Finite Group Databasebeta

← C5×#91904 #72260407

Subgroups — C5×#91904 #72260407

Order 2400.

314 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C2×C5×D5×A4C211yesmaximal, characteristic, complemented
2C2×C52×S4C211yesmaximal, characteristic, complemented
3C5×A4⋊Dic5C211yesmaximal, characteristic, complemented
4C5×#4083—31nomaximal, Hall
5C2×C52×A4C2211yescharacteristic
6C5×D5×A4—21no—
7C52×S4—21no—
8C5×#1201—41nomaximal
9C5×GL(2,ℤ/4)—51nomaximal
10#91904C511yesmaximal, characteristic, direct product factor
11C23×C5×D5D311yescharacteristic, complemented
12C2×C52×D4—31no—
13C2×C5×C22⋊D5—31no—
14C2×C5×C22⋊D5—31no—
15C5×C22⋊Dic5—31no—
16C5×#685—31no—
17C5×#685—31no—
18C2×C52×D3—41no—
19C2×C3×C5×D5—41no—
20C52×A4D411yescharacteristic, complemented
21C5×Dic15—41no—
22C22×C5×A4—51no—
23C2×D5×A4C2×C511yescharacteristic, complemented
24C2×C5×S4—22no—
25C2×C5×S4D511yescharacteristic, complemented
26C2×C5×S4C2×C511yescharacteristic, complemented
27A4⋊Dic5C2×C511yescharacteristic, complemented
28C5×SL(2,ℤ/4)—51no—
29C23×C52—31no—
30C23×C52C2×D311yescharacteristic
31C22×C5×D5—21no—
32C22×C5×D5—31no—
33C22×C5×D5—31no—
34C22×C5×D5—61no—
35C52×D4—61no—
36C52×D4—61no—
37C2×C4×C52—31no—
38C5×C22⋊D5—62no—
39C5×C22⋊D5—61no—
40C5×C22⋊D5—61no—
41C2×C5×Dic5—31no—
42C2×C5×Dic5—31no—
43C5×#120—151no—
44#4083—31no—
45C2×C3×C52—41no—
46C52×D3—81no—
47C3×C5×D5—81no—
48C2×C5×A4—22no—
49C2×C5×A4—101no—
50C2×C5×A4C2×D511yescharacteristic
51C2×C5×A4C22×C511yescharacteristic
52D5×A4—21no—
53C5×S4—24no—
54C5×S4—21no—
55C5×S4—21no—
56C5×#67—201no—
57#1201—41no—
58C22×C52—31no—
59C22×C52—31no—
60C22×C52—61no—
61C22×C52—61no—
62C22×C52—61no—
63C22×C52#6711yescharacteristic, complemented
64C2×C5×D5—32no—
65C2×C5×D5—31no—
66C2×C5×D5—61no—
67C2×C5×D5—61no—
68C2×C5×D5—61no—
69C2×C5×D5S411yescharacteristic, complemented
70C4×C52—61no—
71C5×Dic5—61no—
72C5×Dic5—61no—
73GL(2,ℤ/4)—51noHall
74C24×C5—51no—
75C23×D5C5×D311yescharacteristic, complemented
76C2×C5×D4—31no—
77C2×C5×D4—31no—
78C2×C5×D4—62no—
79C2×C5×D4—151no—
80C2×C5×D4—151no—
81C2×C22⋊D5—31no—
82C2×C22⋊D5—31no—
83C5×C22⋊C4—151no—
84C5×C22⋊C4—151no—
85C5×C22⋊C4—151no—
86C22⋊Dic5—31no—
87#685—31no—
88#685—31no—
89C3×C52—41noHall
90C22×C3×C5—201no—
91C2×C5×D3—41no—
92C2×C5×D3—41no—
93C2×C5×D3—82no—
94C2×C3×D5—41no—
95C5×A4—22no—
96C5×A4C22⋊D511yescharacteristic, complemented
97C5×A4C5×D411yescharacteristic, complemented
98Dic15—41no—
99C5×Dic3—201no—
100C2×C52—31no—
101C2×C52—31no—
102C2×C52—121no—
103C2×C52C2×S411yescharacteristic
104C5×D5—21no—
105C5×D5—61no—
106C22×A4—51no—
107C2×S4C5×D511yescharacteristic, complemented
108SL(2,ℤ/4)—51no—
109C23×C5—22no—
110C23×C5—31no—
111C23×C5—31no—
112C23×C5—62no—
113C23×C5—101no—
114C23×C5—151no—
115C23×C5—151no—
116C23×C5—301no—
117C23×C5D3×D511yescharacteristic
118C23×C5C2×C5×D311yescharacteristic
119C22×D5—21no—
120C22×D5—31no—
121C22×D5—31no—
122C22×D5—61no—
123C5×D4—64no—
124C5×D4—61no—
125C5×D4—61no—
126C5×D4—64no—
127C5×D4—61no—
128C5×D4—61no—
129C5×D4—301no—
130C5×D4—302no—
131C5×D4—301no—
132C2×C4×C5—31no—
133C2×C4×C5—31no—
134C2×C4×C5—62no—
135C2×C4×C5—151no—
136C2×C4×C5—151no—
137C22⋊D5—62no—
138C22⋊D5—61no—
139C22⋊D5—61no—
140C2×Dic5—31no—
141C2×Dic5—31no—
142#120—151noSylow
143C2×C3×C5—41no—
