Finite Group Databasebeta

← #10605966

Subgroups — #10605966

Order 640.

412 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C5×#382C211yesmaximal, characteristic, complemented
2#65276C211yesmaximal, characteristic
3#65276C211yesmaximal, characteristic
4#65274C211yesmaximal, characteristic, complemented
5C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6C4×#685C211yesmaximal, characteristic
7#65217C211yesmaximal, characteristic, complemented
8#64963C212yesmaximal
9#64958C212yesmaximal
10#64949C212yesmaximal, complemented
11#64940C212yesmaximal, complemented
12C22×C4×D5C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C42×C5C2211yescharacteristic
16C2×#685C2211yescharacteristic
17C2×#685C2212yes—
18C2×#685C2211yescharacteristic
19C2×#685C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2212yes—
25C2×C4×Dic5C2211yescharacteristic
26C5×#95C2212yes—
27C5×#95C2212yes—
28#3963C2211yescharacteristic
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2211yescharacteristic
34#3963C2212yes—
35#2495—51nomaximal, Sylow
36C23×D5C2311yescharacteristic
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2312yes—
40C22×C4×C5C2312yes—
41C22×C4×C5C2311yescharacteristic
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2312yes—
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2312yes—
46C22×Dic5C2311yescharacteristic
47C2×C4×D5—22no—
48C2×C4×D5—22no—
49C2×C4×D5—22no—
50C5×C4⋊C4—22no—
51C5×C4⋊C4—22no—
52C42×C5—22no—
53#685—24no—
54#685—22no—
55#685—22no—
56#685—42no—
57C4⋊Dic5—22no—
58C4⋊Dic5—22no—
59C4⋊Dic5—42no—
60#683—42no—
61C4×Dic5—22no—
62C4×Dic5—24no—
63#400—52no—
64#397—52no—
65#389—51no—
66#389—51no—
67#388—52no—
68#383—52no—
69#382D511yescharacteristic, complemented
70#380—51no—
71#380—51no—
72C4×C4⋊C4—51no—
73C4×C22⋊C4—51no—
74C23×C5C2411yescharacteristic
75C22×D5—21no—
76C22×D5—21no—
77C22×D5—21no—
78C22×D5—21no—
79C22×D5—21no—
80C22×D5—21no—
81C22×D5#4112yes—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—21no—
85C2×C4×C5—21no—
86C2×C4×C5—21no—
87C2×C4×C5—21no—
88C2×C4×C5—22no—
89C2×C4×C5—22no—
90C2×C4×C5—21no—
91C2×C4×C5—22no—
92C2×C4×C5—22no—
93C2×C4×C5—21no—
94C2×C4×C5#4112yes—
95C2×C4×C5#4112yes—
96C2×C4×C5#4112yes—
97C2×Dic5—21no—
98C2×Dic5—21no—
99C2×Dic5—21no—
100C2×Dic5—22no—
101C2×Dic5—22no—
102C2×Dic5—22no—
103C2×Dic5—21no—
104C2×Dic5—22no—
105C2×Dic5—21no—
106C2×Dic5—21no—
107C2×Dic5—21no—
108C2×Dic5—22no—
109C2×Dic5—22no—
110C2×Dic5—21no—
111C2×Dic5#4112yes—
112C4×D5—44no—
113C23×C4—51no—
114C2×C4⋊C4—51no—
115C2×C4⋊C4—51no—
116C2×C4⋊C4—52no—
117C2×C4⋊C4—52no—
118C2×C4⋊C4C2×D511yescharacteristic
119C2×C4⋊C4C2×D511yescharacteristic
120C2×C22⋊C4—51no—
121C2×C22⋊C4—51no—
122C2×C22⋊C4—52no—
123C2×C22⋊C4—52no—
124C2×C42—51no—
125C2×C42—52no—
126C2×C42C2×D511yescharacteristic
127#95—51no—
128#95—51no—
129#95—52no—
130#95—52no—
131#95—52no—
132#95—52no—
133#95—52no—
134#95C2×D512yes—
135#95C2×D512yes—
136C22×C5C2×#4111yescharacteristic
