Finite Group Databasebeta

← #12015303

Subgroups — #12015303

Order 1280.

468 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C4×C5×C4⋊Q8C211yesmaximal, characteristic
2C2×#64768C212yesmaximal
3#10602430C212yesmaximal
4#10598135C212yesmaximal
5C2×C5×C4⋊Q8C2211yescharacteristic
6C2×C4×C5×Q8C411yescharacteristic
7C2×C4×C5×Q8C2211yescharacteristic
8C5×#391C411yescharacteristic
9C5×#391C2211yescharacteristic
10C5×#389—22no—
11C5×#384C411yescharacteristic
12C4×C5×C4⋊C4—22no—
13C43×C5C411yescharacteristic
14C2×#3968C2214yes—
15#65213—22no—
16#64768—24no—
17#64750—22no—
18#12802—51nomaximal, Sylow
19C22×C5×Q8C2×C411yescharacteristic
20C22×C5×Q8C2311yescharacteristic
21C5×C4⋊Q8—22no—
22C5×C4⋊Q8D412yes—
23C5×C4⋊Q8Q812yes—
24C4×C5×Q8—41no—
25C4×C5×Q8—41no—
26C4×C5×Q8—41no—
27C4×C5×Q8—41no—
28C2×C5×C4⋊C4—22no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4C2×C411yescharacteristic
33C2×C5×C4⋊C4D412yes—
34C2×C42×C5—21no—
35C2×C42×C5—21no—
36C2×C42×C5—21no—
37C2×C42×C5C2×C411yescharacteristic
38C2×C42×C5C2×C411yescharacteristic
39C2×C42×C5C2×C411yescharacteristic
40C2×C42×C5D412yes—
41C2×#681—24no—
42C5×#95—22no—
43C5×#95—22no—
44#3968—24no—
45#3968—24no—
46C4⋊#168—42no—
47C4⋊#168—42no—
48C4×C4⋊Q8D511yescharacteristic
49C2×#356—52no—
50#1801—52no—
51#1394—52no—
52C2×C5×Q8—21no—
53C2×C5×Q8—41no—
54C2×C5×Q8—41no—
55C2×C5×Q8—41no—
56C2×C5×Q8—41no—
57C2×C5×Q8C4⋊C412yes—
58C2×C5×Q8C2×D411yescharacteristic
59C2×C5×Q8C2×Q811yescharacteristic
60C2×C5×Q8#4112yes—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C22⋊C412yes—
67C22×C4×C5C22⋊C411yescharacteristic
68C22×C4×C5C22×C411yescharacteristic
69C22×C4×C5C2×D412yes—
70C5×C4⋊C4—24no—
71C5×C4⋊C4—24no—
72C5×C4⋊C4—41no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—41no—
75C5×C4⋊C4—44no—
76C5×C4⋊C4—42no—
77C42×C5—24no—
78C42×C5—21no—
79C42×C5—21no—
80C42×C5—22no—
81C42×C5—22no—
82C42×C5—21no—
83C42×C5—41no—
84C42×C5—42no—
85C42×C5—41no—
86C42×C5—41no—
87C42×C5C4⋊C412yes—
88#681—44no—
89#681—44no—
90C2×#168—44no—
91C2×C4⋊Q8C2×D511yescharacteristic
92C2×C4×Q8Dic511yescharacteristic
93C2×C4×Q8C2×D511yescharacteristic
94#423—102no—
95C2×#101—54no—
96#391Dic511yescharacteristic
97#391C2×D511yescharacteristic
98#389—22no—
99#384Dic511yescharacteristic
100C4×C4⋊C4—22no—
101C43Dic511yescharacteristic
102#356—104no—
103#343—102no—
104C23×C5C2×C22⋊C411yescharacteristic
105C5×Q8—41no—
106C5×Q8—41no—
107C5×Q8—42no—
108C5×Q8—42no—
109C5×Q8—42no—
110C2×C4×C5—22no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—21no—
115C2×C4×C5—22no—
116C2×C4×C5—21no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—22no—
120C2×C4×C5—21no—
121C2×C4×C5—22no—
122C2×C4×C5—41no—
123C2×C4×C5—41no—
124C2×C4×C5—41no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5—42no—
128C2×C4×C5—42no—
129C2×C4×C5—41no—
130C2×C4×C5#9911yescharacteristic
131C2×C4×C5#9911yescharacteristic
132C2×C4×C5C2×C4⋊C411yescharacteristic
133C2×C4×C5#11711yescharacteristic
