Finite Group Databasebeta

← #12016184

Subgroups — #12016184

Order 1280.

488 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 135 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×#3407C211yesmaximal, characteristic, complemented
2#10615525C211yesmaximal, characteristic
3#10602800C211yesmaximal, characteristic, complemented
4#10602224C212yesmaximal, complemented
5#10598304C212yesmaximal, complemented
6C2×C5×#131C2211yescharacteristic
7#66162C411yescharacteristic
8C22×C8×D5C2211yescharacteristic
9C5×#504—24no—
10C2×C5×M32—22no—
11C2×C5×M32C2212yes—
12#65523C411yescharacteristic
13#65416—22no—
14#65415C411yescharacteristic
15#65408C411yescharacteristic
16C8×C22⋊D5—22no—
17#65390C2211yescharacteristic
18C2×#3944C2212yes—
19#64773—22no—
20#64727—24no—
21#12981—51nomaximal, Sylow
22C2×C5×#41C2×C411yescharacteristic
23C22×C4×D5C2×C411yescharacteristic
24C5×#131—22no—
25C5×#131—22no—
26C2×C5×M16C2×C411yescharacteristic
27C2×C5×M16D412yes—
28C22×C5×C8D412yes—
29C22×C5×C8C2311yescharacteristic
30#4092—21no—
31#4084—21no—
32#4083C812yes—
33#4081C812yes—
34C4×C22⋊D5—22no—
35#4072C2×C411yescharacteristic
36C2×#681C2×C411yescharacteristic
37C22×#168C2×C411yescharacteristic
38C2×C8×D5—21no—
39C2×C8×D5—21no—
40C2×C8×D5—22no—
41C2×C8×D5—22no—
42C5×M32—24no—
43C5×M32—22no—
44C5×M32—22no—
45C2×C5×C16—24no—
46C2×C5×C16D414yescomplemented
47C22⋊#168—22no—
48#3952—22no—
49#3947—21no—
50#3946—22no—
51C8×Dic5—21no—
52#3944—22no—
53#3944—22no—
54C2×#672—22no—
55#3407D511yescharacteristic, complemented
56#2929—51no—
57#2115—51no—
58#2112—52no—
59#1378—52no—
60C23×D5C2×C811yescharacteristic
61C5×#41—22no—
62C5×#41—22no—
63C2×C5×Q8M1611yescharacteristic
64C2×C5×D4C2×C812yes—
65C2×C5×D4M1611yescharacteristic
66C22×C4×C5C22⋊C412yes—
67C22×C4×C5C22×C411yescharacteristic
68C2×C22⋊D5—22no—
69C22×Dic5C2×C811yescharacteristic
70C2×C4×D5—21no—
71C2×C4×D5—22no—
72C2×C4×D5—21no—
73C2×C4×D5—22no—
74C5×M16—22no—
75C5×M16—24no—
76C2×C5×C8—22no—
77C2×C5×C8—22no—
78C2×C5×C8C22⋊C412yes—
79C2×C5×C8C2×D411yescharacteristic
80C2×C5×C8C2×D412yes—
81C2×C5×C8C2×D411yescharacteristic
82C2×C5×C8#4112yes—
83C22⋊Dic5—22no—
84C22⋊Dic5C2×C812yes—
85#685—22no—
86C4⋊Dic5—21no—
87#683—22no—
88C4×Dic5—21no—
89#681—22no—
90C2×#168—21no—
91C2×#168—21no—
92C2×#168—22no—
93C8×D5—41no—
94C8×D5—42no—
95C8×D5—41no—
96C5×C16—24no—
97#672—42no—
98C2×#131C2×D511yescharacteristic
99C23×C8—51no—
100#525—51no—
101#504—24no—
102C2×M32—22no—
103C2×M32—52no—
104C2×M32C2×D512yes—
105#435—102no—
106C8×D4—102no—
107#424—51no—
108#408—51no—
109#407—51no—
110#405—51no—
111#349—102no—
112C22⋊C16—104no—
113C23×C5C22⋊C812yes—
114C23×C5C22×C811yescharacteristic
115C22×D5—21no—
116C22×D5—22no—
117C22×D5—42no—
118C5×Q8—22no—
119C5×D4—24no—
120C5×D4—22no—
121C2×C4×C5—22no—
122C2×C4×C5—22no—
123C2×C4×C5C22⋊C812yes—
124C2×C4×C5C2×C22⋊C411yescharacteristic
125C2×C4×C5C4×D412yes—
126C2×C4×C5C2×M1611yescharacteristic
127C2×C4×C5#13112yes—
128C22⋊D5—44no—
129C2×Dic5—22no—
130C2×Dic5—21no—
131C2×Dic5—22no—
132C4×D5—41no—
133C4×D5—41no—
134C4×D5—42no—
135C5×C8—22no—
136C5×C8#12011yescharacteristic
137C5×C8#12112yes—
138C5×C8#12311yescharacteristic
139#168—41no—
140#168—41no—
141C2×#41C4×D511yescharacteristic
142C23×C4—51no—
