Finite Group Databasebeta

← #12016138

Subgroups — #12016138

Order 1280.

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

normal subgroups only

This group has 460 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#10614961C211yesmaximal, characteristic
2C5×#1886C211yesmaximal, characteristic
3C2×#64768C211yesmaximal, characteristic
4#10602451C212yesmaximal
5#10598190C212yesmaximal
6C2×C5×C4⋊Q8C2211yescharacteristic
7C5×#519C2211yescharacteristic
8C5×#419—21no—
9C2×C5×#101C2211yescharacteristic
10#65519—21no—
11#65519C2212yes—
12C2×#3968C2212yes—
13#65284—22no—
14#65274—22no—
15C4⋊C4×Dic5—22no—
16C4×C4⋊Dic5—21no—
17C5×#339—21no—
18#64768C414yescomplemented
19#64752—22no—
20#12973—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×C5×Q8C2311yescharacteristic
23C2×C5×M16—21no—
24C5×C4⋊Q8—22no—
25C5×C4⋊Q8C2×C412yes—
26C5×C4⋊Q8Q812yes—
27C4×C5×Q8—41no—
28C4×C5×Q8—41no—
29C5×#117—22no—
30C5×#117—21no—
31C2×C5×C4⋊C4—22no—
32C2×C42×C5D411yescharacteristic
33C2×C4⋊Dic5—21no—
34C2×C4⋊Dic5—21no—
35C2×C4⋊Dic5—21no—
36C2×C4⋊Dic5D412yes—
37C2×C4×Dic5—22no—
38C2×C4×Dic5—21no—
39C2×C4×Dic5D412yes—
40C2×#681—22no—
41C5×#103—41no—
42C5×#103—41no—
43C5×#101—22no—
44#3968—22no—
45#3968C2×C414yes—
46#3963—22no—
47#3963—22no—
48#3963—22no—
49#3940—42no—
50#3939—42no—
51#2369—51no—
52C2×#356—51no—
53#1886D511yescharacteristic
54#1857—52no—
55#1394—52no—
56C2×C5×Q8—21no—
57C2×C5×Q8—41no—
58C2×C5×Q8—41no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C2×C5×Q8C4⋊C412yes—
62C2×C5×Q8C22×C411yescharacteristic
63C2×C5×Q8C2×Q811yescharacteristic
64C2×C5×Q8#4112yes—
65C22×C4×C5—21no—
66C22×C4×C5—21no—
67C22×C4×C5C2×D412yes—
68C22×C4×C5C2×D411yescharacteristic
69C22×Dic5—22no—
70C22×Dic5—22no—
71C5×M16—41no—
72C5×M16—41no—
73C2×C5×C8—41no—
74C5×C4⋊C4—24no—
75C5×C4⋊C4—24no—
76C5×C4⋊C4—22no—
77C5×C4⋊C4—22no—
78C5×C4⋊C4—21no—
79C5×C4⋊C4—21no—
80C5×C22⋊C4—41no—
81C42×C5—21no—
82C42×C5—42no—
83C42×C5—41no—
84C42×C5C4⋊C412yes—
85C4⋊Dic5—24no—
86C4⋊Dic5—42no—
87C4⋊Dic5—42no—
88C4⋊Dic5—81no—
89C4×Dic5—44no—
90C4×Dic5—42no—
91C4×Dic5—81no—
92#681—24no—
93#681—42no—
94C2×#168—42no—
95C2×C4⋊Q8C2×D511yescharacteristic
96#519C2×D511yescharacteristic
97#428—102no—
98#419—21no—
99C2×#101—52no—
100C2×#101C2×D511yescharacteristic
101#391—52no—
102#391—101no—
103#389—102no—
104C4×C4⋊C4—102no—
105C4×C4⋊C4—101no—
106#356—54no—
107#339—21no—
108#339—102no—
109C23×C5#12011yescharacteristic
110C5×Q8—41no—
111C5×Q8—41no—
112C5×Q8—42no—
113C5×Q8—42no—
114C5×Q8—81no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—21no—
120C2×C4×C5—22no—
121C2×C4×C5—21no—
122C2×C4×C5—22no—
123C2×C4×C5—41no—
124C2×C4×C5—41no—
125C2×C4×C5—41no—
126C2×C4×C5—41no—
127C2×C4×C5C2×C4⋊C411yescharacteristic
128C2×C4×C5C4×D412yes—
129C2×C4×C5C22⋊Q812yes—
130C2×C4×C5#12311yescharacteristic
131C2×Dic5—24no—
132C2×Dic5—42no—
133C2×Dic5—42no—
134C2×Dic5—42no—
135C2×Dic5—42no—
136C2×Dic5—42no—
