Finite Group Databasebeta

← #11991078

Subgroups — #11991078

Order 1280.

782 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×C5×#554C211yesmaximal, characteristic
2#10602568C212yesmaximal, complemented
3#10602430C211yesmaximal, characteristic, complemented
4#10602417C211yesmaximal, characteristic, complemented
5#10602391C212yesmaximal
6C5×#554—24no—
7C5×#554—24no—
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×C5×#125C412yes—
10C2×C5×C22⋊Q8—21no—
11C2×C5×C22⋊Q8—22no—
12C2×C4×C5×Q8C2211yescharacteristic
13C2×C4×C5×D4C412yes—
14C2×C4×C5×D4C2211yescharacteristic
15C22×C5×C4⋊C4—21no—
16C2×#3968—21no—
17#65214C2212yes—
18#65213C2212yes—
19#64767—24no—
20D4⋊#168—24no—
21#64750—21no—
22#64748—22no—
23#8398—51nomaximal, Sylow
24C22×C5×Q8—21no—
25C22×C5×Q8D411yescharacteristic
26C22×C5×D4C2×C411yescharacteristic
27C23×C4×C5—21no—
28C23×C4×C5—22no—
29C5×C4⋊Q8—22no—
30C5×C4⋊Q8D414yes—
31C5×#125—24no—
32C5×#125—24no—
33C5×C22⋊Q8—28no—
34C5×C22⋊Q8—216no—
35C4×C5×Q8—22no—
36C4×C5×Q8—22no—
37C4×C5×D4—24no—
38C4×C5×D4—24no—
39C4×C5×D4—22no—
40C4×C5×D4D414yes—
41C2×C5×C4⋊C4—28no—
42C2×C5×C4⋊C4—22no—
43C2×C5×C4⋊C4—22no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4C2×C412yes—
46C2×C5×C4⋊C4C2×C411yescharacteristic
47C2×C5×C4⋊C4D411yescharacteristic
48C2×C5×C4⋊C4D412yes—
49C2×C5×C22⋊C4—22no—
50C2×C5×C22⋊C4—21no—
51C2×C42×C5C2×C412yes—
52C2×C42×C5C2311yescharacteristic
53C2×#681—22no—
54#3968—42no—
55C22⋊#168—22no—
56C4⋊#168—22no—
57C4⋊#168—22no—
58C4⋊#168—22no—
59C4×#168—22no—
60C2×#554D511yescharacteristic
61#1801—51no—
62#1549—52no—
63#1486—52no—
64#1461—51no—
65C24×C5—21no—
66C2×C5×Q8—22no—
67C2×C5×Q8—24no—
68C2×C5×Q8—22no—
69C2×C5×Q8—21no—
70C2×C5×Q8—21no—
71C2×C5×Q8—82no—
72C2×C5×D4—24no—
73C2×C5×D4—21no—
74C2×C5×D4C22⋊C412yes—
75C22×C4×C5—21no—
76C22×C4×C5—22no—
77C22×C4×C5—28no—
78C22×C4×C5—24no—
79C22×C4×C5—42no—
80C22×C4×C5—41no—
81C22×C4×C5—41no—
82C22×C4×C5—84no—
83C22×C4×C5C22⋊C412yes—
84C22×C4×C5C22⋊C412yes—
85C22×C4×C5C22×C411yescharacteristic
86C22×C4×C5C2×D411yescharacteristic
87C22×C4×C5C2×D411yescharacteristic
88C5×C4⋊C4—28no—
89C5×C4⋊C4—28no—
90C5×C4⋊C4—21no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4—22no—
93C5×C4⋊C4—24no—
94C5×C4⋊C4—24no—
95C5×C4⋊C4C22⋊C414yes—
96C5×C4⋊C4C22⋊C412yes—
97C5×C22⋊C4—28no—
98C5×C22⋊C4—24no—
99C42×C5—22no—
100C42×C5—22no—
101C42×C5C2×D412yes—
102C42×C5C2×D412yes—
103#681—42no—
104#681—42no—
105C2×#168—42no—
106C2×#168—42no—
107#554—24no—
108#554—24no—
109C2×C4⋊Q8C2×D511yescharacteristic
110C2×#125Dic512yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C4×Q8C2×D511yescharacteristic
114C2×C4×D4Dic512yes—
115C2×C4×D4C2×D511yescharacteristic
116C22×C4⋊C4—21no—
117#431—52no—
118#423—52no—
119C2×#101—101no—
120#343—101no—
121#332—104no—
122D4⋊C8—104no—
123#324—102no—
124C23×C5—24no—
125C23×C5—22no—
126C23×C5—42no—
127C23×C5—41no—
128C23×C5—41no—
129C23×C5C2×C22⋊C411yescharacteristic
130C5×Q8—42no—
131C5×Q8—42no—
132C5×Q8—84no—
133C5×D4—24no—
134C5×D4—22no—
135C5×D4#10114yescomplemented
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—24no—
141C2×C4×C5—22no—
142C2×C4×C5—22no—
143C2×C4×C5—41no—
144C2×C4×C5—44no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
147C2×C4×C5—44no—
148C2×C4×C5—44no—
149C2×C4×C5—42no—
150C2×C4×C5—88no—
151C2×C4×C5C2×C22⋊C411yescharacteristic
152C2×C4×C5C2×C22⋊C411yescharacteristic
153C2×C4×C5#12012yes—
