Finite Group Databasebeta

← #10614564

Subgroups — #10614564

Order 640.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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, complemented
2C2×#4076C211yesmaximal, characteristic
3#65520C212yesmaximal, complemented
4#65500C211yesmaximal, characteristic, complemented
5#65281C212yesmaximal, complemented
6#65280C212yesmaximal, complemented
7#65272C212yesmaximal, complemented
8#65248C214yesmaximal, complemented
9C22×C5×Q8C2211yescharacteristic
10C23×C4×C5C2211yescharacteristic
11C22×C22⋊D5C2212yes—
12C22×D20C2211yescharacteristic
13C5×C22⋊Q8—24no—
14C2×C5×C4⋊C4C2212yes—
15C2×C5×C4⋊C4C2211yescharacteristic
16C2×C5×C22⋊C4C2212yes—
17C2×C22⋊Dic5C2212yes—
18#4076—24no—
19C2×#685C2212yes—
20C2×#685C2214yes—
21C2×#685C2214yes—
22C2×C4⋊Dic5C2211yescharacteristic
23C2×#683C2214yes—
24#3963C2212yes—
25#3963C2212yes—
26#3963C2214yes—
27#2816—51nomaximal, Sylow
28C24×C5C2311yescharacteristic
29C23×D5C2312yes—
30C2×C5×Q8—22no—
31C2×C5×Q8—42no—
32C22×C4×C5—42no—
33C22×C4×C5D414yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×C4×C5C2311yescharacteristic
38C2×C22⋊D5—24no—
39C2×C22⋊D5—44no—
40C22×Dic5C2314yes—
41C22×Dic5C2312yes—
42C2×D20—22no—
43C2×D20—42no—
44C5×C4⋊C4—22no—
45C5×C4⋊C4—24no—
46C5×C22⋊C4—24no—
47C22⋊Dic5—42no—
48#685—24no—
49#685—44no—
50#685—44no—
51C4⋊Dic5—22no—
52#683—44no—
53C2×C22⋊Q8D511yescharacteristic, complemented
54C2×#121—51no—
55#399—52no—
56#398—52no—
57#394—52no—
58#394—54no—
59#393—52no—
60#381—51no—
61C23×C5—22no—
62C23×C5—22no—
63C23×C5—22no—
64C23×C5C2×D412yes—
65C23×C5C2411yescharacteristic
66C22×D5—22no—
67C22×D5—24no—
68C22×D5—24no—
69C22×D5—24no—
70C5×Q8—44no—
71C2×C4×C5—22no—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—24no—
78C2×C4×C5—44no—
79C2×C4×C5C2×D412yes—
80C2×C4×C5C2×D412yes—
81C22⋊D5—48no—
82C2×Dic5—22no—
83C2×Dic5—24no—
84C2×Dic5—24no—
85C2×Dic5—24no—
86C2×Dic5—24no—
87D20—44no—
88C22×Q8C2×D511yescharacteristic
89C22×D4—52no—
90C22×D4—51no—
91C23×C4C2×D511yescharacteristic
92C22⋊Q8—24no—
93#121—104no—
94C2×C4⋊C4—51no—
95C2×C4⋊C4—54no—
96C2×C4⋊C4C2×D512yes—
97C2×C4⋊C4C2×D511yescharacteristic
98C2×C22⋊C4—52no—
99C2×C22⋊C4—52no—
100C2×C22⋊C4—54no—
101C2×C22⋊C4—54no—
102C2×C22⋊C4C2×D512yes—
103#95—52no—
104#95—52no—
105#95—54no—
106C22×C5—22no—
107C22×C5—22no—
108C22×C5—22no—
109C22×C5—44no—
110C22×C5C22×D411yescharacteristic
111C22×C5H(5,𝔽2)12yes—
112C22×C5H(5,𝔽2)12yes—
113C22×C52−1+412yes—
114C2×D5—42no—
115C2×D5—42no—
116C2×D5—42no—
117C2×D5—48no—
118C4×C5—22no—
119C4×C5—22no—
120C4×C5—42no—
121C4×C5—42no—
122Dic5—42no—
123Dic5—44no—
124C24—52no—
125C24C22×D511yescharacteristic
126C2×Q8—22no—
127C2×Q8—42no—
128C2×D4—104no—
129C2×D4—102no—
130C2×D4—202no—
131C2×D4—204no—
132C22×C4—42no—
133C22×C4—54no—
134C22×C4—52no—
135C22×C4C22⋊D514yes—
136C22×C4C22×D512yes—
137C22×C4C22×D512yes—
138C22×C4C22×D511yescharacteristic
139C22×C4C22×D511yescharacteristic
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—102no—
143C4⋊C4—204no—
144C22⋊C4—24no—
145C22⋊C4—104no—
146C22⋊C4—204no—
147C22⋊C4—204no—
148C22⋊C4—202no—
149C2×C5—42no—
150C2×C5#53411yescharacteristic
151C2×C5#53511yescharacteristic
152C2×C5#53611yescharacteristic
153C2×C5#56214yes—
154D5—82no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—104no—
159C23—104no—
160C23—104no—
161C23—102no—
162C23C2×C22⋊D512yes—
163C23C23×D511yescharacteristic
164Q8—44no—
165D4—204no—
166D4—208no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—104no—
176C2×C4—104no—
177C2×C4—104no—
178C2×C4—104no—
179C2×C4—102no—
180C2×C4C2×C22⋊D512yes—
181C2×C4C2×C22⋊D512yes—
182C5#281611yescharacteristic, complemented, Sylow
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—202no—
188C22—202no—
189C22—202no—
190C22—208no—
191C22#414412yes—
192C22#414712yes—
193C22#414912yes—
194C22C22×C22⋊D511yescharacteristic
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—202no—
200C4—204no—
201C2—42no—
202C2—402no—
203C2#6598114yes—
204C2#6613811yescharacteristic
205C2#6615011yescharacteristic
206C2#6616411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×C22⋊Q8C2
prime radicalC2×#4076C2
hypercenterC2×C22⋊Q8D5
hypocenterC5#2816
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C5×C22⋊Q8C2semidirect
3. #65520C2semidirect
4. #65500C2semidirect
5. #65281C2semidirect
6. #65280C2semidirect
7. #65272C2semidirect
8. #65248C2semidirect
53. C2×C22⋊Q8D5semidirect
182. C5#2816semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C22⋊Q82
C2×#40762
#655202
#655002
#652812
#652802
#652722
#652482
#28165