Finite Group Databasebeta

← #10604344

Subgroups — #10604344

Order 640.

564 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C22×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
2C2×D5×C4⋊C4C211yesmaximal, characteristic
3C2×(C2×AGL(1,𝔽5))⋊C4C212yesmaximal
4C2×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic
5C2×(C2×AGL(1,𝔽5))⋊C4C212yesmaximal
6C23×AGL(1,𝔽5)C2212yes—
7C23×AGL(1,𝔽5)C2212yes—
8C22×C4×D5C2211yescharacteristic
9C22×C4×D5C2211yescharacteristic
10C22×C4×D5C2211yescharacteristic
11C2×C4⋊AGL(1,𝔽5)—24no—
12C2×C4⋊AGL(1,𝔽5)—22no—
13C2×C5×C4⋊C4C411yescharacteristic, complemented
14C2×C4⋊Dic5C411yescharacteristic, complemented
15C2×#683C411yescharacteristic, complemented
16C2×#683C411yescharacteristic, complemented
17D5×C4⋊C4—22no—
18D5×C4⋊C4—22no—
19(C2×AGL(1,𝔽5))⋊C4—24no—
20(C2×AGL(1,𝔽5))⋊C4—22no—
21(C2×AGL(1,𝔽5))⋊C4—24no—
22#1441—51nomaximal, Sylow
23C23×D5C2311yescharacteristic
24C22×AGL(1,𝔽5)—22no—
25C22×AGL(1,𝔽5)—22no—
26C22×AGL(1,𝔽5)—22no—
27C22×AGL(1,𝔽5)—22no—
28C22×AGL(1,𝔽5)—22no—
29C22×AGL(1,𝔽5)—22no—
30C22×AGL(1,𝔽5)—22no—
31C22×AGL(1,𝔽5)—22no—
32C22×AGL(1,𝔽5)—22no—
33C22×AGL(1,𝔽5)—22no—
34C22×AGL(1,𝔽5)—22no—
35C22×AGL(1,𝔽5)—22no—
36C22×AGL(1,𝔽5)D414yes—
37C22×AGL(1,𝔽5)Q814yes—
38C22×C4×C5C2×C411yescharacteristic
39C22×C4×C5C2×C411yescharacteristic
40C22×C4×C5C2×C411yescharacteristic
41C22×Dic5C2×C411yescharacteristic
42C22×Dic5C2×C411yescharacteristic
43C22×Dic5C2×C411yescharacteristic
44C2×C4×D5—22no—
45C2×C4×D5—21no—
46C2×C4×D5—21no—
47C2×C4×D5—21no—
48C2×C4×D5—21no—
49C2×C4×D5—22no—
50C2×C4×D5—22no—
51C2×C4×D5—21no—
52C2×C4×D5—21no—
53C2×C4×D5—22no—
54C2×C4×D5D412yes—
55C2×C4×D5D412yes—
56C2×C4×D5D412yes—
57C2×C4×D5Q812yes—
58C4⋊AGL(1,𝔽5)—28no—
59C5×C4⋊C4—22no—
60C4⋊Dic5—22no—
61#683—22no—
62#683—22no—
63C22×C4⋊C4—51no—
64C22×C4⋊C4—51no—
65C2×#95—51no—
66C2×#95—52no—
67C2×#95—52no—
68C23×C5C22×C411yescharacteristic
69C22×D5C2×D412yes—
70C22×D5C2×D411yescharacteristic
71C22×D5C2×D411yescharacteristic
72C22×D5C2×D411yescharacteristic
73C22×D5C2×Q812yes—
74C22×D5C2×Q811yescharacteristic
75C22×D5#4111yescharacteristic
76C22×D5#4111yescharacteristic
77C22×D5#4112yes—
78C22×D5#4112yes—
79C2×AGL(1,𝔽5)—24no—
80C2×AGL(1,𝔽5)—24no—
81C2×AGL(1,𝔽5)—24no—
82C2×AGL(1,𝔽5)—24no—
83C2×AGL(1,𝔽5)—24no—
84C2×AGL(1,𝔽5)—24no—
85C2×AGL(1,𝔽5)—42no—
86C2×AGL(1,𝔽5)—42no—
87C2×AGL(1,𝔽5)—42no—
