Finite Group Databasebeta

← #10608255

Subgroups — #10608255

Order 640.

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

normal subgroups only

This group has 372 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×#391C211yesmaximal, characteristic
2#65519C211yesmaximal, characteristic
3#65519C211yesmaximal, characteristic
4C4⋊C4×Dic5C211yesmaximal, characteristic
5#65262C211yesmaximal, characteristic
6#65225C211yesmaximal, characteristic
7C4×#683C211yesmaximal, characteristic
8C5⋊#401C212yesmaximal
9#64947C212yesmaximal
10#64941C212yesmaximal
11#64940C212yesmaximal
12C22×C5×Q8C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15C2×C4⋊Dic5C2211yescharacteristic
16C2×C4⋊Dic5C2211yescharacteristic
17C2×#683C2211yescharacteristic
18C2×#683C2212yes—
19C2×#683C2211yescharacteristic
20C2×C4×Dic5C2212yes—
21C2×C4×Dic5C2211yescharacteristic
22C2×C4×Dic5C2211yescharacteristic
23C5×#95C2212yes—
24C5×#95C2212yes—
25#3963C2211yescharacteristic
26#3963C2212yes—
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#3963C2212yes—
33#3963C2212yes—
34#3963C2211yescharacteristic
35#2514—51nomaximal, Sylow
36C2×C5×Q8—22no—
37C2×C5×Q8—22no—
38C2×C5×Q8—22no—
39C22×C4×C5C2311yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C22×C4×C5C2312yes—
42C22×C4×C5C2312yes—
43C22×C4×C5C2311yescharacteristic
44C22×Dic5C2311yescharacteristic
45C22×Dic5C2311yescharacteristic
46C22×Dic5C2312yes—
47C22×Dic5C2312yes—
48C22×Dic5C2311yescharacteristic
49C22×Dic5C2311yescharacteristic
50C5×C4⋊C4—22no—
51C42×C5—22no—
52C4⋊Dic5—22no—
53C4⋊Dic5—22no—
54#683—22no—
55#683—24no—
56#683—22no—
57C4×Dic5—22no—
58C4×Dic5—22no—
59C4×Dic5—24no—
60#401—52no—
61#400—52no—
62#391—51no—
63#391—51no—
64#391D511yescharacteristic
65#389—51no—
66#387—52no—
67#383—52no—
68#382—51no—
69C4×C4⋊C4—51no—
70C4×C4⋊C4—51no—
71C23×C5C2411yescharacteristic
72C5×Q8—44no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—21no—
76C2×C4×C5—21no—
77C2×C4×C5—21no—
78C2×C4×C5—21no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—21no—
84C2×C4×C5—21no—
85C2×C4×C5#4112yes—
86C2×C4×C5#4112yes—
87C2×C4×C5#4112yes—
88C2×Dic5—21no—
89C2×Dic5—21no—
90C2×Dic5—21no—
91C2×Dic5—21no—
92C2×Dic5—21no—
93C2×Dic5—21no—
94C2×Dic5—22no—
95C2×Dic5—21no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5—22no—
99C2×Dic5—21no—
100C2×Dic5—21no—
101C2×Dic5—22no—
102C2×Dic5—21no—
103C2×Dic5—22no—
104C2×Dic5#4112yes—
105C2×Dic5#4112yes—
106C22×Q8C2×D511yescharacteristic
107C2×C4⋊C4—51no—
108C2×C4⋊C4—51no—
109C2×C4⋊C4—51no—
110C2×C4⋊C4—51no—
111C2×C4⋊C4—52no—
112C2×C4⋊C4C2×D511yescharacteristic
113C2×C42—51no—
114C2×C42—52no—
115C2×C42—51no—
116C2×C42C2×D511yescharacteristic
117#95—51no—
118#95—52no—
119#95—51no—
120#95—52no—
121#95—52no—
122#95—52no—
123#95—52no—
124#95—52no—
125#95—52no—
126#95—52no—
127#95C2×D512yes—
128#95C2×D512yes—
129C22×C5C2×#4111yescharacteristic
130C22×C5C2×#4111yescharacteristic
131C22×C5C2×#4111yescharacteristic
132C22×C5C2×#4111yescharacteristic
133C22×C5C2×#4111yescharacteristic
134C22×C52−1+411yescharacteristic
135C22×C52−1+411yescharacteristic
136C4×C5—42no—
137C4×C5—42no—
138C4×C5—41no—
139C4×C5—41no—
140C4×C5#12614yes—
141Dic5—22no—
142Dic5—22no—
143Dic5—41no—
144Dic5—42no—
145Dic5—42no—
146Dic5—41no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C22×C4—51no—
151C22×C4—51no—
152C22×C4—51no—
153C22×C4—52no—
154C22×C4—51no—
155C22×C4—52no—
156C22×C4C22×D512yes—
157C22×C4C22×D511yescharacteristic
158C22×C4C22×D511yescharacteristic
159C22×C4C22×D512yes—
160C22×C4C22×D511yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—102no—
163C4⋊C4—102no—
164C4⋊C4—104no—
165C4⋊C4—102no—
166C4⋊C4—102no—
167C42—22no—
168C42—104no—
169C42—102no—
170C42—102no—
171C2×C5C2×#12611yescharacteristic
172C2×C5#52911yescharacteristic
173C2×C5#54111yescharacteristic
174C2×C5#55211yescharacteristic
175C2×C5#55611yescharacteristic
176C2×C5#55611yescharacteristic
177C2×C5#55611yescharacteristic
178C23C23×D511yescharacteristic
179Q8—44no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—52no—
193C2×C4—52no—
194C2×C4—101no—
195C2×C4—101no—
196C2×C4—101no—
197C2×C4—101no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—101no—
201C2×C4—101no—
202C2×C4—101no—
203C2×C4—101no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4—101no—
207C2×C4—102no—
208C2×C4—102no—
209C2×C4—101no—
210C2×C4#70912yes—
211C2×C4#71112yes—
212C2×C4#71112yes—
213C5#251411yescharacteristic, complemented, Sylow
214C22C2×#70911yescharacteristic
215C22C2×#71111yescharacteristic
216C22C2×#71111yescharacteristic
217C22#414711yescharacteristic
218C22#414711yescharacteristic
219C22D5×#4111yescharacteristic
220C22D5×#4111yescharacteristic
221C4—42no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—102no—
226C4—102no—
227C4—201no—
228C4—202no—
229C4—202no—
230C4—201no—
231C4#402514yes—
232C2C2×#402511yescharacteristic
233C2#6590211yescharacteristic
234C2#6590211yescharacteristic
235C2#6597311yescharacteristic
236C2#6597511yescharacteristic
237C2#6601411yescharacteristic
238C2#6606511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#391C2
hypercenter#391D5
hypocenterC5#2514
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
213. C5#2514semidirect

Maximal subgroups

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

subgroupindex
C5×#3912
#655192
C4⋊C4×Dic52
#652622
#652252
C4×#6832
C5⋊#4012
#649472
#649412
#649402
#25145