Finite Group Databasebeta

← #11987051

Subgroups — #11987051

Order 1280.

302 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 63 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×#1888C211yesmaximal, characteristic
2C2×#64761C211yesmaximal, characteristic
3C2×#64760C211yesmaximal, characteristic
4C2×C5×#124C2211yescharacteristic
5C5×#519C411yescharacteristic, complemented
6C5×#418—21no—
7C2×C5×#101C411yescharacteristic, complemented
8C2×#3966—21no—
9C2×#3935—21no—
10C5×#339—21no—
11#64761—21no—
12#64761C412yescomplemented
13#64760—21no—
14#64760C412yescomplemented
15#7489—51nomaximal, Sylow
16C22×C5×Q8C2×C411yescharacteristic
17C22×C5×D4D411yescharacteristic
18C2×C5×M16—21no—
19C5×#124—41no—
20C5×#124C2×C411yescharacteristic
21C5×#124C2×C411yescharacteristic
22C5×#124D411yescharacteristic
23C5×#124Q811yescharacteristic
24C4×C5×Q8—42no—
25C5×#117—21no—
26C5×#117—41no—
27C2×C5×C22⋊C4—41no—
28C2×C42×C5D411yescharacteristic
29C2×C22⋊Dic5—41no—
30C22×#168—41no—
31C5×#102—42no—
32C5×#101—41no—
33#3966—22no—
34#3966—41no—
35#3935—22no—
36#3935—41no—
37C2×#354—51no—
38#1888D511yescharacteristic
39C2×#329—51no—
40C24×C5—21no—
41C2×C5×Q8—21no—
42C2×C5×Q8—41no—
43C2×C5×Q8—41no—
44C2×C5×Q8C4211yescharacteristic
45C2×C5×Q8C4⋊C411yescharacteristic
46C2×C5×D4—21no—
47C2×C5×D4—41no—
48C2×C5×D4—41no—
49C2×C5×D4C22⋊C411yescharacteristic
50C2×C5×D4C4⋊C411yescharacteristic
51C22×C4×C5—21no—
52C22×C4×C5—21no—
53C22×C4×C5C22⋊C411yescharacteristic
54C22×Dic5—81no—
55C5×M16—41no—
56C5×M16—41no—
57C2×C5×C8—41no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—42no—
60C5×C22⋊C4—41no—
61C5×C22⋊C4—41no—
62C5×C22⋊C4—41no—
63C5×C22⋊C4—41no—
64C5×C22⋊C4—41no—
65C42×C5—21no—
66C42×C5—41no—
67C42×C5—42no—
68C42×C5C22⋊C411yescharacteristic
69C42×C5C4⋊C411yescharacteristic
70C22⋊Dic5—42no—
71C22⋊Dic5—81no—
72C2×#168—42no—
73C2×#168—44no—
74C2×#124C2×D511yescharacteristic
75#519Dic511yescharacteristic, complemented
76#418—21no—
77C2×#101Dic511yescharacteristic, complemented
78C2×#99—101no—
79C2×#97—101no—
80#354—52no—
81#354—101no—
82#339—21no—
83#329—52no—
84#329—101no—
85C23×C5—21no—
86C23×C5—21no—
87C23×C5—41no—
88C23×C5—41no—
89C23×C5—82no—
90C23×C5#9911yescharacteristic
91C5×Q8—41no—
92C5×Q8—41no—
93C5×Q8—42no—
94C5×D4—41no—
95C5×D4—41no—
96C5×D4—81no—
97C2×C4×C5—21no—
98C2×C4×C5—21no—
99C2×C4×C5—21no—
100C2×C4×C5—21no—
101C2×C4×C5—21no—
102C2×C4×C5—21no—
103C2×C4×C5—42no—
104C2×C4×C5—41no—
105C2×C4×C5—41no—
106C2×C4×C5—42no—
107C2×C4×C5#9511yescharacteristic
108C2×C4×C5#9911yescharacteristic
109C2×Dic5—82no—
110C2×Dic5—162no—
111C5×C8—81no—
112#168—44no—
113C22×Q8C2×Dic511yescharacteristic
114C22×D4C22⋊D511yescharacteristic
115C2×M16—21no—
116C22×C8—201no—
117#124—41no—
118#124Dic1011yescharacteristic
119#124C4×D511yescharacteristic
120#124C4×D511yescharacteristic
121#124D2011yescharacteristic
122C4×Q8—42no—
123#117—21no—
124#117—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—201no—
127C2×C42C22⋊D511yescharacteristic
128#102—42no—
129#101—41no—
130#99—102no—
131#99—201no—
132#97—102no—
133#97—201no—
134C22×C5—22no—
135C22×C5—41no—
136C22×C5—41no—
137C22×C5—81no—
138C22×C5—81no—
139C22×C5—82no—
140C22×C5—161no—
141C22×C5#34211yescharacteristic
142C22×C5AΣL(1,𝔽16)11yescharacteristic
143C22×C5#35211yescharacteristic
144C4×C5—21no—
145C4×C5—21no—
146C4×C5—41no—
147C4×C5—41no—
148C4×C5—41no—
149C4×C5—41no—
150C4×C5—82no—
151Dic5—162no—
152C24—21no—
153C2×Q8—21no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8C4×Dic511yescharacteristic
157C2×Q8C4⋊Dic511yescharacteristic
158C2×D4—21no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4#68311yescharacteristic
162C2×D4#68511yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—401no—
166C22×C4C22⋊Dic511yescharacteristic
167M16—41no—
168M16—41no—
169C2×C8—41no—
170C2×C8—202no—
171C2×C8—204no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C22⋊C4—41no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—202no—
180C22⋊C4—401no—
181C42—21no—
182C42—41no—
183C42—42no—
184C42#68311yescharacteristic
185C42#68511yescharacteristic
186C2×C5—21no—
187C2×C5—21no—
188C2×C5—81no—
189C2×C5—81no—
190C2×C5#139011yescharacteristic
191C2×C5#140412yes—
192C23—21no—
193C23—21no—
194C23—41no—
195C23—41no—
196C23—82no—
197C23#396611yescharacteristic
198Q8—41no—
199Q8—41no—
200Q8—42no—
201D4—41no—
202D4—41no—
203D4—81no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—402no—
215C2×C4—802no—
216C2×C4#396311yescharacteristic
217C2×C4#396611yescharacteristic
218C8—81no—
219C8—204no—
220C5#748911yescharacteristic, complemented, Sylow
221C22—22no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—82no—
227C22—161no—
228C22#6474611yescharacteristic
229C22#6475611yescharacteristic
230C22#6475911yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—82no—
238C4—802no—
239C2—21no—
240C2—21no—
241C2—81no—
242C2—81no—
243C2#1059812811yescharacteristic
244C2#1059826212yes—
subgroupG/H
centerC22#64756
commutatorC2×C5×Q8C42
FrattiniC2×#124C2×D5
FittingC5×#1888C2
socleC22×C5AΣL(1,𝔽16)
prime radicalC22×C5×D4D4
cosocleC2×C5×#124C22
hypercenter#1888D5
hypocenterC5#7489
LCSG ▷ C2×C5×Q8 ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ #1888

Complements

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

NH ≅ G/Ntype
5. C5×#519C4semidirect
7. C2×C5×#101C4semidirect
12. #64761C4semidirect
14. #64760C4semidirect
75. #519Dic5semidirect
77. C2×#101Dic5semidirect
220. C5#7489semidirect

Maximal subgroups

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

subgroupindex
C5×#18882
C2×#647612
C2×#647602
#74895