Finite Group Databasebeta

← #10614119

Subgroups — #10614119

Order 640.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×C4×C5×D4C211yesmaximal, characteristic
2C2×C4×Dic10C211yesmaximal, characteristic, complemented
3#65510C212yesmaximal
4#65499C214yesmaximal, complemented
5#65262C212yesmaximal, complemented
6#65238C212yesmaximal, complemented
7#65236C212yesmaximal, complemented
8#65225C211yesmaximal, characteristic, complemented
9C22×C5×D4C2211yescharacteristic
10C23×C4×C5C2212yes—
11C22×Dic10C2211yescharacteristic
12C4×C5×D4—24no—
13C2×C5×C4⋊C4C2211yescharacteristic
14C2×C5×C22⋊C4C2212yes—
15C2×C42×C5C2211yescharacteristic
16C2×C22⋊Dic5C2214yes—
17C2×C22⋊Dic5C2214yes—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×C4⋊Dic5C2212yes—
20C2×#683C2212yes—
21C2×#683C2214yes—
22C2×C4×Dic5C2212yes—
23C4×Dic10—24no—
24#3963C2212yes—
25#3963C2212yes—
26#3963C2214yes—
27#2580—51nomaximal, Sylow
28C24×C5C2312yes—
29C2×C5×D4—22no—
30C2×C5×D4—22no—
31C2×C5×D4Q814yescomplemented
32C22×C4×C5—24no—
33C22×C4×C5—44no—
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2311yescharacteristic
38C22×Dic5C2314yes—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C2×Dic10—22no—
42C2×Dic10—24no—
43C5×C4⋊C4—22no—
44C5×C22⋊C4—24no—
45C42×C5D414yes—
46C22⋊Dic5—28no—
47C22⋊Dic5—44no—
48C4⋊Dic5—22no—
49C4⋊Dic5—24no—
50#683—24no—
51#683—44no—
52C4×Dic5—24no—
53C2×C4×Q8—51no—
54C2×C4×D4D511yescharacteristic
55#399—52no—
56#393—52no—
57#390—52no—
58#389—52no—
59#385—54no—
60#382—51no—
61C23×C5—22no—
62C23×C5—24no—
63C23×C5—22no—
64C23×C5—24no—
65C23×C5C2×Q814yes—
66C23×C5C2411yescharacteristic
67C5×D4—28no—
68C2×C4×C5—22no—
69C2×C4×C5—22no—
70C2×C4×C5—22no—
71C2×C4×C5—24no—
72C2×C4×C5—48no—
73C2×C4×C5C2×D412yes—
74C2×C4×C5C2×D412yes—
75C2×C4×C5C2×D412yes—
76C2×C4×C5C2×Q812yes—
77C2×C4×C5#4112yes—
78C2×Dic5—22no—
79C2×Dic5—22no—
80C2×Dic5—22no—
81C2×Dic5—24no—
82C2×Dic5—24no—
83C2×Dic5—22no—
84C2×Dic5—24no—
85C2×Dic5—22no—
86C2×Dic5#4114yes—
87Dic10—28no—
88C22×Q8—51no—
89C22×D4C2×D511yescharacteristic
90C23×C4C2×D512yes—
91C4×Q8—104no—
92C4×D4—24no—
93C2×C4⋊C4—52no—
94C2×C4⋊C4—51no—
95C2×C4⋊C4—52no—
96C2×C4⋊C4—54no—
97C2×C4⋊C4C2×D511yescharacteristic
98C2×C22⋊C4—54no—
99C2×C22⋊C4—54no—
100C2×C22⋊C4C2×D512yes—
101C2×C42—52no—
102C2×C42C2×D511yescharacteristic
103#95—52no—
104#95—52no—
105#95—54no—
106C22×C5—24no—
107C22×C5—28no—
108C22×C5—42no—
109C22×C5—42no—
110C22×C5—44no—
111C22×C5C22×D411yescharacteristic
112C22×C5C22×Q811yescharacteristic
113C22×C5C2×#4111yescharacteristic
114C22×C5C2×#4112yes—
115C22×C5H(5,𝔽2)12yes—
116C4×C5—22no—
117C4×C5—22no—
118C4×C5—42no—
119C4×C5C22⋊Q814yes—
120Dic5—24no—
121Dic5—42no—
122Dic5—44no—
123C24C22×D512yes—
124C2×Q8—104no—
125C2×Q8—102no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4Dic1014yescomplemented
129C22×C4—24no—
130C22×C4—44no—
131C22×C4—52no—
132C22×C4—54no—
133C22×C4—52no—
134C22×C4C22×D512yes—
135C22×C4C22×D511yescharacteristic
136C22×C4C22×D511yescharacteristic
137C22×C4C22×D511yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—104no—
140C4⋊C4—104no—
141C4⋊C4—102no—
142C4⋊C4—204no—
143C22⋊C4—24no—
144C22⋊C4—108no—
145C22⋊C4—204no—
146C42—104no—
147C42C22⋊D514yes—
148C2×C5—44no—
149C2×C5C2×C22⋊Q811yescharacteristic
150C2×C5#52511yescharacteristic
151C2×C5#53511yescharacteristic
152C2×C5#55412yes—
153C2×C5#55912yes—
154C23—22no—
155C23—24no—
156C23—24no—
157C23—22no—
158C23C2×Dic1014yes—
159C23C23×D511yescharacteristic
160Q8—108no—
161D4—28no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—48no—
167C2×C4—54no—
168C2×C4—102no—
169C2×C4—102no—
170C2×C4—104no—
171C2×C4—102no—
172C2×C4—104no—
173C2×C4—102no—
174C2×C4—104no—
175C2×C4—102no—
176C2×C4C2×Dic1012yes—
177C2×C4#70912yes—
178C2×C4C2×C22⋊D512yes—
179C2×C4C2×C22⋊D512yes—
180C2×C4C2×C22⋊D512yes—
181C5#258011yescharacteristic, complemented, Sylow
182C22—24no—
183C22—28no—
184C22—42no—
185C22—44no—
186C22—42no—
187C22C22×Dic1011yescharacteristic
188C22C2×#70911yescharacteristic
189C22#414412yes—
190C22D5×#4112yes—
191C22C22×C22⋊D511yescharacteristic
192C4—22no—
193C4—22no—
194C4—42no—
195C4—104no—
196C4—202no—
197C4—204no—
198C4#407014yes—
199C2—44no—
200C2#6587312yes—
201C2#6589312yes—
202C2C2×#407011yescharacteristic
203C2#6613911yescharacteristic
204C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C4×C5×D4C2
prime radicalC22×C5×D4C22
hypercenterC2×C4×D4D5
hypocenterC5#2580
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic10C2semidirect
4. #65499C2semidirect
5. #65262C2semidirect
6. #65238C2semidirect
7. #65236C2semidirect
8. #65225C2semidirect
31. C2×C5×D4Q8semidirect
128. C2×D4Dic10semidirect
181. C5#2580semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×D42
C2×C4×Dic102
#655102
#654992
#652622
#652382
#652362
#652252
#25805