Finite Group Databasebeta

← #10604911

Subgroups — #10604911

Order 640.

346 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 64 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×#4084C211yesmaximal, characteristic
2C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3#65756C211yesmaximal, characteristic, complemented
4#64920C214yesmaximal, complemented
5C22×C5×D4C411yescharacteristic, complemented
6C22×C22⋊D5—21no—
7C22×C4×D5C2211yescharacteristic
8C2×C22⋊AGL(1,𝔽5)—21no—
9C2×C4×AGL(1,𝔽5)—21no—
10C2×#700—21no—
11C2×C22⋊Dic5—21no—
12#4084—21no—
13#4084—21no—
14#4084C412yescomplemented
15#4084C2212yes—
16C2×C4⋊Dic5C411yescharacteristic, complemented
17C22⋊#170—21no—
18(C2×AGL(1,𝔽5))⋊C4C2212yescomplemented
19#4003C2212yescomplemented
20#1507—51nomaximal, Sylow
21C24×C5—21no—
22C23×D5D411yescharacteristic
23C22×AGL(1,𝔽5)—22no—
24C2×C5×D4—21no—
25C2×C5×D4—42no—
26C2×C5×D4C2×C412yes—
27C22×C4×C5C2×C411yescharacteristic
28C2×C22⋊D5—22no—
29C2×C22⋊D5—22no—
30C2×C22⋊D5—44no—
31C22×Dic5—21no—
32C22×Dic5D411yescharacteristic
33C2×C4×D5—42no—
34C2×C4×D5C2×C411yescharacteristic
35C2×C4×D5D411yescharacteristic
36C2×C4×D5D411yescharacteristic
37C2×C4×D5C2311yescharacteristic
38C22⋊AGL(1,𝔽5)—24no—
39C2×#170—22no—
40C4×AGL(1,𝔽5)—42no—
41#700—42no—
42C22⋊Dic5—22no—
43C22⋊Dic5—22no—
44C4⋊Dic5—21no—
45C4⋊Dic5C2×C412yes—
46C2×#121—51no—
47#407—51no—
48C4×C22⋊C4—51no—
49#327—54no—
50C23×C5—22no—
51C23×C5—42no—
52C23×C5—42no—
53C23×C5—42no—
54C23×C5C22⋊C411yescharacteristic
55C22×D5—22no—
56C22×D5—22no—
57C22×D5—41no—
58C22×D5C22⋊C411yescharacteristic
59C22×D5C2×D411yescharacteristic
60C2×AGL(1,𝔽5)—24no—
61C2×AGL(1,𝔽5)—42no—
62C2×AGL(1,𝔽5)—42no—
63C5×D4—42no—
64C5×D4—42no—
65C2×C4×C5—22no—
66C2×C4×C5C22⋊C411yescharacteristic
67C2×C4×C5C22×C411yescharacteristic
68C22⋊D5—44no—
69C22⋊D5—44no—
70C2×Dic5—22no—
71C2×Dic5—41no—
72C2×Dic5—42no—
73C2×Dic5C22⋊C411yescharacteristic
74C2×Dic5C2×D411yescharacteristic
75C4×D5—22no—
76C4×D5—22no—
77C4×D5—41no—
78C4×D5—41no—
79#170—42no—
80C22×D4—101no—
81C22×D4AGL(1,𝔽5)11yescharacteristic, complemented
82C23×C4—51no—
83C2×M16—101no—
84#121—52no—
85#121—52no—
86#121—101no—
87#121—101no—
88C2×C4⋊C4—51no—
89C2×C22⋊C4—101no—
90C2×C22⋊C4—101no—
91C2×C42—101no—
92C22⋊C8—52no—
93C22⋊C8—101no—
94#95—52no—
95C22×C5—21no—
96C22×C5—21no—
97C22×C5—42no—
98C22×C5—42no—
99C22×C5—42no—
100C22×C5—84no—
101C22×C5#10412yes—
102C22×C5C2×C22⋊C411yescharacteristic
103C2×D5—21no—
104C2×D5—21no—
105C2×D5—21no—
106C2×D5—21no—
107C2×D5—21no—
108C2×D5—44no—
109C2×D5AGL(1,ℤ/8)12yes—
110AGL(1,𝔽5)—44no—
111C4×C5—21no—
112C4×C5#9912yes—
113Dic5—21no—
114Dic5—21no—
115Dic5—42no—
116C24—21no—
117C24—51no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—102no—
121C2×D4—102no—
122C2×D4—204no—
123C2×D4C2×AGL(1,𝔽5)12yes—
124C22×C4—51no—
125C22×C4—51no—
126C22×C4—51no—
127C22×C4—51no—
128C22×C4—51no—
129C22×C4—102no—
130C22×C4—101no—
131C22×C4—202no—
132C22×C4C2×AGL(1,𝔽5)11yescharacteristic
133M16—202no—
134C2×C8—102no—
135C4⋊C4—52no—
136C4⋊C4—101no—
137C22⋊C4—102no—
138C22⋊C4—102no—
139C22⋊C4—104no—
140C42—202no—
141C2×C5—22no—
142C2×C5—82no—
143C2×C5C2×#9911yescharacteristic
144C2×C5#41811yescharacteristic
145C2×C5C2×#10411yescharacteristic
146D5—22no—
147D5—41no—
148C23—22no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—51no—
153C23—51no—
154C23—102no—
155C23—102no—
156C23—201no—
157C23C22⋊AGL(1,𝔽5)11yescharacteristic
158D4—42no—
159D4—42no—
160D4—204no—
161D4—204no—
162C2×C4—22no—
163C2×C4—51no—
164C2×C4—51no—
165C2×C4—102no—
166C2×C4—102no—
167C2×C4—104no—
168C2×C4—102no—
169C2×C4—201no—
170C2×C4—201no—
171C2×C4—201no—
172C2×C4—202no—
173C2×C4—202no—
174C2×C4—202no—
175C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
176C2×C4C22×AGL(1,𝔽5)11yescharacteristic
177C8—202no—
178C5#150711yescharacteristic, complemented, Sylow
179C22—21no—
180C22—21no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—52no—
185C22—84no—
186C22—101no—
187C22—101no—
188C22—101no—
189C22—101no—
190C22—101no—
191C22—204no—
192C22#400812yes—
193C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
194C4—21no—
195C4—101no—
196C4—101no—
197C4—204no—
198C4—202no—
199C4#401112yes—
200C2—22no—
201C2—82no—
202C2—102no—
203C2—201no—
204C2#6576711yescharacteristic
205C2C2×#400811yescharacteristic
206C2C2×#401111yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC22×C5×D4C4
socleC22×C5C2×C22⋊C4
prime radicalC2×#4084C2
cosocleC2×C4×D5C23
hypercenterC22×D4AGL(1,𝔽5)
hypocenterC5#1507
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊AGL(1,𝔽5)C2semidirect
3. #65756C2semidirect
4. #64920C2semidirect
5. C22×C5×D4C4semidirect
14. #4084C4semidirect
16. C2×C4⋊Dic5C4semidirect
18. (C2×AGL(1,𝔽5))⋊C4C22semidirect
19. #4003C22semidirect
81. C22×D4AGL(1,𝔽5)semidirect
178. C5#1507semidirect

Maximal subgroups

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

subgroupindex
C2×#40842
C4×C22⋊AGL(1,𝔽5)2
#657562
#649202
#15075