Finite Group Databasebeta

← #13019552

Subgroups — #13019552

Order 1280.

498 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 124 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#10617521C211yesmaximal, characteristic
2#10614958C211yesmaximal, characteristic
3#10604976C211yesmaximal, characteristic
4#10604967C212yesmaximal
5#10598695C212yesmaximal
6C2×C5×C4⋊Q8C411yescharacteristic
7C2×Q8×Dic5C2211yescharacteristic
8C2×Q8×Dic5C2211yescharacteristic
9Q8×#170—24no—
10#65748—22no—
11#65748C2212yes—
12#65748C2212yes—
13#65519C411yescharacteristic
14#65519C411yescharacteristic
15#65270—22no—
16#65270—22no—
17#65225C411yescharacteristic
18#65221C2211yescharacteristic
19Q8⋊#170—24no—
20#64913—22no—
21#9100—51nomaximal, Sylow
22C22×C5×Q8C2×C411yescharacteristic
23C22×C5×Q8C2×C411yescharacteristic
24C5×C4⋊Q8—22no—
25C5×C4⋊Q8C814yescomplemented
26C2×C5×C4⋊C4—22no—
27C2×C42×C5C2×C411yescharacteristic
28Q8×Dic5—22no—
29Q8×Dic5—22no—
30Q8×Dic5—22no—
31Q8×Dic5—22no—
32C2×C4⋊Dic5—21no—
33C2×C4⋊Dic5C2×C411yescharacteristic
34C2×C4⋊Dic5C2×C411yescharacteristic
35C2×C4⋊Dic5D412yes—
36C2×C4×Dic5—21no—
37C2×C4×Dic5C2×C411yescharacteristic
38C2×C4×Dic5D412yes—
39C2×C4×Dic5C2311yescharacteristic
40C22⋊#170—22no—
41C22⋊#170—22no—
42C4⋊#170—24no—
43C4⋊#170—22no—
44C4⋊#170—22no—
45C4×#170—24no—
46C4×#170—22no—
47C4×#170D414yes—
48#3963—22no—
49#3963—22no—
50#3963—22no—
51#3963—21no—
52#3963—21no—
53#2982—51no—
54#2327—51no—
55#1479—52no—
56#1461—51no—
57#1274—52no—
58C2×C5×Q8—22no—
59C2×C5×Q8—21no—
60C2×C5×Q8—22no—
61C2×C5×Q8—22no—
62C2×C5×Q8—22no—
63C2×C5×Q8C2×C812yes—
64C2×C5×Q8C2×C812yes—
65C2×C5×Q8M1612yes—
66C22×C4×C5—21no—
67C22×C4×C5—21no—
68C22×C4×C5C22⋊C412yes—
69C22×C4×C5C22×C411yescharacteristic
70C22×Dic5—21no—
71C22×Dic5—21no—
72C22×Dic5C22⋊C412yes—
73C22×Dic5C2×D411yescharacteristic
74C22×Dic5C2×D411yescharacteristic
75C2×#170—24no—
76C2×#170—42no—
77C5×C4⋊C4—28no—
78C42×C5—21no—
79C42×C5C2×C812yes—
80C4⋊Dic5—22no—
81C4⋊Dic5—24no—
82C4⋊Dic5—22no—
83C4⋊Dic5—41no—
84C4⋊Dic5—41no—
85C4×Dic5—22no—
86C4×Dic5—22no—
87C4×Dic5—22no—
88C4×Dic5—42no—
89C4×Dic5C2×D412yes—
90C4×Dic5#4112yes—
91C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
92C2×C4×Q8—51no—
93C2×C4×Q8—51no—
94C8×Q8—104no—
95#431—52no—
96#431—52no—
97#431—102no—
98#391—51no—
99#391—51no—
100#387—102no—
101#387—102no—
102#382—51no—
103#376—51no—
104Q8⋊C8—104no—
105#324—102no—
106C23×C5C2×C22⋊C411yescharacteristic
107C5×Q8—24no—
108C5×Q8—24no—
109C5×Q8—24no—
110C5×Q8—42no—
111C2×C4×C5—22no—
112C2×C4×C5—22no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—41no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—41no—
119C2×C4×C5C22⋊C814yes—
120C2×C4×C5C4×D412yes—
121C2×C4×C5C22×C811yescharacteristic
122C2×C4×C5C2×M1611yescharacteristic
123C2×C4×C5#13112yes—
124C2×Dic5—22no—
125C2×Dic5—41no—
126C2×Dic5—41no—
127C2×Dic5—42no—
128C2×Dic5—41no—
129C2×Dic5—41no—
130C2×Dic5—42no—
131C2×Dic5—41no—
132C2×Dic5—41no—
133C2×Dic5#12011yescharacteristic
134C2×Dic5#12112yes—
135C2×Dic5#12311yescharacteristic
136#170—44no—
137#170—82no—
138C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
