C5×C72D1511yescharacteristic, complemented
47C3×C5×C72D1512yescomplemented
48C3×C5×C72D1511yescharacteristic, complemented
49C3×C5×C72D1511yescharacteristic, complemented
50C3×C5×C72D1511yescharacteristic, complemented
51C5×#3862D1511yescharacteristic, complemented
52C5×#3862D1511yescharacteristic, complemented
53C5×#3862D1511yescharacteristic, complemented
54C5×#3862D1511yescharacteristic, complemented
55C5×#3862D1511yescharacteristic, complemented
56C5×#3862D1512yescomplemented
57C5×#3862D1511yescharacteristic, complemented
58C5×#3862D1511yescharacteristic, complemented
59C5×#3862D1511yescharacteristic, complemented
60C5×#3862D1511yescharacteristic, complemented
61C5×#3862D1512yescomplemented
62C5×#3862D1512yescomplemented
63C5×#3862D1511yescharacteristic, complemented
64C5×#3862D1511yescharacteristic, complemented
65C5×#3862D1511yescharacteristic, complemented
66C3×C52×C7—21no—
67C3×C52×C7—31no—
68C3×C52×C7—61no—
69C52×ΓL(1,𝔽8)—141no—
70C52×ΓL(1,𝔽8)—141no—
71C52×ΓL(1,𝔽8)—141no—
72C7×D35—451no—
73C7×D35—451no—
74C7×D35—451no—
75C7×D35—452no—
76C7×D35—451no—
77C3⋊(C3⋊(C5⋊D5))—491nomaximal, Hall
78C3×#3862C5⋊D511yescharacteristic, complemented, Hall
79C5⋊D35—91no—
80C7×C5⋊D5—631no—
81C3×C5×ΓL(1,𝔽8)—141no—
82C3×C5×ΓL(1,𝔽8)—142no—
83C3×C5×ΓL(1,𝔽8)—141no—
84C3×C5×ΓL(1,𝔽8)—141no—
85C3×C5×ΓL(1,𝔽8)—141no—
86C7×D21—751no—
87#64418—751no—
88#64418—751no—
89#64418—751no—
90C5×C72C3⋊D1511yescharacteristic, complemented
91C5×C72C3⋊D1511yescharacteristic, complemented
92C5×C72C3⋊D1511yescharacteristic, complemented
93C5×C72C3⋊D1512yescomplemented
94C5×C72C3⋊D1511yescharacteristic, complemented
95C32×C52—981noHall
96D105—151no—
97D105—152no—
98D105—151no—
99D105—151no—
100D105—151no—
101C7×D15—1052no—
102C7×D15—1051no—
103C7×D15—1051no—
104C7×D15—1051no—
105C7×D15—1051no—
106C52×C7—21no—
107C52×C7—61no—
108C52×C7—61no—
109C3⋊(C5⋊D5)—211no—
110C3⋊(C5⋊D5)—1471no—
111C3⋊(C5⋊D5)—1471no—
112C3⋊(C5⋊D5)—1471no—
113C3×C72C3⋊(C5⋊D5)11yescharacteristic, complemented
114#3862C3⋊(C5⋊D5)11yescharacteristic, complemented
115#3862C3⋊(C5⋊D5)11yescharacteristic, complemented
116#3862C3⋊(C5⋊D5)11yescharacteristic, complemented
117C3×C5×C7—21no—
118C3×C5×C7—22no—
119C3×C5×C7—21no—
120C3×C5×C7—21no—
121C3×C5×C7—21no—
122C3×C5×C7—31no—
123C3×C5×C7—32no—
124C3×C5×C7—31no—
125C3×C5×C7—61no—
126C3×C5×C7—61no—
127C3×C5×C7—62no—
128C3×C5×C7—61no—
129C3×C5×C7—61no—
130C3×C5×C7—61no—
131C3×C5×C7—61no—
132C5×ΓL(1,𝔽8)—141no—
133C5×ΓL(1,𝔽8)—142no—
134C5×ΓL(1,𝔽8)—141no—
135C5×ΓL(1,𝔽8)—141no—
136C5×ΓL(1,𝔽8)—141no—
137C5×ΓL(1,𝔽8)—141no—
138C5×ΓL(1,𝔽8)—141no—
139C5×ΓL(1,𝔽8)—141no—
140C5×ΓL(1,𝔽8)—141no—
141C5×ΓL(1,𝔽8)—142no—
142C5×ΓL(1,𝔽8)—141no—
143C5×ΓL(1,𝔽8)—141no—
144C5×ΓL(1,𝔽8)—142no—
145C5×ΓL(1,𝔽8)—141no—
146C5×ΓL(1,𝔽8)—141no—
147C7×D7—2251noHall
148C3⋊D15—2451no—
149C3⋊D15—2452no—
150C3⋊D15—2451no—
151C3⋊D15—2451no—
152C3⋊D15—2451no—
153C3×C52—981no—
154C3×C52—981no—
155C3×C52—981no—
156C3×C52#6441811yescharacteristic, complemented
157D35—451no—
158D35—451no—
159D35—452no—
160D35—451no—
161D35—451no—
162C7×D5—3151no—
163C7×D5—3151no—
164C7×D5—3152no—
165C7×D5—3151no—
166C7×D5—3151no—
167C3×ΓL(1,𝔽8)—141no—
168C5⋊D5—631noHall
169C72C3⋊(C3⋊(C5⋊D5))11yescharacteristic, complemented, Sylow
170C32×C5—491no—
171C32×C5—982no—
172C32×C5—981no—
173C32×C5—981no—
174C32×C5—981no—
175D21—751no—
176D3×C7—5251no—
177C5×C7—21no—
178C5×C7—21no—
179C5×C7—22no—
180C5×C7—21no—
181C5×C7—21no—
182C5×C7—31no—
183C5×C7—31no—
184C5×C7—31no—
185C5×C7—31no—
186C5×C7—32no—
187C5×C7—31no—
188C5×C7—62no—
189C5×C7—61no—
190C5×C7—61no—
191C5×C7—61no—
192D15—1051no—
193D15—1052no—
194D15—1051no—
195D15—1051no—
196D15—1051no—
197D15—7351no—
198D15—7352no—
199D15—7351no—
200D15—7351no—
201D15—7351no—
202D15—7351no—
203D15—7351no—
204D15—7351no—
205D15—7351no—
206D15—7352no—
207D15—7351no—
208D15—7351no—
209D15—7351no—
210D15—7352no—
211D15—7351no—
212C52C3⋊#6441811yescharacteristic, complemented, Sylow
213C3×C7—21no—
214C3×C7—61no—
215C3×C7—61no—
216ΓL(1,𝔽8)—141no—
217ΓL(1,𝔽8)—141no—
218ΓL(1,𝔽8)—141no—
219C3⋊D3—12251noHall
220C3×C5—491no—
221C3×C5—492no—
222C3×C5—491no—
223C3×C5—491no—
224C3×C5—981no—
225C3×C5—981no—
226C3×C5—981no—
227C3×C5—981no—
228C3×C5—981no—
229C3×C5—982no—
230C3×C5—981no—
231C3×C5—981no—
232C3×C5—982no—
233C3×C5—981no—
234C3×C5—981no—
235C3×C5C5⋊#6441811yescharacteristic, complemented
236C3×C5C5⋊#6441812yescomplemented
237C3×C5C5⋊#6441811yescharacteristic, complemented
238C3×C5C5⋊#6441811yescharacteristic, complemented
239C3×C5C5⋊#6441811yescharacteristic, complemented
240C2×C7—15751no—
241D7—2251no—
242D5—3151no—
243D5—3151no—
244D5—3151no—
245D5—3152no—
246D5—3151no—
247C32—981noSylow
248C7—21no—
249C7—61no—
250C7—61no—
251D3—5251no—
252D3—36751no—
253D3—36751no—
254D3—36751no—
255C5C3⋊(C5⋊#64418)11yescharacteristic, complemented
256C5C3⋊(C5⋊#64418)11yescharacteristic, complemented
257C5C3⋊(C5⋊#64418)12yescomplemented
258C5C3⋊(C5⋊#64418)11yescharacteristic, complemented
259C5C3⋊(C5⋊#64418)11yescharacteristic, complemented
260C3—981no—
261C3—981no—
262C3—981no—
263C3C5⋊(C5⋊#64418)11yescharacteristic, complemented
264C2—15751noSylow
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC3×C52×#3862C2
Fitting, socleC3×C52×C72D3
LCSG ▷ C3×C52×#3862
UCSC1

