Finite Group Databasebeta

← #12002312

Subgroups — #12002312

Order 1280.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#2551C211yesmaximal, characteristic
2#10600372C211yesmaximal, characteristic
3#10600282C212yesmaximal
4#10598125C211yesmaximal, characteristic
5#10598110C212yesmaximal
6C2×C5×C4⋊Q8—22no—
7C2×#4015C2211yescharacteristic
8C5×#395—22no—
9C5×#389—22no—
10C5×#384C2211yescharacteristic
11C4×C5×C4⋊C4C2212yes—
12C2×#3942—22no—
13C2×C4⋊#168C2212yes—
14C2×C4×#168C2211yescharacteristic
15#64749—22no—
16#64749—22no—
17#10349—51nomaximal, Sylow
18C22×C5×Q8—22no—
19C22×Dic10—21no—
20C5×C4⋊Q8—44no—
21C5×C4⋊Q8—44no—
22C2×C5×C4⋊C4—24no—
23C2×C5×C4⋊C4—22no—
24C2×C5×C4⋊C4D412yes—
25C2×C5×C4⋊C4D412yes—
26C2×C5×C4⋊C4D412yes—
27C2×C42×C5—22no—
28C2×C42×C5C2311yescharacteristic
29C2×C4⋊Dic5—22no—
30C22×#168—21no—
31C22×#168—22no—
32#4015—22no—
33#4015—21no—
34#4015Q812yescomplemented
35C5×#95—22no—
36C5×#95—22no—
37#3942—44no—
38C4⋊#168—22no—
39C4⋊#168—22no—
40C4×#168—22no—
41#2551D511yescharacteristic
42#1987—52no—
43#1956—51no—
44#1296—51no—
45#1279—52no—
46C2×C5×Q8—48no—
47C2×C5×Q8—82no—
48C22×C4×C5—22no—
49C22×C4×C5—22no—
50C22×C4×C5—22no—
51C22×C4×C5C2×D411yescharacteristic
52C22×C4×C5C2×D412yes—
53C22×Dic5—41no—
54C2×Dic10—22no—
55C2×Dic10—42no—
56C2×Dic10—42no—
57C2×Dic10—41no—
58C5×C4⋊C4—22no—
59C5×C4⋊C4—22no—
60C5×C4⋊C4—42no—
61C5×C4⋊C4—48no—
62C5×C4⋊C4—42no—
63C5×C4⋊C4—42no—
64C5×C4⋊C4SD1614yes—
65C5×C4⋊C4Q1614yes—
66C42×C5—42no—
67C42×C5—42no—
68C42×C5C2×Q811yescharacteristic
69C42×C5#4112yes—
70C42×C5#4111yescharacteristic
71C4⋊Dic5—24no—
72C4⋊Dic5—42no—
73C2×#168—22no—
74C2×#168—41no—
75C2×#168—41no—
76C2×#168—42no—
77C2×#168—42no—
78C2×#168—42no—
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8—51no—
81C2×C4⋊C8—52no—
82C2×#103—102no—
83C2×C4×C8—51no—
84#395—22no—
85#389—22no—
86#384C2×D511yescharacteristic
87C4×C4⋊C4C2×D512yes—
88#340—102no—
89#340—102no—
90C23×C5#12011yescharacteristic
91C5×Q8—88no—
92C2×C4×C5—21no—
93C2×C4×C5—21no—
94C2×C4×C5—24no—
95C2×C4×C5—24no—
96C2×C4×C5—42no—
97C2×C4×C5—42no—
98C2×C4×C5—42no—
99C2×C4×C5—42no—
100C2×C4×C5—44no—
101C2×C4×C5—42no—
102C2×C4×C5C22⋊Q811yescharacteristic
103C2×C4×C5C22⋊Q812yes—
104C2×C4×C5#12411yescharacteristic
105C2×C4×C5#12412yes—
106C2×C4×C5C2×SD1612yes—
107C2×C4×C5C2×Q1612yes—
108C2×C4×C5AGL(1,ℤ/8)12yes—
109C2×C4×C5#13712yes—
110C2×Dic5—42no—
111C2×Dic5—81no—
112C2×Dic5—81no—
113Dic10—44no—
114Dic10—82no—
115#168—42no—
116#168—82no—
117C22×Q8—22no—
118C22×Q8—101no—
119C22×C8—101no—
120C22×C8—102no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—52no—
124C4⋊Q8—102no—
