Finite Group Databasebeta

← #10604678

Subgroups — #10604678

Order 640.

336 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 74 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#66059C211yesmaximal, characteristic
2C2×#4010C214yesmaximal, complemented
3C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C4×#700C211yesmaximal, characteristic, complemented
5C2×#713C2212yes—
6C2×C4⋊AGL(1,𝔽5)—21no—
7C2×C4×AGL(1,𝔽5)C2212yes—
8C2×#700C2212yes—
9C5×C4⋊Q8C411yescharacteristic, complemented
10#4093C412yescomplemented
11#4041—21no—
12#4041—21no—
13#4039—21no—
14#4039—21no—
15#4020C411yescharacteristic, complemented
16C42×D5C2211yescharacteristic
17#4010—28no—
18(C2×AGL(1,𝔽5))⋊C4—21no—
19#4002—21no—
20C4×#170—21no—
21#1980—51nomaximal, Sylow
22C22×AGL(1,𝔽5)—22no—
23C2×C5×Q8C2×C412yes—
24#713—24no—
25#713—24no—
26C2×D20—22no—
27C2×D20—21no—
28C2×D20—21no—
29C2×D20C2×C412yes—
30C2×C4×D5—22no—
31C2×C4×D5D411yescharacteristic
32C2×C4×D5D411yescharacteristic
33C2×C4×D5C2311yescharacteristic
34C2×#170—22no—
35C4⋊AGL(1,𝔽5)—24no—
36C4×AGL(1,𝔽5)—24no—
37#700—22no—
38#700D414yescomplemented
39C5×C4⋊C4—21no—
40C5×C4⋊C4—21no—
41C42×C5C2×C411yescharacteristic
42#685—22no—
43#685—22no—
44C4×Dic5C2×C411yescharacteristic
45C4×Dic5D411yescharacteristic
46C4×Dic5D411yescharacteristic
47#532—51no—
48C2×#104—54no—
49C4×M16—51no—
50C4×C4⋊C4—51no—
51C22×D5—22no—
52C22×D5C2×D411yescharacteristic
53C2×AGL(1,𝔽5)—24no—
54C2×AGL(1,𝔽5)—42no—
55C2×AGL(1,𝔽5)—42no—
56C5×Q8—24no—
57C2×C4×C5—22no—
58C2×C4×C5C22⋊C411yescharacteristic
59C2×C4×C5C22⋊C411yescharacteristic
60C2×C4×C5C22×C411yescharacteristic
61C2×Dic5C22⋊C411yescharacteristic
62C2×Dic5C22⋊C411yescharacteristic
63C2×Dic5C2×D411yescharacteristic
64D20—24no—
65D20—42no—
66D20—42no—
67D20—42no—
68D20—42no—
69C4×D5—22no—
70C4×D5—41no—
71C4×D5—44no—
72C4×D5C2×D412yes—
73C4×D5#4112yes—
74#170—24no—
75C2×#41—52no—
76C2×M16—52no—
77C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
78#127—51no—
79#124—52no—
80#121—101no—
81#121—101no—
82C4×D4—101no—
83C4×D4—101no—
84C2×C4⋊C4—101no—
85C2×C42—51no—
86C2×C42—52no—
87#104—108no—
88#97—101no—
89C4×C8—101no—
90#95—101no—
91C22×C5C2×C22⋊C411yescharacteristic
92C2×D5—21no—
93C2×D5—21no—
94C2×D5—42no—
95C2×D5—42no—
96C2×D5—42no—
97C2×D5#12411yescharacteristic
98C2×D5#12711yescharacteristic
99AGL(1,𝔽5)—44no—
100C4×C5—21no—
101C4×C5—42no—
102C4×C5#10412yescomplemented
103C4×C5C4×D412yes—
104Dic5—21no—
105Dic5—21no—
106Dic5#12112yes—
107#41—104no—
108#41—104no—
109C2×Q8C2×AGL(1,𝔽5)12yes—
110C2×D4—52no—
111C2×D4—102no—
112C2×D4—101no—
113C2×D4—101no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—51no—
117C22×C4—102no—
118C22×C4—102no—
119M16—54no—
120M16—102no—
121C2×C8—102no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—104no—
125C22⋊C4—102no—
126C22⋊C4—102no—
127C42—51no—
128C42—51no—
129C42—51no—
130C42—104no—
131C42C2×AGL(1,𝔽5)11yescharacteristic
132C2×C5#39011yescharacteristic
133C2×C5C2×#10411yescharacteristic
134C2×C5#42111yescharacteristic
135D5—21no—
136D5—21no—
137D5—82no—
138C23—51no—
139C23—102no—
140Q8—24no—
141D4—104no—
142D4—202no—
143D4—202no—
144D4—202no—
145D4—202no—
146C2×C4—22no—
147C2×C4—51no—
148C2×C4—52no—
149C2×C4—52no—
150C2×C4—51no—
151C2×C4—51no—
152C2×C4—102no—
153C2×C4—104no—
154C2×C4—201no—
155C2×C4—204no—
156C2×C4—202no—
157C2×C4—202no—
158C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
159C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
160C2×C4C22×AGL(1,𝔽5)11yescharacteristic
161C8—104no—
162C5#198011yescharacteristic, complemented, Sylow
163C22—51no—
164C22—51no—
165C22—101no—
166C22—101no—
167C22—202no—
168C22—202no—
169C22—202no—
170C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
171C4—21no—
172C4—42no—
173C4—52no—
174C4—101no—
175C4—101no—
176C4—204no—
177C4#401012yescomplemented
178C4D4×AGL(1,𝔽5)12yes—
179C2—101no—
180C2—101no—
181C2—402no—
182C2#6578011yescharacteristic
183C2C2×#401011yescharacteristic
184C2#6578411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C4⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radical#66059C2
cosocleC2×C4×D5C23
hypercenterC4⋊Q8AGL(1,𝔽5)
hypocenterC5#1980
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#4010C2semidirect
3. C4×C4⋊AGL(1,𝔽5)C2semidirect
4. C4×#700C2semidirect
9. C5×C4⋊Q8C4semidirect
10. #4093C4semidirect
15. #4020C4semidirect
38. #700D4semidirect
77. C4⋊Q8AGL(1,𝔽5)semidirect
102. C4×C5#104semidirect
162. C5#1980semidirect
177. C4#4010semidirect

Maximal subgroups

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

subgroupindex
#660592
C2×#40102
C4×C4⋊AGL(1,𝔽5)2
C4×#7002
#19805