88C2×AGL(1,𝔽5)—42no—
89C2×AGL(1,𝔽5)—42no—
90C2×AGL(1,𝔽5)—42no—
91C2×AGL(1,𝔽5)—42no—
92C2×AGL(1,𝔽5)—42no—
93C2×C4×C5—21no—
94C2×C4×C5—21no—
95C2×C4×C5—22no—
96C2×C4×C5—21no—
97C2×C4×C5—22no—
98C2×C4×C5C22⋊C412yes—
99C2×C4×C5C4⋊C412yescomplemented
100C2×Dic5—21no—
101C2×Dic5—21no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5C22⋊C412yes—
106C2×Dic5C4⋊C412yescomplemented
107C4×D5—22no—
108C4×D5—22no—
109C4×D5—22no—
110C4×D5—24no—
111C4×D5—22no—
112C4×D5—22no—
113C4×D5—22no—
114C4×D5—41no—
115C4×D5—42no—
116C4×D5—41no—
117C23×C4—51no—
118C23×C4—51no—
119C23×C4—52no—
120C23×C4—52no—
121C23×C4—51no—
122C2×C4⋊C4—51no—
123C2×C4⋊C4—51no—
124C2×C4⋊C4—51no—
125C2×C4⋊C4—102no—
126C2×C4⋊C4—102no—
127C2×C4⋊C4—102no—
128C2×C4⋊C4—104no—
129C2×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
130#95—104no—
131#95—102no—
132#95—104no—
133C22×C5C2×C22⋊C411yescharacteristic
134C22×C5C2×C4⋊C411yescharacteristic
135C22×C5#11711yescharacteristic
136C22×C5C4×D412yes—
137C22×C5C4×Q812yes—
138C2×D5#12011yescharacteristic
139C2×D5#12112yes—
140C2×D5#12111yescharacteristic
141C2×D5#12111yescharacteristic
142C2×D5#12112yes—
143C2×D5C22⋊Q812yes—
144C2×D5C22⋊Q811yescharacteristic
145C2×D5C22⋊Q812yes—
146C2×D5C22⋊Q812yes—
147C2×D5C22⋊Q811yescharacteristic
148C2×D5#12312yes—
149C2×D5#12311yescharacteristic
150C2×D5#12411yescharacteristic
151C2×D5#12512yes—
152C2×D5#12512yes—
153C2×D5#12612yes—
154C2×D5C4⋊Q812yes—
155C2×D5C4⋊Q811yescharacteristic
156AGL(1,𝔽5)—44no—
157AGL(1,𝔽5)—44no—
158C4×C5—22no—
159C4×C5—22no—
160C4×C5—41no—
161Dic5—22no—
162Dic5—22no—
163Dic5—41no—
164C24—51no—
165C22×C4—51no—
166C22×C4—51no—
167C22×C4—52no—
168C22×C4—52no—
169C22×C4—52no—
170C22×C4—52no—
171C22×C4—54no—
172C22×C4—54no—
173C22×C4—51no—
174C22×C4—102no—
175C22×C4—101no—
176C22×C4—101no—
177C22×C4—101no—
178C22×C4—101no—
179C22×C4—102no—
180C22×C4—102no—
181C22×C4—101no—
182C22×C4—101no—
183C22×C4—102no—
184C22×C4—102no—
185C22×C4—102no—
186C22×C4—102no—
187C22×C4—102no—
188C22×C4—102no—
189C22×C4—102no—
190C22×C4—102no—
191C22×C4—102no—
192C22×C4—102no—
193C22×C4—102no—
194C22×C4—102no—
195C22×C4—102no—
196C22×C4C2×AGL(1,𝔽5)11yescharacteristic
197C22×C4C2×AGL(1,𝔽5)11yescharacteristic
198C22×C4C2×AGL(1,𝔽5)11yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—102no—
201C4⋊C4—102no—
202C4⋊C4—108no—
203C4⋊C4—102no—
