Finite Group Databasebeta

← C2×#10604307 #13034364

Subgroups — C2×#10604307 #13034364

Order 1280.

1566 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 280 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×#4038C211yesmaximal, characteristic
2C2×#65764C212yesmaximal
3C2×#64917C214yesmaximal
4#10604307C218yesmaximal, direct product factor
5C23×C4×D5C2211yescharacteristic
6C22×C22⋊AGL(1,𝔽5)C2214yes—
7C22×C5×C4⋊C4C411yescharacteristic, complemented
8C22×#685C2212yes—
9C22×C4⋊Dic5C411yescharacteristic, complemented
10C22×C4×Dic5C412yescomplemented
11C2×#4038—24no—
12C2×#4038—24no—
13C2×#4038C414yescomplemented
14C2×#4038C2214yes—
15#65764—24no—
16#65764C2218yes—
17C2×(C2×AGL(1,𝔽5))⋊C4—24no—
18#64917C22116yes—
19C2×#1820—51nomaximal, Sylow
20C24×D5C2311yescharacteristic
21C23×AGL(1,𝔽5)—24no—
22C23×C4×C5C2×C412yes—
23C23×C4×C5C2×C411yescharacteristic
24C23×Dic5C2×C411yescharacteristic
25C23×Dic5C2×C412yes—
26C22×C4×D5—28no—
27C22×C4×D5C2×C412yes—
28C22×C4×D5D414yes—
29C22×C4×D5D414yes—
30C22×C4×D5C2312yes—
31C2×C22⋊AGL(1,𝔽5)—216no—
32C2×C22⋊AGL(1,𝔽5)—24no—
33C2×C22⋊AGL(1,𝔽5)C2318yes—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4—22no—
36C2×C5×C4⋊C4C2×C414yescomplemented
37C2×#685—24no—
38C2×#685—24no—
39C2×#685C2×C414yes—
40C2×#685C2314yes—
41C2×C4⋊Dic5—22no—
42C2×C4⋊Dic5—22no—
43C2×C4⋊Dic5C2×C414yescomplemented
44C2×C4×Dic5—24no—
45C2×C4×Dic5—24no—
46C2×C4×Dic5C2×C418yescomplemented
47#4038—216no—
48#4038—216no—
49(C2×AGL(1,𝔽5))⋊C4—216no—
50C22×#117—51no—
51C2×#381—52no—
52#1820—58no—
53C2×#342—54no—
54C24×C5C22×C411yescharacteristic
55C23×D5—42no—
56C23×D5—42no—
57C23×D5C22×C411yescharacteristic
58C23×D5C2×D412yes—
59C23×D5C2×D412yes—
60C23×D5#4118yes—
61C23×D5C2411yescharacteristic
62C22×AGL(1,𝔽5)—28no—
63C22×AGL(1,𝔽5)—48no—
64C22×AGL(1,𝔽5)—48no—
65C22×AGL(1,𝔽5)—48no—
66C22×C4×C5—24no—
67C22×C4×C5—22no—
68C22×C4×C5—22no—
69C22×C4×C5—44no—
70C22×C4×C5C22⋊C412yes—
71C22×C4×C5C22⋊C412yes—
72C22×C4×C5C22×C412yes—
73C22×C4×C5C22×C414yes—
74C22×Dic5—24no—
75C22×Dic5—22no—
76C22×Dic5—22no—
77C22×Dic5—44no—
78C22×Dic5C22⋊C412yes—
79C22×Dic5C22⋊C412yes—
80C22×Dic5C22×C412yes—
81C22×Dic5C22×C414yes—
82C2×C4×D5—28no—
83C2×C4×D5—216no—
84C2×C4×D5—216no—
85C2×C4×D5—28no—
86C2×C4×D5C22⋊C418yes—
87C2×C4×D5C2×D418yes—
88C22⋊AGL(1,𝔽5)—216no—
89C22⋊AGL(1,𝔽5)—48no—
90C5×C4⋊C4—28no—
91#685—28no—
92#685—28no—
93C4⋊Dic5—28no—
94C4×Dic5—216no—
95C24×C4—51no—
96C2×#117—54no—
97C2×#117—54no—
98C2×#117—104no—
99C2×#117—104no—
100C22×C4⋊C4—51no—
