Finite Group Databasebeta

← #12002313

Subgroups — #12002313

Order 1280.

504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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#10606066C211yesmaximal, characteristic
2C5×#1956C211yesmaximal, characteristic
3#10602572C212yesmaximal
4#10598146C212yesmaximal
5#10598125C211yesmaximal, characteristic
6C2×C5×C4⋊Q8C2211yescharacteristic
7C2×#4034—22no—
8C2×C5×#103—22no—
9C2×C4×C5×C8C2211yescharacteristic
10C5×#384C2211yescharacteristic
11#65268—22no—
12#65262—22no—
13C4×C4⋊Dic5C2212yes—
14C2×C4⋊#168C2212yes—
15#64749—22no—
16#64749—22no—
17#10349—51nomaximal, Sylow
18C22×C5×Q8—21no—
19C22×Dic10—22no—
20C22×C5×C8—21no—
21C5×C4⋊Q8—22no—
22C5×C4⋊Q8—21no—
23C5×C4⋊Q8Q812yescomplemented
24C2×C5×C4⋊C4—22no—
25C2×C5×C4⋊C4—22no—
26C2×C5×C4⋊C4D412yes—
27C2×C42×C5C2311yescharacteristic
28C2×C4⋊Dic5D412yes—
29C2×C4⋊Dic5D412yes—
30C2×#683—24no—
31C2×C4×Dic5—22no—
32C22×#168—22no—
33#4034—44no—
34#4034—44no—
35C5×#103—44no—
36C4×C5×C8—22no—
37#3963—22no—
38#3963—22no—
39C4⋊#168—22no—
40C4⋊#168—22no—
41#2551—51no—
42#1987—52no—
43#1956D511yescharacteristic
44#1296—51no—
45#1279—52no—
46C2×C5×Q8—22no—
47C2×C5×Q8—42no—
48C2×C5×Q8—42no—
49C2×C5×Q8—41no—
50C22×C4×C5—22no—
51C22×C4×C5—41no—
52C22×C4×C5C2×D412yes—
53C22×C4×C5C2×D411yescharacteristic
54C22×Dic5—22no—
55C22×Dic5—22no—
56C2×Dic10—48no—
57C2×Dic10—82no—
58C2×C5×C8—22no—
59C2×C5×C8—41no—
60C2×C5×C8—41no—
61C5×C4⋊C4—24no—
62C5×C4⋊C4—42no—
63C5×C4⋊C4—42no—
64C5×C4⋊C4—42no—
65C5×C4⋊C4—42no—
66C42×C5C2×Q811yescharacteristic
67C42×C5#4111yescharacteristic
68C42×C5#4112yes—
69C4⋊Dic5—22no—
70C4⋊Dic5—22no—
71C4⋊Dic5SD1614yes—
72C4⋊Dic5Q1614yes—
73#683—48no—
74C4×Dic5—42no—
75C4×Dic5—42no—
76C2×#168—42no—
77C2×#168—42no—
78C2×#168—42no—
79C2×C4⋊Q8—102no—
80C2×C4⋊Q8C2×D511yescharacteristic
81C2×C4⋊C8—52no—
82C2×#103—22no—
83C2×C4×C8C2×D511yescharacteristic
84#395—102no—
85#389—102no—
86#384C2×D511yescharacteristic
87C4×C4⋊C4—52no—
88#340—102no—
89#340—102no—
90C23×C5#12011yescharacteristic
91C5×Q8—44no—
92C5×Q8—82no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—42no—
96C2×C4×C5—44no—
97C2×C4×C5—42no—
98C2×C4×C5—81no—
99C2×C4×C5—81no—
100C2×C4×C5C22⋊Q812yes—
101C2×C4×C5C22⋊Q811yescharacteristic
102C2×C4×C5#12412yes—
103C2×C4×C5#12411yescharacteristic
104C2×C4×C5C2×SD1612yes—
105C2×C4×C5C2×Q1612yes—
106C2×C4×C5AGL(1,ℤ/8)12yes—
107C2×C4×C5#13712yes—
108C2×Dic5—24no—
109C2×Dic5—24no—
110C2×Dic5—42no—
111C2×Dic5—42no—
112C2×Dic5—42no—
113C2×Dic5—42no—
114Dic10—88no—
115C5×C8—42no—
116#168—82no—
117C22×Q8—21no—
118C22×Q8—102no—
119C22×C8—21no—
120C22×C8—102no—
121C4⋊Q8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—204no—
