Finite Group Databasebeta

← C22×#64868 #13030182

Subgroups — C22×#64868 #13030182

Order 1280.

2190 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 2190 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×#4041C211yesmaximal, characteristic
2C22×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
3C22×#4003C211yesmaximal, characteristic, complemented
4C2×#64868C2112yesmaximal, direct product factor
5C24×AGL(1,𝔽5)—21no—
6C23×D20—21no—
7C23×D20C411yescharacteristic, complemented
8C23×C4×D5C2211yescharacteristic
9C23×#170—21no—
10C22×C5×C4⋊C4C411yescharacteristic, complemented
11C22×#685—21no—
12C2×#4041—23no—
13C2×#4041—23no—
14C2×#4041C416yescomplemented
15C2×#4041C2216yes—
16C2×(C2×AGL(1,𝔽5))⋊C4—23no—
17C2×(C2×AGL(1,𝔽5))⋊C4C2216yescomplemented
18C2×#4003—23no—
19C2×#4003C2216yescomplemented
20#64868C22116yesdirect product factor
21C22×#327—51nomaximal, Sylow
22C24×D5—21no—
23C24×D5D411yescharacteristic
24C23×AGL(1,𝔽5)—28no—
25C23×AGL(1,𝔽5)—26no—
26C23×AGL(1,𝔽5)—44no—
27C23×AGL(1,𝔽5)—44no—
28C23×C4×C5—21no—
29C23×C4×C5C2×C411yescharacteristic
30C23×Dic5D411yescharacteristic
31C22×D20—26no—
32C22×D20—26no—
33C22×D20—23no—
34C22×D20—44no—
35C22×D20—44no—
36C22×D20—44no—
37C22×D20C2×C416yescomplemented
38C22×C4×D5—44no—
39C22×C4×D5C2×C413yes—
40C22×C4×D5D413yes—
41C22×C4×D5D413yes—
42C22×C4×D5C2313yes—
43C22×#170—26no—
44C22×#170—44no—
45C2×C5×C4⋊C4—23no—
46C2×C5×C4⋊C4C2×C416yescomplemented
47C2×#685—26no—
48C2×#685—26no—
49#4041—212no—
50#4041—26no—
51#4041—26no—
52#4041C2×C4112yescomplemented
53#4041C2314yes—
54(C2×AGL(1,𝔽5))⋊C4—26no—
55(C2×AGL(1,𝔽5))⋊C4C2314yescomplemented
56#4003—26no—
57#4003C2314yescomplemented
58C22×#121—51no—
59C22×C22⋊C8—51no—
60C22×#95—51no—
61C2×#327—512no—
62C24×C5C22⋊C411yescharacteristic
63C23×D5—24no—
64C23×D5—24no—
65C23×D5—26no—
66C23×D5—44no—
67C23×D5—44no—
68C23×D5—44no—
69C23×D5C22⋊C413yes—
70C23×D5D814yes—
71C23×D5SD1614yes—
72C23×D5C2×D413yes—
73C22×AGL(1,𝔽5)—224no—
74C22×AGL(1,𝔽5)—24no—
75C22×AGL(1,𝔽5)—432no—
76C22×AGL(1,𝔽5)—412no—
77C22×AGL(1,𝔽5)—412no—
78C22×C4×C5—24no—
79C22×C4×C5—26no—
80C22×C4×C5—44no—
81C22×C4×C5C22⋊C413yes—
82C22×C4×C5C22×C413yes—
83C22×Dic5—24no—
84C22×Dic5C22⋊C413yes—
85C22×Dic5C2×D413yes—
86C2×D20—212no—
87C2×D20—24no—
88C2×D20—26no—
89C2×D20—412no—
90C2×D20—412no—
91C2×D20—424no—
92C2×D20—412no—
93C2×D20—412no—
94C2×D20C22×C414yescomplemented
95C2×C4×D5—46no—
96C2×C4×D5—46no—
97C2×C4×D5—46no—
98C2×C4×D5—46no—
99C2×C4×D5C22⋊C416yes—
100C2×C4×D5C22×C413yes—
101C2×C4×D5C2×D413yes—
102C2×C4×D5C2×D413yes—
103C2×C4×D5C2411yescharacteristic
104C2×#170—24no—
105C2×#170—412no—
106C5×C4⋊C4—26no—
107C5×C4⋊C4C22×C414yescomplemented
108#685—212no—
109#685—24no—
110C23×D4—51no—
111C23×D4—101no—
112C24×C4—51no—
113C24×C4—101no—
114C23×C8—101no—
115C2×#121—56no—
116C2×#121—56no—
117C2×#121—103no—
118C2×#121—103no—
