Finite Group Databasebeta

← #13017539

Subgroups — #13017539

Order 1280.

1030 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 152 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×D5×C4⋊Q8C211yesmaximal, characteristic
2C2×#64931C214yesmaximal, complemented
3#10604183C212yesmaximal
4C22×Q8×D5C412yes—
5C2×C5×C4⋊Q8C411yescharacteristic
6C2×#4090C2212yes—
7D5×C4⋊Q8—28no—
8C2×D5×C4⋊C4—22no—
9C2×#4034—22no—
10C2×C42×D5C2211yescharacteristic
11C2×#4015C411yescharacteristic
12C2×#4004C2214yes—
13C2×#4003—22no—
14#64931—28no—
15#64918—22no—
16#8342—51nomaximal, Sylow
17C22×C5×Q8C2×C412yes—
18C2×Q8×D5—24no—
19C2×Q8×D5—24no—
20C2×Q8×D5—44no—
21C2×Q8×D5—44no—
22C22×C4×D5—22no—
23C22×C4×D5C2×C411yescharacteristic
24C22×C4×D5D412yes—
25C22×Dic10—22no—
26C22×Dic10C2×C412yes—
27C22×#170—24no—
28C5×C4⋊Q8—24no—
29C2×C5×C4⋊C4—22no—
30C2×C42×C5C2×C411yescharacteristic
31#4090—24no—
32#4090D418yes—
33C2×C4⋊Dic5—22no—
34C2×#683—24no—
35C2×C4×Dic5D412yes—
36C2×C4×Dic5C2311yescharacteristic
37D5×C4⋊C4—216no—
38#4034—216no—
39C42×D5—24no—
40#4015—24no—
41#4004—28no—
42#4003—44no—
43C22×C4⋊Q8—51no—
44#1797—52no—
45C2×#333—54no—
46C23×D5C22⋊C411yescharacteristic
47C2×C5×Q8—22no—
48C2×C5×Q8—42no—
49C2×C5×Q8—42no—
50C2×C5×Q8C22⋊C414yes—
51C22×C4×C5—22no—
52C22×C4×C5C22⋊C412yes—
53C22×C4×C5C22×C411yescharacteristic
54C22×Dic5—22no—
55C22×Dic5C22⋊C411yescharacteristic
56C22×Dic5C2×D412yes—
57Q8×D5—416no—
58Q8×D5—416no—
59C2×C4×D5—22no—
60C2×C4×D5—22no—
61C2×C4×D5—22no—
62C2×C4×D5—24no—
63C2×C4×D5—24no—
64C2×C4×D5—28no—
65C2×C4×D5—44no—
66C2×C4×D5—44no—
67C2×C4×D5SD1614yes—
68C2×C4×D5Q1614yes—
69C2×Dic10—28no—
70C2×Dic10—22no—
71C2×Dic10—44no—
72C2×Dic10—44no—
73C2×Dic10—42no—
74C2×Dic10—42no—
75C2×Dic10C22⋊C414yes—
76C2×#170—44no—
77C2×#170—44no—
78C2×#170—44no—
79C5×C4⋊C4—28no—
80C42×C5—22no—
81C4⋊Dic5—28no—
82#683—216no—
83C4×Dic5—24no—
84C4×Dic5C2×D414yes—
85C23×Q8—52no—
86C2×C4⋊Q8—52no—
87C2×C4⋊Q8—51no—
88C2×C4⋊Q8—102no—
89C2×C4⋊Q8—108no—
90C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic
91C22×C4⋊C4—102no—
92C22×C42—51no—
93C2×C4⋊C8—54no—
94C2×C22⋊C8—102no—
95#336—102no—
96#333—108no—
97C23×C5C2×C22⋊C411yescharacteristic
98C22×D5—21no—
99C22×D5—21no—
100C22×D5—22no—
101C22×D5#10012yes—
102C22×D5#10314yes—
103C5×Q8—44no—
104C5×Q8—44no—
105C2×C4×C5—22no—
106C2×C4×C5—21no—
107C2×C4×C5—21no—
108C2×C4×C5—22no—
109C2×C4×C5—24no—
110C2×C4×C5—42no—
111C2×C4×C5—42no—
112C2×C4×C5#10314yes—
113C2×C4×C5C2×C22⋊C412yes—
114C2×Dic5—21no—
115C2×Dic5—21no—
116C2×Dic5—24no—
117C2×Dic5—42no—
118C2×Dic5—42no—
119C2×Dic5#10012yes—
120C2×Dic5#12014yes—
121C2×Dic5C2×SD1612yes—
122C2×Dic5C2×Q1612yes—
123C2×Dic5#13714yes—
124C4×D5—28no—
125C4×D5—28no—
126C4×D5—28no—
127C4×D5—416no—
128Dic10—416no—
129Dic10—44no—
130Dic10—44no—
131#170—84no—
132C22×Q8—52no—
133C22×Q8—102no—
134C22×Q8—104no—
135C22×Q8—104no—
136C22×Q8—204no—
137C22×Q8—204no—