144C2×C3×C5—41no—
145C2×C3×C5—82no—
146C2×C3×C5—401no—
147C3×D5—81no—
148C5×D3—81no—
149C5×D3—81no—
150C5×D3—84no—
151C52GL(2,ℤ/4)11yescharacteristic, complemented, Sylow
152C2×A4—101no—
153C2×A4C2×C5×D511yescharacteristic
154S4—21no—
155#67—201no—
156C22×C5—22no—
157C22×C5—31no—
158C22×C5—31no—
159C22×C5—31no—
160C22×C5—31no—
161C22×C5—51no—
162C22×C5—62no—
163C22×C5—62no—
164C22×C5—61no—
165C22×C5—61no—
166C22×C5—61no—
167C22×C5—61no—
168C22×C5—64no—
169C22×C5—61no—
170C22×C5—61no—
171C22×C5—122no—
172C22×C5—122no—
173C22×C5—152no—
174C22×C5—151no—
175C22×C5—301no—
176C22×C5—301no—
177C22×C5—301no—
178C22×C5#120111yescharacteristic, complemented
179C22×C5C5×#6711yescharacteristic, complemented
180C2×D5—32no—
181C2×D5—31no—
182C2×D5—61no—
183C2×D5—61no—
184C2×D5—61no—
185C2×D5C5×S411yescharacteristic, complemented
186C4×C5—61no—
187C4×C5—61no—
188C4×C5—64no—
189C4×C5—301no—
190C4×C5—301no—
191Dic5—61no—
192Dic5—61no—
193C24—51no—
194C2×D4—31no—
195C2×D4—151no—
196C2×D4—151no—
197C22⋊C4—151no—
198C22⋊C4—151no—
199C22⋊C4—151no—
200C3×C5—41no—
201C3×C5—41no—
202C3×C5—82no—
203C22×C3—201no—
204C2×D3—41no—
205A4C5×C22⋊D511yescharacteristic, complemented
206Dic3—201no—
207C2×C5—22no—
208C2×C5—31no—
209C2×C5—31no—
210C2×C5—31no—
211C2×C5—31no—
212C2×C5—62no—
213C2×C5—62no—
214C2×C5—101no—
215C2×C5—124no—
216C2×C5—121no—
217C2×C5—121no—
218C2×C5—301no—
219C2×C5D5×S411yescharacteristic
220C2×C5C2×C5×S411yescharacteristic
221D5—21no—
222D5—61no—
223C23—31no—
224C23—101no—
225C23—151no—
226C23—151no—
227C23—301no—
228C23C5×D3×D511yescharacteristic
229D4—61no—
230D4—61no—
231D4—301no—
232D4—302no—
233D4—301no—
234C2×C4—31no—
235C2×C4—151no—
236C2×C4—151no—
237C2×C3—41no—
238C2×C3—401no—
239D3—81no—
240C5—22no—
241C5#9190411yescharacteristic, direct product factor
242C5C5×GL(2,ℤ/4)11yescharacteristic, complemented
243C22—31no—
244C22—31no—
245C22—51no—
246C22—61no—
247C22—61no—
248C22—61no—
249C22—152no—
250C22—151no—
251C22—301no—
252C22—301no—
253C22—301no—
254C22C5×#120111yescharacteristic, complemented
255C4—61no—
256C4—301no—
257C4—301no—
258C3—41noSylow
259C2—31no—
260C2—31no—
261C2—101no—
262C2—121no—
263C2—301no—
264C2C5×D5×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C5D5×S4
commutator, cosocleC2×C5×A4C22×C5
FrattiniC2C5×D5×S4
Fitting, socleC23×C52C2×D3
hypocenterC5×A4C5×D4
LCSG ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C2×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×D5×A4C2semidirect
2. C2×C52×S4C2semidirect
3. C5×A4⋊Dic5C2semidirect
10. #91904C5direct (N × H)
11. C23×C5×D5D3semidirect
20. C52×A4D4semidirect
23. C2×D5×A4C2×C5semidirect
25. C2×C5×S4D5semidirect
26. C2×C5×S4C2×C5semidirect
27. A4⋊Dic5C2×C5semidirect
63. C22×C52#67semidirect
69. C2×C5×D5S4semidirect
75. C23×D5C5×D3semidirect
96. C5×A4C22⋊D5semidirect
97. C5×A4C5×D4semidirect
107. C2×S4C5×D5semidirect
151. C52GL(2,ℤ/4)semidirect
178. C22×C5#1201semidirect
179. C22×C5C5×#67semidirect
185. C2×D5C5×S4semidirect
205. A4C5×C22⋊D5semidirect
241. C5#91904direct (N × H)
242. C5C5×GL(2,ℤ/4)semidirect
254. C22C5×#1201semidirect

Maximal subgroups

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

subgroupindex
C2×C5×D5×A42
C2×C52×S42
C5×A4⋊Dic52
C5×#40833
C5×#12014
C5×GL(2,ℤ/4)5
#919045