Finite Group Databasebeta

← A4×#6038 #72231281

Subgroups — A4×#6038 #72231281

Order 2400.

342 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 42 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×A4×C5⋊D5C211yesmaximal, characteristic, complemented
2C2×C5×D5×A4C211yesmaximal, characteristic, complemented
3C5×A4×Dic5C211yesmaximal, characteristic, complemented
4C22×#6038C311yesmaximal, characteristic, complemented, Hall
5C2×C52×A4C2211yescharacteristic
6A4×C5⋊D5—21no—
7C5×D5×A4—21no—
8C3×#6038—41nomaximal
9A4×C22⋊D5—51nomaximal
10A4×D20—51nomaximal
11C23×C5⋊D5C2×C311yescharacteristic, complemented
12C23×C5×D5C2×C311yescharacteristic, complemented
13C22×C5×Dic5C2×C311yescharacteristic, complemented
14C2×#6038—31no—
15C2×#6038—31no—
16C2×#6038—31no—
17C2×#6038—31no—
18C2×C3×C5⋊D5—41no—
19C2×C3×C5×D5—41no—
20C52×A4D411yescharacteristic, complemented
21C3×C5×Dic5—41no—
22C22×C5×A4—51no—
23C2×D5×A4—51no—
24C2×D5×A4—51no—
25C2×D5×A4—102no—
26C2×D5×A4D511yescharacteristic, complemented
27C4×C5×A4—51no—
28A4×Dic5D511yescharacteristic, complemented
29C23×C52C22×C311yescharacteristic
30C22×C5⋊D5—21no—
31C22×C5⋊D5—31no—
32C22×C5⋊D5—31no—
33C22×C5⋊D5—61no—
34C22×C5×D5—21no—
35C22×C5×D5—31no—
36C22×C5×D5—31no—
37C22×C5×D5—61no—
38C2×C5×Dic5—31no—
39C2×C5×Dic5—31no—
40#6038—32no—
41#6038—31no—
42#6038—31no—
43#6038—31no—
44#6038A411yescharacteristic, direct product factor
45C22×C22⋊D5—51no—
46C22×D20—51no—
47C2×C3×C52—41no—
48C3×C5⋊D5—81no—
49C3×C5×D5—81no—
50C2×C5×A4—22no—
51C2×C5×A4—101no—
52C2×C5×A4C2×D511yescharacteristic
53C2×C5×A4C2×D511yescharacteristic
54D5×A4—21no—
55D5×A4—104no—
56D5×A4—101no—
57D5×A4—101no—
58C3×C22⋊D5—201no—
59C3×D20—201no—
60C22×C52—31no—
61C22×C52—31no—
62C22×C52C3×D411yescharacteristic, complemented
63C2×C5⋊D5—31no—
64C2×C5⋊D5—61no—
65C2×C5⋊D5—61no—
66C2×C5⋊D5—61no—
67C2×C5⋊D5—61no—
68C2×C5⋊D5C2×A411yescharacteristic, complemented
69C2×C5×D5—31no—
70C2×C5×D5—61no—
71C2×C5×D5—61no—
72C2×C5×D5—61no—
73C2×C5×D5—61no—
74C2×C5×D5C2×A411yescharacteristic, complemented
75C5×Dic5—31no—
76C5×Dic5C2×A411yescharacteristic, complemented
77D4×A4—251noHall
78C24×C5—51no—
79C23×D5—51no—
80C23×D5—51no—
81C23×D5—102no—
82C23×D5C3×D511yescharacteristic, complemented
83C22×C4×C5—51no—
84C2×C22⋊D5—151no—
85C2×C22⋊D5—151no—
86C2×C22⋊D5—151no—
87C2×C22⋊D5—151no—
88C22×Dic5C3×D511yescharacteristic, complemented
89C2×D20—151no—
90C2×D20—151no—
91C2×D20—151no—
92C2×D20—151no—
93C3×C52—41noHall
94C22×C3×C5—201no—
95C2×C3×D5—41no—
96C2×C3×D5—201no—
97C2×C3×D5—201no—
98C2×C3×D5—402no—
99C5×A4—22no—
100C5×A4D2011yescharacteristic, complemented
101C5×A4C22⋊D511yescharacteristic, complemented
102C3×C4×C5—201no—
103C3×Dic5—41no—
104C2×C52—31no—
105C2×C52—31no—
106C2×C52C22×A411yescharacteristic
107C5⋊D5—21no—
108C5⋊D5—61no—
109C5×D5—21no—
110C5×D5—61no—
111C22×A4—51no—
112C22×A4—251no—
113C4×A4—51no—
114C23×C5—22no—
115C23×C5—101no—
116C23×C5—151no—
117C23×C5—151no—
118C23×C5—301no—
119C23×C5C2×C3×D511yescharacteristic
120C23×C5C2×C3×D511yescharacteristic
121C22×D5—21no—
122C22×D5—31no—
123C22×D5—31no—
124C22×D5—61no—
125C22×D5—104no—
126C22×D5—101no—
127C22×D5—101no—
128C22×D5—151no—
129C22×D5—151no—
130C22×D5—151no—
131C22×D5—151no—
132C22×D5—302no—
133C22×D5—302no—
134C22×D5—304no—
135C22×D5—301no—
136C22×D5—301no—
137C2×C4×C5—151no—
138C2×C4×C5—151no—
139C22⋊D5—51no—
140C22⋊D5—151no—
141C22⋊D5—151no—
142C22⋊D5—152no—
143C22⋊D5—151no—
144C2×Dic5—31no—
145C2×Dic5—31no—
146D20—51no—
147D20—151no—
148D20—151no—
149D20—152no—
150D20—151no—