134C2×C4×C5#12112yes—
135C2×C4×C5C22⋊Q812yes—
136#168—84no—
137C22×Q8C2×Dic511yescharacteristic
138C22×Q8C22×D511yescharacteristic
139C2×M16—104no—
140C4⋊Q8—22no—
141C4⋊Q8Dic1012yes—
142C4⋊Q8D2012yes—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×Dic511yescharacteristic
151C2×C4⋊C4C2×Dic511yescharacteristic
152C2×C4⋊C4C22⋊D512yes—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42C2×Dic511yescharacteristic
157C2×C42C2×Dic511yescharacteristic
158C2×C42C2×Dic511yescharacteristic
159C2×C42C22⋊D512yes—
160C4⋊C8—202no—
161C4⋊C8—202no—
162#101—104no—
163#101—104no—
164#95—22no—
165#95—22no—
166C22×C5—21no—
167C22×C5—21no—
168C22×C5#38011yescharacteristic
169C22×C5C2×#9911yescharacteristic
170C22×C5#41011yescharacteristic
171C4×C5—24no—
172C4×C5—22no—
173C4×C5—21no—
174C4×C5—21no—
175C4×C5—22no—
176C4×C5—21no—
177C4×C5—42no—
178C4×C5—42no—
179C4×C5—41no—
180C4×C5—42no—
181C4×C5—81no—
182C4×C5#35612yescomplemented
183C2×Q8—21no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8C4⋊Dic512yes—
189C2×Q8C2×Dic1011yescharacteristic
190C2×Q8C2×D2011yescharacteristic
191C2×Q8#70912yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4C22⋊Dic512yes—
197C22×C4C22⋊Dic511yescharacteristic
198C22×C4C22⋊Dic511yescharacteristic
199C22×C4C22×Dic511yescharacteristic
200C22×C4C2×C22⋊D512yes—
201M16—204no—
202M16—204no—
203C2×C8—204no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C42—24no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—41no—
218C42—42no—
219C42—41no—
220C42—41no—
221C42C4⋊Dic512yes—
222C2×C5—21no—
223C2×C5—21no—
224C2×C5#179911yescharacteristic
225C2×C5C2×#35611yescharacteristic
226C2×C5#213211yescharacteristic
227C23C2×C22⋊Dic511yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231Q8—42no—
232Q8—42no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4#396611yescharacteristic
254C2×C4#396611yescharacteristic
255C2×C4#407012yes—
256C2×C4C2×C4⋊Dic511yescharacteristic
257C2×C4#407211yescharacteristic
258C2×C4#407612yes—
259C8—404no—
260C5#1280211yescharacteristic, complemented, Sylow
261C22—21no—
262C22—21no—
263C22#6550111yescharacteristic
264C22C2×#396611yescharacteristic
265C22#6552911yescharacteristic
266C4—24no—
267C4—21no—
268C4—22no—
269C4—22no—
270C4—21no—
271C4—21no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—42no—
276C4—81no—
277C4#6476812yescomplemented
278C2—21no—
279C2—21no—
280C2#1060241211yescharacteristic
281C2C2×#6476811yescharacteristic
282C2#1060261311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC22×Q8C22×D5
FittingC4×C5×C4⋊Q8C2
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC22×C5×Q8C23
hypercenterC4×C4⋊Q8D5
hypocenterC5#12802
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
182. C4×C5#356semidirect
260. C5#12802semidirect
277. C4#64768semidirect

Maximal subgroups

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

subgroupindex
C4×C5×C4⋊Q82
C2×#647682
#106024302
#105981352
#128025