Finite Group Databasebeta

← #13016777

Subgroups — #13016777

Order 1280.

614 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 138 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×#65123C211yesmaximal, characteristic
2C8×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
3C4×#3999C211yesmaximal, characteristic
4#10604160C211yesmaximal, characteristic
5#10598766C212yesmaximal, complemented
6C22×C8×D5C411yescharacteristic, complemented
7C22×C8×D5C2211yescharacteristic
8C2×C4×D20C2211yescharacteristic
9C4×C22⋊AGL(1,𝔽5)C2212yes—
10D5⋊(C22⋊C8)C2212yes—
11C2×#3999C412yes—
12C2×C8×AGL(1,𝔽5)—22no—
13C2×C5×C4⋊C8C411yescharacteristic, complemented
14C2×#3952C411yescharacteristic, complemented
15C2×#3952C2211yescharacteristic
16C2×C4×#168C411yescharacteristic, complemented
17#65123—22no—
18#65123—22no—
19#64917C412yes—
20#64915C412yescomplemented
21#64913C412yescomplemented
22(C2×AGL(1,𝔽5))⋊C8—22no—
23C8×#99—51nomaximal, Sylow
24C22×D20C2×C411yescharacteristic
25C22×C4×D5C2×C411yescharacteristic
26C22×C4×D5C2311yescharacteristic
27C2×C22⋊AGL(1,𝔽5)C2×C414yes—
28C2×C4×AGL(1,𝔽5)—22no—
29C2×D5⋊C8—22no—
30C22×C5×C8C2×C411yescharacteristic
31C22×C5×C8C2×C411yescharacteristic
32C2×C42×C5C2×C411yescharacteristic
33C2×#685C2×C411yescharacteristic
34C2×#685C2×C411yescharacteristic
35C2×C4⋊Dic5C2×C411yescharacteristic
36C22×#168C2×C411yescharacteristic
37C22×#168C2×C411yescharacteristic
38C2×C8×D5—21no—
39C2×C8×D5—21no—
40C2×C8×D5—42no—
41C2×C8×D5—42no—
42C2×C8×D5D412yes—
43C2×C8×D5D412yes—
44C4×D20—22no—
45C4×D20—22no—
46C22⋊#170C2×C414yes—
47(C2×AGL(1,𝔽5))⋊C4—22no—
48#4003—22no—
49#3999C818yescomplemented
50C8×AGL(1,𝔽5)—44no—
51C5×C4⋊C8—22no—
52#3952—22no—
53#3952—22no—
54C4×#168—21no—
55C4×#168—21no—
56C2×C8×D4—51no—
57C4×#99—51no—
58C8×C22⋊C4—52no—
59#1459—51no—
60#1282—52no—
61C23×D5C4211yescharacteristic
62C23×D5C22×C411yescharacteristic
63C22×AGL(1,𝔽5)—24no—
64C22×C4×C5C4211yescharacteristic
65C22×C4×C5C4211yescharacteristic
66C22×C4×C5C22×C411yescharacteristic
67C22×Dic5C4211yescharacteristic
68C22×Dic5C22×C411yescharacteristic
69C2×D20—22no—
70C2×D20—22no—
71C2×D20C2×C814yes—
72C2×C4×D5—21no—
73C2×C4×D5—21no—
74C2×C4×D5—41no—
75C2×C4×D5—41no—
76C2×C4×D5—41no—
77C2×C4×D5—41no—
78C2×C4×D5C2×D412yes—
79C2×C4×D5#4112yes—
80C22⋊AGL(1,𝔽5)—24no—
81C22⋊AGL(1,𝔽5)C2×C818yescomplemented
82C2×#170—24no—
83C4×AGL(1,𝔽5)—42no—
84C4×AGL(1,𝔽5)—42no—
85D5⋊C8—42no—
86D5⋊C8—42no—
87C2×C5×C8—21no—
88C2×C5×C8—41no—
89C2×C5×C8—41no—
90C2×C5×C8C22⋊C412yes—
91C42×C5—21no—
92C42×C5—21no—
93#685—22no—
94#685—22no—
95C4⋊Dic5—22no—
96C2×#168—22no—
97C2×#168—21no—
98C2×#168—41no—
99C2×#168C22⋊C411yescharacteristic