137C5×C8—81no—
138#168—44no—
139C22×Q8C22⋊D511yescharacteristic
140C22×Q8C22×D511yescharacteristic
141C2×M16—21no—
142C2×M16—102no—
143C4⋊Q8—22no—
144C4⋊Q8Dic1012yes—
145C4⋊Q8C4×D512yes—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—52no—
152C2×C4⋊C4—101no—
153C2×C4⋊C4—101no—
154C2×C4⋊C4—101no—
155C2×C42—52no—
156C2×C42—102no—
157C2×C42—101no—
158C2×C42C22⋊D511yescharacteristic
159#107—202no—
160#106—202no—
161#103—41no—
162#103—41no—
163#101—22no—
164#101—54no—
165#101—102no—
166#95—102no—
167#95—102no—
168#95—102no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5#38511yescharacteristic
172C22×C5B(4,𝔽2)11yescharacteristic
173C22×C5#45811yescharacteristic
174C4×C5—24no—
175C4×C5—22no—
176C4×C5—21no—
177C4×C5—21no—
178C4×C5—22no—
179C4×C5—42no—
180C4×C5—81no—
181C4×C5—81no—
182Dic5—44no—
183Dic5—82no—
184C2×Q8—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#68312yes—
190C2×Q8C2×Dic1011yescharacteristic
191C2×Q8C2×C4×D511yescharacteristic
192C2×Q8#71112yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—102no—
196C22×C4—102no—
197C22×C4D4×D512yes—
198C22×C4C2×C22⋊D511yescharacteristic
199M16—41no—
200M16—41no—
201M16—104no—
202M16—202no—
203C2×C8—41no—
204C2×C8—202no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—104no—
212C4⋊C4—202no—
213C4⋊C4—202no—
214C4⋊C4—401no—
215C22⋊C4—41no—
216C42—21no—
217C42—42no—
218C42—41no—
219C42—202no—
220C42—204no—
221C42—401no—
222C42#68312yes—
223C2×C5—21no—
224C2×C5—21no—
225C2×C5#186711yescharacteristic
226C2×C5#220411yescharacteristic
227C2×C5#220511yescharacteristic
228C23#408311yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—81no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—104no—
247C2×C4—202no—
248C2×C4—202no—
249C2×C4—202no—
250C2×C4—202no—
251C2×C4—202no—
252C2×C4#402412yes—
253C2×C4#402712yes—
254C2×C4C2×#68311yescharacteristic
255C2×C4#408111yescharacteristic
256C8—81no—
257C8—204no—
258C5#1297311yescharacteristic, complemented, Sylow
259C22—21no—
260C22—21no—
261C22#6523511yescharacteristic
262C22#6532111yescharacteristic
263C22#6534211yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—22no—
268C4—21no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4—204no—
273C4—402no—
274C2—21no—
275C2—21no—
276C2#1060039011yescharacteristic
277C2#1060178711yescharacteristic
278C2#1060178811yescharacteristic
subgroupG/H
centerC22#65321
commutatorC2×C5×Q8C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#1886C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
cosocleC22×C5×Q8C23
hypercenter#1886D5
hypocenterC5#12973
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1886

Complements

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

NH ≅ G/Ntype
18. #64768C4semidirect
258. C5#12973semidirect

Maximal subgroups

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

subgroupindex
#106149612
C5×#18862
C2×#647682
#106024512
#105981902
#129735