204C2×C5#38011yescharacteristic
205C2×C5#38611yescharacteristic
206C2×C5#38712yes—
207C2×C5#38912yes—
208C2×C5#39111yescharacteristic
209D5#39411yescharacteristic
210D5#39511yescharacteristic
211D5#39612yes—
212D5#39811yescharacteristic
213D5#39812yes—
214D5#40011yescharacteristic
215C23—51no—
216C23—51no—
217C23—51no—
218C23—51no—
219C23—52no—
220C23—52no—
221C23—52no—
222C23—52no—
223C23—51no—
224C23—51no—
225C23C22×AGL(1,𝔽5)11yescharacteristic
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—52no—
232C2×C4—52no—
233C2×C4—102no—
234C2×C4—102no—
235C2×C4—102no—
236C2×C4—102no—
237C2×C4—102no—
238C2×C4—104no—
239C2×C4—102no—
240C2×C4—101no—
241C2×C4—102no—
242C2×C4—101no—
243C2×C4—104no—
244C2×C4—104no—
245C2×C4—104no—
246C2×C4—104no—
247C2×C4—104no—
248C2×C4—104no—
249C2×C4—102no—
250C2×C4—101no—
251C2×C4—201no—
252C2×C4—202no—
253C2×C4—201no—
254C2×C4—202no—
255C2×C4—202no—
256C2×C4—202no—
257C2×C4—202no—
258C2×C4—202no—
259C2×C4—202no—
260C2×C4—202no—
261C2×C4—202no—
262C2×C4C4⋊AGL(1,𝔽5)12yescomplemented
263C2×C4C22⋊AGL(1,𝔽5)12yes—
264C5#144111yescharacteristic, complemented, Sylow
265C22—52no—
266C22—52no—
267C22—52no—
268C22—52no—
269C22—52no—
270C22—52no—
271C22—52no—
272C22—52no—
273C22—51no—
274C22—51no—
275C22—51no—
276C22—51no—
277C22—51no—
278C22—52no—
279C22—51no—
280C22—52no—
281C22—51no—
282C22—51no—
283C22C2×C4⋊AGL(1,𝔽5)11yescharacteristic
284C22#413011yescharacteristic
285C22D4×AGL(1,𝔽5)12yes—
286C22Q8×AGL(1,𝔽5)12yes—
287C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—102no—
292C4—102no—
293C4—204no—
294C4—204no—
295C4—201no—
296C2—51no—
297C2—52no—
298C2—51no—
299C2—52no—
300C2—51no—
301C2—51no—
302C2#6571112yes—
303C2#6571212yes—
304C2#6576511yescharacteristic
305C2#6577911yescharacteristic
306C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C22×AGL(1,𝔽5)
commutator, socleC23×C5C22×C4
FittingC2×C5×C4⋊C4C4
prime radical, cosocleC23×D5C23
hypercenterC2×C4⋊C4AGL(1,𝔽5)
hypocenterC5#1441
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
13. C2×C5×C4⋊C4C4semidirect
14. C2×C4⋊Dic5C4semidirect
15. C2×#683C4semidirect
16. C2×#683C4semidirect
99. C2×C4×C5C4⋊C4semidirect
106. C2×Dic5C4⋊C4semidirect
129. C2×C4⋊C4AGL(1,𝔽5)semidirect
262. C2×C4C4⋊AGL(1,𝔽5)semidirect
264. C5#1441semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,𝔽5)2
C2×D5×C4⋊C42
C2×(C2×AGL(1,𝔽5))⋊C42
#14415