137C22×C5C2×#4111yescharacteristic
138C22×C5C2×#4111yescharacteristic
139C22×C5C2×#4111yescharacteristic
140C22×C5C2×#4111yescharacteristic
141C22×C5H(5,𝔽2)11yescharacteristic
142C22×C52−1+411yescharacteristic
143C2×D5—22no—
144C2×D5—22no—
145C2×D5—22no—
146C2×D5—41no—
147C2×D5—41no—
148C2×D5—41no—
149C2×D5—41no—
150C4×C5—42no—
151C4×C5—42no—
152C4×C5—41no—
153C4×C5—41no—
154C4×C5#12614yes—
155Dic5—22no—
156Dic5—41no—
157Dic5—41no—
158Dic5—42no—
159Dic5—42no—
160C24—51no—
161C22×C4—51no—
162C22×C4—51no—
163C22×C4—52no—
164C22×C4—51no—
165C22×C4—52no—
166C22×C4—102no—
167C22×C4—102no—
168C22×C4—102no—
169C22×C4C22×D511yescharacteristic
170C22×C4C22×D512yes—
171C22×C4C22×D512yes—
172C22×C4C22×D511yescharacteristic
173C22×C4C22×D511yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—102no—
177C4⋊C4—102no—
178C4⋊C4—202no—
179C4⋊C4—202no—
180C22⋊C4—102no—
181C22⋊C4—102no—
182C22⋊C4—104no—
183C22⋊C4—202no—
184C42—22no—
185C42—104no—
186C42—102no—
187C2×C5C2×#12611yescharacteristic
188C2×C5#52911yescharacteristic
189C2×C5#54211yescharacteristic
190C2×C5#55211yescharacteristic
191C2×C5#55211yescharacteristic
192C2×C5#55311yescharacteristic
193C2×C5#55511yescharacteristic
194D5—42no—
195C23—52no—
196C23—101no—
197C23—101no—
198C23—101no—
199C23—101no—
200C23—101no—
201C23—101no—
202C23C23×D511yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—52no—
216C2×C4—101no—
217C2×C4—101no—
218C2×C4—102no—
219C2×C4—101no—
220C2×C4—101no—
221C2×C4—101no—
222C2×C4—102no—
223C2×C4—101no—
224C2×C4—102no—
225C2×C4—101no—
226C2×C4—102no—
227C2×C4—102no—
228C2×C4—102no—
229C2×C4—101no—
230C2×C4—204no—
231C2×C4#70912yes—
232C2×C4#71112yes—
233C2×C4#71312yes—
234C5#249511yescharacteristic, complemented, Sylow
235C22—102no—
236C22—102no—
237C22—102no—
238C22—201no—
239C22—201no—
240C22—201no—
241C22—201no—
242C22C2×#70911yescharacteristic
243C22C2×#71111yescharacteristic
244C22C2×#71311yescharacteristic
245C22D5×#4111yescharacteristic
246C22D5×#4111yescharacteristic
247C22#414911yescharacteristic
248C22#415011yescharacteristic
249C4—41no—
250C4—41no—
251C4—42no—
252C4—42no—
253C4—102no—
254C4—201no—
255C4—202no—
256C4—201no—
257C4—202no—
258C4#404414yes—
259C2—202no—
260C2C2×#404411yescharacteristic
261C2#6586311yescharacteristic
262C2#6586611yescharacteristic
263C2#6601411yescharacteristic
264C2#6601611yescharacteristic
265C2#6602811yescharacteristic
266C2#6602911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#382C2
prime radicalC23×D5C23
hypercenter#382D5
hypocenterC5#2495
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C5×#382C2semidirect
4. #65274C2semidirect
5. C4⋊C4×Dic5C2semidirect
7. #65217C2semidirect
10. #64949C2semidirect
11. #64940C2semidirect
69. #382D5semidirect
234. C5#2495semidirect

Maximal subgroups

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

subgroupindex
C5×#3822
#652762
#652742
C4⋊C4×Dic52
C4×#6852
#652172
#649632
#649582
#649492
#649402
#24955