Finite Group Databasebeta

← #11991038

Subgroups — #11991038

Order 1280.

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

normal subgroups only

This group has 706 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×#557C211yesmaximal, characteristic
2#10602566C212yesmaximal
3#10602430C211yesmaximal, characteristic
4#10602417C211yesmaximal, characteristic
5#10602415C212yesmaximal
6C5×#557—24no—
7C5×#557—24no—
8C2×C5×C4⋊Q8—21no—
9C2×C5×C4⋊Q8C2211yescharacteristic
10C2×C5×#125—22no—
11C2×C5×#125C412yes—
12C2×C4×C5×Q8—21no—
13C2×C4×C5×Q8C412yes—
14C2×C4×C5×Q8C2211yescharacteristic
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#3968—21no—
17#65214C2212yes—
18#65213C2212yes—
19#64767—24no—
20Q8⋊#168—24no—
21#64750—21no—
22#64748—22no—
23#8390—51nomaximal, Sylow
24C22×C5×Q8—21no—
25C22×C5×Q8C2×C411yescharacteristic
26C22×C5×Q8D411yescharacteristic
27C5×C4⋊Q8—28no—
28C5×C4⋊Q8—22no—
29C5×C4⋊Q8D414yes—
30C5×#125—216no—
31C5×#125—24no—
32C5×#125—24no—
33C4×C5×Q8—28no—
34C4×C5×Q8—24no—
35C4×C5×Q8—24no—
36C4×C5×Q8—22no—
37C4×C5×Q8—22no—
38C4×C5×Q8—22no—
39C4×C5×Q8D414yes—
40C2×C5×C4⋊C4—22no—
41C2×C5×C4⋊C4—22no—
42C2×C5×C4⋊C4—22no—
43C2×C5×C4⋊C4—21no—
44C2×C5×C4⋊C4—21no—
45C2×C5×C4⋊C4C2×C412yes—
46C2×C5×C4⋊C4C2×C411yescharacteristic
47C2×C5×C4⋊C4D411yescharacteristic
48C2×C5×C4⋊C4D412yes—
49C2×C42×C5—22no—
50C2×C42×C5—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×#557D511yescharacteristic
61#1801—51no—
62#1546—52no—
63#1487—52no—
64#1461—51no—
65C2×C5×Q8—24no—
66C2×C5×Q8—21no—
67C2×C5×Q8—22no—
68C2×C5×Q8—24no—
69C2×C5×Q8—22no—
70C2×C5×Q8—21no—
71C2×C5×Q8—21no—
72C2×C5×Q8—42no—
73C2×C5×Q8—41no—
74C2×C5×Q8—41no—
75C2×C5×Q8C22⋊C412yes—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—22no—
79C22×C4×C5C22⋊C412yes—
80C22×C4×C5C22⋊C412yes—
81C22×C4×C5C22×C411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C5×C4⋊C4—28no—
85C5×C4⋊C4—28no—
86C5×C4⋊C4—28no—
87C5×C4⋊C4—21no—
88C5×C4⋊C4—22no—
89C5×C4⋊C4—24no—
90C5×C4⋊C4—24no—
91C5×C4⋊C4—22no—
92C5×C4⋊C4—24no—
93C5×C4⋊C4C22⋊C414yes—
94C5×C4⋊C4C22⋊C412yes—
95C42×C5—28no—
96C42×C5—22no—
97C42×C5—22no—
98C42×C5—24no—
99C42×C5C2×D412yes—
100C42×C5C2×D412yes—
101#681—42no—
102#681—42no—
103C2×#168—42no—
104C2×#168—42no—
105#557—24no—
106#557—24no—
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8C2×D511yescharacteristic
109C2×#125—22no—
110C2×#125Dic512yes—
111C2×C4×Q8—21no—
112C2×C4×Q8Dic512yes—
113C2×C4×Q8C2×D511yescharacteristic
114C2×C4×Q8C2×D511yescharacteristic
115#431—52no—
116#423—52no—
117C2×#101—101no—
118#343—101no—
119#332—104no—
120Q8⋊C8—104no—
121#324—102no—
122C23×C5C2×C22⋊C411yescharacteristic
123C5×Q8—24no—
124C5×Q8—22no—
125C5×Q8—42no—
126C5×Q8—44no—
127C5×Q8—42no—
128C5×Q8—44no—
129C5×Q8#10114yescomplemented
130C2×C4×C5—24no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—22no—
135C2×C4×C5—22no—
136C2×C4×C5—24no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C2×C4×C5—41no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—42no—
144C2×C4×C5C2×C22⋊C411yescharacteristic
145C2×C4×C5C2×C22⋊C411yescharacteristic
146C2×C4×C5#12012yes—
147C2×C4×C5#12012yes—
