Finite Group Databasebeta

← #13016781

Subgroups — #13016781

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×#65014C211yesmaximal, characteristic
2C4×#4011C211yesmaximal, characteristic
3#10604878C211yesmaximal, characteristic
4C8×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
5#10598766C212yesmaximal, complemented
6C2×C4×C22⋊D5C2211yescharacteristic
7C23×#168C411yescharacteristic, complemented
8C22×C8×D5C2211yescharacteristic
9C2×#4011C412yes—
10C4×C22⋊AGL(1,𝔽5)C2212yes—
11D5⋊(C22⋊C8)C2212yes—
12C2×C8×AGL(1,𝔽5)—22no—
13C2×C5×C22⋊C8C411yescharacteristic, complemented
14C2×#3952C2211yescharacteristic
15C2×#3946C411yescharacteristic, complemented
16C2×C8×Dic5C411yescharacteristic, complemented
17#65014—22no—
18#65014—22no—
19#64933C412yescomplemented
20#64917C412yes—
21#64914C412yescomplemented
22(C2×AGL(1,𝔽5))⋊C8—22no—
23C8×#99—51nomaximal, Sylow
24C23×C4×C5C2×C411yescharacteristic
25C22×C22⋊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×C22⋊Dic5C2×C411yescharacteristic
33C4×C22⋊D5—22no—
34C4×C22⋊D5—22no—
35C2×#685C2×C411yescharacteristic
36C2×#683C2×C411yescharacteristic
37C2×C4×Dic5C2×C411yescharacteristic
38C22×#168—21no—
39C22×#168—21no—
40C22×#168—42no—
41C22×#168C2×C411yescharacteristic
42C22×#168C2×C411yescharacteristic
43C2×C8×D5—42no—
44C2×C8×D5D412yes—
45C2×C8×D5D412yes—
46C22⋊#170C2×C414yes—
47#4011C818yescomplemented
48(C2×AGL(1,𝔽5))⋊C4—22no—
49#4003—22no—
50C8×AGL(1,𝔽5)—44no—
51C5×C22⋊C8—22no—
52#3952—22no—
53#3946—22no—
54C8×Dic5—21no—
55C8×Dic5—21no—
56C2×C8×D4—51no—
57C4×#99—51no—
58C8×C22⋊C4—52no—
59#1459—51no—
60#1282—52no—
61C24×C5C4211yescharacteristic
62C23×D5C22×C411yescharacteristic
63C22×AGL(1,𝔽5)—24no—
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—41no—
67C22×C4×C5—41no—
68C22×C4×C5C4211yescharacteristic
69C22×C4×C5C22×C411yescharacteristic
70C2×C22⋊D5—22no—
71C2×C22⋊D5—22no—
72C2×C22⋊D5C2×C814yes—
73C22×Dic5C4211yescharacteristic
74C22×Dic5C4211yescharacteristic
75C22×Dic5C22×C411yescharacteristic
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—
91C22⋊Dic5—22no—
92#685—22no—
93#683—22no—
94C4×Dic5—21no—
95C4×Dic5—21no—
96C2×#168—22no—
97C2×#168—21no—
98C2×#168—42no—
99C2×#168—42no—
100C2×#168—41no—
101C2×#168C22⋊C411yescharacteristic
102C2×#168C22⋊C411yescharacteristic
103C8×D5—24no—
104C8×D5—42no—
105C23×C8—51no—
106C23×C8—51no—
107C2×C4×D4—51no—
108C8×D4—102no—
109C8×D4—102no—
110C2×C4⋊C8—51no—
111C2×#99—52no—
112C2×C22⋊C8—52no—
113C2×C22⋊C8—51no—
114C2×C22⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
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×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5C4×C812yescomplemented
127C23×C5C2×C4211yescharacteristic
128C22×D5—41no—
129C22×D5—41no—
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—42no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5C2×C22⋊C411yescharacteristic
144C2×C4×C5#11711yescharacteristic
145C22⋊D5—28no—
146C2×Dic5—21no—
147C2×Dic5—21no—
148C2×Dic5—21no—
149C2×Dic5—41no—
150C2×Dic5—41no—
