Finite Group Databasebeta

← #10608274

Subgroups — #10608274

Order 640.

500 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 185 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×Q8×Dic5C211yesmaximal, characteristic
2C2×#4034C211yesmaximal, characteristic
3C5×#391C211yesmaximal, characteristic
4#65519C211yesmaximal, characteristic
5#65265C211yesmaximal, characteristic
6#65224C211yesmaximal, characteristic
7C4×#683C211yesmaximal, characteristic
8#64945C214yesmaximal
9#64936C214yesmaximal
10C22×C5×Q8C2211yescharacteristic
11C22×Dic10C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C42×C5C2211yescharacteristic
14Q8×Dic5—24no—
15C2×C4⋊Dic5C2211yescharacteristic
16C2×C4⋊Dic5C2212yes—
17C2×C4⋊Dic5C2211yescharacteristic
18C2×C4⋊Dic5C2214yes—
19C2×#683C2212yes—
20C2×#683C2214yes—
21C2×#683C2212yes—
22C2×C4×Dic5C2212yes—
23C2×C4×Dic5C2211yescharacteristic
24C2×C4×Dic5C2211yescharacteristic
25#4034—24no—
26C5×#95C2214yes—
27#3963C2214yes—
28#3963C2212yes—
29#2567—51nomaximal, Sylow
30C2×C5×Q8—22no—
31C2×C5×Q8—22no—
32C2×C5×Q8Q814yescomplemented
33C22×C4×C5C2312yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×Dic5C2312yes—
38C22×Dic5C2312yes—
39C22×Dic5C2311yescharacteristic
40C22×Dic5C2312yes—
41C22×Dic5C2311yescharacteristic
42C2×Dic10—24no—
43C2×Dic10—44no—
44C5×C4⋊C4—22no—
45C42×C5—22no—
46C4⋊Dic5—22no—
47C4⋊Dic5—22no—
48C4⋊Dic5—24no—
49C4⋊Dic5—44no—
50#683—24no—
51#683—44no—
52#683Q818yescomplemented
53C4×Dic5—22no—
54C4×Dic5—24no—
55C4×Dic5D414yes—
56C2×C4⋊Q8—51no—
57C2×C4×Q8—51no—
58#395—54no—
59#391—51no—
60#391D511yescharacteristic
61#389—54no—
62#384—51no—
63#384—51no—
64C4×C4⋊C4—51no—
65C23×C5C2411yescharacteristic
66C5×Q8—28no—
67C2×C4×C5—22no—
68C2×C4×C5—22no—
69C2×C4×C5—22no—
70C2×C4×C5—22no—
71C2×C4×C5—22no—
72C2×C4×C5—24no—
73C2×C4×C5C2×D412yes—
74C2×C4×C5C2×Q814yes—
75C2×C4×C5C2×Q812yes—
76C2×C4×C5C2×Q814yes—
77C2×C4×C5#4112yes—
78C2×Dic5—22no—
79C2×Dic5—22no—
80C2×Dic5—22no—
81C2×Dic5—22no—
82C2×Dic5—22no—
83C2×Dic5—24no—
84C2×Dic5—24no—
85C2×Dic5C2×D412yes—
86C2×Dic5C2×D412yes—
87C2×Dic5C2×Q814yes—
88C2×Dic5C2×Q814yes—
89Dic10—48no—
90C22×Q8—52no—
91C22×Q8C2×D511yescharacteristic
92C4⋊Q8—104no—
93C4×Q8—104no—
94C2×C4⋊C4—52no—
95C2×C4⋊C4—51no—
96C2×C4⋊C4—52no—
97C2×C4⋊C4—51no—
98C2×C4⋊C4—52no—
99C2×C4⋊C4—54no—
100C2×C4⋊C4—54no—
101C2×C4⋊C4C2×D511yescharacteristic
102C2×C42—51no—
103C2×C42—52no—
104C2×C42—51no—
105C2×C42C2×D511yescharacteristic
106#95—52no—
107#95—54no—
108#95C2×D514yes—
109C22×C5C22×D411yescharacteristic
110C22×C5C22×Q811yescharacteristic
111C22×C5C22×Q812yes—
112C22×C5C2×#4111yescharacteristic
113C22×C5H(5,𝔽2)12yes—
114C4×C5—24no—
115C4×C5—24no—
116C4×C5—42no—
117C4×C5C22⋊Q814yes—
118Dic5—22no—
119Dic5—24no—
120Dic5—42no—
121Dic5—42no—
122Dic5C4⋊Q814yes—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—104no—
126C2×Q8—204no—
127C2×Q8Dic1014yescomplemented
128C22×C4—51no—
129C22×C4—52no—
130C22×C4—51no—
131C22×C4—52no—
132C22×C4—52no—
133C22×C4C22×D512yes—
134C22×C4C22×D512yes—
135C22×C4C22×D512yes—
136C22×C4C22×D511yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—58no—
139C4⋊C4—104no—
140C4⋊C4—104no—
141C4⋊C4—102no—
142C4⋊C4—102no—
143C4⋊C4—204no—
144C4⋊C4—204no—
145C42—22no—
146C42—54no—
147C42—104no—
148C42—102no—
149C2×C5C2×C22⋊Q811yescharacteristic
150C2×C5C2×C4⋊Q811yescharacteristic
151C2×C5#53511yescharacteristic
152C2×C5#55412yes—
153C2×C5Q8212yes—
154C23C23×D511yescharacteristic
155Q8—28no—
156Q8—208no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—52no—
164C2×C4—54no—
165C2×C4—52no—
166C2×C4—54no—
167C2×C4—102no—
168C2×C4—102no—
169C2×C4—102no—
170C2×C4—102no—
171C2×C4—102no—
172C2×C4—104no—
173C2×C4—104no—
174C2×C4C2×Dic1014yes—
175C2×C4C2×Dic1012yes—
176C2×C4D4×D512yes—
177C2×C4#71112yes—
178C2×C4Q8×D514yes—
179C5#256711yescharacteristic, complemented, Sylow
180C22C22×Dic1011yescharacteristic
181C22C2×D4×D511yescharacteristic
182C22C2×#71111yescharacteristic
183C22C2×Q8×D512yes—
184C22#414912yes—
185C4—24no—
186C4—24no—
187C4—42no—
188C4—54no—
189C4—104no—
190C4—102no—
191C4—202no—
192C4—202no—
193C4#402414yes—
194C2C2×#402411yescharacteristic
195C2Q8×Dic1012yes—
196C2#6596912yes—
197C2#6600811yescharacteristic
198C2D5×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, prime radical, cosocleC23×C5C24
FittingC5×#391C2
hypercenter#391D5
hypocenterC5#2567
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
32. C2×C5×Q8Q8semidirect
52. #683Q8semidirect
127. C2×Q8Dic10semidirect
179. C5#2567semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic52
C2×#40342
C5×#3912
#655192
#652652
#652242
C4×#6832
#649452
#649362
#25675