Finite Group Databasebeta

← #10604484

Subgroups — #10604484

Order 640.

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

normal subgroups only

This group has 246 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#65930C211yesmaximal, characteristic
2#65759C211yesmaximal, characteristic, complemented
3#65748C211yesmaximal, characteristic, complemented
4D4⋊#170C212yesmaximal, complemented
5#64871C212yesmaximal, complemented
6C2×#711—21no—
7C5×#121C411yescharacteristic
8#4086C411yescharacteristic
9#4085—21no—
10D4×Dic5C2211yescharacteristic
11#4076C411yescharacteristic
12C2×C4×Dic5C2211yescharacteristic
13#4039C2211yescharacteristic
14#4034C411yescharacteristic
15#4031—21no—
16#4027—21no—
17C22⋊#170—21no—
18C22⋊#170—21no—
19#4002—21no—
20C4⋊#170—21no—
21C4⋊#170C2212yes—
22C4×#170C2212yes—
23#1496—51nomaximal, Sylow
24C2×C5×D4—21no—
25C2×C5×D4C2×C411yescharacteristic
26C22×C4×C5C2×C411yescharacteristic
27C2×C22⋊D5—21no—
28C2×C22⋊D5—21no—
29C2×C22⋊D5—21no—
30C22×Dic5—21no—
31C22×Dic5D411yescharacteristic
32C22×Dic5D411yescharacteristic
33#711—44no—
34C2×D20C2×C411yescharacteristic
35C2×C4×D5—21no—
36C2×Dic10—21no—
37C2×#170—22no—
38C2×#170—22no—
39C5×C4⋊C4C2×C411yescharacteristic
40C5×C22⋊C4—21no—
41C22⋊Dic5—21no—
42#685—21no—
43#685—21no—
44C4⋊Dic5C2×C411yescharacteristic
45#683—21no—
46C4×Dic5C2×C411yescharacteristic
47C4×Dic5D411yescharacteristic
48C4×Dic5D411yescharacteristic
49C4×Dic5C2311yescharacteristic
50#532—51no—
51#432—51no—
52#431—51no—
53D4⋊C8—52no—
54D4⋊C8—52no—
55C23×C5—21no—
56C23×C5C22⋊C411yescharacteristic
57C22×D5—21no—
58C5×D4—22no—
59C5×D4—42no—
60C2×C4×C5—22no—
61C2×C4×C5—21no—
62C2×C4×C5C22⋊C411yescharacteristic
63C2×C4×C5C22×C411yescharacteristic
64C22⋊D5—42no—
65C22⋊D5—42no—
66C22⋊D5—42no—
67C2×Dic5—22no—
68C2×Dic5—21no—
69C2×Dic5—41no—
70C2×Dic5—42no—
71C2×Dic5C22⋊C411yescharacteristic
72C2×Dic5C22⋊C411yescharacteristic
73C2×Dic5C2×D411yescharacteristic
74C2×Dic5C2×D411yescharacteristic
75D20—22no—
76C4×D5—42no—
77Dic10—42no—
78#170—42no—
79#170—42no—
80C2×#41—101no—
81C4⋊Q8—51no—
82#127—51no—
83#124—101no—
84#121—51no—
85#121—101no—
86#121AGL(1,𝔽5)11yescharacteristic
87C4×D4—51no—
88C4×D4—51no—
89C4×D4—101no—
90C2×C42—51no—
91C4⋊C8—52no—
92C4⋊C8—101no—
93C22⋊C8—101no—
94C22⋊C8—101no—
95#97—101no—
96C4×C8—52no—
97C22×C5—21no—
98C22×C5—21no—
99C22×C5—21no—
100C22×C5—41no—
101C22×C5—41no—
102C22×C5—41no—
103C22×C5C2×C22⋊C411yescharacteristic
104C2×D5—41no—
105C2×D5—41no—
106C2×D5—41no—
107C4×C5—21no—
108C4×C5—41no—
109C4×C5#13112yes—
110Dic5—21no—
111Dic5—21no—
112Dic5—21no—
113Dic5—41no—
114Dic5AGL(1,ℤ/8)12yes—
115#41—204no—
116C2×Q8—101no—
117C2×D4—21no—
118C2×D4—51no—
119C2×D4—101no—
120C2×D4—101no—
121C2×D4—101no—
122C2×D4C2×AGL(1,𝔽5)11yescharacteristic
123C22×C4—51no—
124C22×C4—51no—
125C22×C4—101no—
126C22×C4—101no—
127C22×C4C2×AGL(1,𝔽5)11yescharacteristic
128C2×C8—102no—
129C2×C8—102no—
130C4⋊C4—51no—
131C4⋊C4—101no—
132C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—101no—
135C22⋊C4—101no—
136C22⋊C4—101no—
137C42—51no—
138C42—51no—
139C42—51no—
140C42—51no—
141C2×C5—41no—
142C2×C5—81no—
143C2×C5#40811yescharacteristic
144C2×C5#41811yescharacteristic
145C2×C5#42111yescharacteristic
146D5—81no—
147C23—21no—
148C23—101no—
149C23C22⋊AGL(1,𝔽5)11yescharacteristic
150Q8—202no—
151D4—22no—
152D4—42no—
153D4—102no—
154D4—202no—
155D4—202no—
156D4—202no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—51no—
160C2×C4—51no—
161C2×C4—51no—
162C2×C4—51no—
163C2×C4—102no—
164C2×C4—101no—
165C2×C4—202no—
166C2×C4—202no—
167C2×C4—201no—
168C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
169C2×C4C22×AGL(1,𝔽5)11yescharacteristic
170C8—202no—
171C8—202no—
172C5#149611yescharacteristic, complemented, Sylow
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—201no—
180C22—201no—
181C22—201no—
182C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
183C4—21no—
184C4—41no—
185C4—52no—
186C4—101no—
187C4—101no—
188C4—101no—
189C4—201no—
190C4#413112yes—
191C2—41no—
192C2—81no—
193C2—401no—
194C2#6576711yescharacteristic
195C2#6577511yescharacteristic
196C2#6579511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×#121C4
socleC22×C5C2×C22⋊C4
prime radical#65930C2
cosocleC4×Dic5C23
hypercenter#121AGL(1,𝔽5)
hypocenterC5#1496
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #65759C2semidirect
3. #65748C2semidirect
4. D4⋊#170C2semidirect
5. #64871C2semidirect
172. C5#1496semidirect

Maximal subgroups

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

subgroupindex
#659302
#657592
#657482
D4⋊#1702
#648712
#14965