154C2×C4×C5#12012yes—
155C2×C4×C5#13512yes—
156C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C5#13711yescharacteristic
158#168—82no—
159#168—82no—
160C22×Q8—21no—
161C22×Q8C22⋊D511yescharacteristic
162C22×D4C2×Dic511yescharacteristic
163C23×C4—21no—
164C23×C4—22no—
165C2×M16—102no—
166C4⋊Q8—22no—
167C4⋊Q8C22⋊D514yes—
168#125—24no—
169#125—24no—
170C22⋊Q8—28no—
171C22⋊Q8—216no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—22no—
177C4×D4C22⋊D514yes—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic512yes—
183C2×C4⋊C4C2×Dic511yescharacteristic
184C2×C4⋊C4C22⋊D511yescharacteristic
185C2×C4⋊C4C22⋊D512yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42C2×Dic512yes—
189C2×C42C22×D511yescharacteristic
190C4⋊C8—102no—
191C4⋊C8—102no—
192C4⋊C8—102no—
193#101—202no—
194C22⋊C8—102no—
195C4×C8—102no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—24no—
199C22×C5—44no—
200C22×C5—44no—
201C22×C5—42no—
202C22×C5—42no—
203C22×C5#37911yescharacteristic
204C22×C5#41611yescharacteristic
205C22×C5#41711yescharacteristic
206C4×C5—22no—
207C4×C5—21no—
208C4×C5—44no—
209C4×C5—42no—
210C4×C5—42no—
211C4×C5—42no—
212C4×C5C2×#10112yes—
213C4×C5#44912yes—
214C4×C5#45212yes—
215C24—21no—
216C2×Q8—22no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—21no—
221C2×Q8—82no—
222C2×D4—24no—
223C2×D4—21no—
224C2×D4C22⋊Dic512yes—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—84no—
233C22×C4C22⋊Dic512yes—
234C22×C4C22⋊Dic512yes—
235C22×C4C22×Dic511yescharacteristic
236C22×C4C2×C22⋊D511yescharacteristic
237C22×C4C2×C22⋊D511yescharacteristic
238M16—202no—
239M16—202no—
240C2×C8—202no—
241C2×C8—202no—
242C4⋊C4—28no—
243C4⋊C4—28no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4C22⋊Dic514yes—
250C4⋊C4C22⋊Dic512yes—
251C22⋊C4—28no—
252C22⋊C4—24no—
253C42—22no—
254C42—22no—
255C42C2×C22⋊D512yes—
256C42C2×C22⋊D512yes—
257C2×C5—21no—
258C2×C5—21no—
259C2×C5—44no—
260C2×C5#178411yescharacteristic
261C2×C5#178811yescharacteristic
262C2×C5#179511yescharacteristic
263C23—24no—
264C23—22no—
265C23—42no—
266C23—41no—
267C23—41no—
268C23C2×C22⋊Dic511yescharacteristic
269Q8—42no—
270Q8—42no—
271Q8—84no—
272D4—24no—
273D4—22no—
274D4#396814yescomplemented
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—88no—
290C2×C4#409511yescharacteristic
291C2×C4#409612yes—
292C2×C4#409711yescharacteristic
293C2×C4C2×C22⋊Dic511yescharacteristic
294C2×C4C2×C22⋊Dic511yescharacteristic
295C2×C4#409912yes—
296C2×C4#409912yes—
297C8—402no—
298C8—402no—
299C5#839811yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22#6552111yescharacteristic
308C22#6552211yescharacteristic
309C22#6553611yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4C2×#396812yes—
317C4#6552712yes—
318C4#6552812yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2#1060292611yescharacteristic
323C2#1060292811yescharacteristic
324C2#1060293411yescharacteristic
subgroupG/H
centerC22#65536
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC2×C5×#554C2
socleC22×C5#379
prime radicalC22×C5×D4C2×C4
cosocleC2×C42×C5C23
hypercenterC2×#554D5
hypocenterC5#8398
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #10602568C2semidirect
3. #10602430C2semidirect
4. #10602417C2semidirect
135. C5×D4#101semidirect
274. D4#3968semidirect
299. C5#8398semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5542
#106025682
#106024302
#106024172
#106023912
#83985