119C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
120C22×C22⋊C4—101no—
121C2×C22⋊C8—56no—
122C2×C22⋊C8—103no—
123C2×#95—56no—
124C2×#95—103no—
125#327—516no—
126C23×C5#9914yes—
127C23×C5#10214yes—
128C23×C5#10414yes—
129C23×C5C2×C22⋊C413yes—
130C22×D5—216no—
131C22×D5—216no—
132C22×D5—26no—
133C22×D5—26no—
134C22×D5—26no—
135C22×D5—26no—
136C22×D5—24no—
137C22×D5—412no—
138C22×D5—412no—
139C22×D5—412no—
140C22×D5—816no—
141C22×D5#102112yes—
142C22×D5C2×C22⋊C413yes—
143C22×D5C2×D816yes—
144C22×D5C2×SD1616yes—
145C22×D5C22×D411yescharacteristic
146C2×AGL(1,𝔽5)—28no—
147C2×AGL(1,𝔽5)—44no—
148C2×AGL(1,𝔽5)—448no—
149C2×AGL(1,𝔽5)—44no—
150C2×C4×C5—26no—
151C2×C4×C5—26no—
152C2×C4×C5—24no—
153C2×C4×C5—412no—
154C2×C4×C5C2×C22⋊C413yes—
155C2×C4×C5C23×C411yescharacteristic
156C2×Dic5—26no—
157C2×Dic5—26no—
158C2×Dic5C2×C22⋊C413yes—
159C2×Dic5C22×D411yescharacteristic
160D20—44no—
161D20—412no—
162D20—44no—
163D20—44no—
164D20—424no—
165C4×D5—46no—
166C4×D5—43no—
167C4×D5—43no—
168C4×D5—43no—
169C4×D5—41no—
170#170—44no—
171C25—51no—
172C25—101no—
173C22×D4—56no—
174C22×D4—106no—
175C22×D4—106no—
176C22×D4—103no—
177C22×D4—204no—
178C22×D4—204no—
179C22×D4—204no—
180C23×C4—21no—
181C23×C4—51no—
182C23×C4—53no—
183C23×C4—53no—
184C23×C4—53no—
185C23×C4—53no—
186C23×C4—106no—
187C23×C4—108no—
188C23×C4—204no—
189C23×C4—204no—
190C23×C4—204no—
191C23×C4C2×AGL(1,𝔽5)11yescharacteristic
192C22×C8—106no—
193C22×C8—204no—
194#121—54no—
195#121—512no—
196#121—106no—
197#121—106no—
198#121—1012no—
199C2×C4⋊C4—23no—
200C2×C4⋊C4C2×AGL(1,𝔽5)16yescomplemented
201C2×C22⋊C4—106no—
202C2×C22⋊C4—106no—
203C22⋊C8—54no—
204C22⋊C8—106no—
205#95—54no—
206#95—106no—
207C22×C5#327116yescomplemented
208C22×C5C2×#9916yes—
209C22×C5C2×#10216yes—
210C22×C5C2×#10416yes—
211C22×C5C22×C22⋊C411yescharacteristic
212C2×D5—21no—
213C2×D5—23no—
214C2×D5—21no—
215C2×D5—23no—
216C2×D5—212no—
217C2×D5—224no—
218C2×D5—212no—
219C2×D5—44no—
220C2×D5—44no—
221C2×D5—44no—
222C2×D5—824no—
223C2×D5C2×#10216yes—
224C2×D5C22×D811yescharacteristic
225C2×D5C22×SD1611yescharacteristic
226AGL(1,𝔽5)—48no—
227C4×C5—23no—
228C4×C5—21no—
229C4×C5—44no—
230Dic5—21no—
231Dic5—23no—
232C24—53no—
233C24—53no—
234C24—54no—
235C24—54no—
236C24—104no—
237C24—104no—
238C24—106no—
239C24—204no—
240C24—204no—
241C24—204no—
242C24C22⋊AGL(1,𝔽5)11yescharacteristic
243C2×D4—54no—
244C2×D4—104no—
245C2×D4—1012no—
246C2×D4—106no—
247C2×D4—2012no—
248C2×D4—2012no—
249C2×D4—2012no—
250C2×D4—2024no—
251C2×D4—2012no—
252C22×C4—24no—
253C22×C4—26no—
254C22×C4—44no—
255C22×C4—53no—
256C22×C4—53no—
257C22×C4—53no—
258C22×C4—53no—
259C22×C4—56no—
260C22×C4—51no—
261C22×C4—53no—
262C22×C4—104no—
263C22×C4—104no—
264C22×C4—1024no—
265C22×C4—206no—
266C22×C4—206no—
267C22×C4—206no—
268C22×C4—206no—
269C22×C4—2012no—
270C22×C4—2032no—
271C22×C4—2012no—