101C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
102C22×C22⋊C4—52no—
103C22×C22⋊C4—54no—
104C22×C42—52no—
105#381—58no—
106#381—104no—
107C2×#95—104no—
108#342—516no—
109C23×C5—22no—
110C23×C5—22no—
111C23×C5C2×C22⋊C411yescharacteristic
112C23×C5C2×C22⋊C411yescharacteristic
113C23×C5#11714yes—
114C23×C5C23×C411yescharacteristic
115C22×D5—24no—
116C22×D5—24no—
117C22×D5—41no—
118C22×D5—41no—
119C22×D5—42no—
120C22×D5—42no—
121C22×D5—416no—
122C22×D5—41no—
123C22×D5—41no—
124C22×D5C2×C22⋊C412yes—
125C22×D5#11714yes—
126C22×D5#12318yes—
127C22×D5#12318yes—
128C22×D5C22×D412yes—
129C22×D5C2×#4114yes—
130C2×AGL(1,𝔽5)—48no—
131C2×AGL(1,𝔽5)—48no—
132C2×AGL(1,𝔽5)—48no—
133C2×AGL(1,𝔽5)—816no—
134C2×C4×C5—24no—
135C2×C4×C5—24no—
136C2×C4×C5—24no—
137C2×C4×C5—24no—
138C2×C4×C5—44no—
139C2×C4×C5—44no—
140C2×C4×C5—44no—
141C2×C4×C5C2×C22⋊C414yes—
142C2×Dic5—24no—
143C2×Dic5—24no—
144C2×Dic5—24no—
145C2×Dic5—24no—
146C2×Dic5—44no—
147C2×Dic5—44no—
148C2×Dic5—44no—
149C2×Dic5C2×C22⋊C414yes—
150C4×D5—216no—
151C4×D5—216no—
152C25—51no—
153C23×C4—51no—
154C23×C4—54no—
155C23×C4—54no—
156C23×C4—52no—
157C23×C4—52no—
158C23×C4—52no—
159C23×C4—108no—
160C23×C4—104no—
161C23×C4C2×AGL(1,𝔽5)12yes—
162C23×C4C2×AGL(1,𝔽5)11yescharacteristic
163#117—1016no—
164#117—1016no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—54no—
168C2×C4⋊C4—102no—
169C2×C4⋊C4—102no—
170C2×C4⋊C4C2×AGL(1,𝔽5)14yescomplemented
171C2×C22⋊C4—54no—
172C2×C22⋊C4—54no—
173C2×C22⋊C4—58no—
174C2×C22⋊C4—104no—
175C2×C22⋊C4—104no—
176C2×C22⋊C4—1016no—
177C2×C22⋊C4—104no—
178C2×C42—58no—
179C2×C42—104no—
180C2×C42—104no—
181#95—1016no—
182C22×C5—28no—
183C22×C5—21no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5#38112yes—
188C22×C5#38112yes—
189C22×C5#41012yes—
190C22×C5#41012yes—
191C22×C5C22×C22⋊C411yescharacteristic
192C22×C5C2×#11712yes—
193C2×D5—216no—
194C2×D5—22no—
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—22no—
198C2×D5—44no—
199C2×D5—48no—
200C2×D5—44no—
201C2×D5#38114yes—
202C2×D5C2×#12314yes—
203AGL(1,𝔽5)—88no—
204C4×C5—24no—
205C4×C5—44no—
206Dic5—24no—
207Dic5—44no—
208C24—52no—
209C24—58no—
210C24—52no—
211C24—51no—
212C24—51no—
213C24—202no—
214C24—202no—
215C24C22×AGL(1,𝔽5)11yescharacteristic
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—44no—
220C22×C4—52no—
221C22×C4—52no—
222C22×C4—52no—
223C22×C4—58no—
224C22×C4—58no—
225C22×C4—54no—
226C22×C4—108no—
227C22×C4—108no—
228C22×C4—1016no—
229C22×C4—1016no—
230C22×C4—104no—
231C22×C4—102no—
232C22×C4—102no—
233C22×C4—108no—
234C22×C4—204no—
