Finite Group Databasebeta

← #10602136

Subgroups — #10602136

Order 640.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#66060C211yesmaximal, characteristic
2C5×#498C211yesmaximal, characteristic, complemented
3#65376C211yesmaximal, characteristic, complemented
4#65372C211yesmaximal, characteristic, complemented
5#64767C211yesmaximal, characteristic, complemented
6#64679C211yesmaximal, characteristic, complemented
7#64679C211yesmaximal, characteristic, complemented
8C2×#713—21no—
9C5×C4⋊Q8C2211yescharacteristic
10C5×C4⋊Q8C2211yescharacteristic
11#4092—21no—
12C2×#688—21no—
13#4041—21no—
14#4040—22no—
15D5×C4⋊C4—21no—
16C4×D20C2211yescharacteristic
17C4×D20C2211yescharacteristic
18C5×#106—22no—
19C4×C5×C8C2211yescharacteristic
20#3967—21no—
21#3941—21no—
22#3939—21no—
23C4⋊#168C2211yescharacteristic
24C4⋊#168C2211yescharacteristic
25#1654—51nomaximal, Sylow
26C2×C5×Q8—21no—
27C2×C5×Q8—21no—
28#713—42no—
29#713—42no—
30C2×D20—22no—
31C2×D20D411yescharacteristic
32C2×D20D411yescharacteristic
33C2×C4×D5—22no—
34C2×C4×D5—21no—
35C2×C4×D5—21no—
36C2×C5×C8—21no—
37C5×C4⋊C4—21no—
38C5×C4⋊C4—21no—
39C5×C4⋊C4—21no—
40C5×C4⋊C4D412yes—
41C42×C5C2311yescharacteristic
42#688—42no—
43#685—22no—
44#685—21no—
45#685—21no—
46C4⋊Dic5D411yescharacteristic
47C4⋊Dic5D411yescharacteristic
48#683—22no—
49C2×#168—21no—
50C2×#168—21no—
51#547—51no—
52#498D511yescharacteristic, complemented
53#474—51no—
54#461—51no—
55#332—51no—
56D4⋊C8—51no—
57D4⋊C8—51no—
58C22×D5—21no—
59C22×D5—21no—
60C5×Q8—41no—
61C5×Q8—41no—
62C5×Q8—41no—
63C5×Q8—41no—
64C2×C4×C5—22no—
65C2×C4×C5—41no—
66C2×C4×C5C2×D411yescharacteristic
67C2×C4×C5C2×D411yescharacteristic
68C2×C4×C5C2×D411yescharacteristic
69C2×Dic5—21no—
70C2×Dic5—21no—
71D20—21no—
72D20—21no—
73D20—21no—
74D20—44no—
75D20SD1612yescomplemented
76C4×D5—44no—
77C4×D5—41no—
78C4×D5—41no—
79C4×D5—41no—
80C4×D5—41no—
81C5×C8—22no—
82#168—41no—
83#168—41no—
84C2×#41—101no—
85C2×SD16—101no—
86C4⋊Q8C2×D511yescharacteristic
87C4⋊Q8C2×D511yescharacteristic
88#123—102no—
89C22⋊Q8—101no—
90#121—101no—
91C4×D4—51no—
92C4×D4—51no—
93C2×C4⋊C4—101no—
94#107—101no—
95#106—22no—
96C4⋊C8—51no—
97C4⋊C8—51no—
98#103—101no—
99#102—101no—
100C4×C8C2×D511yescharacteristic
101C22×C5#12011yescharacteristic
102C2×D5—22no—
103C2×D5—41no—
104C2×D5—41no—
105C2×D5—41no—
106C2×D5—41no—
107C2×D5—41no—
108C4×C5—21no—
109C4×C5—42no—
110C4×C5—81no—
111C4×C5C2×SD1611yescharacteristic
112C4×C5#13511yescharacteristic
113C4×C5AGL(1,ℤ/8)11yescharacteristic
114C4×C5AGL(1,ℤ/8)11yescharacteristic
115Dic5—41no—
116Dic5—41no—
117#41—202no—
118#41—202no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×D4—51no—
122C2×D4—51no—
123C2×D4—102no—
124C22×C4—102no—
125C22×C4—101no—
126C22×C4—101no—
127SD16—202no—
128C2×C8—21no—
129C2×C8—101no—
130C2×C8—101no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—51no—
135C4⋊C4—51no—
136C4⋊C4—102no—
137C4⋊C4C22⋊D512yes—
138C22⋊C4—102no—
139C22⋊C4—101no—
140C22⋊C4—101no—
141C42C22×D511yescharacteristic
142C2×C5#44911yescharacteristic
143C2×C5#45011yescharacteristic
144C2×C5#45611yescharacteristic
145D5—42no—
146D5—81no—
147C23—101no—
148C23—101no—
149Q8—41no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153D4—52no—
154D4—101no—
155D4—101no—
156D4—101no—
157D4—204no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—101no—
161C2×C4—101no—
162C2×C4—204no—
163C2×C4—201no—
164C2×C4—201no—
165C2×C4—201no—
166C2×C4—201no—
167C2×C4D4×D511yescharacteristic
168C2×C4D4×D511yescharacteristic
169C2×C4C2×C22⋊D511yescharacteristic
170C8—22no—
171C8—201no—
172C8—201no—
173C5#165411yescharacteristic, complemented, Sylow
174C22—102no—
175C22—201no—
176C22—201no—
177C22—201no—
178C22—201no—
179C22—201no—
180C22#408311yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—201no—
185C4—201no—
186C4D5×SD1611yescharacteristic
187C4#406011yescharacteristic
188C4#406011yescharacteristic
189C4#406211yescharacteristic
190C2—202no—
191C2—401no—
192C2#6538411yescharacteristic
193C2#6545811yescharacteristic
194C2#6546111yescharacteristic
subgroupG/H
centerC22#4083
commutator, cosocleC42×C5C23
FrattiniC42C22×D5
FittingC5×#498C2
socleC22×C5#120
prime radical#66060C2
hypercenter#498D5
hypocenterC5#1654
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. C5×#498C2semidirect
3. #65376C2semidirect
4. #65372C2semidirect
5. #64767C2semidirect
6. #64679C2semidirect
7. #64679C2semidirect
52. #498D5semidirect
75. D20SD16semidirect
173. C5#1654semidirect

Maximal subgroups

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

subgroupindex
#660602
C5×#4982
#653762
#653722
#647672
#646792
#16545