Finite Group Databasebeta

← #10604992

Subgroups — #10604992

Order 640.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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#66162C211yesmaximal, characteristic
2#65764C211yesmaximal, characteristic, complemented
3C2×#4003C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64920C212yesmaximal, complemented
6C2×C5×#41C411yescharacteristic, complemented
7C22×C4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—21no—
9C22×#170—21no—
10#4092C411yescharacteristic, complemented
11#4092C2211yescharacteristic
12#4084C411yescharacteristic, complemented
13#4084C2211yescharacteristic
14#4083—21no—
15#4081—21no—
16C4×C22⋊D5—21no—
17C4×C22⋊D5—21no—
18#4072C411yescharacteristic, complemented
19(C2×AGL(1,𝔽5))⋊C4—21no—
20(C2×AGL(1,𝔽5))⋊C4C2212yes—
21#4003—21no—
22#4003C2212yes—
23#1516—51nomaximal, Sylow
24C23×D5D411yescharacteristic
25C22×AGL(1,𝔽5)—22no—
26C5×#41—42no—
27C2×C5×Q8C2×C411yescharacteristic
28C2×C5×D4—21no—
29C2×C5×D4C2×C411yescharacteristic
30C22×C4×C5—21no—
31C22×C4×C5C2×C411yescharacteristic
32C2×C22⋊D5—21no—
33C2×C22⋊D5—21no—
34C22×Dic5D411yescharacteristic
35C2×C4×D5—42no—
36C2×C4×D5C2×C411yescharacteristic
37C2×C4×D5D411yescharacteristic
38C2×C4×D5D411yescharacteristic
39C2×C4×D5C2311yescharacteristic
40C22⋊AGL(1,𝔽5)—42no—
41C2×#170—22no—
42C2×#170—42no—
43C22⋊Dic5—21no—
44C22⋊Dic5—21no—
45C22⋊Dic5—21no—
46#685—21no—
47#685—21no—
48C4⋊Dic5C2×C411yescharacteristic
49C4⋊Dic5C2×C411yescharacteristic
50#683—21no—
51#683—21no—
52C4×Dic5—21no—
53#525—51no—
54C2×C22⋊C8—51no—
55#381—51no—
56#328—52no—
57#327—52no—
58C23×C5—21no—
59C23×C5C22⋊C411yescharacteristic
60C22×D5—22no—
61C22×D5—22no—
62C22×D5—41no—
63C22×D5C22⋊C411yescharacteristic
64C22×D5C2×D411yescharacteristic
65C2×AGL(1,𝔽5)—42no—
66C2×AGL(1,𝔽5)—42no—
67C2×AGL(1,𝔽5)—42no—
68C5×Q8—41no—
69C5×D4—41no—
70C5×D4—42no—
71C2×C4×C5—22no—
72C2×C4×C5—21no—
73C2×C4×C5—42no—
74C2×C4×C5C22⋊C411yescharacteristic
75C2×C4×C5C22×C411yescharacteristic
76C22⋊D5—42no—
77C22⋊D5—42no—
78C2×Dic5—22no—
79C2×Dic5—21no—
80C2×Dic5—21no—
81C2×Dic5C22⋊C411yescharacteristic
82C2×Dic5C2×D411yescharacteristic
83C4×D5—41no—
84C4×D5—41no—
85C4×D5—41no—
86C4×D5—41no—
87#170—42no—
88C2×#41AGL(1,𝔽5)11yescharacteristic, complemented
89C23×C4—51no—
90C22×C8—101no—
91#123—101no—
92C22⋊Q8—51no—
93C22⋊Q8—51no—
94#121—51no—
95#121—51no—
96#120—101no—
97C4×D4—101no—
98C4×D4—101no—
99#117—51no—
100C2×C22⋊C4—101no—
101C22⋊C8—52no—
102C22⋊C8—101no—
103#95—52no—
104#95—101no—
105C22×C5—21no—
106C22×C5—21no—
107C22×C5—41no—
108C22×C5—41no—
109C22×C5—41no—
110C22×C5#9912yes—
111C22×C5C2×C22⋊C411yescharacteristic
112C2×D5—21no—
113C2×D5—21no—
114C2×D5—21no—
115C2×D5—21no—
116C2×D5—21no—
117C2×D5—44no—
118C2×D5#13512yes—
119AGL(1,𝔽5)—82no—
120C4×C5—21no—
121C4×C5—21no—
122C4×C5—41no—
123Dic5—21no—
124Dic5—21no—
125Dic5—41no—
126Dic5—41no—
127C24—51no—
128#41—42no—
129C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
130C2×D4—21no—
131C2×D4—101no—
132C2×D4—101no—
133C2×D4C2×AGL(1,𝔽5)11yescharacteristic
134C22×C4—21no—
135C22×C4—51no—
136C22×C4—51no—
137C22×C4—51no—
138C22×C4—51no—
139C22×C4—51no—
140C22×C4—102no—
141C22×C4—202no—
142C22×C4C2×AGL(1,𝔽5)11yescharacteristic
143C2×C8—102no—
144C2×C8—202no—
145C4⋊C4—51no—
146C4⋊C4—51no—
147C4⋊C4—101no—
148C4⋊C4—101no—
149C22⋊C4—101no—
150C22⋊C4—101no—
151C22⋊C4—101no—
152C22⋊C4—101no—
153C22⋊C4—101no—
154C22⋊C4—202no—
155C42—101no—
156C2×C5—22no—
157C2×C5—81no—
158C2×C5C2×#9911yescharacteristic
159C2×C5#41611yescharacteristic
160C2×C5#42111yescharacteristic
161D5—22no—
162D5—41no—
163C23—21no—
164C23—51no—
165C23—51no—
166C23—102no—
167C23—102no—
168C23—201no—
169C23C22⋊AGL(1,𝔽5)11yescharacteristic
170Q8—41no—
171D4—41no—
172D4—42no—
173D4—202no—
174D4—202no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—51no—
179C2×C4—51no—
180C2×C4—102no—
181C2×C4—101no—
182C2×C4—101no—
183C2×C4—201no—
184C2×C4—201no—
185C2×C4—201no—
186C2×C4—201no—
187C2×C4—202no—
188C2×C4—202no—
189C2×C4—202no—
190C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
191C2×C4C22×AGL(1,𝔽5)11yescharacteristic
192C8—202no—
193C5#151611yescharacteristic, complemented, Sylow
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—52no—
200C22—101no—
201C22—101no—
202C22—101no—
203C22—101no—
204C22—101no—
205C22—204no—
206C22#401112yes—
207C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—101no—
212C4—101no—
213C4—201no—
214C4—201no—
215C4—402no—
216C2—22no—
217C2—81no—
218C2—102no—
219C2—201no—
220C2#6579411yescharacteristic
221C2#6579511yescharacteristic
222C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC2×C5×#41C4
socleC22×C5C2×C22⋊C4
prime radical#66162C2
cosocleC2×C4×D5C23
hypercenterC2×#41AGL(1,𝔽5)
hypocenterC5#1516
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #65764C2semidirect
3. C2×#4003C2semidirect
4. #64926C2semidirect
5. #64920C2semidirect
6. C2×C5×#41C4semidirect
10. #4092C4semidirect
12. #4084C4semidirect
18. #4072C4semidirect
88. C2×#41AGL(1,𝔽5)semidirect
193. C5#1516semidirect

Maximal subgroups

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

subgroupindex
#661622
#657642
C2×#40032
#649262
#649202
#15165