Finite Group Databasebeta

← #10609603

Subgroups — #10609603

Order 640.

456 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 149 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×#4034C212yesmaximal
2C5×#398C211yesmaximal, characteristic
3#65272C211yesmaximal, characteristic
4#65268C212yesmaximal
5#65263C211yesmaximal, characteristic
6#65261C212yesmaximal
7#64945C212yesmaximal
8#64944C212yesmaximal
9#64941C212yesmaximal
10C22×Dic10C2212yes—
11C22×Dic10C2211yescharacteristic
12C2×C5×C4⋊C4C2212yes—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic5C2212yes—
16C2×C4⋊Dic5C2211yescharacteristic
17C2×#683C2212yes—
18C2×#683C2212yes—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×C4×Dic5C2212yes—
22C2×C4×Dic5C2211yescharacteristic
23#4034—28no—
24C5×#95C2212yes—
25C5×#95C2211yescharacteristic
26#3963C2211yescharacteristic
27#3963C2212yes—
28#3963C2212yes—
29#3963C2212yes—
30#3963C2212yes—
31#3963C2212yes—
32#2734—51nomaximal, Sylow
33C22×C4×C5C2312yes—
34C22×C4×C5C2311yescharacteristic
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2312yes—
38C22×Dic5C2312yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2312yes—
42C2×Dic10—24no—
43C2×Dic10—41no—
44C2×Dic10—41no—
45C2×Dic10—42no—
46C2×Dic10—42no—
47C2×Dic10D414yes—
48C5×C4⋊C4—24no—
49C5×C4⋊C4—41no—
50C5×C4⋊C4—41no—
51C4⋊Dic5—24no—
52C4⋊Dic5—41no—
53#683—24no—
54#683—24no—
55#683—42no—
56#683Q818yescomplemented
57C4×Dic5—24no—
58C4×Dic5—41no—
59C2×C4⋊Q8—52no—
60#398—51no—
61#398D511yescharacteristic
62#395—52no—
63#395—52no—
64#391—51no—
65#389—52no—
66#387—52no—
67#387—52no—
68C23×C5C2411yescharacteristic
69C2×C4×C5—22no—
70C2×C4×C5—21no—
71C2×C4×C5—21no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—21no—
76C2×C4×C5—21no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5—22no—
80C2×C4×C5C2×D412yes—
81C2×C4×C5C2×Q814yes—
82C2×Dic5—22no—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—22no—
89C2×Dic5—22no—
90C2×Dic5—22no—
91C2×Dic5C2×D414yes—
92C2×Dic5C2×Q814yes—
93C2×Dic5C2×Q814yes—
94Dic10—44no—
95Dic10—48no—
96C22×Q8—51no—
97C22×Q8—52no—
98C4⋊Q8—108no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—52no—
103C2×C4⋊C4—52no—
104C2×C4⋊C4—51no—
105C2×C4⋊C4C2×D512yes—
106C2×C4⋊C4C2×D511yescharacteristic
107C2×C4⋊C4C2×D511yescharacteristic
108C2×C42—52no—
109C2×C42—51no—
110#95—51no—
111#95—52no—
112#95—52no—
113#95—52no—
114#95—52no—
115#95—52no—
116#95C2×D511yescharacteristic
117#95C2×D512yes—
118C22×C5C22×D411yescharacteristic
119C22×C5C22×Q812yes—
120C22×C5H(5,𝔽2)11yescharacteristic
121C22×C5H(5,𝔽2)11yescharacteristic
122C22×C52−1+412yes—
123C4×C5—22no—
124C4×C5—24no—
125C4×C5—42no—
126C4×C5—41no—
127C4×C5—41no—
128Dic5—24no—
129Dic5—24no—
130Dic5—24no—
131Dic5—42no—
132C2×Q8—54no—
133C2×Q8—104no—
134C2×Q8—201no—
135C2×Q8—201no—
136C2×Q8—202no—
137C2×Q8—202no—
138C22×C4—52no—
139C22×C4—52no—
140C22×C4—52no—
141C22×C4—52no—
142C22×C4C22×D511yescharacteristic
143C22×C4C22×D511yescharacteristic
144C22×C4C22×D512yes—
145C22×C4C22×D511yescharacteristic
146C22×C4C22×D512yes—
147C4⋊C4—24no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—58no—
151C4⋊C4—104no—
152C4⋊C4—104no—
153C4⋊C4—104no—
154C4⋊C4—202no—
155C4⋊C4—201no—
156C42—104no—
157C42—201no—
158C2×C5#53411yescharacteristic
159C2×C5#54412yes—
160C2×C5D4×Q812yes—
161C2×C5Q8211yescharacteristic
162C2×C5#56311yescharacteristic
163C23C23×D511yescharacteristic
164Q8—204no—
165Q8—208no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—54no—
178C2×C4—54no—
179C2×C4—54no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—102no—
186C2×C4—102no—
187C2×C4—102no—
188C2×C4—102no—
189C2×C4D4×D512yes—
190C2×C4Q8×D514yes—
191C5#273411yescharacteristic, complemented, Sylow
192C22C2×D4×D511yescharacteristic
193C22#414411yescharacteristic
194C22C2×Q8×D512yes—
195C22#414911yescharacteristic
196C22#415012yes—
197C4—22no—
198C4—24no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—104no—
203C4—104no—
204C4—104no—
205C4—202no—
206C2#6593911yescharacteristic
207C2#6596612yes—
208C2#6598511yescharacteristic
209C2#6602312yes—
210C2#6605611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#398C2
hypercenter#398D5
hypocenterC5#2734
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
56. #683Q8semidirect
191. C5#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#40342
C5×#3982
#652722
#652682
#652632
#652612
#649452
#649442
#649412
#27345