Finite Group Databasebeta

← #10604982

Subgroups — #10604982

Order 640.

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

normal subgroups only

This group has 324 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
1C2×#4092C211yesmaximal, characteristic
2#65765C211yesmaximal, characteristic
3C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#64917C214yesmaximal
5C22×C5×Q8C411yescharacteristic, complemented
6C22×C4×D5C2211yescharacteristic
7C2×C22⋊AGL(1,𝔽5)C2212yes—
8C2×C22⋊AGL(1,𝔽5)C2212yes—
9C2×C4⋊AGL(1,𝔽5)—21no—
10C2×C4×AGL(1,𝔽5)—21no—
11#4092—21no—
12#4092—21no—
13#4092—21no—
14#4092—21no—
15C2×#685C2212yes—
16C2×C4⋊Dic5C411yescharacteristic, complemented
17C2×#683C412yescomplemented
18(C2×AGL(1,𝔽5))⋊C4—21no—
19(C2×AGL(1,𝔽5))⋊C4—21no—
20#1996—51nomaximal, Sylow
21C23×D5C2311yescharacteristic
22C22×AGL(1,𝔽5)—22no—
23C22×AGL(1,𝔽5)—22no—
24C2×C5×Q8—21no—
25C2×C5×Q8—21no—
26C2×C5×Q8—42no—
27C22×C4×C5C2×C412yes—
28C22×C4×C5C2×C411yescharacteristic
29C22×Dic5C2×C412yes—
30C22×Dic5C2×C411yescharacteristic
31C2×C4×D5—42no—
32C2×C4×D5D411yescharacteristic
33C2×C4×D5D411yescharacteristic
34C2×C4×D5D411yescharacteristic
35C2×C4×D5D411yescharacteristic
36C22⋊AGL(1,𝔽5)—22no—
37C22⋊AGL(1,𝔽5)—24no—
38C22⋊AGL(1,𝔽5)Q814yescomplemented
39C4⋊AGL(1,𝔽5)—42no—
40C4×AGL(1,𝔽5)—42no—
41#685—22no—
42#685—22no—
43C4⋊Dic5—21no—
44C4⋊Dic5—21no—
45#683—22no—
46#683—22no—
47C2×C22⋊Q8—51no—
48#380—51no—
49C4×C22⋊C4—51no—
50#342—54no—
51C23×C5C22×C411yescharacteristic
52C22×D5—41no—
53C22×D5—41no—
54C22×D5C2×D411yescharacteristic
55C22×D5C2×D411yescharacteristic
56C22×D5C2×Q812yes—
57C22×D5#4112yes—
58C2×AGL(1,𝔽5)—24no—
59C2×AGL(1,𝔽5)—42no—
60C2×AGL(1,𝔽5)—42no—
61C2×AGL(1,𝔽5)—42no—
62C2×AGL(1,𝔽5)—42no—
63C2×AGL(1,𝔽5)—42no—
64C5×Q8—42no—
65C5×Q8—42no—
66C2×C4×C5—22no—
67C2×C4×C5—22no—
68C2×C4×C5—42no—
69C2×C4×C5C22⋊C411yescharacteristic
70C2×C4×C5C22⋊C411yescharacteristic
71C2×Dic5—22no—
72C2×Dic5—41no—
73C2×Dic5—42no—
74C2×Dic5C22⋊C411yescharacteristic
75C2×Dic5C22⋊C411yescharacteristic
76C4×D5—22no—
77C4×D5—22no—
78C4×D5—41no—
79C4×D5—41no—
80C22×Q8AGL(1,𝔽5)11yescharacteristic, complemented
81C23×C4—51no—
82C22⋊Q8—101no—
83C22⋊Q8—101no—
84C22⋊Q8—101no—
85C22⋊Q8—101no—
86C2×C4⋊C4—52no—
87C2×C4⋊C4—51no—
88C2×C4⋊C4—101no—
89C2×C22⋊C4—52no—
90C2×C22⋊C4—52no—
91C2×C22⋊C4—52no—
92C2×C42—101no—
93#95—101no—
94#95—101no—
95C22×C5—21no—
96C22×C5—21no—
97C22×C5C2×C22⋊C411yescharacteristic
98C22×C5C4×Q812yes—
99C2×D5—21no—
100C2×D5—21no—
101C2×D5—21no—
