Finite Group Databasebeta

← C22×#64869 #13030183

Subgroups — C22×#64869 #13030183

Order 1280.

1846 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 1846 conjugacy classes of subgroups — too many to draw. Only its 360 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
1C22×#4043C211yesmaximal, characteristic
2C22×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic
3C22×#4003C211yesmaximal, characteristic
4C2×#64869C2112yesmaximal, direct product factor
5C24×AGL(1,𝔽5)—21no—
6C23×C4×D5C2211yescharacteristic
7C23×Dic10C411yescharacteristic, complemented
8C23×#170—21no—
9C22×C5×C4⋊C4C411yescharacteristic, complemented
10C22×#685—21no—
11C22×C4⋊Dic5—21no—
12C2×#4043—23no—
13C2×#4043—23no—
14C2×#4043C416yescomplemented
15C2×#4043C2216yes—
16C2×(C2×AGL(1,𝔽5))⋊C4—23no—
17C2×(C2×AGL(1,𝔽5))⋊C4C2216yes—
18C2×#4003—23no—
19C2×#4003C2216yes—
20#64869C22116yesdirect product factor
21C22×#328—51nomaximal, Sylow
22C24×D5D411yescharacteristic
23C23×AGL(1,𝔽5)—28no—
24C23×AGL(1,𝔽5)—26no—
25C23×AGL(1,𝔽5)—44no—
26C23×AGL(1,𝔽5)—44no—
27C23×C4×C5—21no—
28C23×C4×C5C2×C411yescharacteristic
29C23×Dic5—21no—
30C23×Dic5D411yescharacteristic
31C22×C4×D5—44no—
32C22×C4×D5C2×C413yes—
33C22×C4×D5D413yes—
34C22×C4×D5D413yes—
35C22×C4×D5C2313yes—
36C22×Dic10—23no—
37C22×Dic10—44no—
38C22×Dic10C2×C416yescomplemented
39C22×#170—26no—
40C22×#170—44no—
41C2×C5×C4⋊C4—23no—
42C2×C5×C4⋊C4C2×C416yescomplemented
43C2×#685—26no—
44C2×#685—26no—
45C2×C4⋊Dic5—26no—
46C2×C4⋊Dic5—26no—
47#4043—212no—
48#4043—26no—
49#4043—26no—
50#4043C2×C4112yescomplemented
51#4043C2314yes—
52(C2×AGL(1,𝔽5))⋊C4—26no—
53(C2×AGL(1,𝔽5))⋊C4C2314yes—
54#4003—26no—
55#4003C2314yes—
56C22×C22⋊Q8—51no—
57C22×C22⋊C8—51no—
58C22×#95—51no—
59C2×#328—512no—
60C24×C5C22⋊C411yescharacteristic
61C23×D5—24no—
62C23×D5—24no—
63C23×D5C22⋊C413yes—
64C23×D5SD1614yes—
65C23×D5Q1614yes—
66C23×D5C2×D413yes—
67C22×AGL(1,𝔽5)—224no—
68C22×AGL(1,𝔽5)—24no—
69C22×AGL(1,𝔽5)—432no—
70C22×AGL(1,𝔽5)—412no—
71C22×AGL(1,𝔽5)—412no—
72C22×C4×C5—24no—
73C22×C4×C5—26no—
74C22×C4×C5—44no—
75C22×C4×C5C22⋊C413yes—
76C22×C4×C5C22×C413yes—
77C22×Dic5—24no—
78C22×Dic5—26no—
79C22×Dic5—44no—
80C22×Dic5C22⋊C413yes—
81C22×Dic5C2×D413yes—
82C2×C4×D5—46no—
83C2×C4×D5—46no—
84C2×C4×D5—46no—
85C2×C4×D5—46no—
86C2×C4×D5C22⋊C416yes—
87C2×C4×D5C22×C413yes—
88C2×C4×D5C2×D413yes—
89C2×C4×D5C2×D413yes—
90C2×C4×D5C2411yescharacteristic
91C2×Dic10—26no—
92C2×Dic10—412no—
93C2×Dic10—412no—
94C2×Dic10C22×C414yescomplemented
95C2×#170—24no—
96C2×#170—412no—
97C5×C4⋊C4—26no—
98C5×C4⋊C4C22×C414yescomplemented
99#685—212no—
100#685—24no—
101C4⋊Dic5—212no—
102C4⋊Dic5—24no—
103C23×Q8—51no—
