Finite Group Databasebeta

← #13014989

Subgroups — #13014989

Order 1280.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#10615000C211yesmaximal, characteristic
2#10604182C211yesmaximal, characteristic
3#10604171C211yesmaximal, characteristic
4#10598714C214yesmaximal, complemented
5C22×D5⋊C8—21no—
6C2×C5×C4⋊Q8C411yescharacteristic
7C2×#4092—21no—
8C2×C42×D5C2211yescharacteristic
9C2×#4003—21no—
10C2×C4⋊#170—21no—
11#65519—21no—
12#65276—21no—
13#65276C412yes—
14#65262—21no—
15#65262C2212yes—
16#65224C411yescharacteristic
17#64918—21no—
18#64918C2212yes—
19#64918C2212yes—
20#7665—51nomaximal, Sylow
21C22×C5×Q8—21no—
22C22×C4×D5C2×C411yescharacteristic
23C22×C4×D5D411yescharacteristic
24C22×C4×D5D411yescharacteristic
25C22×#170—22no—
26C22×#170—22no—
27C2×D5⋊C8—24no—
28C2×D5⋊C8—42no—
29C2×D5⋊C8—42no—
30C5×C4⋊Q8—42no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C42×C5C2×C411yescharacteristic
34#4092—42no—
35#4092—42no—
36C2×#685—22no—
37C2×C4⋊Dic5—21no—
38C2×C4⋊Dic5—21no—
39C2×C4⋊Dic5C2×C412yes—
40C2×#683—22no—
41C2×C4×Dic5D411yescharacteristic
42C2×C4×Dic5D411yescharacteristic
43C2×C4×Dic5C2311yescharacteristic
44C42×D5—42no—
45#4003—42no—
46C4⋊#170—42no—
47#3963—22no—
48#3963—22no—
49#2857—51no—
50#1797—51no—
51C2×#336—51no—
52#1271—54no—
53C23×D5C22⋊C411yescharacteristic
54C2×C5×Q8—42no—
55C2×C5×Q8—42no—
56C2×C5×Q8—42no—
57C22×C4×C5—22no—
58C22×C4×C5C22⋊C411yescharacteristic
59C22×C4×C5C22⋊C411yescharacteristic
60C22×C4×C5C22×C411yescharacteristic
61C22×Dic5—22no—
62C22×Dic5C22⋊C411yescharacteristic
63C22×Dic5C2×D411yescharacteristic
64C22×Dic5C2×D411yescharacteristic
65C2×C4×D5—22no—
66C2×C4×D5—22no—
67C2×C4×D5—22no—
68C2×C4×D5—22no—
69C2×C4×D5—41no—
70C2×C4×D5—41no—
71C2×C4×D5—41no—
72C2×C4×D5D812yes—
73C2×C4×D5SD1612yes—
74C2×C4×D5SD1612yes—
75C2×C4×D5Q1612yes—
76C2×#170—24no—
77C2×#170—42no—
78C2×#170—42no—
79C2×#170—42no—
80C2×#170—42no—
81C2×#170—42no—
82D5⋊C8—48no—
83C5×C4⋊C4—42no—
84C5×C4⋊C4—42no—
85C42×C5—22no—
86#685—44no—
87C4⋊Dic5—24no—
88C4⋊Dic5—42no—
89C4⋊Dic5—42no—
90#683—42no—
91#683—42no—
92C4×Dic5—22no—
93C4×Dic5—41no—
94C4×Dic5—41no—
95C23×C8—101no—
96C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
97C2×C22⋊Q8—101no—
98C22×C42—51no—
99C2×C4⋊C8—101no—
100C2×C22⋊C8—101no—
101#391—101no—
102#389—52no—
103#389—101no—
104#384—51no—
105#380—52no—
106#380—101no—
107#336—52no—
108#336—52no—
109#336—101no—
110C23×C5C2×C22⋊C411yescharacteristic
111C22×D5—21no—
112C22×D5—21no—
113C22×D5—21no—
114C22×D5—21no—
115C22×D5#10112yes—
116C22×D5#10212yes—
117C22×D5#10312yes—
118C5×Q8—84no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—42no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5#9912yes—
127C2×C4×C5#10212yes—
128C2×C4×C5#10312yes—
129C2×C4×C5#10412yes—
130C2×C4×C5#13112yes—
131C2×Dic5—21no—
132C2×Dic5—21no—
133C2×Dic5—21no—
134C2×Dic5—21no—
135C2×Dic5—21no—
136C2×Dic5—21no—
137C2×Dic5—21no—
138C2×Dic5—42no—
139C2×Dic5—42no—
140C2×Dic5—42no—
141C2×Dic5C2×D811yescharacteristic
142C2×Dic5C2×SD1611yescharacteristic
143C2×Dic5C2×SD1611yescharacteristic
144C2×Dic5C2×Q1611yescharacteristic
145C4×D5—24no—
146C4×D5—24no—
147C4×D5—42no—
148C4×D5—42no—
149C4×D5—44no—
150#170—44no—
151#170—82no—
152C22×Q8—21no—
153C23×C4—51no—
154C23×C4—51no—
155C23×C4—51no—
156C22×C8—102no—
157C22×C8—104no—
158C22×C8—102no—
159C22×C8—202no—
160C22×C8—202no—
161C4⋊Q8—42no—
162C22⋊Q8—202no—
163C22⋊Q8—202no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—52no—
166C2×C4⋊C4—102no—
167C2×C4⋊C4—101no—
168C2×C4⋊C4—101no—
