Finite Group Databasebeta

← Dic5×#356 #12014484

Subgroups — Dic5×#356 #12014484

Order 1280.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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
1Dic5×C4⋊Q8C211yesmaximal, characteristic
2C2×C5×#356C211yesmaximal, characteristic
3C2×#64768C211yesmaximal, characteristic
4Dic5×#101C212yesmaximal
5#10598190C212yesmaximal
6C2×C5×C4⋊Q8C2211yescharacteristic
7C2×Q8×Dic5C411yescharacteristic
8C2×Q8×Dic5C2211yescharacteristic
9C2×C5×#101C2212yes—
10#65519C411yescharacteristic
11#65519C2211yescharacteristic
12C2×#3968C2212yes—
13M16×Dic5—22no—
14#65274—22no—
15C4⋊C4×Dic5—22no—
16#65224C411yescharacteristic
17C42×Dic5C411yescharacteristic
18C5×#356C414yescomplemented
19#64774—22no—
20#64768C414yescomplemented
21C4×#356—51nomaximal, Sylow
22C22×C5×Q8C2×C411yescharacteristic
23C22×C5×Q8C2311yescharacteristic
24C2×C5×M16—22no—
25C5×C4⋊Q8—22no—
26C5×C4⋊Q8C2×C412yes—
27C5×C4⋊Q8C2×C412yes—
28C2×C5×C4⋊C4—22no—
29C2×C42×C5C2×C411yescharacteristic
30Q8×Dic5—41no—
31Q8×Dic5—41no—
32Q8×Dic5—41no—
33Q8×Dic5—41no—
34C2×C4⋊Dic5—21no—
35C2×C4⋊Dic5—22no—
36C2×C4⋊Dic5C2×C411yescharacteristic
37C2×C4⋊Dic5C2×C411yescharacteristic
38C2×C4⋊Dic5D412yes—
39C2×C4×Dic5—21no—
40C2×C4×Dic5—21no—
41C2×C4×Dic5—21no—
42C2×C4×Dic5C2×C411yescharacteristic
43C2×C4×Dic5C2×C411yescharacteristic
44C2×C4×Dic5D412yes—
45C2×#681—22no—
46C5×#101—22no—
47C5×#101C2×C414yes—
48#3968—22no—
49#3968C2×C414yes—
50#3963—22no—
51#3963—22no—
52#3947—42no—
53C8×Dic5—42no—
54C4×C4⋊Q8—51no—
55C2×#356—51no—
56C2×#356D511yescharacteristic
57C4×#101—52no—
58#1394—52no—
59C2×C5×Q8—21no—
60C2×C5×Q8—41no—
61C2×C5×Q8—41no—
62C2×C5×Q8—41no—
63C2×C5×Q8—41no—
64C2×C5×Q8C4212yes—
65C2×C5×Q8C22×C411yescharacteristic
66C2×C5×Q8C22×C411yescharacteristic
67C2×C5×Q8#4112yes—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5C22×C411yescharacteristic
71C22×C4×C5C2×D412yes—
72C22×Dic5—21no—
73C22×Dic5—21no—
74C22×Dic5C22⋊C411yescharacteristic
75C22×Dic5C22⋊C412yes—
76C22×Dic5C22⋊C411yescharacteristic
77C5×M16—24no—
78C5×M16—42no—
79C2×C5×C8—42no—
80C5×C4⋊C4—24no—
81C5×C4⋊C4—24no—
82C42×C5—21no—
83C42×C5C4212yes—
84C4⋊Dic5—41no—
85C4⋊Dic5—42no—
86C4⋊Dic5—42no—
87C4⋊Dic5—44no—
88C4⋊Dic5—41no—
89C4×Dic5—24no—
90C4×Dic5—22no—
91C4×Dic5—22no—
92C4×Dic5—21no—
93C4×Dic5—21no—
94C4×Dic5—42no—
95C4×Dic5—41no—
96C4×Dic5—41no—
97C4×Dic5—41no—
98#681—24no—
99#681—42no—
100C2×#168—42no—
101C2×C4⋊Q8C2×D511yescharacteristic
102C2×C4×Q8—51no—
103C2×C4×Q8—51no—
104C2×#101—52no—
105C2×#101C2×D512yes—
106C4×M16—102no—
107#391—51no—
108#391—51no—
109#389—102no—
110#384—51no—
111C4×C4⋊C4—102no—
112C43—51no—
113#356—54no—
114#356Dic514yesdirect product factor
115#343—102no—
116C23×C5C2×C22⋊C411yescharacteristic
117C5×Q8—41no—
118C5×Q8—41no—
119C5×Q8—42no—
120C5×Q8—42no—
121C5×Q8—42no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—22no—
125C2×C4×C5—22no—
126C2×C4×C5—21no—
127C2×C4×C5—22no—
128C2×C4×C5—21no—
129C2×C4×C5—22no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5C2×C4211yescharacteristic
133C2×C4×C5#11711yescharacteristic
134C2×C4×C5C4×D412yes—
135C2×C4×C5C4×D412yes—
136C2×Dic5—22no—
137C2×Dic5—22no—
138C2×Dic5—21no—
139C2×Dic5—22no—
140C2×Dic5—41no—
141C2×Dic5—41no—
142C2×Dic5—41no—
143C2×Dic5—42no—
144C2×Dic5—42no—
145C2×Dic5—41no—
146C2×Dic5#9911yescharacteristic
