Finite Group Databasebeta

← #10604552

Subgroups — #10604552

Order 640.

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

normal subgroups only

This group has 212 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#66014C211yesmaximal, characteristic
2#65680C211yesmaximal, characteristic, complemented
3#65680C211yesmaximal, characteristic, complemented
4Q8⋊#170C212yesmaximal, complemented
5D4⋊#170C212yesmaximal, complemented
6C5×#124C411yescharacteristic
7#4092C411yescharacteristic
8Q8×Dic5C2211yescharacteristic
9#4084C411yescharacteristic
10D4×Dic5C2211yescharacteristic
11#4029—21no—
12#4027—21no—
13#4025—21no—
14#4023—21no—
15C42×D5C2211yescharacteristic
16#4016C411yescharacteristic
17#4003—21no—
18#4003—21no—
19C4⋊#170—21no—
20C4⋊#170C2212yes—
21C4×#170—21no—
22C4×#170C2212yes—
23#1482—51nomaximal, Sylow
24C2×C5×Q8C2×C411yescharacteristic
25C2×C5×D4C2×C411yescharacteristic
26C2×C22⋊D5—21no—
27C22×Dic5—21no—
28C2×C4×D5C2×C411yescharacteristic
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×#170—22no—
32C2×#170—22no—
33C5×C22⋊C4—21no—
34C5×C22⋊C4—21no—
35C42×C5C2×C411yescharacteristic
36C22⋊Dic5—21no—
37C22⋊Dic5—21no—
38#685—21no—
39C4⋊Dic5—21no—
40C4⋊Dic5C2×C411yescharacteristic
41C4⋊Dic5C2×C411yescharacteristic
42#683—21no—
43#683—21no—
44#683—21no—
45C4×Dic5—21no—
46C4×Dic5D411yescharacteristic
47C4×Dic5D411yescharacteristic
48C4×Dic5C2311yescharacteristic
49#529—51no—
50#431—51no—
51#431—51no—
52Q8⋊C8—52no—
53D4⋊C8—52no—
54C23×C5—21no—
55C22×D5C22⋊C411yescharacteristic
56C5×Q8—22no—
57C5×D4—22no—
58C2×C4×C5—21no—
59C2×C4×C5C22⋊C411yescharacteristic
60C2×C4×C5C22⋊C411yescharacteristic
61C2×C4×C5C22×C411yescharacteristic
62C22⋊D5—42no—
63C2×Dic5—21no—
64C2×Dic5—21no—
65C2×Dic5—42no—
66C2×Dic5C22⋊C411yescharacteristic
67C2×Dic5C2×D411yescharacteristic
68C2×Dic5C2×D411yescharacteristic
69C4×D5—22no—
70C4×D5—22no—
71C4×D5—41no—
72#170—42no—
73#170—42no—
74#126—101no—
75#125—51no—
76#124AGL(1,𝔽5)11yescharacteristic
77#123—101no—
78C22⋊Q8—51no—
79#121—51no—
80C4×Q8—51no—
81C4×D4—51no—
82C4×D4—101no—
83#117—101no—
84C2×C42—51no—
85C4⋊C8—52no—
86C4⋊C8—101no—
87C22⋊C8—101no—
88C22⋊C8—101no—
89C4×C8—52no—
90C4×C8—101no—
91C22×C5—41no—
92C22×C5—41no—
93C22×C5—41no—
94C22×C5C2×C22⋊C411yescharacteristic
95C2×D5—21no—
96C2×D5—21no—
97C2×D5—21no—
98C4×C5—21no—
99C4×C5—21no—
100C4×C5—41no—
101C4×C5#13112yes—
102Dic5—21no—
103Dic5—21no—
104Dic5—41no—
105Dic5—41no—
106Dic5#13512yes—
107C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
108C2×D4—101no—
109C2×D4C2×AGL(1,𝔽5)11yescharacteristic
110C22×C4—51no—
111C22×C4—51no—
112C22×C4—51no—
113C22×C4—101no—
114C2×C8—102no—
115C2×C8—102no—
116C4⋊C4—51no—
117C4⋊C4—51no—
118C4⋊C4—101no—
119C4⋊C4—101no—
120C4⋊C4—101no—
121C4⋊C4—101no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—101no—
125C22⋊C4—101no—
126C22⋊C4—101no—
127C42—51no—
128C42—51no—
129C42—51no—
130C42—101no—
131C42C2×AGL(1,𝔽5)11yescharacteristic
132C2×C5—81no—
133C2×C5#40811yescharacteristic
134C2×C5#41611yescharacteristic
135C2×C5#42111yescharacteristic
136D5—41no—
137C23—21no—
138C23—51no—
139Q8—22no—
140D4—22no—
141D4—202no—
142C2×C4—21no—
143C2×C4—51no—
144C2×C4—51no—
145C2×C4—51no—
146C2×C4—102no—
147C2×C4—102no—
148C2×C4—101no—
149C2×C4—101no—
150C2×C4—201no—
151C2×C4—202no—
152C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
153C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
154C2×C4C22×AGL(1,𝔽5)11yescharacteristic
155C8—202no—
156C8—202no—
157C5#148211yescharacteristic, complemented, Sylow
158C22—41no—
159C22—41no—
160C22—41no—
161C22—101no—
162C22—101no—
163C22—101no—
164C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
165C4—21no—
166C4—21no—
167C4—41no—
168C4—52no—
169C4—101no—
170C4—101no—
171C4—201no—
172C4—201no—
173C4#413112yes—
174C2—81no—
175C2—201no—
176C2#6577611yescharacteristic
177C2#6579411yescharacteristic
178C2#6579511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#124C4
socleC22×C5C2×C22⋊C4
prime radical#4084C4
cosocleC4×Dic5C23
hypercenter#124AGL(1,𝔽5)
hypocenterC5#1482
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #65680C2semidirect
3. #65680C2semidirect
4. Q8⋊#170C2semidirect
5. D4⋊#170C2semidirect
157. C5#1482semidirect

Maximal subgroups

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

subgroupindex
#660142
#656802
Q8⋊#1702
D4⋊#1702
#14825