169C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
170C2×C22⋊C4—102no—
171C2×C42—51no—
172C2×C42—51no—
173C2×C42—51no—
174C2×C42—202no—
175C2×C42C2×AGL(1,𝔽5)11yescharacteristic
176C4⋊C8—202no—
177C22⋊C8—202no—
178#95—102no—
179#95—102no—
180C22×C5#40811yescharacteristic
181C22×C5C2×#9911yescharacteristic
182C22×C5C2×#10111yescharacteristic
183C22×C5C2×#10211yescharacteristic
184C22×C5C2×#10311yescharacteristic
185C22×C5C2×#10411yescharacteristic
186C22×C5#42111yescharacteristic
187C2×D5—22no—
188C2×D5—22no—
189C2×D5—22no—
190C2×D5—22no—
191C2×D5—22no—
192C2×D5—22no—
193C2×D5#33214yes—
194C4×C5—22no—
195C4×C5—22no—
196C4×C5—82no—
197C4×C5#32814yes—
198Dic5—21no—
199Dic5—21no—
200Dic5—21no—
201Dic5—21no—
202Dic5—41no—
203Dic5—82no—
204C24—51no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—22no—
209C22×C4—51no—
210C22×C4—51no—
211C22×C4—52no—
212C22×C4—52no—
213C22×C4—52no—
214C22×C4—52no—
215C22×C4—51no—
216C22×C4—102no—
217C22×C4—102no—
218C22×C4—102no—
219C22×C4—102no—
220C22×C4—102no—
221C22×C4—201no—
222C22×C4—201no—
223C22×C4—201no—
224C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
225C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
226C22×C4C22×AGL(1,𝔽5)11yescharacteristic
227C2×C8—104no—
228C2×C8—202no—
229C2×C8—208no—
230C2×C8—202no—
231C2×C8—202no—
232C2×C8—202no—
233C2×C8—202no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—104no—
237C4⋊C4—202no—
238C4⋊C4—202no—
239C4⋊C4—202no—
240C4⋊C4—202no—
241C22⋊C4—204no—
242C42—22no—
243C42—102no—
244C42—201no—
245C42—201no—
246C2×C5#147211yescharacteristic
247C2×C5#148111yescharacteristic
248C2×C5#149211yescharacteristic
249C2×C5C2×#32811yescharacteristic
250C2×C5#151911yescharacteristic
251C2×C5#153111yescharacteristic
252C2×C5C2×#33211yescharacteristic
253D5—24no—
254C23—52no—
255C23—52no—
256C23—52no—
257C23—101no—
258C23—101no—
259C23—101no—
260C23—101no—
261C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
262Q8—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—51no—
271C2×C4—51no—
272C2×C4—51no—
273C2×C4—51no—
274C2×C4—101no—
275C2×C4—101no—
276C2×C4—101no—
277C2×C4—101no—
278C2×C4—104no—
279C2×C4—104no—
280C2×C4—101no—
281C2×C4—101no—
282C2×C4—101no—
283C2×C4—202no—
284C2×C4—202no—
285C2×C4—204no—
286C2×C4—202no—
287C2×C4—202no—
288C2×C4—202no—
289C2×C4#400712yes—
290C2×C4#400912yes—
291C2×C4#401012yes—
292C2×C4#401112yes—
293C2×C4#413312yes—
294C8—204no—
295C8—402no—
296C5#766511yescharacteristic, complemented, Sylow
297C22—54no—
298C22—102no—
299C22—102no—
300C22—102no—
301C22—102no—
302C22—102no—
303C22—102no—
304C22C2×#400711yescharacteristic
305C22#6576911yescharacteristic
306C22C2×#400911yescharacteristic
307C22C2×#401011yescharacteristic
308C22#6578711yescharacteristic
309C22D5⋊#10111yescharacteristic
310C22C2×#401111yescharacteristic
311C4—22no—
312C4—22no—
313C4—82no—
314C4—101no—
315C4—101no—
316C4—101no—
317C4—101no—
318C4—201no—
319C4—402no—
320C4#6492614yes—
321C2—104no—
322C2#1060463711yescharacteristic
323C2#1060464611yescharacteristic
324C2#1060465011yescharacteristic
325C2D5⋊#33211yescharacteristic
326C2#1060491311yescharacteristic
327C2C2×#6492611yescharacteristic
328C2#1060497911yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC23×D5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#7665
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #10598714C2semidirect
296. C5#7665semidirect

Maximal subgroups

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

subgroupindex
#106150002
#106041822
#106041712
#105987142
#76655