235C22×C4—208no—
236C22×C4—208no—
237C22×C4—208no—
238C22×C4C22⋊AGL(1,𝔽5)12yes—
239C22×C4C22⋊AGL(1,𝔽5)12yes—
240C22×C4C22×AGL(1,𝔽5)12yes—
241C22×C4C22×AGL(1,𝔽5)14yes—
242C4⋊C4—28no—
243C4⋊C4—108no—
244C22⋊C4—108no—
245C22⋊C4—108no—
246C22⋊C4—1016no—
247C22⋊C4—208no—
248C42—1016no—
249C2×C5—22no—
250C2×C5—22no—
251C2×C5#182014yescomplemented
252C2×C5C2×#38111yescharacteristic
253C2×C5C2×#41011yescharacteristic
254C2×C5C2×#41011yescharacteristic
255D5—24no—
256D5—24no—
257C23—22no—
258C23—22no—
259C23—52no—
260C23—52no—
261C23—58no—
262C23—54no—
263C23—54no—
264C23—58no—
265C23—104no—
266C23—104no—
267C23—2016no—
268C23—202no—
269C23—201no—
270C23—201no—
271C23—202no—
272C23—201no—
273C23—201no—
274C23#413014yes—
275C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
276C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
277C23C23×AGL(1,𝔽5)11yescharacteristic
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—54no—
286C2×C4—1016no—
287C2×C4—1016no—
288C2×C4—104no—
289C2×C4—104no—
290C2×C4—104no—
291C2×C4—104no—
292C2×C4—204no—
293C2×C4—204no—
294C2×C4—204no—
295C2×C4—208no—
296C2×C4—208no—
297C2×C4—208no—
298C2×C4—4016no—
299C2×C4C2×C22⋊AGL(1,𝔽5)14yes—
300C5C2×#182011yescharacteristic, complemented, Sylow
301C22—28no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—54no—
307C22—54no—
308C22—1016no—
309C22—102no—
310C22—102no—
311C22—102no—
312C22—102no—
313C22—204no—
314C22—204no—
315C22—208no—
316C22#6576412yes—
317C22#6576412yes—
318C22#6577112yes—
319C22#6578512yes—
320C22C2×#413012yes—
321C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
322C4—24no—
323C4—44no—
324C4—104no—
325C4—204no—
326C4—408no—
327C2—22no—
328C2—22no—
329C2—104no—
330C2—104no—
331C2#1060430714yesdirect product factor
332C2C2×#6576411yescharacteristic
333C2C2×#6577111yescharacteristic
334C2C2×#6578511yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
prime radicalC24×D5C23
cosocleC23×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5C2×#1820
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #10604307C2direct (N × H)
7. C22×C5×C4⋊C4C4semidirect
9. C22×C4⋊Dic5C4semidirect
10. C22×C4×Dic5C4semidirect
13. C2×#4038C4semidirect
36. C2×C5×C4⋊C4C2×C4semidirect
43. C2×C4⋊Dic5C2×C4semidirect
46. C2×C4×Dic5C2×C4semidirect
101. C22×C4⋊C4AGL(1,𝔽5)semidirect
170. C2×C4⋊C4C2×AGL(1,𝔽5)semidirect
251. C2×C5#1820semidirect
300. C5C2×#1820semidirect
331. C2#10604307direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40382
C2×#657642
C2×#649172
#106043072
C2×#18205