Finite Group Databasebeta

← #10605974

Subgroups — #10605974

Order 640.

540 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#4018C212yesmaximal, complemented
2C5×#382C211yesmaximal, characteristic, complemented
3#65283C212yesmaximal, complemented
4#65280C211yesmaximal, characteristic
5#65272C211yesmaximal, characteristic, complemented
6#64959C212yesmaximal, complemented
7#64957C212yesmaximal, complemented
8#64954C212yesmaximal
9#64942C212yesmaximal, complemented
10C22×D20C2211yescharacteristic
11C22×C4×D5C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15C2×#685C2212yes—
16C2×#685C2212yes—
17C2×#685C2212yes—
18C2×#685C2212yes—
19C2×#685C2212yes—
20C2×C4⋊Dic5C2212yes—
21C2×C4⋊Dic5C2211yescharacteristic
22C2×#683C2212yes—
23C2×#683C2212yes—
24C2×C4×Dic5C2212yes—
25#4018—28no—
26C5×#95C2212yes—
27C5×#95C2212yes—
28#3963C2211yescharacteristic
29#3963C2212yes—
30#3963C2212yes—
31#3963C2211yescharacteristic
32#2609—51nomaximal, Sylow
33C23×D5C2312yes—
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2311yescharacteristic
38C22×C4×C5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C2×D20—24no—
43C2×D20—41no—
44C2×C4×D5—24no—
45C2×C4×D5—24no—
46C2×C4×D5—24no—
47C5×C4⋊C4—41no—
48C5×C4⋊C4—41no—
49C42×C5D414yescomplemented
50#685—24no—
51#685—24no—
52#685—24no—
53#685—24no—
54#685—42no—
55C4⋊Dic5—24no—
56C4⋊Dic5—41no—
57#683—24no—
58#683—24no—
59C4×Dic5—24no—
60C2×#117—52no—
61#398—51no—
62#396—52no—
63#394—51no—
64#388—52no—
65#387—52no—
66#386—52no—
67#385—52no—
68#382D511yescharacteristic, complemented
69C23×C5C2411yescharacteristic
70C22×D5—22no—
71C22×D5—22no—
72C22×D5—22no—
73C22×D5—22no—
74C22×D5—22no—
75C22×D5—22no—
76C22×D5#4114yes—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—21no—
80C2×C4×C5—21no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—22no—
85C2×C4×C5—21no—
86C2×C4×C5—22no—
87C2×C4×C5—21no—
88C2×C4×C5C2×D414yes—
89C2×C4×C5C2×D412yes—
90C2×Dic5—22no—
91C2×Dic5—22no—
92C2×Dic5—22no—
93C2×Dic5—22no—
94C2×Dic5—22no—
95C2×Dic5—22no—
96C2×Dic5—22no—
97C2×Dic5—22no—
98C2×Dic5#4114yes—
99D20—44no—
100C4×D5—216no—
101C22×D4—51no—
102C23×C4—52no—
103#117—108no—
104C2×C4⋊C4—52no—
105C2×C4⋊C4—52no—
106C2×C4⋊C4—52no—
107C2×C4⋊C4—51no—
108C2×C4⋊C4C2×D511yescharacteristic
109C2×C4⋊C4C2×D511yescharacteristic
110C2×C22⋊C4—52no—
111C2×C22⋊C4—52no—
112C2×C22⋊C4—52no—
113C2×C22⋊C4—52no—
114C2×C22⋊C4—52no—
115C2×C42—52no—
116C2×C42C2×D511yescharacteristic
117#95—51no—
118#95—52no—
119#95—51no—
120#95—52no—
121#95C2×D512yes—
122#95C2×D512yes—
123C22×C5C22×D411yescharacteristic
124C22×C5C2×#4112yes—
125C22×C5C2×#4112yes—
126C22×C5H(5,𝔽2)11yescharacteristic
127C22×C52−1+411yescharacteristic
128C2×D5—24no—
129C2×D5—24no—
130C2×D5—24no—
131C2×D5—42no—
132C2×D5—42no—
133C2×D5—42no—
134C2×D5—42no—
135C4×C5—24no—
136C4×C5—22no—
137C4×C5—41no—
138C4×C5—41no—
139C4×C5—42no—
140Dic5—24no—
141Dic5—42no—
142Dic5—42no—
143C24—52no—
144C2×D4—104no—
145C2×D4—201no—
146C22×C4—52no—
147C22×C4—52no—
148C22×C4—52no—
149C22×C4—104no—
150C22×C4—104no—
151C22×C4—104no—
152C22×C4C22×D511yescharacteristic
153C22×C4C22×D512yes—
154C22×C4C22×D511yescharacteristic
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—104no—
160C4⋊C4—104no—
161C4⋊C4—104no—
162C4⋊C4—201no—
163C22⋊C4—104no—
164C22⋊C4—104no—
165C22⋊C4—104no—
166C22⋊C4—104no—
167C22⋊C4—202no—
168C42—104no—
169C42D2014yescomplemented
170C2×C5#52512yes—
171C2×C5#53711yescharacteristic
172C2×C5#55111yescharacteristic
173C2×C5#55212yes—
174C2×C5#55511yescharacteristic
175D5—44no—
176C23—54no—
177C23—102no—
178C23—102no—
179C23—102no—
180C23—102no—
181C23—102no—
182C23—102no—
183C23C23×D511yescharacteristic
184D4—204no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—54no—
197C2×C4—1016no—
198C2×C4—102no—
199C2×C4—102no—
200C2×C4—102no—
201C2×C4—102no—
202C2×C4—102no—
203C2×C4—102no—
204C2×C4—102no—
205C2×C4—102no—
206C2×C4C2×D2014yes—
207C2×C4C2×D2012yes—
208C5#260911yescharacteristic, complemented, Sylow
209C22—104no—
210C22—104no—
211C22—104no—
212C22—202no—
213C22—202no—
214C22—202no—
215C22—202no—
216C22C22×D2011yescharacteristic
217C22#414411yescharacteristic
218C22#414711yescharacteristic
219C22D5×#4112yes—
220C22D5×#4112yes—
221C4—22no—
222C4—24no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—104no—
227C4—202no—
228C4—202no—
229C2—204no—
230C2#6584411yescharacteristic
231C2#6585812yes—
232C2#6600411yescharacteristic
233C2#6601311yescharacteristic
234C2#6602712yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#382C2
prime radicalC22×D20C22
hypercenter#382D5
hypocenterC5#2609
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#4018C2semidirect
2. C5×#382C2semidirect
3. #65283C2semidirect
5. #65272C2semidirect
6. #64959C2semidirect
7. #64957C2semidirect
9. #64942C2semidirect
49. C42×C5D4semidirect
68. #382D5semidirect
169. C42D20semidirect
208. C5#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#40182
C5×#3822
#652832
#652802
#652722
#649592
#649572
#649542
#649422
#26095