Finite Group Databasebeta

← #10604427

Subgroups — #10604427

Order 640.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#65914C211yesmaximal, characteristic
2#65689C211yesmaximal, characteristic, complemented
3#65684C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64868C212yesmaximal
6C2×C4⋊AGL(1,𝔽5)—21no—
7C4×C5×Q8C411yescharacteristic, complemented
8#4093C411yescharacteristic, complemented
9#4092C2211yescharacteristic
10#4044—21no—
11#4041C2211yescharacteristic
12#4040—21no—
13#4036C411yescharacteristic, complemented
14C4×D20—21no—
15C4×D20C411yescharacteristic, complemented
16#4018—21no—
17C42×D5C2211yescharacteristic
18(C2×AGL(1,𝔽5))⋊C4—21no—
19(C2×AGL(1,𝔽5))⋊C4C2212yes—
20#4004—21no—
21#4003C2212yes—
22#4002—21no—
23#1512—51nomaximal, Sylow
24C22×AGL(1,𝔽5)—22no—
25C2×C5×Q8C2×C411yescharacteristic
26C2×D20—21no—
27C2×D20C2×C411yescharacteristic
28C2×C4×D5—21no—
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×C4×D5C2311yescharacteristic
32C2×#170—22no—
33C4⋊AGL(1,𝔽5)—42no—
34C5×C4⋊C4—21no—
35C5×C4⋊C4C2×C411yescharacteristic
36C42×C5—21no—
37C42×C5C2×C411yescharacteristic
38#685—21no—
39#685—21no—
40#685—21no—
41#685—21no—
42#685—21no—
43C4⋊Dic5—21no—
44C4⋊Dic5C2×C411yescharacteristic
45#683—21no—
46#683—21no—
47C4×Dic5C2×C411yescharacteristic
48C4×Dic5D411yescharacteristic
49C4×Dic5D411yescharacteristic
50#529—51no—
51#432—51no—
52#382—51no—
53#328—52no—
54#327—52no—
55C22×D5—21no—
56C22×D5C2×D411yescharacteristic
57C2×AGL(1,𝔽5)—42no—
58C2×AGL(1,𝔽5)—42no—
59C2×AGL(1,𝔽5)—42no—
60C5×Q8—41no—
61C2×C4×C5—21no—
62C2×C4×C5—21no—
63C2×C4×C5C22⋊C411yescharacteristic
64C2×C4×C5C22⋊C411yescharacteristic
65C2×C4×C5C22×C411yescharacteristic
66C2×Dic5—21no—
67C2×Dic5C22⋊C411yescharacteristic
68C2×Dic5C22⋊C411yescharacteristic
69C2×Dic5C2×D411yescharacteristic
70D20—41no—
71D20—41no—
72D20—41no—
73C4×D5—21no—
74C4×D5—21no—
75C4×D5—41no—
76C4×D5—41no—
77C4×D5—42no—
78#170—42no—
79#126—101no—
80#125—51no—
81#124—51no—
82#123—101no—
83C22⋊Q8—51no—
84#121—51no—
85C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
86C4×D4—51no—
87C4×D4—101no—
88#117—101no—
89C2×C4⋊C4—101no—
90C2×C42—51no—
91C4⋊C8—101no—
92C22⋊C8—52no—
93#97—101no—
94#95—52no—
95#95—101no—
96C22×C5C2×C22⋊C411yescharacteristic
97C2×D5—21no—
98C2×D5—21no—
99C2×D5—41no—
100C2×D5—41no—
101C2×D5—41no—
102C2×D5AGL(1,ℤ/8)11yescharacteristic
103C2×D5#13711yescharacteristic
104AGL(1,𝔽5)—82no—
105C4×C5—21no—
106C4×C5—21no—
107C4×C5—21no—
108C4×C5—41no—
109C4×C5—41no—
110Dic5—21no—
111Dic5—21no—
112Dic5—21no—
113Dic5—41no—
114C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
115C2×D4—51no—
116C2×D4—101no—
117C22×C4—51no—
118C22×C4—51no—
119C22×C4—51no—
120C22×C4—101no—
121C22×C4—102no—
122C2×C8—102no—
123C4⋊C4—21no—
124C4⋊C4—51no—
125C4⋊C4—101no—
126C4⋊C4—101no—
127C4⋊C4—101no—
128C4⋊C4—202no—
129C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
130C22⋊C4—101no—
131C22⋊C4—101no—
132C22⋊C4—101no—
133C22⋊C4—101no—
134C22⋊C4—101no—
135C42—21no—
136C42—51no—
137C42—51no—
138C42—51no—
139C42C2×AGL(1,𝔽5)11yescharacteristic
140C2×C5#41011yescharacteristic
141C2×C5#41711yescharacteristic
142C2×C5#42111yescharacteristic
143D5—21no—
144D5—21no—
145D5—81no—
146C23—51no—
147C23—101no—
148Q8—41no—
149D4—201no—
150D4—201no—
151D4—201no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—51no—
155C2×C4—51no—
156C2×C4—51no—
157C2×C4—101no—
158C2×C4—101no—
159C2×C4—101no—
160C2×C4—201no—
161C2×C4—201no—
162C2×C4—202no—
163C2×C4—202no—
164C2×C4—202no—
165C2×C4—202no—
166C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
167C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
168C2×C4C22×AGL(1,𝔽5)11yescharacteristic
169C8—202no—
170C5#151211yescharacteristic, complemented, Sylow
171C22—51no—
172C22—51no—
173C22—101no—
174C22—101no—
175C22—201no—
176C22—201no—
177C22—201no—
178C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—101no—
185C4—101no—
186C4—101no—
187C4—201no—
188C4—402no—
189C2—101no—
190C2—101no—
191C2—401no—
192C2#6574511yescharacteristic
193C2#6578411yescharacteristic
194C2#6579311yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC4×C5×Q8C4
socleC22×C5C2×C22⋊C4
prime radical#4041C22
cosocleC2×C4×D5C23
hypercenterC4×Q8AGL(1,𝔽5)
hypocenterC5#1512
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. #65689C2semidirect
3. #65684C2semidirect
4. #64926C2semidirect
7. C4×C5×Q8C4semidirect
8. #4093C4semidirect
13. #4036C4semidirect
15. C4×D20C4semidirect
85. C4×Q8AGL(1,𝔽5)semidirect
170. C5#1512semidirect

Maximal subgroups

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

subgroupindex
#659142
#656892
#656842
#649262
#648682
#15125