Complements

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

NH ≅ G/Ntype
1. C3×C52×#3862C2semidirect
11. C52×#3862D3semidirect
12. C52×#3862D3semidirect
13. C52×#3862D3semidirect
14. C3×C52×C72D3semidirect
16. C3×C5×#3862D5semidirect
17. C3×C5×#3862D5semidirect
18. C3×C5×#3862D5semidirect
19. C3×C5×#3862D5semidirect
20. C3×C5×#3862D5semidirect
42. C52×C72C3⋊D3semidirect
46. C3×C5×C72D15semidirect
47. C3×C5×C72D15semidirect
48. C3×C5×C72D15semidirect
49. C3×C5×C72D15semidirect
50. C3×C5×C72D15semidirect
51. C5×#3862D15semidirect
52. C5×#3862D15semidirect
53. C5×#3862D15semidirect
54. C5×#3862D15semidirect
55. C5×#3862D15semidirect
56. C5×#3862D15semidirect
57. C5×#3862D15semidirect
58. C5×#3862D15semidirect
59. C5×#3862D15semidirect
60. C5×#3862D15semidirect
61. C5×#3862D15semidirect
62. C5×#3862D15semidirect
63. C5×#3862D15semidirect
64. C5×#3862D15semidirect
65. C5×#3862D15semidirect
78. C3×#3862C5⋊D5semidirect
90. C5×C72C3⋊D15semidirect
91. C5×C72C3⋊D15semidirect
92. C5×C72C3⋊D15semidirect
93. C5×C72C3⋊D15semidirect
94. C5×C72C3⋊D15semidirect
113. C3×C72C3⋊(C5⋊D5)semidirect
114. #3862C3⋊(C5⋊D5)semidirect
115. #3862C3⋊(C5⋊D5)semidirect
116. #3862C3⋊(C5⋊D5)semidirect
156. C3×C52#64418semidirect
169. C72C3⋊(C3⋊(C5⋊D5))semidirect
212. C52C3⋊#64418semidirect
235. C3×C5C5⋊#64418semidirect
236. C3×C5C5⋊#64418semidirect
237. C3×C5C5⋊#64418semidirect
238. C3×C5C5⋊#64418semidirect
239. C3×C5C5⋊#64418semidirect
255. C5C3⋊(C5⋊#64418)semidirect
256. C5C3⋊(C5⋊#64418)semidirect
257. C5C3⋊(C5⋊#64418)semidirect
258. C5C3⋊(C5⋊#64418)semidirect
259. C5C3⋊(C5⋊#64418)semidirect
263. C3C5⋊(C5⋊#64418)semidirect

Maximal subgroups

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

subgroupindex
C3×C52×#38622
C7×C3⋊(C5⋊D35)3
C5⋊(C5⋊#64418)3
C3⋊(C5⋊#64418)5
C3⋊(C3⋊(C5⋊D5))49