Finite Group Databasebeta

← #10615090

Subgroups — #10615090

Order 640.

370 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C5×#533C211yesmaximal, characteristic, complemented
2#65811C211yesmaximal, characteristic
3#65377C211yesmaximal, characteristic, complemented
4#65377C211yesmaximal, characteristic, complemented
5#65368C211yesmaximal, characteristic, complemented
6#65367C211yesmaximal, characteristic, complemented
7#65355C214yesmaximal, complemented
8#65292C214yesmaximal, complemented
9#65214C211yesmaximal, characteristic, complemented
10C2×#709—21no—
11C5×C4⋊Q8C2211yescharacteristic
12C5×C4⋊Q8C2211yescharacteristic
13C5×#125C2212yes—
14C5×C22⋊Q8—21no—
15C5×C22⋊Q8—21no—
16C4×C5×Q8—21no—
17C4×C5×Q8—21no—
18C5×#117C2212yes—
19C2×C42×C5C2211yescharacteristic
20#4076—21no—
21#4076C2212yes—
22#4021—21no—
23#4020C2211yescharacteristic
24C4×D20—21no—
25C4×D20C2212yes—
26#4015C2211yescharacteristic
27C22⋊#168C2212yes—
28#3941C2214yes—
29#3941C2214yes—
30#3940C2214yes—
31#3939C2214yes—
32C4⋊#168C2212yes—
33C4×#168C2211yescharacteristic
34C4×#168C2211yescharacteristic
35#3228—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×Q8—21no—
38C22×C4×C5D411yescharacteristic
39C22×C4×C5D411yescharacteristic
40C22×C4×C5C2311yescharacteristic
41C2×C22⋊D5—22no—
42#709—44no—
43C2×D20—21no—
44C2×D20—21no—
45C2×D20C2312yes—
46C2×C4×D5—22no—
47C2×Dic10—21no—
48C5×C4⋊C4—22no—
49C5×C4⋊C4—22no—
50C5×C4⋊C4—21no—
51C5×C4⋊C4—21no—
52C5×C4⋊C4C2312yes—
53C5×C4⋊C4C2312yes—
54C5×C22⋊C4—22no—
55C42×C5—22no—
56C42×C5D411yescharacteristic
57C42×C5D411yescharacteristic
58C42×C5C2311yescharacteristic
59C42×C5C2311yescharacteristic
60#685—22no—
61#685—22no—
62C4⋊Dic5—21no—
63C4⋊Dic5C2312yes—
64C2×#168C2312yes—
65C2×#168C2312yes—
66#533D511yescharacteristic, complemented
67#532—51no—
68#500—51no—
69#498—51no—
70#486—51no—
71#486—51no—
72#482—54no—
73#478—54no—
74#431—51no—
75C23×C5C2×D411yescharacteristic
76C22×D5—22no—
77C5×Q8—42no—
78C5×Q8—42no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5—22no—
82C2×C4×C5—22no—
83C2×C4×C5—22no—
84C2×C4×C5C2×D411yescharacteristic
85C2×C4×C5C2×D411yescharacteristic
86C2×C4×C5C2×D411yescharacteristic
87C2×C4×C5C2×D411yescharacteristic
88C2×C4×C5C2×D411yescharacteristic
89C2×C4×C5C2411yescharacteristic
90C22⋊D5—44no—
91C2×Dic5—22no—
92D20—24no—
93D20—42no—
94D20—42no—
95C4×D5—44no—
96Dic10—42no—
97#168—22no—
98#168—22no—
99C2×#41—101no—
100C4⋊Q8—51no—
101C4⋊Q8C2×D511yescharacteristic
102C4⋊Q8C2×D511yescharacteristic
103#127—51no—
104#125C2×D512yes—
105#124—101no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#121—52no—
109#121—101no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×D4—52no—
113C4×D4—101no—
114#117C2×D512yes—
115C2×C42C2×D511yescharacteristic
116#107—54no—
117#106—54no—
118C4⋊C8—52no—
119#102—54no—
120#102—54no—
121C22⋊C8—52no—
122C4×C8—51no—
123C4×C8—51no—
124C22×C5—21no—
125C22×C5—21no—
126C22×C5—21no—
127C22×C5C22×D411yescharacteristic
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—42no—
131C4×C5—21no—
132C4×C5—21no—
133C4×C5—21no—
134C4×C5—42no—
135C4×C5—42no—
136C4×C5#13512yes—
137C4×C5AGL(1,ℤ/8)12yes—
138C4×C52−1+412yes—
139Dic5—42no—
140#41—204no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—101no—
144C2×D4—52no—
145C2×D4—102no—
146C2×D4—101no—
147C2×D4—101no—
148C22×C4—102no—
149C22×C4C22⋊D511yescharacteristic
150C22×C4C22⋊D511yescharacteristic
151C22×C4C22×D511yescharacteristic
152C2×C8—52no—
153C2×C8—52no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—52no—
159C4⋊C4—101no—
160C4⋊C4C22×D512yes—
161C4⋊C4C22×D512yes—
162C22⋊C4—22no—
163C22⋊C4—102no—
164C22⋊C4—102no—
165C42—22no—
166C42C22⋊D511yescharacteristic
167C42C22⋊D511yescharacteristic
168C42C22×D511yescharacteristic
169C42C22×D511yescharacteristic
170C2×C5—41no—
171C2×C5#53611yescharacteristic
172C2×C5C2×#13511yescharacteristic
173C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
174D5—82no—
175C23—102no—
176C23C2×C22⋊D511yescharacteristic
177Q8—42no—
178Q8—42no—
179Q8—202no—
180D4—104no—
181D4—204no—
182D4—202no—
183D4—202no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—102no—
190C2×C4—204no—
191C2×C4C2×C22⋊D511yescharacteristic
192C2×C4C2×C22⋊D511yescharacteristic
193C2×C4C2×C22⋊D511yescharacteristic
194C2×C4C2×C22⋊D511yescharacteristic
195C2×C4C2×C22⋊D511yescharacteristic
196C2×C4C23×D511yescharacteristic
197C8—102no—
198C8—102no—
199C5#322811yescharacteristic, complemented, Sylow
200C22—21no—
201C22—21no—
202C22—21no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22C22×C22⋊D511yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—202no—
213C4#409512yes—
214C4#409612yes—
215C4#414712yes—
216C2—41no—
217C2—402no—
218C2#6615011yescharacteristic
219C2C2×#409511yescharacteristic
220C2C2×#409611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#533C2
socleC22×C5C22×D4
prime radical#65811C2
hypercenter#533D5
hypocenterC5#3228
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C5×#533C2semidirect
3. #65377C2semidirect
4. #65377C2semidirect
5. #65368C2semidirect
6. #65367C2semidirect
7. #65355C2semidirect
8. #65292C2semidirect
9. #65214C2semidirect
66. #533D5semidirect
199. C5#3228semidirect

Maximal subgroups

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

subgroupindex
C5×#5332
#658112
#653772
#653682
#653672
#653552
#652922
#652142
#32285