272C22×C4C22⋊AGL(1,𝔽5)13yes—
273C22×C4C22×AGL(1,𝔽5)13yes—
274C2×C8—104no—
275C2×C8—2012no—
276C4⋊C4—26no—
277C4⋊C4C22×AGL(1,𝔽5)14yescomplemented
278C22⋊C4—104no—
279C22⋊C4—1012no—
280C2×C5C2×#327112yescomplemented
281C2×C5C22×#9911yescharacteristic
282C2×C5C22×#10211yescharacteristic
283C2×C5C22×#10411yescharacteristic
284D5—21no—
285D5—21no—
286D5—26no—
287D5—84no—
288C23—51no—
289C23—53no—
290C23—56no—
291C23—512no—
292C23—56no—
293C23—1016no—
294C23—1016no—
295C23—106no—
296C23—106no—
297C23—106no—
298C23—106no—
299C23—104no—
300C23—2012no—
301C23—2012no—
302C23—2012no—
303C23—4016no—
304C23#399914yes—
305C23#400714yes—
306C23#401014yes—
307C23C2×C22⋊AGL(1,𝔽5)13yes—
308D4—204no—
309D4—2012no—
310D4—204no—
311D4—204no—
312D4—2024no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—24no—
316C2×C4—412no—
317C2×C4—51no—
318C2×C4—53no—
319C2×C4—106no—
320C2×C4—106no—
321C2×C4—108no—
322C2×C4—203no—
323C2×C4—203no—
324C2×C4—206no—
325C2×C4—201no—
326C2×C4—203no—
327C2×C4—204no—
328C2×C4—204no—
329C2×C4—2048no—
330C2×C4C2×C22⋊AGL(1,𝔽5)13yes—
331C2×C4C23×AGL(1,𝔽5)11yescharacteristic
332C8—204no—
333C5C22×#32711yescharacteristic, complemented, Sylow
334C22—51no—
335C22—51no—
336C22—56no—
337C22—101no—
338C22—103no—
339C22—101no—
340C22—103no—
341C22—1012no—
342C22—1024no—
343C22—1012no—
344C22—204no—
345C22—204no—
346C22—204no—
347C22—4024no—
348C22#64868116yesdirect product factor
349C22C2×#399916yes—
350C22C2×#400716yes—
351C22C2×#401016yes—
352C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
353C4—23no—
354C4—21no—
355C4—44no—
356C4—101no—
357C4—103no—
358C4—208no—
359C2—101no—
360C2—101no—
361C2—106no—
362C2—404no—
363C2C2×#64868112yesdirect product factor
364C2C22×#399911yescharacteristic
365C2C22×#400711yescharacteristic
366C2C22×#401011yescharacteristic
subgroupG/H
centerC24C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC24×C5C22⋊C4
prime radicalC22×#4041C2
cosocleC2×C4×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C22×#327
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
2. C22×(C2×AGL(1,𝔽5))⋊C4C2semidirect
3. C22×#4003C2semidirect
4. C2×#64868C2direct (N × H)
7. C23×D20C4semidirect
10. C22×C5×C4⋊C4C4semidirect
14. C2×#4041C4semidirect
17. C2×(C2×AGL(1,𝔽5))⋊C4C22semidirect
19. C2×#4003C22semidirect
20. #64868C22direct (N × H)
37. C22×D20C2×C4semidirect
46. C2×C5×C4⋊C4C2×C4semidirect
52. #4041C2×C4semidirect
55. (C2×AGL(1,𝔽5))⋊C4C23semidirect
57. #4003C23semidirect
94. C2×D20C22×C4semidirect
107. C5×C4⋊C4C22×C4semidirect
119. C22×C4⋊C4AGL(1,𝔽5)semidirect
200. C2×C4⋊C4C2×AGL(1,𝔽5)semidirect
207. C22×C5#327semidirect
277. C4⋊C4C22×AGL(1,𝔽5)semidirect
280. C2×C5C2×#327semidirect
333. C5C22×#327semidirect
348. C22#64868direct (N × H)
363. C2C2×#64868direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40412
C22×(C2×AGL(1,𝔽5))⋊C42
C22×#40032
C2×#648682
C22×#3275