Finite Group Databasebeta

← #10614453

Subgroups — #10614453

Order 640.

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

normal subgroups only

This group has 644 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×#121C211yesmaximal, characteristic, complemented
2C2×#4076C211yesmaximal, characteristic, complemented
3#65512C212yesmaximal, complemented
4#65512C214yesmaximal, complemented
5#65500C211yesmaximal, characteristic, complemented
6#65282C212yesmaximal, complemented
7#65249C212yesmaximal, complemented
8#65242C212yesmaximal, complemented
9C22×C5×D4C2212yes—
10C22×C5×D4C2211yescharacteristic
11C23×C4×C5C2211yescharacteristic
12C22×C22⋊D5C2212yes—
13C22×C22⋊D5C2214yes—
14C22×D20C2211yescharacteristic
15C5×#121—24no—
16C2×C5×C4⋊C4C2211yescharacteristic
17C2×C5×C22⋊C4C2212yes—
18C2×C22⋊Dic5C2214yes—
19C2×C22⋊Dic5C2212yes—
20#4076—24no—
21C2×#685C2212yes—
22C2×#685C2214yes—
23C2×C4⋊Dic5C2211yescharacteristic
24#3963C2214yes—
25#3963C2212yes—
26#3963C2212yes—
27#2815—51nomaximal, Sylow
28C24×C5C2312yes—
29C24×C5C2311yescharacteristic
30C23×D5C2312yes—
31C2×C5×D4—24no—
32C2×C5×D4—22no—
33C2×C5×D4—42no—
34C2×C5×D4—44no—
35C22×C4×C5—42no—
36C22×C4×C5D414yescomplemented
37C22×C4×C5C2312yes—
38C22×C4×C5C2311yescharacteristic
39C22×C4×C5C2311yescharacteristic
40C2×C22⋊D5—24no—
41C2×C22⋊D5—44no—
42C2×C22⋊D5—44no—
43C2×C22⋊D5—44no—
44C2×C22⋊D5—44no—
45C22×Dic5C2314yes—
46C22×Dic5C2312yes—
47C2×D20—22no—
48C2×D20—42no—
49C5×C4⋊C4—22no—
50C5×C22⋊C4—24no—
51C22⋊Dic5—42no—
52C22⋊Dic5—44no—
53#685—24no—
54#685—44no—
55C4⋊Dic5—22no—
56C2×#121—51no—
57C2×#121D511yescharacteristic, complemented
58#397—52no—
59#397—52no—
60#392—52no—
61#392—54no—
62#392—52no—
63#381—51no—
64C23×C5—22no—
65C23×C5—22no—
66C23×C5—22no—
67C23×C5—22no—
68C23×C5—22no—
69C23×C5—22no—
70C23×C5—24no—
71C23×C5—24no—
72C23×C5C2×D412yes—
73C23×C5C2411yescharacteristic
74C22×D5—22no—
75C22×D5—22no—
76C22×D5—24no—
77C22×D5—22no—
78C22×D5—24no—
79C5×D4—44no—
80C5×D4—48no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5—24no—
85C2×C4×C5—44no—
86C2×C4×C5C2×D412yes—
87C2×C4×C5C2×D412yes—
88C22⋊D5—48no—
89C22⋊D5—416no—
90C2×Dic5—22no—
91C2×Dic5—24no—
92C2×Dic5—24no—
93C2×Dic5—24no—
94C2×Dic5—24no—
95D20—44no—
96C22×D4—52no—
97C22×D4—54no—
98C22×D4—51no—
99C22×D4C2×D512yes—
100C22×D4C2×D511yescharacteristic
101C23×C4C2×D511yescharacteristic
102#121—24no—
103#121—104no—
104C2×C4⋊C4—51no—
105C2×C4⋊C4C2×D511yescharacteristic
106C2×C22⋊C4—52no—
107C2×C22⋊C4—52no—
108C2×C22⋊C4—54no—
109C2×C22⋊C4—54no—
110C2×C22⋊C4C2×D512yes—
111#95—52no—
112#95—52no—
113#95—54no—
114C22×C5—22no—
115C22×C5—22no—
116C22×C5—22no—
117C22×C5—44no—
118C22×C5—42no—
119C22×C5—42no—
120C22×C5—42no—
121C22×C5—44no—
122C22×C5—44no—
123C22×C5C22×D411yescharacteristic
124C22×C5H(5,𝔽2)12yes—
125C22×C5H(5,𝔽2)12yes—
126C22×C5H(5,𝔽2)12yes—
127C2×D5—42no—
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—44no—
131C2×D5—44no—
132C4×C5—22no—
133C4×C5—22no—
134C4×C5—42no—
135Dic5—42no—
136Dic5—44no—
137C24—52no—
138C24C22×D512yes—
139C24C22×D511yescharacteristic
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4—104no—
145C2×D4—102no—
146C2×D4—202no—
147C2×D4—204no—
148C2×D4—204no—
149C2×D4—204no—
150C2×D4—204no—
151C22×C4—42no—
152C22×C4—54no—
153C22×C4—52no—
154C22×C4C22⋊D514yescomplemented
155C22×C4C22×D512yes—
156C22×C4C22×D511yescharacteristic
157C22×C4C22×D511yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—102no—
160C22⋊C4—24no—
161C22⋊C4—104no—
162C22⋊C4—204no—
163C22⋊C4—202no—
164C22⋊C4—204no—
165C2×C5—42no—
166C2×C5—82no—
167C2×C5#53411yescharacteristic
168C2×C5#53511yescharacteristic
169C2×C5#53511yescharacteristic
170C2×C5#56014yes—
171D5—82no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—24no—
176C23—24no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—102no—
181C23—104no—
182C23—102no—
183C23—104no—
184C23—102no—
185C23C2×C22⋊D512yes—
186C23C23×D511yescharacteristic
187D4—44no—
188D4—48no—
189D4—204no—
190D4—208no—
191D4—2016no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—44no—
197C2×C4—104no—
198C2×C4—104no—
199C2×C4—104no—
200C2×C4—104no—
201C2×C4—102no—
202C2×C4C2×C22⋊D512yes—
203C2×C4C2×C22⋊D512yes—
204C5#281511yescharacteristic, complemented, Sylow
205C22—22no—
206C22—22no—
207C22—22no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—202no—
215C22—202no—
216C22—202no—
217C22—204no—
218C22—204no—
219C22#414412yes—
220C22#414412yes—
221C22#414912yes—
222C22C22×C22⋊D511yescharacteristic
223C4—22no—
224C4—22no—
225C4—42no—
226C4—202no—
227C4—204no—
228C2—42no—
229C2—82no—
230C2—402no—
231C2#6595414yes—
232C2#6613811yescharacteristic
233C2#6613911yescharacteristic
234C2#6616411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D5
commutator, socle, cosocleC23×C5C24
FittingC2×C5×#121C2
hypercenterC2×#121D5
hypocenterC5#2815
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C5×#121C2semidirect
2. C2×#4076C2semidirect
3. #65512C2semidirect
4. #65512C2semidirect
5. #65500C2semidirect
6. #65282C2semidirect
7. #65249C2semidirect
8. #65242C2semidirect
36. C22×C4×C5D4semidirect
57. C2×#121D5semidirect
154. C22×C4C22⋊D5semidirect
204. C5#2815semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#1212
C2×#40762
#655122
#655002
#652822
#652492
#652422
#28155