151C22×D4—251noSylow
152C2×C3×C5—41no—
153C2×C3×C5—41no—
154C2×C3×C5—82no—
155C2×C3×C5—401no—
156C3×D5—81no—
157C3×D5—401no—
158C3×D5—401no—
159C3×D5—404no—
160C52D4×A411yescharacteristic, complemented, Sylow
161C2×A4—101no—
162C2×A4—501no—
163C2×A4D5211yescharacteristic
164C3×D4—1001no—
165C22×C5—22no—
166C22×C5—31no—
167C22×C5—31no—
168C22×C5—31no—
169C22×C5—31no—
170C22×C5—51no—
171C22×C5—62no—
172C22×C5—62no—
173C22×C5—151no—
174C22×C5—301no—
175C22×C5—301no—
176C22×C5—301no—
177C22×C5—301no—
178C22×C5C3×D2011yescharacteristic, complemented
179C22×C5C3×C22⋊D511yescharacteristic, complemented
180C2×D5—31no—
181C2×D5—51no—
182C2×D5—51no—
183C2×D5—61no—
184C2×D5—61no—
185C2×D5—61no—
186C2×D5—61no—
187C2×D5—102no—
188C2×D5—151no—
189C2×D5—151no—
190C2×D5—302no—
191C2×D5—301no—
192C2×D5—301no—
193C2×D5—301no—
194C2×D5—301no—
195C2×D5—304no—
196C2×D5—304no—
197C2×D5—304no—
198C2×D5—301no—
199C2×D5—301no—
200C2×D5—301no—
201C2×D5—301no—
202C2×D5—304no—
203C2×D5D5×A411yescharacteristic, complemented
204C4×C5—51no—
205C4×C5—151no—
206Dic5—31no—
207Dic5D5×A411yescharacteristic, complemented
208C24—51no—
209C24—251no—
210C2×D4—751no—
211C2×D4—751no—
212C2×D4—751no—
213C2×D4—751no—
214C22×C4—51no—
215C3×C5—41no—
216C3×C5—41no—
217C3×C5—82no—
218C22×C3—201no—
219C22×C3—1001no—
220A4#603811yescharacteristic, direct product factor
221C3×C4—201no—
222C2×C5—22no—
223C2×C5—31no—
224C2×C5—31no—
225C2×C5—31no—
226C2×C5—31no—
227C2×C5—62no—
228C2×C5—62no—
229C2×C5—101no—
230C2×C5—301no—
231C2×C5C2×D5×A411yescharacteristic
232C2×C5C2×D5×A411yescharacteristic
233D5—21no—
234D5—61no—
235D5—104no—
236D5—101no—
237D5—101no—
238D5—304no—
239D5—301no—
240D5—301no—
241C23—101no—
242C23—151no—
243C23—151no—
244C23—301no—
245C23—501no—
246C23—751no—
247C23—751no—
248C23—1501no—
249C23C3×D5211yescharacteristic
250D4—251no—
251D4—751no—
252D4—751no—
253D4—751no—
254D4—752no—
255C2×C4—151no—
256C2×C4—151no—
257C2×C3—41no—
258C2×C3—401no—
259C2×C3—2001no—
260C5—22no—
261C5A4×D2011yescharacteristic, complemented
262C5A4×C22⋊D511yescharacteristic, complemented
263C22—31no—
264C22—31no—
265C22—51no—
266C22—151no—
267C22—251no—
268C22—301no—
269C22—301no—
270C22—301no—
271C22—301no—
272C22—751no—
273C22—1501no—
274C22—1501no—
275C22—1501no—
276C22—1501no—
277C22C3×#603811yescharacteristic, complemented
278C4—51no—
279C4—151no—
280C3—41noSylow
281C2—31no—
282C2—31no—
283C2—101no—
284C2—301no—
285C2—501no—
286C2—1501no—
287C2D52×A411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D52×A4
commutator, Fitting, socle, cosocleC23×C52C22×C3
hypocenterC22×C52C3×D4
LCSG ▷ C23×C52 ▷ C22×C52
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×A4×C5⋊D5C2semidirect
2. C2×C5×D5×A4C2semidirect
3. C5×A4×Dic5C2semidirect
4. C22×#6038C3semidirect
11. C23×C5⋊D5C2×C3semidirect
12. C23×C5×D5C2×C3semidirect
13. C22×C5×Dic5C2×C3semidirect
20. C52×A4D4semidirect
26. C2×D5×A4D5semidirect
28. A4×Dic5D5semidirect
44. #6038A4direct (N × H)
62. C22×C52C3×D4semidirect
68. C2×C5⋊D5C2×A4semidirect
74. C2×C5×D5C2×A4semidirect
76. C5×Dic5C2×A4semidirect
82. C23×D5C3×D5semidirect
88. C22×Dic5C3×D5semidirect
100. C5×A4D20semidirect
101. C5×A4C22⋊D5semidirect
160. C52D4×A4semidirect
178. C22×C5C3×D20semidirect
179. C22×C5C3×C22⋊D5semidirect
203. C2×D5D5×A4semidirect
207. Dic5D5×A4semidirect
220. A4#6038direct (N × H)
261. C5A4×D20semidirect
262. C5A4×C22⋊D5semidirect
277. C22C3×#6038semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C5⋊D52
C2×C5×D5×A42
C5×A4×Dic52
C22×#60383
C3×#60384
A4×C22⋊D55
A4×D205