Finite Group Databasebeta

← #10604516

Subgroups — #10604516

Order 640.

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

normal subgroups only

This group has 202 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#65956C211yesmaximal, characteristic
2#65759C211yesmaximal, characteristic
3#65748C211yesmaximal, characteristic
4Q8⋊#170C212yesmaximal, complemented
5#64872C212yesmaximal, complemented
6C5×C22⋊Q8C411yescharacteristic
7Q8×Dic5C2211yescharacteristic
8#4090C411yescharacteristic
9#4070C411yescharacteristic
10C2×C4×Dic5C2211yescharacteristic
11#4035—21no—
12#4034C411yescharacteristic
13#4033—21no—
14#4033C2211yescharacteristic
15#4024—21no—
16#4023—21no—
17C22⋊#170—21no—
18C22⋊#170—21no—
19#4002—21no—
20C4⋊#170—21no—
21C4⋊#170C2212yes—
22C4×#170C2212yes—
23#1497—51nomaximal, Sylow
24C2×C5×Q8C2×C411yescharacteristic
25C22×C4×C5C2×C411yescharacteristic
26C22×Dic5D411yescharacteristic
27C22×Dic5D411yescharacteristic
28C2×Dic10—21no—
29C2×Dic10C2×C411yescharacteristic
30C2×#170—22no—
31C2×#170—22no—
32C5×C4⋊C4—21no—
33C5×C4⋊C4C2×C411yescharacteristic
34C5×C22⋊C4—21no—
35C22⋊Dic5—21no—
36C4⋊Dic5—21no—
37C4⋊Dic5C2×C411yescharacteristic
38#683—21no—
39#683—21no—
40#683—21no—
41#683—21no—
42#683—21no—
43C4×Dic5—21no—
44C4×Dic5—21no—
45C4×Dic5C2×C411yescharacteristic
46C4×Dic5D411yescharacteristic
47C4×Dic5D411yescharacteristic
48C4×Dic5C2311yescharacteristic
49#533—51no—
50#432—51no—
51#431—51no—
52Q8⋊C8—52no—
53Q8⋊C8—52no—
54C23×C5C22⋊C411yescharacteristic
55C5×Q8—22no—
56C2×C4×C5—22no—
57C2×C4×C5—21no—
58C2×C4×C5—21no—
59C2×C4×C5C22⋊C411yescharacteristic
60C2×C4×C5C22×C411yescharacteristic
61C2×Dic5—22no—
62C2×Dic5—21no—
63C2×Dic5—21no—
64C2×Dic5—41no—
65C2×Dic5C22⋊C411yescharacteristic
66C2×Dic5C22⋊C411yescharacteristic
67C2×Dic5C2×D411yescharacteristic
68C2×Dic5C2×D411yescharacteristic
69Dic10—22no—
70Dic10—42no—
71#170—42no—
72#170—42no—
73C4⋊Q8—51no—
74C4⋊Q8—51no—
75#125—101no—
76C22⋊Q8—51no—
77C22⋊Q8—101no—
78C22⋊Q8AGL(1,𝔽5)11yescharacteristic
79C4×Q8—51no—
80C4×Q8—51no—
81C4×Q8—101no—
82#117—101no—
83C2×C42—51no—
84C4⋊C8—52no—
85C4⋊C8—101no—
86C22⋊C8—101no—
87C22⋊C8—101no—
88#97—101no—
89C4×C8—52no—
90C22×C5—21no—
91C22×C5—21no—
92C22×C5—21no—
93C22×C5C2×C22⋊C411yescharacteristic
94C4×C5—21no—
95C4×C5—41no—
96C4×C5—41no—
97C4×C5#13112yes—
98Dic5—21no—
99Dic5—21no—
100Dic5—21no—
101Dic5—41no—
102Dic5—41no—
103Dic5#13712yes—
104C2×Q8—51no—
105C2×Q8—101no—
106C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
107C22×C4—51no—
108C22×C4—51no—
109C22×C4C2×AGL(1,𝔽5)11yescharacteristic
110C2×C8—102no—
111C2×C8—102no—
112C4⋊C4—21no—
113C4⋊C4—51no—
114C4⋊C4—101no—
115C4⋊C4—101no—
116C4⋊C4—101no—
117C4⋊C4—101no—
118C4⋊C4—101no—
119C4⋊C4—101no—
120C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
121C22⋊C4—21no—
122C22⋊C4—101no—
123C42—51no—
124C42—51no—
125C42—51no—
126C42—51no—
127C42—101no—
128C42—101no—
129C2×C5—41no—
130C2×C5#40811yescharacteristic
131C2×C5#41911yescharacteristic
132C2×C5#42111yescharacteristic
133C23C22⋊AGL(1,𝔽5)11yescharacteristic
134Q8—22no—
135Q8—102no—
136Q8—202no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—51no—
141C2×C4—51no—
142C2×C4—51no—
143C2×C4—51no—
144C2×C4—102no—
145C2×C4—101no—
146C2×C4—101no—
147C2×C4—201no—
148C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
149C2×C4C22×AGL(1,𝔽5)11yescharacteristic
150C8—202no—
151C8—202no—
152C5#149711yescharacteristic, complemented, Sylow
153C22—21no—
154C22—21no—
155C22—21no—
156C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
157C4—21no—
158C4—41no—
159C4—41no—
160C4—52no—
161C4—101no—
162C4—101no—
163C4—101no—
164C4—201no—
165C4—201no—
166C4#413112yes—
167C2—41no—
168C2#6577511yescharacteristic
169C2#6578211yescharacteristic
170C2#6579511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C22⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radicalC23×C5C22⋊C4
cosocleC4×Dic5C23
hypercenterC22⋊Q8AGL(1,𝔽5)
hypocenterC5#1497
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. Q8⋊#170C2semidirect
5. #64872C2semidirect
152. C5#1497semidirect

Maximal subgroups

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

subgroupindex
#659562
#657592
#657482
Q8⋊#1702
#648722
#14975