Finite Group Databasebeta

← #10613673

Subgroups — #10613673

Order 640.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#66062C211yesmaximal, characteristic, complemented
2#66059C211yesmaximal, characteristic
3C5×#500C211yesmaximal, characteristic, complemented
4#65377C211yesmaximal, characteristic, complemented
5#65368C211yesmaximal, characteristic, complemented
6#65180C214yesmaximal, complemented
7#65174C214yesmaximal, complemented
8#64983C211yesmaximal, characteristic, complemented
9#64974C211yesmaximal, characteristic, complemented
10C2×#713—21no—
11C5×C4⋊Q8C2211yescharacteristic
12C5×C4⋊Q8C2211yescharacteristic
13#4093—21no—
14#4092—21no—
15Q8×Dic5—21no—
16#4043—21no—
17#4041—21no—
18#4041C2212yes—
19#4039—21no—
20#4039C2212yes—
21#4038C2212yes—
22#4036C2212yes—
23#4033—21no—
24#4020C2211yescharacteristic
25C42×D5C2211yescharacteristic
26#4015C2211yescharacteristic
27C5×#107C2214yes—
28C4×C5×C8C2211yescharacteristic
29#3953C2214yes—
30#3952C2212yes—
31#3946C2212yes—
32#3941C2214yes—
33#3940C2214yes—
34C4×#168C2211yescharacteristic
35#3228—51nomaximal, Sylow
36C2×C5×Q8—21no—
37C2×C5×Q8—21no—
38#713—44no—
39C2×D20—22no—
40C2×D20—21no—
41C2×D20—21no—
42C2×D20C2312yes—
43C2×C4×D5—22no—
44C2×C4×D5D411yescharacteristic
45C2×C4×D5D411yescharacteristic
46C2×C4×D5C2311yescharacteristic
47C2×Dic10—21no—
48C2×C5×C8C2312yes—
49C5×C4⋊C4—21no—
50C5×C4⋊C4—21no—
51C5×C4⋊C4C2312yes—
52C5×C4⋊C4C2312yes—
53C42×C5C2311yescharacteristic
54#685—22no—
55#685—22no—
56#685—22no—
57C4⋊Dic5—21no—
58C4⋊Dic5—22no—
59C4⋊Dic5C2312yes—
60#683—22no—
61C4×Dic5—22no—
62C4×Dic5D411yescharacteristic
63C4×Dic5D411yescharacteristic
64C4×Dic5C2311yescharacteristic
65C2×#168C2312yes—
66#533—51no—
67#532—51no—
68#500D511yescharacteristic, complemented
69#498—51no—
70#486—51no—
71#486—51no—
72#482—54no—
73#478—54no—
74#431—51no—
75C22×D5—22no—
76C22×D5C2×D411yescharacteristic
77C5×Q8—42no—
78C5×Q8—42no—
79C2×C4×C5—22no—
80C2×C4×C5—22no—
81C2×C4×C5C2×D411yescharacteristic
82C2×C4×C5C2×D411yescharacteristic
83C2×C4×C5C2411yescharacteristic
84C2×Dic5—22no—
85C2×Dic5C2×D411yescharacteristic
86C2×Dic5C2×D411yescharacteristic
87C2×Dic5C2×D411yescharacteristic
88D20—24no—
89D20—44no—
90D20—42no—
91D20—42no—
92C4×D5—22no—
93C4×D5—22no—
94C4×D5—22no—
95C4×D5—44no—
96Dic10—42no—
97C5×C8—22no—
98#168—22no—
99C2×#41—101no—
100C4⋊Q8—51no—
101C4⋊Q8C2×D511yescharacteristic
102C4⋊Q8C2×D511yescharacteristic
103#127—51no—
104#125—52no—
105#124—101no—
106C22⋊Q8—101no—
107C22⋊Q8—101no—
108#121—52no—
109#121—101no—
110C4×Q8—101no—
111C4×Q8—101no—
112C4×D4—52no—
113C4×D4—101no—
114#117—52no—
115C2×C42—51no—
116#107C2×D514yes—
117#106—54no—
118C4⋊C8—52no—
119#102—54no—
120#102—54no—
121C22⋊C8—52no—
122C4×C8—51no—
123C4×C8C2×D511yescharacteristic
124C22×C5C22×D411yescharacteristic
125C2×D5—21no—
126C2×D5—21no—
127C2×D5—21no—
128C2×D5—42no—
129C2×D5—42no—
130C2×D5—42no—
131C4×C5—42no—
132C4×C5—42no—
133C4×C5#13512yes—
134C4×C5AGL(1,ℤ/8)12yes—
135C4×C52−1+412yes—
136Dic5—21no—
137Dic5—21no—
138Dic5—21no—
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—51no—
149C22×C4—51no—
150C22×C4—51no—
151C22×C4—102no—
152C2×C8—52no—
153C2×C8C22×D512yes—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—52no—
157C4⋊C4—102no—
158C4⋊C4—102no—
159C4⋊C4—101no—
160C4⋊C4C22×D512yes—
161C4⋊C4C22×D512yes—
162C22⋊C4—102no—
163C22⋊C4—102no—
164C22⋊C4—102no—
165C42—51no—
166C42—51no—
167C42—51no—
168C42—102no—
169C42C22×D511yescharacteristic
170C2×C5#53611yescharacteristic
171C2×C5C2×#13511yescharacteristic
172C2×C5C2×AGL(1,ℤ/8)11yescharacteristic
173D5—41no—
174D5—82no—
175C23—51no—
176C23—102no—
177Q8—42no—
178Q8—42no—
179Q8—202no—
180D4—104no—
181D4—204no—
182D4—202no—
183D4—202no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—51no—
187C2×C4—51no—
188C2×C4—51no—
189C2×C4—102no—
190C2×C4—102no—
191C2×C4—102no—
192C2×C4—102no—
193C2×C4—204no—
194C2×C4D4×D511yescharacteristic
195C2×C4D4×D511yescharacteristic
196C2×C4C23×D511yescharacteristic
197C8—22no—
198C8—102no—
199C5#322811yescharacteristic, complemented, Sylow
200C22—101no—
201C22—101no—
202C22—101no—
203C22—202no—
204C22—202no—
205C22—202no—
206C22C2×D4×D511yescharacteristic
207C4—42no—
208C4—42no—
209C4—101no—
210C4—101no—
211C4—101no—
212C4—202no—
213C4#405712yes—
214C4#406512yes—
215C4#414712yes—
216C2—201no—
217C2—402no—
218C2#6605811yescharacteristic
219C2C2×#405711yescharacteristic
220C2C2×#406511yescharacteristic
subgroupG/H
centerC22C2×D4×D5
commutator, cosocleC2×C4×C5C24
FrattiniC2×C4C23×D5
FittingC5×#500C2
socleC22×C5C22×D4
prime radical#66059C2
hypercenter#500D5
hypocenterC5#3228
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #66062C2semidirect
3. C5×#500C2semidirect
4. #65377C2semidirect
5. #65368C2semidirect
6. #65180C2semidirect
7. #65174C2semidirect
8. #64983C2semidirect
9. #64974C2semidirect
68. #500D5semidirect
199. C5#3228semidirect

Maximal subgroups

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

subgroupindex
#660622
#660592
C5×#5002
#653772
#653682
#651802
#651742
#649832
#649742
#32285