Finite Group Databasebeta

← #10610034

Subgroups — #10610034

Order 640.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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×#400C211yesmaximal, characteristic, complemented
2#65283C212yesmaximal, complemented
3#65280C211yesmaximal, characteristic
4#65265C211yesmaximal, characteristic, complemented
5#64961C212yesmaximal
6#64959C212yesmaximal, complemented
7#64959C212yesmaximal, complemented
8#64958C212yesmaximal, complemented
9#64943C212yesmaximal, complemented
10C22×D20C2211yescharacteristic
11C2×C5×C4⋊C4C2211yescharacteristic
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×#685C2212yes—
14C2×#685C2212yes—
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×C4⋊Dic5C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×C4×Dic5C2211yescharacteristic
24C2×C4×Dic5C2212yes—
25C5×#95C2212yes—
26C5×#95C2211yescharacteristic
27C5×#95C2212yes—
28#3963C2211yescharacteristic
29#3963C2212yes—
30#3963C2212yes—
31#2795—51nomaximal, Sylow
32C23×D5C2312yes—
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2312yes—
36C22×C4×C5C2312yes—
37C22×C4×C5C2311yescharacteristic
38C22×Dic5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C2×D20—42no—
42C2×D20—41no—
43C5×C4⋊C4—41no—
44C5×C4⋊C4—41no—
45#685—24no—
46#685—24no—
47#685—24no—
48#685—42no—
49#685—42no—
50#685—42no—
51C4⋊Dic5—24no—
52C4⋊Dic5—41no—
53#683—42no—
54#683—42no—
55C4×Dic5—22no—
56C4×Dic5—42no—
57#400D511yescharacteristic, complemented
58#396—52no—
59#394—51no—
60#394—52no—
61#388—52no—
62#388—52no—
63#388—52no—
64#387—52no—
65#384—51no—
66C23×C5C2411yescharacteristic
67C22×D5—22no—
68C22×D5—22no—
69C22×D5—22no—
70C22×D5—22no—
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C2×C4×C5—21no—
75C2×C4×C5—21no—
76C2×C4×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
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—22no—
86C2×C4×C5#4112yes—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5#4114yes—
96D20—44no—
97C22×D4—51no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—51no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4C2×D511yescharacteristic
103C2×C4⋊C4C2×D511yescharacteristic
104C2×C22⋊C4—52no—
105C2×C22⋊C4—52no—
106C2×C22⋊C4—52no—
107C2×C22⋊C4—52no—
108C2×C22⋊C4—52no—
109C2×C22⋊C4—52no—
110C2×C42—51no—
111C2×C42—52no—
112#95—51no—
113#95—52no—
114#95—52no—
115#95C2×D512yes—
116#95C2×D511yescharacteristic
117#95C2×D512yes—
118C22×C5C2×#4111yescharacteristic
119C22×C5C2×#4112yes—
120C22×C5H(5,𝔽2)11yescharacteristic
121C22×C5H(5,𝔽2)12yes—
122C22×C52−1+411yescharacteristic
123C2×D5—42no—
124C2×D5—42no—
125C2×D5—42no—
126C2×D5—42no—
127C2×D5—42no—
128C2×D5—42no—
129C2×D5—42no—
130C4×C5—22no—
131C4×C5—42no—
132C4×C5—42no—
133C4×C5—41no—
134C4×C5—41no—
135Dic5—24no—
136Dic5—42no—
137Dic5—42no—
138C24—52no—
139C2×D4—201no—
140C2×D4—202no—
141C22×C4—52no—
142C22×C4—52no—
143C22×C4—52no—
144C22×C4C22×D512yes—
145C22×C4C22×D512yes—
146C22×C4C22×D511yescharacteristic
147C22×C4C22×D511yescharacteristic
148C22×C4C22×D511yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—104no—
152C4⋊C4—202no—
153C4⋊C4—202no—
154C4⋊C4—201no—
155C22⋊C4—104no—
156C22⋊C4—104no—
157C22⋊C4—104no—
158C22⋊C4—202no—
159C22⋊C4—202no—
160C22⋊C4—202no—
161C42—102no—
162C42—202no—
163C2×C5#53911yescharacteristic
164C2×C5#54212yes—
165C2×C5#55312yes—
166C2×C5#55511yescharacteristic
167C2×C5#56011yescharacteristic
168D5—82no—
169C23—102no—
170C23—102no—
171C23—102no—
172C23—102no—
173C23—102no—
174C23—102no—
175C23—102no—
176C23C23×D511yescharacteristic
177D4—204no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—54no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—102no—
195C2×C4—102no—
196C2×C4—102no—
197C2×C4—102no—
198C2×C4—102no—
199C2×C4#71312yes—
200C5#279511yescharacteristic, complemented, Sylow
201C22—202no—
202C22—202no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22—202no—
207C22—202no—
208C22#414411yescharacteristic
209C22C2×#71311yescharacteristic
210C22D5×#4112yes—
211C22#414912yes—
212C22#415011yescharacteristic
213C4—22no—
214C4—41no—
215C4—41no—
216C4—42no—
217C4—42no—
218C4—104no—
219C4—202no—
220C4—202no—
221C2—402no—
222C2#6597911yescharacteristic
223C2#6600011yescharacteristic
224C2#6600411yescharacteristic
225C2#6602912yes—
226C2#6604412yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#400C2
prime radicalC22×D20C22
hypercenter#400D5
hypocenterC5#2795
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C5×#400C2semidirect
2. #65283C2semidirect
4. #65265C2semidirect
6. #64959C2semidirect
7. #64959C2semidirect
8. #64958C2semidirect
9. #64943C2semidirect
57. #400D5semidirect
200. C5#2795semidirect

Maximal subgroups

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

subgroupindex
C5×#4002
#652832
#652802
#652652
#649612
#649592
#649582
#649432
#27955