Finite Group Databasebeta

← #10604498

Subgroups — #10604498

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#65961C211yesmaximal, characteristic
2C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C2×#4003C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64868C212yesmaximal, complemented
6C2×#713C411yescharacteristic, complemented
7C22×C4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—21no—
9C22×#170—21no—
10C2×C4×AGL(1,𝔽5)—21no—
11C5×C22⋊Q8C411yescharacteristic, complemented
12#4092C2211yescharacteristic
13#4076C411yescharacteristic, complemented
14#4041C2211yescharacteristic
15#4040—21no—
16#4039—21no—
17#4038C411yescharacteristic, complemented
18#4028—21no—
19#4027—21no—
20(C2×AGL(1,𝔽5))⋊C4C2212yes—
21#4003—21no—
22#4003C2212yes—
23#1501—51nomaximal, Sylow
24C23×D5D411yescharacteristic
25C22×AGL(1,𝔽5)—22no—
26C2×C5×Q8C2×C411yescharacteristic
27C22×C4×C5C2×C411yescharacteristic
28C2×C22⋊D5—21no—
29C22×Dic5D411yescharacteristic
30#713—42no—
31C2×D20—21no—
32C2×D20—21no—
33C2×D20C2×C411yescharacteristic
34C2×C4×D5—21no—
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—
42C2×#170—42no—
43C4×AGL(1,𝔽5)—42no—
44C5×C4⋊C4—21no—
45C5×C4⋊C4C2×C411yescharacteristic
46C5×C22⋊C4—21no—
47#685—21no—
48#685—21no—
49#685—21no—
50#685—21no—
51C4⋊Dic5C2×C411yescharacteristic
52#683—21no—
53C4×Dic5—21no—
54#525—51no—
55C2×C22⋊C8—51no—
56C4×C22⋊C4—51no—
57#328—52no—
58#327—52no—
59C23×C5C22⋊C411yescharacteristic
60C22×D5—22no—
61C22×D5—22no—
62C22×D5—21no—
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—
70C2×C4×C5—21no—
71C2×C4×C5—21no—
72C2×C4×C5—41no—
73C2×C4×C5C22⋊C411yescharacteristic
74C2×C4×C5C22×C411yescharacteristic
75C22⋊D5—42no—
76C2×Dic5—22no—
77C2×Dic5—21no—
78C2×Dic5C22⋊C411yescharacteristic
79C2×Dic5C2×D411yescharacteristic
80D20—41no—
81D20—42no—
82D20—42no—
83C4×D5—22no—
84C4×D5—22no—
85C4×D5—41no—
86C4×D5—41no—
87C4×D5—42no—
88#170—42no—
89C2×#41—51no—
90C23×C4—51no—
91C22×C8—101no—
92#123—101no—
93C22⋊Q8—51no—
94C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
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#10412yes—
109C22×C5C2×C22⋊C411yescharacteristic
110C2×D5—21no—
111C2×D5—21no—
112C2×D5—21no—
113C2×D5—21no—
114C2×D5—21no—
115C2×D5—44no—
116C2×D5—41no—
117C2×D5—41no—
118C2×D5—41no—
119C2×D5#13512yes—
120AGL(1,𝔽5)—44no—
121C4×C5—21no—
122C4×C5—21no—
123C4×C5—41no—
124C4×C5—41no—
125Dic5—21no—
126Dic5—41no—
127Dic5#9912yes—
128C24—51no—
129#41—202no—
130C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
131C2×D4—51no—
132C2×D4—101no—
133C2×D4—101no—
134C2×D4—101no—
135C22×C4—51no—
136C22×C4—51no—
137C22×C4—51no—
138C22×C4—51no—
139C22×C4—51no—
140C22×C4—101no—
141C22×C4—102no—
142C22×C4—202no—
143C22×C4C2×AGL(1,𝔽5)11yescharacteristic
144C2×C8—102no—
145C2×C8—202no—
146C4⋊C4—21no—
147C4⋊C4—51no—
148C4⋊C4—101no—
149C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
150C22⋊C4—21no—
151C22⋊C4—101no—
152C22⋊C4—101no—
153C22⋊C4—101no—
154C22⋊C4—104no—
155C22⋊C4—101no—
156C42—101no—
157C42—202no—
158C2×C5—22no—
159C2×C5C2×#9911yescharacteristic
160C2×C5#41611yescharacteristic
161C2×C5C2×#10411yescharacteristic
162D5—22no—
163D5—41no—
164D5—81no—
165C23—51no—
166C23—51no—
167C23—102no—
168C23—102no—
169C23—101no—
170C23—201no—
171C23C22⋊AGL(1,𝔽5)11yescharacteristic
172Q8—41no—
173D4—201no—
174D4—202no—
175D4—202no—
176D4—202no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—51no—
181C2×C4—51no—
182C2×C4—102no—
183C2×C4—102no—
184C2×C4—102no—
185C2×C4—104no—
186C2×C4—101no—
187C2×C4—201no—
188C2×C4—201no—
189C2×C4—202no—
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—52no—
199C22—101no—
200C22—101no—
201C22—101no—
202C22—101no—
203C22—101no—
204C22—204no—
205C22—201no—
206C22—201no—
207C22—201no—
208C22#401012yes—
209C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—52no—
215C4—101no—
216C4—204no—
217C4—201no—
218C2—22no—
219C2—102no—
220C2—201no—
221C2—401no—
222C2D5⋊#9911yescharacteristic
223C2C2×#401011yescharacteristic
224C2#6579411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C22⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radical#65961C2
cosocleC2×C4×D5C23
hypercenterC22⋊Q8AGL(1,𝔽5)
hypocenterC5#1501
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊AGL(1,𝔽5)C2semidirect
3. C2×#4003C2semidirect
4. #64926C2semidirect
5. #64868C2semidirect
6. C2×#713C4semidirect
11. C5×C22⋊Q8C4semidirect
13. #4076C4semidirect
17. #4038C4semidirect
94. C22⋊Q8AGL(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
#659612
C4×C22⋊AGL(1,𝔽5)2
C2×#40032
#649262
#648682
#15015