124C4⋊Q8—204no—
125C4⋊Q8Dic1012yescomplemented
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—52no—
129C2×C4⋊C4—52no—
130C2×C4⋊C4—104no—
131C2×C4⋊C4C22⋊D512yes—
132C2×C42—102no—
133C2×C42C22×D511yescharacteristic
134C4⋊C8—102no—
135C4⋊C8—102no—
136#103—44no—
137C4×C8—22no—
138#95—102no—
139#95—102no—
140C22×C5#39311yescharacteristic
141C22×C5#45012yes—
142C22×C5#45112yes—
143C22×C5#45311yescharacteristic
144C22×C5#45411yescharacteristic
145C4×C5—21no—
146C4×C5—21no—
147C4×C5—82no—
148C4×C5—82no—
149C4×C5#47512yes—
150C4×C5#47712yes—
151C4×C5#48712yes—
152C4×C5#48912yes—
153Dic5—44no—
154Dic5—44no—
155C2×Q8—22no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—208no—
160C2×Q8—402no—
161C22×C4—22no—
162C22×C4—41no—
163C22×C4—102no—
164C22×C4—102no—
165C22×C4D4×D512yes—
166C22×C4C2×C22⋊D511yescharacteristic
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—202no—
171C2×C8—202no—
172C2×C8—202no—
173C4⋊C4—24no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—54no—
179C4⋊C4—54no—
180C4⋊C4—102no—
181C4⋊C4—102no—
182C4⋊C4—208no—
183C42—202no—
184C42—202no—
185C42C2×Dic1011yescharacteristic
186C42#70911yescharacteristic
187C42#71112yes—
188C2×C5#162911yescharacteristic
189C2×C5#163311yescharacteristic
190C2×C5#163712yes—
191C2×C5#202412yes—
192C2×C5#202711yescharacteristic
193C23#408311yescharacteristic
194Q8—44no—
195Q8—82no—
196Q8—408no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—104no—
205C2×C4—104no—
206C2×C4—202no—
207C2×C4—202no—
208C2×C4—202no—
209C2×C4—202no—
210C2×C4#402412yes—
211C2×C4#403112yes—
212C2×C4D5×SD1612yes—
213C2×C4#406012yes—
214C2×C4D5×Q1612yes—
215C2×C4#406412yes—
216C2×C4#407011yescharacteristic
217C2×C4#408211yescharacteristic
218C8—42no—
219C8—402no—
220C5#1034911yescharacteristic, complemented, Sylow
221C22#6523611yescharacteristic
222C22#6535111yescharacteristic
223C22#6536611yescharacteristic
224C22#6545812yes—
225C22#6547512yes—
226C4—21no—
227C4—21no—
228C4—82no—
229C4—82no—
230C4—204no—
231C4—204no—
232C4#6508012yes—
233C4#6508112yes—
234C4#6508312yes—
235C4#6508412yes—
236C2#1060043611yescharacteristic
237C2#1060102412yes—
238C2#1060203012yes—
239C2#1060205911yescharacteristic
240C2#1060210011yescharacteristic
subgroupG/H
centerC23#4083
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1956C2
socle, prime radicalC23×C5#120
hypercenter#1956D5
hypocenterC5#10349
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
23. C5×C4⋊Q8Q8semidirect
125. C4⋊Q8Dic10semidirect
220. C5#10349semidirect

Maximal subgroups

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

subgroupindex
#106060662
C5×#19562
#106025722
#105981462
#105981252
#103495