Finite Group Databasebeta

← #10604989

Subgroups — #10604989

Order 640.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 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
2C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3D5⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64920C212yesmaximal, complemented
6C2×C5×#41C411yescharacteristic, complemented
7C22×C4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—21no—
9C2×C4×AGL(1,𝔽5)—21no—
10C2×D5⋊C8—21no—
11#4092C411yescharacteristic, complemented
12#4092C2211yescharacteristic
13#4084C411yescharacteristic, complemented
14#4084C2211yescharacteristic
15#4083—21no—
16#4081—21no—
17C4×C22⋊D5—21no—
18C4×C22⋊D5—21no—
19#4072C411yescharacteristic, complemented
20C22⋊#170—21no—
21(C2×AGL(1,𝔽5))⋊C4C2212yes—
22#4003C2212yes—
23#1501—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)—24no—
41C2×#170—22no—
42C4×AGL(1,𝔽5)—42no—
43D5⋊C8—42no—
44C22⋊Dic5—21no—
45C22⋊Dic5—21no—
46C22⋊Dic5—21no—
47#685—21no—
48#685—21no—
49C4⋊Dic5C2×C411yescharacteristic
50C4⋊Dic5C2×C411yescharacteristic
51#683—21no—
52#683—21no—
53C4×Dic5—21no—
54#525—51no—
55C2×C22⋊C8—51no—
56C4×C22⋊C4—51no—
57#328—52no—
58#327—52no—
59C23×C5—21no—
60C23×C5C22⋊C411yescharacteristic
61C22×D5—22no—
62C22×D5—22no—
63C22×D5—41no—
64C22×D5C22⋊C411yescharacteristic
65C22×D5C2×D411yescharacteristic
66C2×AGL(1,𝔽5)—24no—
67C2×AGL(1,𝔽5)—42no—
68C2×AGL(1,𝔽5)—42no—
69C5×Q8—41no—
70C5×D4—41no—
71C5×D4—42no—
72C2×C4×C5—22no—
73C2×C4×C5—21no—
74C2×C4×C5—42no—
75C2×C4×C5C22⋊C411yescharacteristic
76C2×C4×C5C22×C411yescharacteristic
77C22⋊D5—42no—
78C22⋊D5—42no—
79C2×Dic5—21no—
80C2×Dic5—21no—
81C2×Dic5—41no—
82C2×Dic5C22⋊C411yescharacteristic
83C2×Dic5C2×D411yescharacteristic
84C4×D5—22no—
85C4×D5—22no—
86C4×D5—41no—
87C4×D5—41no—
88#170—42no—
89C2×#41AGL(1,𝔽5)11yescharacteristic, complemented
90C23×C4—51no—
91C22×C8—101no—
92#123—101no—
93C22⋊Q8—51no—
94C22⋊Q8—51no—
95#121—51no—
96#121—51no—
97#120—101no—
98C4×D4—101no—
99C4×D4—101no—
100#117—51no—
101C2×C22⋊C4—101no—
102C2×C42—101no—
103C22⋊C8—52no—
104C22⋊C8—101no—
105#95—52no—
106C22×C5—21no—
107C22×C5—21no—
108C22×C5—21no—
109C22×C5—41no—
110C22×C5—41no—
111C22×C5—41no—
112C22×C5C2×C22⋊C411yescharacteristic
113C2×D5—21no—
114C2×D5—21no—
115C2×D5—21no—
116C2×D5—21no—
117C2×D5—44no—
118C2×D5#10412yes—
119C2×D5#13512yes—
120AGL(1,𝔽5)—44no—
121C4×C5—21no—
122C4×C5—41no—
123C4×C5#9912yes—
124Dic5—21no—
125Dic5—21no—
126Dic5—41no—
127Dic5—41no—
128C24—51no—
129#41—42no—
130C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
131C2×D4—21no—
132C2×D4—101no—
133C2×D4—101no—
134C2×D4C2×AGL(1,𝔽5)11yescharacteristic
135C22×C4—21no—
136C22×C4—51no—
137C22×C4—51no—
138C22×C4—51no—
139C22×C4—51no—
140C22×C4—51no—
141C22×C4—102no—
142C22×C4—202no—
143C22×C4C2×AGL(1,𝔽5)11yescharacteristic
144C2×C8—102no—
145C2×C8—202no—
146C4⋊C4—51no—
147C4⋊C4—51no—
148C4⋊C4—101no—
149C4⋊C4—101no—
150C22⋊C4—101no—
151C22⋊C4—101no—
152C22⋊C4—101no—
153C22⋊C4—101no—
154C22⋊C4—104no—
155C22⋊C4—101no—
156C42—101no—
157C42—202no—
158C2×C5—41no—
159C2×C5—81no—
160C2×C5C2×#9911yescharacteristic
161C2×C5#41611yescharacteristic
162C2×C5C2×#10411yescharacteristic
163D5—22no—
164D5—22no—
165C23—21no—
166C23—51no—
167C23—51no—
168C23—102no—
169C23—102no—
170C23—201no—
171C23C22⋊AGL(1,𝔽5)11yescharacteristic
172Q8—41no—
173D4—41no—
174D4—42no—
175D4—202no—
176D4—202no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—51no—
181C2×C4—51no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—104no—
185C2×C4—101no—
186C2×C4—101no—
187C2×C4—201no—
188C2×C4—201no—
189C2×C4—201no—
190C2×C4—202no—
191C2×C4—202no—
192C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
193C2×C4C22×AGL(1,𝔽5)11yescharacteristic
194C8—202no—
195C5#150111yescharacteristic, complemented, Sylow
196C22—21no—
197C22—21no—
198C22—21no—
199C22—41no—
200C22—41no—
201C22—41no—
202C22—52no—
203C22—52no—
204C22—101no—
205C22—101no—
206C22—101no—
207C22—101no—
208C22—204no—
209C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
210C4—21no—
211C4—41no—
212C4—101no—
213C4—101no—
214C4—204no—
215C4—201no—
216C4—201no—
217C4#401112yes—
218C2—41no—
219C2—81no—
220C2—102no—
221C2—102no—
222C2D5⋊#10411yescharacteristic
223C2#6579411yescharacteristic
224C2C2×#401111yescharacteristic
subgroupG/H
centerC2×C4C22⋊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#1501
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊AGL(1,𝔽5)C2semidirect
3. D5⋊(C22⋊C8)C2semidirect
4. #64926C2semidirect
5. #64920C2semidirect
6. C2×C5×#41C4semidirect
11. #4092C4semidirect
13. #4084C4semidirect
19. #4072C4semidirect
89. C2×#41AGL(1,𝔽5)semidirect
195. C5#1501semidirect

Maximal subgroups

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

subgroupindex
#661622
C4×C22⋊AGL(1,𝔽5)2
D5⋊(C22⋊C8)2
#649262
#649202
#15015