104C24×C4—51no—
105C24×C4—101no—
106C23×C8—101no—
107C2×C22⋊Q8—56no—
108C2×C22⋊Q8—56no—
109C2×C22⋊Q8—103no—
110C2×C22⋊Q8—103no—
111C22×C4⋊C4—101no—
112C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
113C22×C22⋊C4—101no—
114C2×C22⋊C8—56no—
115C2×C22⋊C8—103no—
116C2×#95—56no—
117C2×#95—103no—
118#328—516no—
119C23×C5#9914yes—
120C23×C5#10314yes—
121C23×C5#10414yes—
122C23×C5C2×C22⋊C413yes—
123C22×D5—216no—
124C22×D5—216no—
125C22×D5—26no—
126C22×D5—26no—
127C22×D5—26no—
128C22×D5—26no—
129C22×D5#103112yes—
130C22×D5C2×C22⋊C413yes—
131C22×D5C2×SD1616yes—
132C22×D5C2×Q1616yes—
133C22×D5C22×D411yescharacteristic
134C2×AGL(1,𝔽5)—28no—
135C2×AGL(1,𝔽5)—44no—
136C2×AGL(1,𝔽5)—448no—
137C2×AGL(1,𝔽5)—44no—
138C2×C4×C5—26no—
139C2×C4×C5—26no—
140C2×C4×C5—24no—
141C2×C4×C5—412no—
142C2×C4×C5C2×C22⋊C413yes—
143C2×C4×C5C23×C411yescharacteristic
144C2×Dic5—26no—
145C2×Dic5—26no—
146C2×Dic5—24no—
147C2×Dic5—412no—
148C2×Dic5C2×C22⋊C413yes—
149C2×Dic5C22×D411yescharacteristic
150C4×D5—46no—
151C4×D5—43no—
152C4×D5—43no—
153C4×D5—43no—
154C4×D5—41no—
155Dic10—412no—
156Dic10—44no—
157#170—44no—
158C25—51no—
159C22×Q8—56no—
160C22×Q8—103no—
161C22×Q8—204no—
162C23×C4—21no—
163C23×C4—51no—
164C23×C4—53no—
165C23×C4—53no—
166C23×C4—53no—
167C23×C4—53no—
168C23×C4—106no—
169C23×C4—108no—
170C23×C4—101no—
171C23×C4—204no—
172C23×C4—204no—
173C23×C4—204no—
174C23×C4C2×AGL(1,𝔽5)11yescharacteristic
175C22×C8—106no—
176C22×C8—204no—
177C22⋊Q8—54no—
178C22⋊Q8—512no—
179C22⋊Q8—106no—
180C22⋊Q8—106no—
181C22⋊Q8—1012no—
182C2×C4⋊C4—23no—
183C2×C4⋊C4—106no—
184C2×C4⋊C4—106no—
185C2×C4⋊C4C2×AGL(1,𝔽5)16yescomplemented
186C2×C22⋊C4—106no—
187C2×C22⋊C4—106no—
188C22⋊C8—54no—
189C22⋊C8—106no—
190#95—54no—
191#95—106no—
192C22×C5#328116yescomplemented
193C22×C5C2×#9916yes—
194C22×C5C2×#10316yes—
195C22×C5C2×#10416yes—
196C22×C5C22×C22⋊C411yescharacteristic
197C2×D5—21no—
198C2×D5—23no—
199C2×D5—21no—
200C2×D5—23no—
201C2×D5—212no—
202C2×D5—224no—
203C2×D5—212no—
204C2×D5C2×#10316yes—
205C2×D5C22×SD1611yescharacteristic
206C2×D5C22×Q1611yescharacteristic
207AGL(1,𝔽5)—48no—
208C4×C5—23no—
209C4×C5—21no—
210C4×C5—44no—
211Dic5—21no—
212Dic5—23no—
213Dic5—44no—
214C24—53no—
215C24—53no—
216C24—54no—
217C24—54no—
218C24—104no—
219C24—104no—
220C24C22⋊AGL(1,𝔽5)11yescharacteristic
221C2×Q8—54no—
222C2×Q8—106no—
223C2×Q8—2012no—
224C2×Q8—2012no—
225C22×C4—24no—
226C22×C4—26no—
227C22×C4—44no—
228C22×C4—53no—
229C22×C4—53no—
230C22×C4—53no—
231C22×C4—53no—
232C22×C4—56no—
233C22×C4—51no—
234C22×C4—53no—
235C22×C4—104no—
236C22×C4—104no—
237C22×C4—1024no—
238C22×C4—106no—
239C22×C4—206no—
240C22×C4—206no—
241C22×C4—206no—