148C2×C4×C5#13512yes—
149C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C5#13711yescharacteristic
151#168—82no—
152#168—82no—
153C22×Q8—21no—
154C22×Q8C2×Dic511yescharacteristic
155C22×Q8C22⋊D511yescharacteristic
156C2×M16—102no—
157C4⋊Q8—28no—
158C4⋊Q8—22no—
159C4⋊Q8C22⋊D514yes—
160#125—216no—
161#125—24no—
162#125—24no—
163C4×Q8—28no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8C22⋊D514yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C2×Dic512yes—
176C2×C4⋊C4C2×Dic511yescharacteristic
177C2×C4⋊C4C22⋊D511yescharacteristic
178C2×C4⋊C4C22⋊D512yes—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C2×Dic512yes—
182C2×C42C22×D511yescharacteristic
183C4⋊C8—102no—
184C4⋊C8—102no—
185C4⋊C8—102no—
186#101—202no—
187C22⋊C8—102no—
188C4×C8—102no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5#37911yescharacteristic
192C22×C5#41611yescharacteristic
193C22×C5#41711yescharacteristic
194C4×C5—22no—
195C4×C5—24no—
196C4×C5—21no—
197C4×C5—42no—
198C4×C5—42no—
199C4×C5—42no—
200C4×C5—44no—
201C4×C5C2×#10112yes—
202C4×C5#44912yes—
203C4×C5#45212yes—
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—24no—
210C2×Q8—21no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8C22⋊Dic512yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4C22⋊Dic512yes—
219C22×C4C22⋊Dic512yes—
220C22×C4C22×Dic511yescharacteristic
221C22×C4C2×C22⋊D511yescharacteristic
222C22×C4C2×C22⋊D511yescharacteristic
223M16—202no—
224M16—202no—
225C2×C8—202no—
226C2×C8—202no—
227C4⋊C4—28no—
228C4⋊C4—28no—
229C4⋊C4—28no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—24no—
236C4⋊C4C22⋊Dic514yes—
237C4⋊C4C22⋊Dic512yes—
238C42—28no—
239C42—22no—
240C42—22no—
241C42—24no—
242C42C2×C22⋊D512yes—
243C42C2×C22⋊D512yes—
244C2×C5—21no—
245C2×C5—21no—
246C2×C5#178411yescharacteristic
247C2×C5#178811yescharacteristic
248C2×C5#179611yescharacteristic
249C23C2×C22⋊Dic511yescharacteristic
250Q8—24no—
251Q8—22no—
252Q8—42no—
253Q8—44no—
254Q8—42no—
255Q8—44no—
256Q8#396814yescomplemented
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4#409511yescharacteristic
272C2×C4#409612yes—
273C2×C4#409711yescharacteristic
274C2×C4C2×C22⋊Dic511yescharacteristic
275C2×C4C2×C22⋊Dic511yescharacteristic
276C2×C4#409912yes—
277C2×C4#409912yes—
278C8—402no—
279C8—402no—
280C5#839011yescharacteristic, complemented, Sylow
281C22—21no—
282C22—21no—
283C22#6552111yescharacteristic
284C22#6552211yescharacteristic
285C22#6553611yescharacteristic
286C4—24no—
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4C2×#396812yes—
294C4#6552712yes—
295C4#6552812yes—
296C2—21no—
297C2—21no—
298C2#1060292611yescharacteristic
299C2#1060292811yescharacteristic
300C2#1060293711yescharacteristic
subgroupG/H
centerC22#65536
commutatorC22×C4×C5C22×C4
FrattiniC2×C42C22×D5
FittingC2×C5×#557C2
socleC22×C5#379
prime radicalC23×C5C2×C22⋊C4
cosocleC2×C42×C5C23
hypercenterC2×#557D5
hypocenterC5#8390
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
129. C5×Q8#101semidirect
256. Q8#3968semidirect
280. C5#8390semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5572
#106025662
#106024302
#106024172
#106024152
#83905