147C2×Dic5#9911yescharacteristic
148C5×C8—44no—
149#168—44no—
150C22×Q8C4×D511yescharacteristic
151C22×Q8C22×D511yescharacteristic
152C2×M16—22no—
153C2×M16—102no—
154C4⋊Q8—22no—
155C4⋊Q8C4×D512yes—
156C4⋊Q8C2×Dic512yes—
157C4×Q8—201no—
158C4×Q8—201no—
159C4×Q8—201no—
160C4×Q8—201no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—51no—
163C2×C4⋊C4—52no—
164C2×C4⋊C4—51no—
165C2×C4⋊C4—101no—
166C2×C4⋊C4—102no—
167C2×C42—51no—
168C2×C42—52no—
169C2×C42—51no—
170C2×C42—101no—
171C2×C42—101no—
172C2×C42—101no—
173C2×C42C4×D511yescharacteristic
174#101—22no—
175#101—54no—
176#101—102no—
177#101C2×Dic514yes—
178#97—202no—
179C4×C8—202no—
180#95—102no—
181#95—102no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5C4×C22⋊C411yescharacteristic
185C22×C5C2×#9911yescharacteristic
186C22×C5#41011yescharacteristic
187C4×C5—24no—
188C4×C5—22no—
189C4×C5—22no—
190C4×C5—21no—
191C4×C5—21no—
192C4×C5—42no—
193Dic5—21no—
194Dic5—42no—
195Dic5—41no—
196Dic5—42no—
197Dic5—81no—
198Dic5#35612yesdirect product factor
199C2×Q8—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8C4×Dic512yes—
205C2×Q8C2×C4×D511yescharacteristic
206C2×Q8#71112yes—
207C2×Q8C22×Dic511yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—51no—
211C22×C4—52no—
212C22×C4—51no—
213C22×C4—101no—
214C22×C4—101no—
215C22×C4C2×C4×D511yescharacteristic
216C22×C4D4×D512yes—
217M16—24no—
218M16—42no—
219M16—104no—
220M16—202no—
221C2×C8—42no—
222C2×C8—202no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—201no—
226C4⋊C4—202no—
227C4⋊C4—201no—
228C4⋊C4—204no—
229C4⋊C4—202no—
230C42—21no—
231C42—102no—
232C42—102no—
233C42—101no—
234C42—101no—
235C42—104no—
236C42—201no—
237C42—202no—
238C42—201no—
239C42—201no—
240C42C4×Dic512yes—
241C2×C5—21no—
242C2×C5—21no—
243C2×C5C4×#9911yescharacteristic
244C2×C5C2×#35611yescharacteristic
245C2×C5#213211yescharacteristic
246C23D5×C22⋊C411yescharacteristic
247Q8—41no—
248Q8—42no—
249Q8—41no—
250Q8—42no—
251Q8—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—51no—
263C2×C4—51no—
264C2×C4—102no—
265C2×C4—102no—
266C2×C4—101no—
267C2×C4—102no—
268C2×C4—201no—
269C2×C4—201no—
270C2×C4—201no—
271C2×C4—202no—
272C2×C4—202no—
273C2×C4—201no—
274C2×C4#402311yescharacteristic
275C2×C4#402712yes—
276C2×C4C2×C4×Dic511yescharacteristic
277C2×C4D4×Dic512yes—
278C8—44no—
279C8—204no—
280C5C4×#35611yescharacteristic, complemented, Sylow
281C22—21no—
282C22—21no—
283C22#6502911yescharacteristic
284C22D5×#9911yescharacteristic
285C22C22⋊C4×Dic511yescharacteristic
286C4—24no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—52no—
293C4—101no—
294C4—202no—
295C4—201no—
296C4—202no—
297C4—401no—
298C2—21no—
299C2—21no—
300C2D5×#35611yescharacteristic
301C2#1059996511yescharacteristic
302C2Dic5×#9911yescharacteristic
subgroupG/H
centerC22D5×#99
commutatorC2×C4×C5C2×C42
FrattiniC22×Q8C22×D5
FittingC2×C5×#356C2
socleC22×C5C2×#99
prime radicalC23×C5C2×C22⋊C4
cosocleC22×C5×Q8C23
hypercenterC2×#356D5
hypocenterC5C4×#356
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
18. C5×#356C4semidirect
20. #64768C4semidirect
114. #356Dic5direct (N × H)
198. Dic5#356direct (N × H)
280. C5C4×#356semidirect

Maximal subgroups

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

subgroupindex
Dic5×C4⋊Q82
C2×C5×#3562
C2×#647682
Dic5×#1012
#105981902
C4×#3565