Finite Group Databasebeta

← #10608094

Subgroups — #10608094

Order 640.

548 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×Dic5C211yesmaximal, characteristic
2C2×#4036C211yesmaximal, characteristic, complemented
3C5×#390C211yesmaximal, characteristic
4#65510C211yesmaximal, characteristic
5#65265C211yesmaximal, characteristic, complemented
6#65237C214yesmaximal, complemented
7#65235C214yesmaximal, complemented
8#65225C211yesmaximal, characteristic, complemented
9C4×#683C211yesmaximal, characteristic, complemented
10C22×C5×D4C2211yescharacteristic
11C23×Dic5C2212yes—
12C2×C5×C4⋊C4C2211yescharacteristic
13C2×C5×C22⋊C4C2214yes—
14C2×C42×C5C2211yescharacteristic
15C2×C22⋊Dic5C2212yes—
16C2×C22⋊Dic5C2214yes—
17D4×Dic5—24no—
18C2×C4⋊Dic5C2211yescharacteristic
19C2×C4⋊Dic5C2212yes—
20C2×C4⋊Dic5C2211yescharacteristic
21C2×#683C2212yes—
22C2×#683C2212yes—
23C2×#683C2214yes—
24C2×C4×Dic5C2211yescharacteristic
25C2×C4×Dic5C2211yescharacteristic
26#4036—24no—
27#3963C2212yes—
28#3963C2214yes—
29#2566—51nomaximal, Sylow
30C24×C5C2312yes—
31C2×C5×D4—22no—
32C2×C5×D4—22no—
33C2×C5×D4Q814yes—
34C22×C4×C5C2312yes—
35C22×C4×C5C2312yes—
36C22×C4×C5C2311yescharacteristic
37C22×Dic5—24no—
38C22×Dic5—44no—
39C22×Dic5C2312yes—
40C22×Dic5C2312yes—
41C22×Dic5C2311yescharacteristic
42C22×Dic5C2311yescharacteristic
43C22×Dic5C2312yes—
44C5×C4⋊C4—22no—
45C5×C22⋊C4—44no—
46C42×C5—22no—
47C22⋊Dic5—24no—
48C22⋊Dic5—28no—
49C4⋊Dic5—22no—
50C4⋊Dic5—22no—
51C4⋊Dic5—24no—
52#683—24no—
53#683—24no—
54#683—44no—
55C4×Dic5—22no—
56C4×Dic5D414yes—
57C2×#125—51no—
58C2×C4×D4—51no—
59#396—54no—
60#390—51no—
61#390D511yescharacteristic
62#385—54no—
63#384—51no—
64#382—51no—
65C4×C4⋊C4—51no—
66C23×C5—22no—
67C23×C5—24no—
68C23×C5—22no—
69C23×C5—24no—
70C23×C5C2×Q814yes—
71C23×C5C2411yescharacteristic
72C5×D4—28no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—24no—
77C2×C4×C5—22no—
78C2×C4×C5C2×D412yes—
79C2×C4×C5C2×Q812yes—
80C2×C4×C5#4112yes—
81C2×Dic5—22no—
82C2×Dic5—22no—
83C2×Dic5—22no—
84C2×Dic5—22no—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—24no—
88C2×Dic5—24no—
89C2×Dic5—48no—
90C2×Dic5C2×D412yes—
91C2×Dic5C2×D412yes—
92C2×Dic5#4114yes—
93C22×D4C2×D511yescharacteristic
94C23×C4—52no—
95#125—104no—
96C4×D4—104no—
97C2×C4⋊C4—51no—
98C2×C4⋊C4—51no—
99C2×C4⋊C4—52no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—54no—
103C2×C4⋊C4C2×D511yescharacteristic
104C2×C22⋊C4—52no—
105C2×C22⋊C4—54no—
106C2×C22⋊C4C2×D514yes—
107C2×C42—51no—
108C2×C42—51no—
109C2×C42C2×D511yescharacteristic
110#95—52no—
111#95—54no—
112C22×C5—24no—
113C22×C5—28no—
114C22×C5—42no—
115C22×C5—42no—
116C22×C5—44no—
117C22×C5C22×D411yescharacteristic
118C22×C5C22×Q811yescharacteristic
119C22×C5C2×#4111yescharacteristic
120C22×C5C2×#4112yes—
121C22×C5H(5,𝔽2)12yes—
122C4×C5—42no—
123C4×C5—42no—
124C4×C5C22⋊Q814yes—
125Dic5—22no—
126Dic5—22no—
127Dic5—24no—
128Dic5—42no—
129Dic5—42no—
130C24C22×D512yes—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4Dic1014yes—
134C22×C4—51no—
135C22×C4—52no—
136C22×C4—51no—
137C22×C4—52no—
138C22×C4—52no—
139C22×C4—104no—
140C22×C4—204no—
141C22×C4C22×D512yes—
142C22×C4C22×D512yes—
143C22×C4C22×D511yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—104no—
146C4⋊C4—104no—
147C4⋊C4—104no—
148C4⋊C4—102no—
149C4⋊C4—102no—
150C4⋊C4—204no—
151C22⋊C4—44no—
152C22⋊C4—104no—
153C22⋊C4—108no—
154C42—22no—
155C42—54no—
156C42—102no—
157C2×C5—44no—
158C2×C5C2×C22⋊Q811yescharacteristic
159C2×C5#53211yescharacteristic
160C2×C5#53511yescharacteristic
161C2×C5#55312yes—
162C2×C5#55412yes—
163C23—22no—
164C23—22no—
165C23—24no—
166C23—24no—
167C23C2×Dic1014yes—
168C23C23×D511yescharacteristic
169D4—28no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—52no—
176C2×C4—52no—
177C2×C4—54no—
178C2×C4—102no—
179C2×C4—102no—
180C2×C4—102no—
181C2×C4—102no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—104no—
185C2×C4—104no—
186C2×C4—208no—
187C2×C4C2×Dic1012yes—
188C2×C4D4×D512yes—
189C2×C4#71112yes—
190C5#256611yescharacteristic, complemented, Sylow
191C22—24no—
192C22—28no—
193C22—42no—
194C22—44no—
195C22—42no—
196C22C22×Dic1011yescharacteristic
197C22C2×D4×D511yescharacteristic
198C22C2×#71111yescharacteristic
199C22#414412yes—
200C22D5×#4112yes—
201C4—42no—
202C4—42no—
203C4—102no—
204C4—104no—
205C4—102no—
206C4—202no—
207C4—202no—
208C4#402414yes—
209C2—44no—
210C2C2×#402411yescharacteristic
211C2#6587312yes—
212C2#6594812yes—
213C2#6600211yescharacteristic
214C2#6604911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC5×#390C2
prime radicalC22×C5×D4C22
hypercenter#390D5
hypocenterC5#2566
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#4036C2semidirect
5. #65265C2semidirect
6. #65237C2semidirect
7. #65235C2semidirect
8. #65225C2semidirect
9. C4×#683C2semidirect
190. C5#2566semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic52
C2×#40362
C5×#3902
#655102
#652652
#652372
#652352
#652252
C4×#6832
#25665