Finite Group Databasebeta

← #11988458

Subgroups — #11988458

Order 1280.

350 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C5×#2805C211yesmaximal, characteristic
2#10600463C211yesmaximal, characteristic
3#10600404C211yesmaximal, characteristic
4#10600400C211yesmaximal, characteristic
5#10598133C211yesmaximal, characteristic
6#10598129C211yesmaximal, characteristic
7#10598129C211yesmaximal, characteristic
8C5×#400—21no—
9C5×#400—21no—
10C5×#395—21no—
11C5×#391C2211yescharacteristic
12C5×#389—21no—
13C5×#389C2211yescharacteristic
14C5×#389C2211yescharacteristic
15C5×#387—21no—
16C5×#387—21no—
17C2×#3967—21no—
18C2×#3939—21no—
19#65224C2211yescharacteristic
20#64749—21no—
21#64749—21no—
22#64749—21no—
23#64749—21no—
24#64744C2211yescharacteristic
25#64744C2211yescharacteristic
26#64744C2211yescharacteristic
27#7806—51nomaximal, Sylow
28C22×C5×Q8D411yescharacteristic
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4D411yescharacteristic
34C2×C5×C4⋊C4D411yescharacteristic
35C2×C5×C4⋊C4D411yescharacteristic
36C2×C5×C4⋊C4D411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C42×C5—21no—
39C2×C42×C5C2311yescharacteristic
40C2×C4⋊Dic5—21no—
41C2×C4⋊Dic5—21no—
42C2×C4⋊Dic5—21no—
43C22×#168—21no—
44C22×#168—21no—
45C22×#168—21no—
46C5×#95—21no—
47C5×#95—21no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95—21no—
55#3967—42no—
56#3939—42no—
57#2805D511yescharacteristic
58#2099—51no—
59#2083—51no—
60#2072—51no—
61#1272—51no—
62#1271—51no—
63#1271—51no—
64C2×C5×Q8—21no—
65C2×C5×Q8—21no—
66C2×C5×Q8—41no—
67C2×C5×Q8—41no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5C2×D411yescharacteristic
75C22×C4×C5C2×D411yescharacteristic
76C22×C4×C5C2×D411yescharacteristic
77C22×Dic5—41no—
78C5×C4⋊C4—21no—
79C5×C4⋊C4—21no—
80C5×C4⋊C4—21no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—81no—
89C5×C4⋊C4—81no—
90C5×C4⋊C4Q1612yes—
91C42×C5—41no—
92C42×C5—81no—
93C4⋊Dic5—42no—
94C4⋊Dic5—41no—
95C4⋊Dic5—41no—
96C4⋊Dic5—41no—
97C4⋊Dic5—41no—
98C2×#168—22no—
99C2×#168—41no—
100C2×#168—41no—
101C2×#168—41no—
102C2×#168—41no—
103C2×#168—41no—
104C2×#168—41no—
105C2×#168—41no—
106C2×#168—41no—
107C2×#107—101no—
108C2×#103—101no—
109#400—21no—
110#400—21no—
111#395—21no—
112#391C2×D511yescharacteristic
113#389—21no—
114#389C2×D511yescharacteristic
115#389C2×D511yescharacteristic
116#387—21no—
117#387—21no—
118#384—51no—
119#340—101no—
120#340—101no—
121#340—101no—
122#340—101no—
123#336—51no—
124#336—51no—
125#336—51no—
126C23×C5#12011yescharacteristic
127C5×Q8—42no—
128C5×Q8—81no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—22no—
136C2×C4×C5—41no—
137C2×C4×C5—41no—
138C2×C4×C5—41no—
139C2×C4×C5—41no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5C2×Q1611yescharacteristic
154C2×C4×C5#13511yescharacteristic
155C2×C4×C5#13511yescharacteristic
156C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C5#13711yescharacteristic
159C2×Dic5—81no—
160C2×Dic5—81no—
161C2×Dic5—81no—
162#168—42no—
163#168—81no—
164#168—81no—
165C22×Q8C22⋊D511yescharacteristic