102C2×D5—21no—
103C2×D5—42no—
104C2×D5—42no—
105C2×D5C22⋊Q812yes—
106C2×D5#12411yescharacteristic
107C2×D5C4⋊Q811yescharacteristic
108AGL(1,𝔽5)—44no—
109AGL(1,𝔽5)—82no—
110C4×C5—21no—
111C4×C5—42no—
112C4×C5#9912yes—
113Dic5—21no—
114Dic5—21no—
115Dic5—42no—
116C24—51no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—42no—
120C22×C4—51no—
121C22×C4—51no—
122C22×C4—51no—
123C22×C4—51no—
124C22×C4—51no—
125C22×C4—52no—
126C22×C4—102no—
127C22×C4—102no—
128C22×C4—202no—
129C22×C4C2×AGL(1,𝔽5)12yes—
130C22×C4C2×AGL(1,𝔽5)11yescharacteristic
131C4⋊C4—102no—
132C4⋊C4—102no—
133C4⋊C4—101no—
134C4⋊C4—101no—
135C4⋊C4—202no—
136C22⋊C4—54no—
137C22⋊C4—102no—
138C22⋊C4—102no—
139C22⋊C4—102no—
140C22⋊C4—104no—
141C42—202no—
142C2×C5—22no—
143C2×C5#39111yescharacteristic
144C2×C5C2×#9911yescharacteristic
145C2×C5#41011yescharacteristic
146D5—22no—
147D5—21no—
148D5—21no—
149C23—51no—
150C23—51no—
151C23—52no—
152C23—52no—
153C23—201no—
154C23—201no—
155C23C22×AGL(1,𝔽5)11yescharacteristic
156Q8—42no—
157Q8—42no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—51no—
162C2×C4—51no—
163C2×C4—102no—
164C2×C4—102no—
165C2×C4—104no—
166C2×C4—102no—
167C2×C4—201no—
168C2×C4—201no—
169C2×C4—201no—
170C2×C4—202no—
171C2×C4—202no—
172C2×C4—202no—
173C2×C4—202no—
174C2×C4—202no—
175C2×C4—202no—
176C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
177C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
178C5#199611yescharacteristic, complemented, Sylow
179C22—21no—
180C22—21no—
181C22—52no—
182C22—51no—
183C22—51no—
184C22—101no—
185C22—101no—
186C22—101no—
187C22—101no—
188C22—202no—
189C22—202no—
190C22Q8×AGL(1,𝔽5)12yes—
191C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
192C4—21no—
193C4—42no—
194C4—101no—
195C4—101no—
196C4—204no—
197C4—202no—
198C4—402no—
199C4#401112yes—
200C2—22no—
201C2—102no—
202C2—101no—
203C2—101no—
204C2#6578511yescharacteristic
205C2(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
206C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC22×C5×Q8C4
socleC22×C5C2×C22⋊C4
prime radical, cosocleC23×D5C23
hypercenterC22×Q8AGL(1,𝔽5)
hypocenterC5#1996
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
5. C22×C5×Q8C4semidirect
16. C2×C4⋊Dic5C4semidirect
17. C2×#683C4semidirect
38. C22⋊AGL(1,𝔽5)Q8semidirect
80. C22×Q8AGL(1,𝔽5)semidirect
178. C5#1996semidirect

Maximal subgroups

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

subgroupindex
C2×#40922
#657652
C4×C22⋊AGL(1,𝔽5)2
#649172
#19965