143#131—22no—
144#131—22no—
145C2×M16—51no—
146C2×M16C4×D511yescharacteristic
147C2×M16D2012yes—
148C22×C8—51no—
149C22×C8—101no—
150C22×C8—102no—
151C22×C8—102no—
152C22×C8—101no—
153C22×C8C22⋊D512yes—
154C22×C8C22×D511yescharacteristic
155#123—52no—
156C22⋊Q8—101no—
157#121—101no—
158#120—52no—
159C4×D4—102no—
160#117—51no—
161M32—24no—
162M32—22no—
163M32—22no—
164M32—102no—
165M32—102no—
166C2×C16—24no—
167C2×C16—102no—
168C2×C16C22⋊D514yescomplemented
169C4⋊C8—102no—
170C22⋊C8—102no—
171C22⋊C8—102no—
172#97—101no—
173C4×C8—101no—
174C22×C5—22no—
175C22×C5—41no—
176C22×C5—42no—
177C22×C5C2×C22⋊C811yescharacteristic
178C22×C5C8×D412yes—
179C2×D5—22no—
180C2×D5—24no—
181C2×D5—41no—
182C2×D5—42no—
183C2×D5—41no—
184C4×C5—22no—
185C4×C5#38611yescharacteristic
186C4×C5#40811yescharacteristic
187C4×C5#43512yes—
188Dic5—42no—
189Dic5—41no—
190C24—51no—
191#41—22no—
192#41—22no—
193C2×Q8#67611yescharacteristic
194C2×D4—102no—
195C2×D4C8×D512yes—
196C2×D4#67611yescharacteristic
197C22×C4—51no—
198C22×C4—101no—
199C22×C4—102no—
200C22×C4—101no—
201C22×C4—102no—
202C22×C4#68512yes—
203C22×C4C2×C4×D511yescharacteristic
204M16—22no—
205M16—24no—
206M16—102no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—101no—
210C2×C8—101no—
211C2×C8—102no—
212C2×C8—201no—
213C2×C8—201no—
214C2×C8—202no—
215C2×C8#68512yes—
216C2×C8C2×D2011yescharacteristic
217C2×C8#70912yes—
218C2×C8C2×C22⋊D511yescharacteristic
219C2×C8C2×C22⋊D512yes—
220C4⋊C4—102no—
221C4⋊C4—101no—
222C22⋊C4—52no—
223C22⋊C4—102no—
224C22⋊C4—102no—
225C42—101no—
226C16—24no—
227C16—202no—
228C2×C5—21no—
229C2×C5—22no—
230C2×C5—42no—
231C2×C5#186911yescharacteristic
232D5—42no—
233C23—101no—
234C23—102no—
235C23—202no—
236C23#395212yes—
237C23C2×C8×D511yescharacteristic
238Q8—22no—
239D4—24no—
240D4—22no—
241D4—204no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—102no—
245C2×C4—101no—
246C2×C4—102no—
247C2×C4—201no—
248C2×C4—202no—
249C2×C4—201no—
250C2×C4#395212yes—
251C2×C4C2×#67611yescharacteristic
252C2×C4#404712yes—
253C2×C4C2×#68511yescharacteristic
254C2×C4C4×C22⋊D512yes—
255C8—22no—
256C8—201no—
257C8—201no—
258C8#407511yescharacteristic
259C8#407612yes—
260C8#409911yescharacteristic
261C5#1298111yescharacteristic, complemented, Sylow
262C22—22no—
263C22—41no—
264C22—42no—
265C22—102no—
266C22—104no—
267C22—201no—
268C22—202no—
269C22—201no—
270C22C2×#395211yescharacteristic
271C22C8×C22⋊D512yes—
272C4—22no—
273C4—202no—
274C4—201no—
275C4#6539911yescharacteristic
276C4#6540012yes—
277C4#6550211yescharacteristic
278C2—21no—
279C2—22no—
280C2—42no—
281C2—202no—
282C2#1060276811yescharacteristic
subgroupG/H
centerC8#4099
commutatorC23×C5C22×C8
FrattiniC22×C8C22×D5
FittingC5×#3407C2
socleC2×C5#1869
prime radical#66162C4
cosocleC22×C5×C8C23
hypercenter#3407D5
hypocenterC5#12981
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
1. C5×#3407C2semidirect
3. #10602800C2semidirect
4. #10602224C2semidirect
5. #10598304C2semidirect
46. C2×C5×C16D4semidirect
55. #3407D5semidirect
168. C2×C16C22⋊D5semidirect
261. C5#12981semidirect

Maximal subgroups

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

subgroupindex
C5×#34072
#106155252
#106028002
#106022242
#105983042
#129815