Finite Group Databasebeta

← #10604499

Subgroups — #10604499

Order 640.

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

normal subgroups only

This group has 334 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
2C2×(C2×AGL(1,𝔽5))⋊C4C211yesmaximal, characteristic, complemented
3#65756C211yesmaximal, characteristic, complemented
4#64926C212yesmaximal, complemented
5#64868C212yesmaximal, complemented
6C23×AGL(1,𝔽5)—21no—
7C2×#713C411yescharacteristic, complemented
8C22×C4×D5C2211yescharacteristic
9C2×#700—21no—
10C5×C22⋊Q8C411yescharacteristic, complemented
11#4092C2211yescharacteristic
12#4076C411yescharacteristic, complemented
13#4041C2211yescharacteristic
14#4040—21no—
15#4039—21no—
16#4038C411yescharacteristic, complemented
17#4028—21no—
18#4027—21no—
19C22⋊#170—21no—
20(C2×AGL(1,𝔽5))⋊C4—21no—
21(C2×AGL(1,𝔽5))⋊C4C2212yes—
22#4003C2212yes—
23#1504—51nomaximal, Sylow
24C23×D5D411yescharacteristic
25C22×AGL(1,𝔽5)—24no—
26C22×AGL(1,𝔽5)—22no—
27C22×AGL(1,𝔽5)—42no—
28C22×AGL(1,𝔽5)—42no—
29C2×C5×Q8C2×C411yescharacteristic, complemented
30C22×C4×C5C2×C411yescharacteristic
31C2×C22⋊D5—21no—
32C22×Dic5D411yescharacteristic
33#713—42no—
34C2×D20—21no—
35C2×D20—21no—
36C2×D20C2×C411yescharacteristic, complemented
37C2×C4×D5—21no—
38C2×C4×D5—42no—
39C2×C4×D5C2×C411yescharacteristic
40C2×C4×D5D411yescharacteristic
41C2×C4×D5D411yescharacteristic
42C2×C4×D5C2311yescharacteristic
43C2×#170—22no—
44#700—42no—
45C5×C4⋊C4—21no—
46C5×C4⋊C4C2×C411yescharacteristic, complemented
47C5×C22⋊C4—21no—
48#685—21no—
49#685—21no—
50#685—21no—
51#685—21no—
52C4⋊Dic5C2×C411yescharacteristic, complemented
53#683—21no—
54C4×Dic5—21no—
55#525—51no—
56#407—51no—
57C2×#95—51no—
58#328—52no—
59#327—52no—
60C23×C5C22⋊C411yescharacteristic
61C22×D5—22no—
62C22×D5—22no—
63C22×D5—21no—
64C22×D5—21no—
65C22×D5—21no—
66C22×D5C22⋊C411yescharacteristic
67C22×D5C2×D411yescharacteristic
68C2×AGL(1,𝔽5)—24no—
69C2×AGL(1,𝔽5)—48no—
70C2×AGL(1,𝔽5)—42no—
71C2×AGL(1,𝔽5)—42no—
72C5×Q8—41no—
73C2×C4×C5—21no—
74C2×C4×C5—21no—
75C2×C4×C5—41no—
76C2×C4×C5C22⋊C411yescharacteristic
77C2×C4×C5C22×C411yescharacteristic
78C22⋊D5—42no—
79C2×Dic5—21no—
80C2×Dic5—41no—
81C2×Dic5C22⋊C411yescharacteristic
82C2×Dic5C2×D411yescharacteristic
83D20—41no—
84D20—42no—
85D20—41no—
86D20—41no—
87C4×D5—22no—
88C4×D5—22no—
89C4×D5—41no—
90C4×D5—41no—
91C4×D5—42no—
92#170—42no—
93C2×#41—51no—
94C23×C4—51no—
95C23×C4—101no—
96C2×M16—101no—
97#123—101no—
98C22⋊Q8—51no—
99C22⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
100#121—51no—
101#121—51no—
102#120—101no—
103C4×D4—101no—
104C4×D4—101no—
105#117—51no—
106C22⋊C8—52no—
107C22⋊C8—101no—
108#95—52no—
109#95—101no—
110C22×C5—21no—
111C22×C5—21no—
112C22×C5#10412yes—
113C22×C5C2×C22⋊C411yescharacteristic
114C2×D5—24no—
115C2×D5—24no—
116C2×D5—21no—
117C2×D5—21no—
118C2×D5—21no—