151C2×Dic5C4×C812yescomplemented
152C2×Dic5C4×D412yes—
153C4×D5—22no—
154C4×D5—22no—
155C4×D5—42no—
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×C4C2×AGL(1,𝔽5)11yescharacteristic
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—51no—
183C2×C42—102no—
184C4⋊C8—102no—
185#99—58no—
186C22⋊C8—22no—
187C22⋊C8—54no—
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×C5—22no—
197C22×C5—42no—
198C22×C5—42no—
199C22×C5—41no—
200C22×C5—41no—
201C22×C5C4×C22⋊C411yescharacteristic
202C22×C5C2×C4×C811yescharacteristic
203C22×C5#40511yescharacteristic
204C2×D5—22no—
205C2×D5—22no—
206C2×D5—42no—
207C2×D5—42no—
208C2×D5C8×D414yes—
209AGL(1,𝔽5)—48no—
210C4×C5—21no—
211C4×C5—41no—
212C4×C5C2×#9911yescharacteristic
213C4×C5#41011yescharacteristic
214Dic5—22no—
215Dic5—22no—
216Dic5—41no—
217C24—51no—
218C24C4×AGL(1,𝔽5)11yescharacteristic
219C2×D4—54no—
220C2×D4—102no—
221C2×D4—102no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—52no—
229C22×C4—52no—
230C22×C4—51no—
231C22×C4—104no—
232C22×C4—201no—
233C22×C4—201no—
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—101no—
258C42—101no—
259C42—202no—
260C42—202no—
261C2×C5—21no—
262C2×C5—21no—
263C2×C5—42no—
264C2×C5C8×C22⋊C411yescharacteristic
265C2×C5C4×#9911yescharacteristic
266C2×C5#175711yescharacteristic
267D5—24no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—52no—
273C23—52no—
274C23—52no—
275C23—201no—
276C23—201no—
277C23C8×AGL(1,𝔽5)12yescomplemented
278C23C2×C4×AGL(1,𝔽5)11yescharacteristic
279D4—108no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—52no—
288C2×C4—52no—
289C2×C4—101no—
290C2×C4—101no—
291C2×C4—102no—
292C2×C4—102no—
293C2×C4—108no—
294C2×C4—101no—
295C2×C4—202no—
296C2×C4—201no—
297C2×C4—204no—
298C2×C4—204no—
299C2×C4—201no—
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—22no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—54no—
317C22—102no—
318C22—102no—
319C22—202no—
320C22—202no—
321C22C2×C8×AGL(1,𝔽5)11yescharacteristic
322C22#6571811yescharacteristic
323C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
324C4—21no—
325C4—41no—
326C4—102no—
327C4—102no—
328C4—208no—
329C4—201no—
330C4D5⋊#9911yescharacteristic
331C4#6577111yescharacteristic
332C2—21no—
333C2—21no—
334C2—42no—
335C2—104no—
336C2#1060421611yescharacteristic
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×C22⋊C8C4
socleC22×C5C4×C22⋊C4
prime radicalC22×C22⋊D5C2×C4
cosocleC22×C4×D5C23
hypercenterC2×C22⋊C8AGL(1,𝔽5)
hypocenterC5C8×#99
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C8×C22⋊AGL(1,𝔽5)C2semidirect
5. #10598766C2semidirect
7. C23×#168C4semidirect
13. C2×C5×C22⋊C8C4semidirect
15. C2×#3946C4semidirect
16. C2×C8×Dic5C4semidirect
19. #64933C4semidirect
21. #64914C4semidirect
47. #4011C8semidirect
81. C22⋊AGL(1,𝔽5)C2×C8semidirect
114. C2×C22⋊C8AGL(1,𝔽5)semidirect
126. C23×C5C4×C8semidirect
151. C2×Dic5C4×C8semidirect
161. #168#99semidirect
277. C23C8×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×#650142
C4×#40112
#106048782
C8×C22⋊AGL(1,𝔽5)2
#105987662
C8×#995