Finite Group Databasebeta

← #10614516

Subgroups — #10614516

Order 640.

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

normal subgroups only

This group has 544 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×C5×C22⋊Q8C211yesmaximal, characteristic
2C22×#683C211yesmaximal, characteristic
3#65518C212yesmaximal, complemented
4C4×C22⋊Dic5C211yesmaximal, characteristic
5#65269C212yesmaximal, complemented
6#65261C214yesmaximal, complemented
7#65235C212yesmaximal
8#65235C212yesmaximal
9C22×C5×Q8C2211yescharacteristic
10C23×C4×C5C2211yescharacteristic
11C23×Dic5C2211yescharacteristic
12C23×Dic5C2211yescharacteristic
13C5×C22⋊Q8—24no—
14C2×C5×C4⋊C4C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18C2×#683—24no—
19C2×#683C2212yes—
20C2×#683C2212yes—
21C2×#683C2214yes—
22C2×#683C2214yes—
23C2×C4×Dic5C2212yes—
24#3963C2214yes—
25#3963C2214yes—
26#3963C2212yes—
27#2583—51nomaximal, Sylow
28C24×C5C2311yescharacteristic
29C2×C5×Q8—22no—
30C2×C5×Q8—42no—
31C22×C4×C5—42no—
32C22×C4×C5D414yes—
33C22×C4×C5C2312yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×C4×C5C2311yescharacteristic
37C22×Dic5—22no—
38C22×Dic5—24no—
39C22×Dic5—22no—
40C22×Dic5—42no—
41C22×Dic5C2312yes—
42C22×Dic5C2314yes—
43C22×Dic5C2312yes—
44C5×C4⋊C4—22no—
45C5×C4⋊C4—24no—
46C5×C22⋊C4—24no—
47C22⋊Dic5Q818yescomplemented
48#683—24no—
49#683—24no—
50#683—28no—
51#683—44no—
52C4×Dic5—42no—
53C2×C22⋊Q8D511yescharacteristic
54C22×C4⋊C4—51no—
55#398—52no—
56#395—52no—
57#389—54no—
58#385—52no—
59#385—52no—
60C4×C22⋊C4—51no—
61C23×C5—22no—
62C23×C5—22no—
63C23×C5C2×D412yes—
64C23×C5C2×Q814yes—
65C23×C5C2411yescharacteristic
66C5×Q8—44no—
67C2×C4×C5—22no—
68C2×C4×C5—22no—
69C2×C4×C5—22no—
70C2×C4×C5—22no—
71C2×C4×C5—24no—
72C2×C4×C5—24no—
73C2×C4×C5—44no—
74C2×C4×C5C2×D412yes—
75C2×C4×C5C2×D412yes—
76C2×Dic5—28no—
77C2×Dic5—22no—
78C2×Dic5—24no—
79C2×Dic5—22no—
80C2×Dic5—24no—
81C2×Dic5—22no—
82C2×Dic5—24no—
83C2×Dic5—44no—
84C2×Dic5C2×Q818yes—
85C2×Dic5#4114yes—
86C22×Q8C2×D511yescharacteristic
87C23×C4—51no—
88C23×C4—51no—
89C23×C4C2×D511yescharacteristic
90C22⋊Q8—24no—
91C2×C4⋊C4—52no—
92C2×C4⋊C4—52no—
93C2×C4⋊C4—54no—
94C2×C4⋊C4—54no—
95C2×C4⋊C4—104no—
96C2×C4⋊C4C2×D512yes—
97C2×C4⋊C4C2×D511yescharacteristic
98C2×C22⋊C4—52no—
99C2×C22⋊C4C2×D512yes—
100C2×C42—52no—
101#95—52no—
102#95—54no—
103#95—54no—
104C22×C5—22no—
105C22×C5—22no—
106C22×C5—28no—
107C22×C5C4⋊Q814yes—
108C22×C5C22×D411yescharacteristic
109C22×C5C22×Q812yes—
110C22×C5C2×#4112yes—
111C22×C5H(5,𝔽2)12yes—
112C4×C5—22no—
113C4×C5—22no—
114C4×C5—42no—
115C4×C5—42no—
116Dic5—24no—
117Dic5—28no—
118Dic5—42no—
119C24C22×D511yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C22×C4—42no—
123C22×C4—54no—
124C22×C4—52no—
125C22×C4—52no—
126C22×C4—104no—
127C22×C4—102no—
128C22×C4—102no—
129C22×C4—202no—
130C22×C4C22⋊D514yes—
131C22×C4C22×D512yes—
132C22×C4C22×D512yes—
133C22×C4C22×D511yescharacteristic
134C22×C4C22×D511yescharacteristic
135C4⋊C4—22no—
136C4⋊C4—24no—
137C4⋊C4—108no—
138C4⋊C4—104no—
139C4⋊C4—104no—
140C4⋊C4—204no—
141C22⋊C4—24no—
142C22⋊C4—58no—
143C42—202no—
144C2×C5—24no—
145C2×C5#52511yescharacteristic
146C2×C5C2×C4⋊Q811yescharacteristic
147C2×C5#53411yescharacteristic
148C2×C5#55414yes—
149C23—22no—
150C23—22no—
151C23Q8×D514yes—
152C23C2×C22⋊D512yes—
153C23C23×D511yescharacteristic
154Q8—44no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—58no—
163C2×C4—54no—
164C2×C4—108no—
165C2×C4—104no—
166C2×C4—104no—
167C2×C4—102no—
168C2×C4—102no—
169C2×C4—104no—
170C2×C4—102no—
171C2×C4—204no—
172C2×C4C2×C22⋊D512yes—
173C2×C4C2×C22⋊D512yes—
174C5#258311yescharacteristic, complemented, Sylow
175C22—22no—
176C22—22no—
177C22—28no—
178C22#409014yes—
179C22#414412yes—
180C22C2×Q8×D512yes—
181C22D5×#4112yes—
182C22C22×C22⋊D511yescharacteristic
183C4—22no—
184C4—22no—
185C4—42no—
186C4—42no—
187C4—104no—
188C4—108no—
189C4—202no—
190C2—24no—
191C2#6597014yes—
192C2#6613811yescharacteristic
193C2C2×#409011yescharacteristic
194C2#6616211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC24×C5C23
hypercenterC2×C22⋊Q8D5
hypocenterC5#2583
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #65518C2semidirect
5. #65269C2semidirect
6. #65261C2semidirect
47. C22⋊Dic5Q8semidirect
174. C5#2583semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C22×#6832
#655182
C4×C22⋊Dic52
#652692
#652612
#652352
#25835