Finite Group Databasebeta

← #10605451

Subgroups — #10605451

Order 640.

548 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 165 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×D20C211yesmaximal, characteristic
2C2×#4016C211yesmaximal, characteristic, complemented
3C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
4#65283C214yesmaximal, complemented
5#65228C211yesmaximal, characteristic
6#65228C211yesmaximal, characteristic
7#65225C211yesmaximal, characteristic, complemented
8#65224C211yesmaximal, characteristic, complemented
9#64957C214yesmaximal, complemented
10C22×D20C2211yescharacteristic
11C22×C4×D5C2212yes—
12C2×C5×C4⋊C4C2212yes—
13C2×C42×C5C2211yescharacteristic
14C2×C42×C5C2211yescharacteristic
15C2×C42×C5C2211yescharacteristic
16C2×#685C2212yes—
17C2×#685C2214yes—
18C2×#685C2214yes—
19C2×C4⋊Dic5C2211yescharacteristic
20C2×C4⋊Dic5C2211yescharacteristic
21C2×C4⋊Dic5C2211yescharacteristic
22C2×C4⋊Dic5C2214yes—
23C2×#683C2212yes—
24C2×#683C2212yes—
25C4×D20—24no—
26#4016—24no—
27C5×#95C2212yes—
28#3963C2214yes—
29#2566—51nomaximal, Sylow
30C23×D5C2312yes—
31C22×C4×C5C2312yes—
32C22×C4×C5C2311yescharacteristic
33C22×C4×C5C2311yescharacteristic
34C22×C4×C5C2312yes—
35C22×C4×C5C2311yescharacteristic
36C22×Dic5C2312yes—
37C22×Dic5C2312yes—
38C22×Dic5C2312yes—
39C2×D20—22no—
40C2×D20—22no—
41C2×D20Q814yes—
42C2×C4×D5—24no—
43C2×C4×D5—44no—
44C5×C4⋊C4—24no—
45C42×C5—22no—
46C42×C5—22no—
47C42×C5D414yes—
48#685—24no—
49#685—28no—
50#685—44no—
51C4⋊Dic5—22no—
52C4⋊Dic5—22no—
53C4⋊Dic5—22no—
54C4⋊Dic5—44no—
55#683—24no—
56#683—24no—
57C2×#125—51no—
58C2×C4×D4—51no—
59#396—54no—
60#390—51no—
61#390—51no—
62#385—54no—
63#384—51no—
64#382—51no—
65C4×C4⋊C4D511yescharacteristic, complemented
66C23×C5C2411yescharacteristic
67C22×D5—22no—
68C22×D5—24no—
69C22×D5—22no—
70C22×D5—24no—
71C22×D5C2×Q814yes—
72C2×C4×C5—22no—
73C2×C4×C5—22no—
74C2×C4×C5—22no—
75C2×C4×C5—22no—
76C2×C4×C5—22no—
77C2×C4×C5—22no—
78C2×C4×C5—22no—
79C2×C4×C5C2×D412yes—
80C2×C4×C5C2×D412yes—
81C2×C4×C5C2×D412yes—
82C2×C4×C5C2×Q812yes—
83C2×C4×C5#4114yes—
84C2×C4×C5#4112yes—
85C2×Dic5—22no—
86C2×Dic5—22no—
87C2×Dic5—22no—
88C2×Dic5—24no—
89C2×Dic5—24no—
90C2×Dic5—24no—
91D20—28no—
92C4×D5—48no—
93C22×D4—51no—
94C23×C4—52no—
95#125—104no—
96C4×D4—104no—
97C2×C4⋊C4—51no—
98C2×C4⋊C4—51no—
99C2×C4⋊C4—51no—
100C2×C4⋊C4—52no—
101C2×C4⋊C4—52no—
102C2×C4⋊C4—54no—
103C2×C4⋊C4C2×D512yes—
104C2×C22⋊C4—52no—
105C2×C22⋊C4—54no—
106C2×C22⋊C4—54no—
107C2×C42C2×D511yescharacteristic
108C2×C42C2×D511yescharacteristic
109C2×C42C2×D511yescharacteristic
110#95—54no—
111#95C2×D512yes—
112C22×C5C22×D411yescharacteristic
113C22×C5C22×Q811yescharacteristic
114C22×C5C2×#4112yes—
115C22×C5C2×#4111yescharacteristic
116C22×C5H(5,𝔽2)12yes—
117C2×D5—24no—
118C2×D5—28no—
119C2×D5—42no—
120C2×D5—42no—
121C2×D5—44no—
122C4×C5—22no—
123C4×C5—22no—
124C4×C5—24no—
125C4×C5—42no—
126C4×C5C22⋊Q814yes—
127Dic5—42no—
128Dic5—42no—
129Dic5—42no—
130C24—52no—
131C2×D4—54no—
132C2×D4—102no—
133C2×D4—102no—
134C22×C4—52no—
135C22×C4—52no—
136C22×C4—52no—
137C22×C4—104no—
138C22×C4—204no—
139C22×C4C22×D512yes—
140C22×C4C22×D511yescharacteristic
141C22×C4C22×D511yescharacteristic
142C22×C4C22×D512yes—
143C22×C4C22×D511yescharacteristic
144C4⋊C4—24no—
145C4⋊C4—102no—
146C4⋊C4—104no—
147C4⋊C4—104no—
148C4⋊C4—102no—
149C4⋊C4—102no—
150C4⋊C4—204no—
151C22⋊C4—104no—
152C22⋊C4—108no—
153C22⋊C4—204no—
154C42—22no—
155C42—22no—
156C42D2014yes—
157C2×C5C2×C22⋊Q811yescharacteristic
158C2×C5#53211yescharacteristic
159C2×C5#53511yescharacteristic
160C2×C5#55312yes—
161C2×C5#55412yes—
162D5—44no—
163C23—54no—
164C23—102no—
165C23—104no—
166C23—102no—
167C23—104no—
168C23C23×D511yescharacteristic
169D4—108no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—102no—
178C2×C4—102no—
179C2×C4—102no—
180C2×C4—104no—
181C2×C4—104no—
182C2×C4—104no—
183C2×C4—208no—
184C2×C4C2×D2012yes—
185C2×C4C2×D2012yes—
186C2×C4C2×D2012yes—
187C2×C4#70914yes—
188C2×C4#71112yes—
189C2×C4Q8×D512yes—
190C5#256611yescharacteristic, complemented, Sylow
191C22—104no—
192C22—108no—
193C22—202no—
194C22—202no—
195C22—204no—
196C22C22×D2011yescharacteristic
197C22C2×#70912yes—
198C22C2×#71111yescharacteristic
199C22C2×Q8×D511yescharacteristic
200C22#414912yes—
201C4—22no—
202C4—22no—
203C4—24no—
204C4—42no—
205C4—202no—
206C4—202no—
207C4—202no—
208C4#404314yes—
209C2—204no—
210C2#6581111yescharacteristic
211C2C2×#404311yescharacteristic
212C2#6585911yescharacteristic
213C2#6589912yes—
214C2#6591312yes—
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC4×C5×C4⋊C4C2
prime radicalC22×D20C22
hypercenterC4×C4⋊C4D5
hypocenterC5#2566
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#4016C2semidirect
3. C4×C5×C4⋊C4C2semidirect
4. #65283C2semidirect
7. #65225C2semidirect
8. #65224C2semidirect
9. #64957C2semidirect
65. C4×C4⋊C4D5semidirect
190. C5#2566semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D202
C2×#40162
C4×C5×C4⋊C42
#652832
#652282
#652252
#652242
#649572
#25665