Finite Group Databasebeta

← #12003800

Subgroups — #12003800

Order 1280.

374 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#10605949C211yesmaximal, characteristic
2C5×#1847C211yesmaximal, characteristic
3#10600368C212yesmaximal
4#10598150C212yesmaximal
5#10598121C211yesmaximal, characteristic
6C2×C5×#107—21no—
7C2×C5×#106—21no—
8C2×C5×#97C2211yescharacteristic
9C5×#384C2211yescharacteristic
10C5×#382C2211yescharacteristic
11#65275—21no—
12#65268—21no—
13#65225C2212yes—
14C2×#3935C2212yes—
15#64949—22no—
16#64945—22no—
17#64749—22no—
18#64749—22no—
19#10584—51nomaximal, Sylow
20C22×Dic10—21no—
21C22×C5×C8—21no—
22C2×C5×C4⋊C4—21no—
23C2×C5×C4⋊C4—22no—
24C2×C5×C4⋊C4D411yescharacteristic
25C2×C5×C4⋊C4D411yescharacteristic
26C2×C42×C5C2311yescharacteristic
27C2×C4⋊Dic5—21no—
28C2×C4⋊Dic5D412yes—
29C2×C4⋊Dic5D412yes—
30C2×#683—22no—
31C22×#168—22no—
32C5×#107—42no—
33C5×#106—42no—
34C5×#97—21no—
35C5×#97Q812yescomplemented
36C5×#95—22no—
37#3963—22no—
38#3963—21no—
39#3963—21no—
40#3963—22no—
41#3963—22no—
42#3935—22no—
43#3935Q814yescomplemented
44#2844—51no—
45#1847D511yescharacteristic
46#1836—52no—
47#1295—52no—
48#1295—51no—
49C22×C4×C5—21no—
50C22×C4×C5—21no—
51C22×C4×C5—41no—
52C22×C4×C5C2×D412yes—
53C22×C4×C5C2×D411yescharacteristic
54C22×Dic5—22no—
55C22×Dic5—22no—
56C2×Dic10—81no—
57C2×Dic10—82no—
58C2×C5×C8—22no—
59C2×C5×C8—22no—
60C2×C5×C8—22no—
61C5×C4⋊C4—41no—
62C5×C4⋊C4—41no—
63C5×C4⋊C4—44no—
64C5×C4⋊C4—41no—
65C5×C4⋊C4—41no—
66C42×C5C2×Q811yescharacteristic
67C42×C5C2×Q812yes—
68C42×C5#4111yescharacteristic
69C4⋊Dic5—42no—
70C4⋊Dic5—42no—
71C4⋊Dic5—81no—
72#683—82no—
73C2×#168—24no—
74C2×#168—42no—
75C2×#168—42no—
76C2×#107—21no—
77C2×#106—21no—
78C2×#97—52no—
79C2×#97C2×D511yescharacteristic
80#400—102no—
81#400—101no—
82#395—102no—
83#395—101no—
84#384C2×D511yescharacteristic
85#382—52no—
86#382C2×D511yescharacteristic
87#340—102no—
88#340—102no—
89C23×C5#12011yescharacteristic
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5—41no—
95C2×C4×C5—41no—
96C2×C4×C5—42no—
97C2×C4×C5—42no—
98C2×C4×C5—81no—
99C2×C4×C5—82no—
100C2×C4×C5C22⋊Q812yes—
101C2×C4×C5C22⋊Q811yescharacteristic
102C2×C4×C5C4⋊Q812yes—
103C2×C4×C5C4⋊Q811yescharacteristic
104C2×Dic5—42no—
105C2×Dic5—42no—
106C2×Dic5—42no—
107C2×Dic5—42no—
108C2×Dic5—42no—
109C2×Dic5—42no—
110Dic10—84no—
111C5×C8—24no—
112#168—44no—
113C22×Q8—101no—
114C22×C8—21no—
115C22×C8—102no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—52no—
119C2×C4⋊C4—52no—
120C2×C4⋊C4—102no—
121C2×C4⋊C4—101no—
122C2×C4⋊C4C22⋊D511yescharacteristic
123C2×C4⋊C4C22⋊D511yescharacteristic
124C2×C42C22×D511yescharacteristic
125#107—42no—
126#106—42no—
127#97—21no—
128#97—54no—
129#97—102no—
130#97Dic1012yescomplemented
131#95—22no—
132#95—102no—
133#95—101no—
134#95—101no—
135#95—102no—
136#95—102no—
137C22×C5#39511yescharacteristic
138C22×C5#45311yescharacteristic
139C22×C5#45412yes—
140C22×C5#45512yes—
141C22×C5#45611yescharacteristic
142C4×C5—21no—
143C4×C5—21no—
144C4×C5—22no—
145C4×C5—22no—
146C4×C5—81no—
147C4×C5—81no—
148C4×C5—161no—
149Dic5—82no—
150Dic5—82no—
151C2×Q8—401no—
152C2×Q8—402no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—102no—
157C22×C4—102no—
158C22×C4D4×D512yes—
159C22×C4C2×C22⋊D511yescharacteristic
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—104no—
164C2×C8—202no—
165C2×C8—202no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—44no—
171C4⋊C4—202no—
172C4⋊C4—202no—
173C4⋊C4—402no—
174C4⋊C4—401no—
175C42C2×Dic1011yescharacteristic
176C42#70911yescharacteristic
177C42Q8×D512yes—
178C2×C5#165711yescharacteristic
179C2×C5#165911yescharacteristic
180C2×C5#166412yes—
181C2×C5#206011yescharacteristic
182C2×C5#206112yes—
183C23#408311yescharacteristic
184Q8—404no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—202no—
196C2×C4—202no—
197C2×C4—202no—
198C2×C4—202no—
199C2×C4—202no—
200C2×C4—202no—
201C2×C4#403412yes—
202C2×C4#404212yes—
203C2×C4#407011yescharacteristic
204C2×C4#409011yescharacteristic
205C8—24no—
206C8—204no—
207C5#1058411yescharacteristic, complemented, Sylow
208C22#6526811yescharacteristic
209C22#6536611yescharacteristic
210C22#6538411yescharacteristic
211C22#6549012yes—
212C22#6549112yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—161no—
220C4—402no—
221C4—402no—
222C2#1060036611yescharacteristic
223C2#1060146012yes—
224C2#1060212111yescharacteristic
225C2#1060212211yescharacteristic
226C2#1060218712yes—
subgroupG/H
centerC23#4083
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1847C2
socle, prime radicalC23×C5#120
hypercenter#1847D5
hypocenterC5#10584
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
35. C5×#97Q8semidirect
43. #3935Q8semidirect
130. #97Dic10semidirect
207. C5#10584semidirect

Maximal subgroups

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

subgroupindex
#106059492
C5×#18472
#106003682
#105981502
#105981212
#105845