138C22×Q8C2×AGL(1,𝔽5)12yes—
139C23×C4—52no—
140C23×C4—51no—
141C23×C4—102no—
142C22×C8—104no—
143C4⋊Q8—24no—
144C4⋊Q8—58no—
145C4⋊Q8—104no—
146C4⋊Q8—104no—
147C4⋊Q8—1016no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—1016no—
150C2×C4⋊C4—104no—
151C2×C4⋊C4—102no—
152C2×C42—51no—
153C2×C42—52no—
154C2×C42—104no—
155C2×C42C2×AGL(1,𝔽5)11yescharacteristic
156C4⋊C8—108no—
157C22⋊C8—204no—
158C22×C5#37911yescharacteristic
159C22×C5C2×#10011yescharacteristic
160C22×C5C2×#10011yescharacteristic
161C22×C5C2×#10312yes—
162C22×C5#41912yes—
163C2×D5—24no—
164C2×D5—22no—
165C2×D5—22no—
166C2×D5—24no—
167C2×D5#33314yescomplemented
168C4×C5—22no—
169C4×C5—24no—
170C4×C5—44no—
171Dic5—22no—
172Dic5—44no—
173Dic5#44814yes—
174Dic5#45114yes—
175C24—51no—
176C2×Q8—22no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—54no—
180C2×Q8—108no—
181C2×Q8—102no—
182C2×Q8—202no—
183C2×Q8—202no—
184C2×Q8—204no—
185C2×Q8—204no—
186C2×Q8—2016no—
187C2×Q8—2016no—
188C2×Q8C22⋊AGL(1,𝔽5)14yes—
189C22×C4—22no—
190C22×C4—52no—
191C22×C4—51no—
192C22×C4—54no—
193C22×C4—54no—
194C22×C4—104no—
195C22×C4—102no—
196C22×C4—104no—
197C22×C4—102no—
198C22×C4—108no—
199C22×C4—102no—
200C22×C4—102no—
201C22×C4—204no—
202C22×C4—204no—
203C22×C4C22⋊AGL(1,𝔽5)12yes—
204C22×C4C22×AGL(1,𝔽5)11yescharacteristic
205C2×C8—204no—
206C2×C8—204no—
207C2×C8—204no—
208C4⋊C4—28no—
209C4⋊C4—1016no—
210C4⋊C4—108no—
211C42—22no—
212C42—54no—
213C42—104no—
214C2×C5C2×#33311yescharacteristic
215C2×C5#154812yes—
216C2×C5#155811yescharacteristic
217C2×C5#178311yescharacteristic
218C2×C5#178712yes—
219D5—24no—
220C23—52no—
221C23—54no—
222C23—101no—
223C23—101no—
224C23—102no—
225C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
226Q8—44no—
227Q8—44no—
228Q8—204no—
229Q8—204no—
230Q8—2016no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—54no—
239C2×C4—52no—
240C2×C4—52no—
241C2×C4—54no—
242C2×C4—52no—
243C2×C4—108no—
244C2×C4—108no—
245C2×C4—108no—
246C2×C4—101no—
247C2×C4—101no—
248C2×C4—104no—
249C2×C4—2016no—
250C2×C4—202no—
251C2×C4—202no—
252C2×C4#400914yes—
253C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
254C8—404no—
255C5#834211yescharacteristic, complemented, Sylow
256C22—54no—
257C22—104no—
258C22—102no—
259C22—102no—
260C22—104no—
261C22D5⋊#10011yescharacteristic
262C22D5⋊#10011yescharacteristic
263C22C2×#400912yes—
264C22#6578212yes—
265C22#6580011yescharacteristic
266C4—22no—
267C4—24no—
268C4—44no—
269C4—54no—
270C4—54no—
271C4—102no—
272C4—204no—
273C2—104no—
274C2D5⋊#33311yescharacteristic
275C2#1060465912yes—
276C2#1060466211yescharacteristic
277C2#1060495011yescharacteristic
278C2#1060497312yes—
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#8342
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
2. C2×#64931C2semidirect
167. C2×D5#333semidirect
255. C5#8342semidirect

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#649312
#106041832
#83425