119C2×D5—21no—
120C2×D5—41no—
121C2×D5—41no—
122C2×D5—41no—
123C2×D5#9912yes—
124C2×D5AGL(1,ℤ/8)11yescharacteristic
125C2×D5#13711yescharacteristic
126AGL(1,𝔽5)—44no—
127C4×C5—21no—
128C4×C5—21no—
129C4×C5—41no—
130C4×C5—41no—
131Dic5—21no—
132Dic5—21no—
133Dic5—41no—
134C24—51no—
135#41—202no—
136C2×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
137C2×D4—51no—
138C2×D4—101no—
139C2×D4—101no—
140C2×D4—101no—
141C22×C4—51no—
142C22×C4—51no—
143C22×C4—51no—
144C22×C4—51no—
145C22×C4—51no—
146C22×C4—101no—
147C22×C4—102no—
148C22×C4—104no—
149C22×C4—202no—
150C22×C4—202no—
151C22×C4—202no—
152C22×C4C2×AGL(1,𝔽5)11yescharacteristic
153M16—202no—
154C2×C8—102no—
155C4⋊C4—21no—
156C4⋊C4—51no—
157C4⋊C4—101no—
158C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
159C22⋊C4—21no—
160C22⋊C4—101no—
161C22⋊C4—101no—
162C22⋊C4—101no—
163C22⋊C4—101no—
164C42—101no—
165C2×C5—22no—
166C2×C5C2×#9911yescharacteristic
167C2×C5#41711yescharacteristic
168C2×C5C2×#10411yescharacteristic
169D5—21no—
170D5—22no—
171D5—21no—
172D5—81no—
173C23—51no—
174C23—51no—
175C23—101no—
176C23—102no—
177C23—101no—
178C23—102no—
179C23—101no—
180C23C22⋊AGL(1,𝔽5)11yescharacteristic
181Q8—41no—
182D4—201no—
183D4—202no—
184D4—201no—
185D4—201no—
186D4—202no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—51no—
191C2×C4—51no—
192C2×C4—102no—
193C2×C4—102no—
194C2×C4—104no—
195C2×C4—101no—
196C2×C4—201no—
197C2×C4—201no—
198C2×C4—202no—
199C2×C4—201no—
200C2×C4—202no—
201C2×C4—208no—
202C2×C4—202no—
203C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
204C2×C4C22×AGL(1,𝔽5)11yescharacteristic
205C8—202no—
206C5#150411yescharacteristic, complemented, Sylow
207C22—21no—
208C22—21no—
209C22—51no—
210C22—51no—
211C22—52no—
212C22—104no—
213C22—104no—
214C22—101no—
215C22—101no—
216C22—101no—
217C22—101no—
218C22—201no—
219C22—201no—
220C22—201no—
221C22#401012yes—
222C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—101no—
228C4—101no—
229C4—204no—
230C4—201no—
231C2—22no—
232C2—101no—
233C2—101no—
234C2—102no—
235C2—401no—
236C2D5⋊#9911yescharacteristic
237C2C2×#401011yescharacteristic
238C2#6579311yescharacteristic
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#1504
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. C2×(C2×AGL(1,𝔽5))⋊C4C2semidirect
3. #65756C2semidirect
4. #64926C2semidirect
5. #64868C2semidirect
7. C2×#713C4semidirect
10. C5×C22⋊Q8C4semidirect
12. #4076C4semidirect
16. #4038C4semidirect
29. C2×C5×Q8C2×C4semidirect
36. C2×D20C2×C4semidirect
46. C5×C4⋊C4C2×C4semidirect
52. C4⋊Dic5C2×C4semidirect
99. C22⋊Q8AGL(1,𝔽5)semidirect
136. C2×Q8C2×AGL(1,𝔽5)semidirect
158. C4⋊C4C2×AGL(1,𝔽5)semidirect
206. C5#1504semidirect

Maximal subgroups

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

subgroupindex
#659612
C2×(C2×AGL(1,𝔽5))⋊C42
#657562
#649262
#648682
#15045