Finite Group Databasebeta

← #10604515

Subgroups — #10604515

Order 640.

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

normal subgroups only

This group has 216 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#65958C211yesmaximal, characteristic
2#65758C211yesmaximal, characteristic, complemented
3#65748C211yesmaximal, characteristic, complemented
4Q8⋊#170C212yesmaximal, complemented
5#64871C212yesmaximal, complemented
6C2×#704—21no—
7C5×C22⋊Q8C411yescharacteristic
8#4093C411yescharacteristic
9Q8×Dic5C2211yescharacteristic
10#4076C411yescharacteristic
11C2×C4×Dic5C2211yescharacteristic
12#4044—21no—
13#4039C2211yescharacteristic
14#4038—21no—
15#4036C411yescharacteristic
16#4032—21no—
17#4027—21no—
18C22⋊#170—21no—
19#4004—21no—
20C4⋊#170—21no—
21C4⋊#170C2212yes—
22C4×#170C2212yes—
23#1495—51nomaximal, Sylow
24C2×C5×Q8C2×C411yescharacteristic
25C22×C4×C5C2×C411yescharacteristic
26C2×C22⋊D5—21no—
27C22×Dic5D411yescharacteristic
28C22×Dic5D411yescharacteristic
29C2×D20C2×C411yescharacteristic
30C2×C4×D5—21no—
31#704—42no—
32C2×#170—22no—
33C2×#170—22no—
34C5×C4⋊C4—21no—
35C5×C4⋊C4C2×C411yescharacteristic
36C5×C22⋊C4—21no—
37#685—21no—
38#685—21no—
39#685—21no—
40#685—21no—
41C4⋊Dic5—21no—
42C4⋊Dic5—21no—
43C4⋊Dic5C2×C411yescharacteristic
44#683—21no—
45C4×Dic5—21no—
46C4×Dic5C2×C411yescharacteristic
47C4×Dic5D411yescharacteristic
48C4×Dic5D411yescharacteristic
49C4×Dic5C2311yescharacteristic
50#529—51no—
51#431—51no—
52#423—51no—
53Q8⋊C8—52no—
54D4⋊C8—52no—
55C23×C5C22⋊C411yescharacteristic
56C22×D5—21no—
57C5×Q8—22no—
58C2×C4×C5—22no—
59C2×C4×C5—21no—
60C2×C4×C5—21no—
61C2×C4×C5C22⋊C411yescharacteristic
62C2×C4×C5C22×C411yescharacteristic
63C22⋊D5—42no—
64C2×Dic5—21no—
65C2×Dic5—21no—
66C2×Dic5—21no—
67C2×Dic5—41no—
68C2×Dic5C22⋊C411yescharacteristic
69C2×Dic5C22⋊C411yescharacteristic
70C2×Dic5C2×D411yescharacteristic
71C2×Dic5C2×D411yescharacteristic
72D20—22no—
73C4×D5—41no—
74C4×D5—41no—
75#170—42no—
76#170—42no—
77C2×M16—101no—
78#126—101no—
79#125—51no—
80#124—51no—
81#123—101no—
82C22⋊Q8AGL(1,𝔽5)11yescharacteristic
83#121—51no—
84C4×Q8—51no—
85C4×D4—51no—
86C4×D4—101no—
87#117—101no—
88C2×C42—51no—
89C4⋊C8—52no—
90C4⋊C8—101no—
91C4⋊C8—101no—
92C22⋊C8—101no—
93C4×C8—52no—
94C22×C5—21no—
95C22×C5—21no—
96C22×C5—21no—
97C22×C5C2×C22⋊C411yescharacteristic
98C2×D5—41no—
99C2×D5—41no—
100C2×D5—41no—
101C4×C5—21no—
102C4×C5—41no—
103C4×C5—41no—
104C4×C5#13112yes—
105Dic5—21no—
106Dic5—21no—
107Dic5—21no—
108Dic5—41no—
109Dic5AGL(1,ℤ/8)11yescharacteristic
110Dic5#13711yescharacteristic
111C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
112C2×D4—51no—
113C2×D4—101no—
114C22×C4—51no—
115C22×C4—51no—
116C22×C4—101no—
117C22×C4C2×AGL(1,𝔽5)11yescharacteristic
118M16—202no—
119C2×C8—102no—
120C2×C8—102no—
121C4⋊C4—21no—
122C4⋊C4—51no—
123C4⋊C4—101no—
124C4⋊C4—101no—
125C4⋊C4—101no—
126C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
127C22⋊C4—21no—
128C22⋊C4—101no—
129C22⋊C4—101no—
130C22⋊C4—101no—
131C22⋊C4—101no—
132C42—51no—
133C42—51no—
134C42—51no—
135C42—51no—
136C42—101no—
137C2×C5—41no—
138C2×C5#40811yescharacteristic
139C2×C5#41711yescharacteristic
140C2×C5#42111yescharacteristic
141D5—81no—
142C23—101no—
143C23C22⋊AGL(1,𝔽5)11yescharacteristic
144Q8—22no—
145D4—102no—
146D4—202no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—51no—
151C2×C4—51no—
152C2×C4—51no—
153C2×C4—51no—
154C2×C4—101no—
155C2×C4—101no—
156C2×C4—101no—
157C2×C4—201no—
158C2×C4—201no—
159C2×C4—201no—
160C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
161C2×C4C22×AGL(1,𝔽5)11yescharacteristic
162C8—202no—
163C8—202no—
164C5#149511yescharacteristic, complemented, Sylow
165C22—21no—
166C22—21no—
167C22—21no—
168C22—201no—
169C22—201no—
170C22—201no—
171C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
172C4—21no—
173C4—41no—
174C4—41no—
175C4—51no—
176C4—51no—
177C4—101no—
178C4—101no—
179C4—101no—
180C4—201no—
181C4#413112yes—
182C2—41no—
183C2—401no—
184C2#6577511yescharacteristic
185C2#6578411yescharacteristic
186C2#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#4076C4
cosocleC4×Dic5C23
hypercenterC22⋊Q8AGL(1,𝔽5)
hypocenterC5#1495
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
2. #65758C2semidirect
3. #65748C2semidirect
4. Q8⋊#170C2semidirect
5. #64871C2semidirect
164. C5#1495semidirect

Maximal subgroups

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

subgroupindex
#659582
#657582
#657482
Q8⋊#1702
#648712
#14955