242C22×C4—206no—
243C22×C4—2012no—
244C22×C4—2032no—
245C22×C4—2012no—
246C22×C4—204no—
247C22×C4C22⋊AGL(1,𝔽5)13yes—
248C22×C4C22×AGL(1,𝔽5)13yes—
249C2×C8—104no—
250C2×C8—2012no—
251C4⋊C4—26no—
252C4⋊C4—104no—
253C4⋊C4—1012no—
254C4⋊C4C22×AGL(1,𝔽5)14yescomplemented
255C22⋊C4—104no—
256C22⋊C4—1012no—
257C2×C5C2×#328112yescomplemented
258C2×C5C22×#9911yescharacteristic
259C2×C5C22×#10311yescharacteristic
260C2×C5C22×#10411yescharacteristic
261D5—21no—
262D5—21no—
263D5—26no—
264C23—51no—
265C23—53no—
266C23—56no—
267C23—512no—
268C23—56no—
269C23—1016no—
270C23—1016no—
271C23—106no—
272C23—106no—
273C23—106no—
274C23—106no—
275C23#399914yes—
276C23#400814yes—
277C23#400914yes—
278C23C2×C22⋊AGL(1,𝔽5)13yes—
279Q8—204no—
280Q8—2012no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4—24no—
284C2×C4—412no—
285C2×C4—51no—
286C2×C4—53no—
287C2×C4—106no—
288C2×C4—106no—
289C2×C4—108no—
290C2×C4—104no—
291C2×C4—203no—
292C2×C4—203no—
293C2×C4—206no—
294C2×C4—201no—
295C2×C4—203no—
296C2×C4—204no—
297C2×C4—204no—
298C2×C4—2048no—
299C2×C4—2012no—
300C2×C4C2×C22⋊AGL(1,𝔽5)13yes—
301C2×C4C23×AGL(1,𝔽5)11yescharacteristic
302C8—204no—
303C5C22×#32811yescharacteristic, complemented, Sylow
304C22—51no—
305C22—51no—
306C22—56no—
307C22—101no—
308C22—103no—
309C22—101no—
310C22—103no—
311C22—1012no—
312C22—1024no—
313C22—1012no—
314C22#64869116yesdirect product factor
315C22C2×#399916yes—
316C22C2×#400816yes—
317C22C2×#400916yes—
318C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
319C4—23no—
320C4—21no—
321C4—44no—
322C4—101no—
323C4—103no—
324C4—208no—
325C4—204no—
326C2—101no—
327C2—101no—
328C2—106no—
329C2C2×#64869112yesdirect product factor
330C2C22×#399911yescharacteristic
331C2C22×#400811yescharacteristic
332C2C22×#400911yescharacteristic
subgroupG/H
centerC24C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC24×C5C22⋊C4
prime radicalC24×D5D4
cosocleC2×C4×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C22×#328
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#64869C2direct (N × H)
7. C23×Dic10C4semidirect
9. C22×C5×C4⋊C4C4semidirect
14. C2×#4043C4semidirect
20. #64869C22direct (N × H)
38. C22×Dic10C2×C4semidirect
42. C2×C5×C4⋊C4C2×C4semidirect
50. #4043C2×C4semidirect
94. C2×Dic10C22×C4semidirect
98. C5×C4⋊C4C22×C4semidirect
112. C22×C4⋊C4AGL(1,𝔽5)semidirect
185. C2×C4⋊C4C2×AGL(1,𝔽5)semidirect
192. C22×C5#328semidirect
254. C4⋊C4C22×AGL(1,𝔽5)semidirect
257. C2×C5C2×#328semidirect
303. C5C22×#328semidirect
314. C22#64869direct (N × H)
329. C2C2×#64869direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40432
C22×(C2×AGL(1,𝔽5))⋊C42
C22×#40032
C2×#648692
C22×#3285