100C2×#168C22⋊C411yescharacteristic
101C8×D5—24no—
102C8×D5—42no—
103C8×D5—42no—
104C8×D5—42no—
105C23×C8—51no—
106C23×C8—51no—
107C2×C4×D4—51no—
108C8×D4—102no—
109C8×D4—102no—
110C2×C4⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
111C2×#99—52no—
112C2×C22⋊C8—52no—
113C2×C22⋊C8—51no—
114C2×C22⋊C8—51no—
115C2×C4×C8—51no—
116C2×C4×C8—102no—
117C4×C22⋊C4—52no—
118#342—52no—
119#336—102no—
120#324—52no—
121#323—52no—
122C23×C5C2×C4211yescharacteristic
123C22×D5—21no—
124C22×D5—21no—
125C22×D5—41no—
126C22×D5—41no—
127C22×D5—41no—
128C22×D5—41no—
129C22×D5C4×C812yescomplemented
130C22×D5C4×D412yes—
131C22×D5C22×C812yes—
132C22×D5#13112yes—
133C2×AGL(1,𝔽5)—28no—
134C2×AGL(1,𝔽5)—44no—
135C2×AGL(1,𝔽5)—44no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—41no—
142C2×C4×C5C4×C812yescomplemented
143C2×C4×C5C2×C22⋊C411yescharacteristic
144C2×C4×C5#11711yescharacteristic
145C2×Dic5—21no—
146C2×Dic5—41no—
147C2×Dic5—41no—
148C2×Dic5C4×D412yes—
149D20—28no—
150C4×D5—22no—
151C4×D5—22no—
152C4×D5—42no—
153C4×D5—42no—
154C4×D5—41no—
155C4×D5—41no—
156#170—44no—
157C5×C8—22no—
158C5×C8—41no—
159#168—21no—
160#168—41no—
161#168#9912yescomplemented
162C22×D4—51no—
163C23×C4—51no—
164C23×C4—51no—
165C22×C8—51no—
166C22×C8—52no—
167C22×C8—52no—
168C22×C8—51no—
169C22×C8—101no—
170C22×C8—101no—
171C22×C8—102no—
172C22×C8—202no—
173C22×C8—202no—
174C22×C8C2×AGL(1,𝔽5)11yescharacteristic
175C22×C8C2×AGL(1,𝔽5)11yescharacteristic
176C4×D4—102no—
177C4×D4—102no—
178C2×C4⋊C4—51no—
179C2×C22⋊C4—54no—
180C2×C22⋊C4—51no—
181C2×C22⋊C4—51no—
182C2×C42—102no—
183C2×C42C2×AGL(1,𝔽5)11yescharacteristic
184C4⋊C8—22no—
185#99—58no—
186C22⋊C8—54no—
187C22⋊C8—102no—
188C22⋊C8—102no—
189C22⋊C8—102no—
190C4×C8—101no—
191C4×C8—101no—
192C4×C8—204no—
193#95—102no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5C4×C22⋊C411yescharacteristic
197C22×C5C2×C4×C811yescharacteristic
198C22×C5#40511yescharacteristic
199C2×D5—22no—
200C2×D5—22no—
201C2×D5—22no—
202C2×D5—42no—
203C2×D5—42no—
204C2×D5—42no—
205C2×D5—41no—
206C2×D5—41no—
207C2×D5—42no—
208C2×D5C8×D414yes—
209AGL(1,𝔽5)—48no—
210C4×C5—22no—
211C4×C5—21no—
212C4×C5—41no—
213C4×C5C2×#9911yescharacteristic
214C4×C5#41011yescharacteristic
215Dic5—22no—
216Dic5—41no—
217C24—51no—
218C24—51no—
219C2×D4—54no—
220C2×D4—102no—
221C2×D4—102no—
222C22×C4—51no—
223C22×C4—52no—
224C22×C4—52no—
225C22×C4—51no—
226C22×C4—104no—
227C22×C4—101no—
228C22×C4—101no—
229C22×C4—201no—
230C22×C4—201no—
231C22×C4—201no—
232C22×C4—201no—
233C22×C4C4×AGL(1,𝔽5)11yescharacteristic
234C22×C4C4×AGL(1,𝔽5)11yescharacteristic
235C22×C4C22×AGL(1,𝔽5)11yescharacteristic
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—51no—