125C4⋊Q8—101no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—102no—
129C2×C4⋊C4C22⋊D512yes—
130C2×C4⋊C4C22⋊D512yes—
131C2×C4⋊C4C22⋊D512yes—
132C2×C42—22no—
133C2×C42C22×D511yescharacteristic
134C4⋊C8—102no—
135C4⋊C8—102no—
136#103—204no—
137C4×C8—102no—
138#95—22no—
139#95—22no—
140C22×C5#39311yescharacteristic
141C22×C5#45012yes—
142C22×C5#45112yes—
143C22×C5#45311yescharacteristic
144C22×C5#45411yescharacteristic
145C4×C5—21no—
146C4×C5—21no—
147C4×C5—44no—
148C4×C5—44no—
149C4×C5—82no—
150C4×C5#47512yes—
151C4×C5#47712yes—
152C4×C5#48712yes—
153C4×C5#48912yes—
154Dic5—82no—
155C2×Q8—48no—
156C2×Q8—82no—
157C2×Q8—102no—
158C2×Q8—202no—
159C2×Q8—202no—
160C2×Q8—201no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—201no—
165C22×C4C2×C22⋊D511yescharacteristic
166C22×C4C2×C22⋊D512yes—
167C2×C8—102no—
168C2×C8—201no—
169C2×C8—201no—
170C2×C8—202no—
171C2×C8—202no—
172C2×C8—202no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—48no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—104no—
180C4⋊C4—202no—
181C4⋊C4#68714yes—
182C4⋊C4#68914yes—
183C42—42no—
184C42—42no—
185C42Q8×D511yescharacteristic
186C42#71312yes—
187C42#71311yescharacteristic
188C2×C5#162911yescharacteristic
189C2×C5#163311yescharacteristic
190C2×C5#163712yes—
191C2×C5#202412yes—
192C2×C5#202711yescharacteristic
193C23#409911yescharacteristic
194Q8—88no—
195Q8—204no—
196Q8—402no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—202no—
208C2×C4—401no—
209C2×C4—401no—
210C2×C4#407812yes—
211C2×C4C2×#68712yes—
212C2×C4#408812yes—
213C2×C4C2×#68912yes—
214C2×C4#409211yescharacteristic
215C2×C4#409212yes—
216C2×C4#409311yescharacteristic
217C2×C4#409312yes—
218C8—202no—
219C8—402no—
220C5#1034911yescharacteristic, complemented, Sylow
221C22#6550712yes—
222C22#6551712yes—
223C22#6552011yescharacteristic
224C22#6553011yescharacteristic
225C22#6553411yescharacteristic
226C4—21no—
227C4—21no—
228C4—44no—
229C4—44no—
230C4—82no—
231C4—402no—
232C4#6537812yes—
233C4#6538012yes—
234C4#6538212yes—
235C4#6538312yes—
236C2#1060258712yes—
237C2#1060269611yescharacteristic
238C2#1060270011yescharacteristic
239C2#1060270812yes—
240C2#1060273311yescharacteristic
subgroupG/H
centerC23#4099
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#2551C2
socle, prime radicalC23×C5#120
hypercenter#2551D5
hypocenterC5#10349
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2551

Complements

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

NH ≅ G/Ntype
34. #4015Q8semidirect
220. C5#10349semidirect

Maximal subgroups

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

subgroupindex
C5×#25512
#106003722
#106002822
#105981252
#105981102
#103495