139C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
140C4⋊Q8—22no—
141C4⋊Q8#17014yescomplemented
142C4×Q8—102no—
143C4×Q8—102no—
144C4×Q8—102no—
145C4×Q8—102no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—52no—
148C2×C4⋊C4—51no—
149C2×C4⋊C4—51no—
150C2×C4⋊C4—101no—
151C2×C42—51no—
152C2×C42—52no—
153C2×C42—51no—
154C2×C42—101no—
155C2×C42C2×AGL(1,𝔽5)11yescharacteristic
156C4⋊C8—102no—
157C4⋊C8—102no—
158C4⋊C8—104no—
159C22⋊C8—102no—
160C22⋊C8—102no—
161C4×C8—54no—
162C4×C8—102no—
163C4×C8—104no—
164#95—101no—
165#95—101no—
166#95—102no—
167#95—102no—
168#95—102no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5#38611yescharacteristic
172C22×C5C2×C22⋊C811yescharacteristic
173C22×C5#40811yescharacteristic
174C4×C5—24no—
175C4×C5—42no—
176C4×C5—42no—
177C4×C5C8×D412yes—
178C4×C5#43512yes—
179Dic5—22no—
180Dic5—21no—
181Dic5—21no—
182Dic5—42no—
183Dic5—81no—
184Dic5—81no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—21no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C2×#17012yes—
191C2×Q8C2×#17012yes—
192C2×Q8#70412yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—51no—
196C22×C4—51no—
197C22×C4—52no—
198C22×C4—101no—
199C22×C4—101no—
200C22×C4C22⋊AGL(1,𝔽5)12yes—
201C22×C4C22×AGL(1,𝔽5)11yescharacteristic
202C2×C8—104no—
203C2×C8—202no—
204C4⋊C4—28no—
205C4⋊C4—104no—
206C4⋊C4—102no—
207C4⋊C4—102no—
208C4⋊C4—201no—
209C4⋊C4—201no—
210C42—21no—
211C42—52no—
212C42—52no—
213C42—102no—
214C42—102no—
215C42—102no—
216C42—202no—
217C42C2×#17012yes—
218C2×C5—21no—
219C2×C5—21no—
220C2×C5#186911yescharacteristic
221C2×C5#190911yescharacteristic
222C2×C5#191211yescharacteristic
223C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
224Q8—24no—
225Q8—24no—
226Q8—24no—
227Q8—42no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—51no—
237C2×C4—51no—
238C2×C4—52no—
239C2×C4—102no—
240C2×C4—201no—
241C2×C4—201no—
242C2×C4—202no—
243C2×C4—201no—
244C2×C4—201no—
245C2×C4—202no—
246C2×C4—201no—
247C2×C4—201no—
248C2×C4C22⋊#17014yes—
249C2×C4#413112yes—
250C2×C4D4×AGL(1,𝔽5)12yes—
251C2×C4C22×#17011yescharacteristic
252C2×C4C2×#70411yescharacteristic
253C8—204no—
254C8—402no—
255C5#910011yescharacteristic, complemented, Sylow
256C22—21no—
257C22—21no—
258C22#6577511yescharacteristic
259C22#6577911yescharacteristic
260C22C2×C22⋊#17011yescharacteristic
261C4—24no—
262C4—42no—
263C4—42no—
264C4—101no—
265C4—101no—
266C4—102no—
267C4—202no—
268C4—401no—
269C4—401no—
270C4D4×#17012yes—
271C4#6577312yes—
272C2—21no—
273C2—21no—
274C2#1060467111yescharacteristic
275C2#1060467711yescharacteristic
276C2#1060494611yescharacteristic
subgroupG/H
centerC22#65779
commutatorC2×C4×C5C22×C8
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC22×C5#386
prime radicalC23×C5C2×C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#9100
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
25. C5×C4⋊Q8C8semidirect
141. C4⋊Q8#170semidirect
255. C5#9100semidirect

Maximal subgroups

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

subgroupindex
#106175212
#106149582
#106049762
#106049672
#105986952
#91005