240C2×C8—51no—
241C2×C8—104no—
242C2×C8—102no—
243C2×C8—104no—
244C2×C8—101no—
245C2×C8—202no—
246C2×C8—202no—
247C2×C8—202no—
248C2×C8—202no—
249C2×C8—202no—
250C2×C8—201no—
251C2×C8C22⋊AGL(1,𝔽5)12yes—
252C4⋊C4—102no—
253C22⋊C4—58no—
254C22⋊C4—102no—
255C22⋊C4—102no—
256C22⋊C4—104no—
257C42—21no—
258C42—21no—
259C42—202no—
260C42—202no—
261C2×C5—21no—
262C2×C5—21no—
263C2×C5C8×C22⋊C411yescharacteristic
264C2×C5C4×#9911yescharacteristic
265C2×C5#175711yescharacteristic
266D5—24no—
267D5—42no—
268C23—52no—
269C23—52no—
270C23—52no—
271C23—52no—
272C23—101no—
273C23—101no—
274C23—201no—
275C23—201no—
276C23—201no—
277C23—201no—
278C23C2×C4×AGL(1,𝔽5)11yescharacteristic
279D4—108no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—52no—
287C2×C4—102no—
288C2×C4—102no—
289C2×C4—108no—
290C2×C4—101no—
291C2×C4—202no—
292C2×C4—202no—
293C2×C4—201no—
294C2×C4—204no—
295C2×C4—204no—
296C2×C4—201no—
297C2×C4—201no—
298C2×C4—201no—
299C2×C4C8×AGL(1,𝔽5)12yescomplemented
300C2×C4#413011yescharacteristic
301C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
302C8—22no—
303C8—41no—
304C8—52no—
305C8—101no—
306C8—204no—
307C8—201no—
308C5C8×#9911yescharacteristic, complemented, Sylow
309C22—21no—
310C22—21no—
311C22—54no—
312C22—102no—
313C22—102no—
314C22—102no—
315C22—202no—
316C22—202no—
317C22—202no—
318C22—202no—
319C22—201no—
320C22—201no—
321C22C2×C8×AGL(1,𝔽5)11yescharacteristic
322C22#6571811yescharacteristic
323C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
324C4—21no—
325C4—22no—
326C4—41no—
327C4—102no—
328C4—208no—
329C4—201no—
330C4D5⋊#9911yescharacteristic
331C4#6578511yescharacteristic
332C2—21no—
333C2—21no—
334C2—104no—
335C2—202no—
336C2#1060430511yescharacteristic
337C2C8×C22⋊AGL(1,𝔽5)11yescharacteristic
338C2#1060476411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC22×C5C2×C4×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊C8C4
socleC22×C5C4×C22⋊C4
prime radicalC22×D20C2×C4
cosocleC22×C4×D5C23
hypercenterC2×C4⋊C8AGL(1,𝔽5)
hypocenterC5C8×#99
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C8×C22⋊AGL(1,𝔽5)C2semidirect
5. #10598766C2semidirect
6. C22×C8×D5C4semidirect
13. C2×C5×C4⋊C8C4semidirect
14. C2×#3952C4semidirect
16. C2×C4×#168C4semidirect
20. #64915C4semidirect
21. #64913C4semidirect
49. #3999C8semidirect
81. C22⋊AGL(1,𝔽5)C2×C8semidirect
110. C2×C4⋊C8AGL(1,𝔽5)semidirect
129. C22×D5C4×C8semidirect
142. C2×C4×C5C4×C8semidirect
161. #168#99semidirect
299. C2×C4C8×AGL(1,𝔽5)semidirect
308. C5C8×#99semidirect

Maximal subgroups

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

subgroupindex
C2×#651232
C8×C22⋊AGL(1,𝔽5)2
C4×#39992
#106041602
#105987662
C8×#995