166C22×C8—101no—
167C22×C8—101no—
168C22×C8—101no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—101no—
174C2×C4⋊C4—101no—
175C2×C4⋊C4—101no—
176C2×C4⋊C4C22⋊D511yescharacteristic
177C2×C4⋊C4C22⋊D511yescharacteristic
178C2×C4⋊C4C22⋊D511yescharacteristic
179C2×C4⋊C4C22⋊D511yescharacteristic
180C2×C4⋊C4C22⋊D511yescharacteristic
181C2×C42—21no—
182C2×C42C22×D511yescharacteristic
183#107—202no—
184#103—202no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194C22×C5#44911yescharacteristic
195C22×C5#44911yescharacteristic
196C22×C5#45111yescharacteristic
197C22×C5#45411yescharacteristic
198C22×C5#45511yescharacteristic
199C22×C5#45611yescharacteristic
200C22×C5#45811yescharacteristic
201C4×C5—21no—
202C4×C5—21no—
203C4×C5—21no—
204C4×C5—21no—
205C4×C5—21no—
206C4×C5—21no—
207C4×C5—42no—
208C4×C5—81no—
209C4×C5—81no—
210C4×C5—81no—
211C4×C5—81no—
212C4×C5—81no—
213Dic5—161no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—201no—
225C22×C4C2×C22⋊D511yescharacteristic
226C22×C4C2×C22⋊D511yescharacteristic
227C22×C4C2×C22⋊D511yescharacteristic
228C2×C8—102no—
229C2×C8—201no—
230C2×C8—201no—
231C2×C8—201no—
232C2×C8—201no—
233C2×C8—201no—
234C2×C8—201no—
235C2×C8—201no—
236C2×C8—201no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—81no—
248C4⋊C4—81no—
249C4⋊C4—202no—
250C4⋊C4—201no—
251C4⋊C4—201no—
252C4⋊C4—201no—
253C4⋊C4—201no—
254C4⋊C4#68912yes—
255C42—41no—
256C42—81no—
257C2×C5#160911yescharacteristic
258C2×C5#161211yescharacteristic
259C2×C5#161711yescharacteristic
260C2×C5#162811yescharacteristic
261C2×C5#164411yescharacteristic
262C2×C5#164811yescharacteristic
263C2×C5#166411yescharacteristic
264C23#409911yescharacteristic
265Q8—42no—
266Q8—81no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—401no—
292C2×C4—401no—
293C2×C4—401no—
294C2×C4#408811yescharacteristic
295C2×C4C2×#68911yescharacteristic
296C2×C4#409511yescharacteristic
297C2×C4#409511yescharacteristic
298C2×C4#409611yescharacteristic
299C2×C4#409611yescharacteristic
300C8—202no—
301C8—401no—
302C8—401no—
303C5#780611yescharacteristic, complemented, Sylow
304C22#6551711yescharacteristic
305C22#6552711yescharacteristic
306C22#6552711yescharacteristic
307C22#6553111yescharacteristic
308C22#6553211yescharacteristic
309C22#6553411yescharacteristic
310C22#6553511yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—801no—
324C2#1060267511yescharacteristic
325C2#1060267911yescharacteristic
326C2#1060269211yescharacteristic
327C2#1060271911yescharacteristic
328C2#1060272411yescharacteristic
329C2#1060273011yescharacteristic
330C2#1060275011yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2805C2
socle, prime radicalC23×C5#120
hypercenter#2805D5
hypocenterC5#7806
LCSG ▷ C2×C42×C5 ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2805

Complements

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

NH ≅ G/Ntype
303. C5#7806semidirect

Maximal subgroups

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

subgroupindex
C5×#28052
#106004632
#106004042
#106004002
#105981332
#105981292
#78065