Finite Group Databasebeta

← #10604434

Subgroups — #10604434

Order 640.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#65914C211yesmaximal, characteristic
2#65684C211yesmaximal, characteristic, complemented
3#65683C211yesmaximal, characteristic, complemented
4#64927C212yesmaximal, complemented
5#64870C212yesmaximal, complemented
6C4×C5×Q8C411yescharacteristic
7#4093C2211yescharacteristic
8#4092C411yescharacteristic
9#4044—21no—
10#4041C411yescharacteristic
11#4040—21no—
12#4036C2211yescharacteristic
13C4×D20—21no—
14C4×D20C411yescharacteristic
15#4018—21no—
16C42×D5C2211yescharacteristic
17#4004—21no—
18#4003—21no—
19#4003—21no—
20#4002C2212yes—
21#4002C2212yes—
22C4⋊#170—21no—
23#1535—51nomaximal, Sylow
24C2×C5×Q8C2×C411yescharacteristic
25C2×D20—21no—
26C2×D20C2×C411yescharacteristic
27C2×C4×D5—21no—
28C2×C4×D5C2×C411yescharacteristic
29C2×C4×D5D411yescharacteristic
30C2×C4×D5D411yescharacteristic
31C2×#170—22no—
32C2×#170—22no—
33C5×C4⋊C4—21no—
34C5×C4⋊C4C2×C411yescharacteristic
35C42×C5—21no—
36C42×C5C2×C411yescharacteristic
37#685—21no—
38#685—21no—
39#685—21no—
40#685—21no—
41#685—21no—
42C4⋊Dic5—21no—
43C4⋊Dic5C2×C411yescharacteristic
44#683—21no—
45#683—21no—
46C4×Dic5D411yescharacteristic
47C4×Dic5D411yescharacteristic
48C4×Dic5C2311yescharacteristic
49#529—51no—
50#432—51no—
51#432—51no—
52#330—52no—
53#329—52no—
54C22×D5—21no—
55C22×D5C22⋊C411yescharacteristic
56C5×Q8—41no—
57C2×C4×C5—21no—
58C2×C4×C5—21no—
59C2×C4×C5C22⋊C411yescharacteristic
60C2×C4×C5C22⋊C411yescharacteristic
61C2×C4×C5C22×C411yescharacteristic
62C2×Dic5—21no—
63C2×Dic5C22⋊C411yescharacteristic
64C2×Dic5C2×D411yescharacteristic
65C2×Dic5C2×D411yescharacteristic
66D20—41no—
67D20—42no—
68C4×D5—41no—
69C4×D5—41no—
70C4×D5—41no—
71C4×D5—42no—
72#170—42no—
73#170—42no—
74#126—101no—
75#125—51no—
76#124—51no—
77#123—101no—
78C22⋊Q8—51no—
79#121—51no—
80C4×Q8AGL(1,𝔽5)11yescharacteristic
81C4×D4—51no—
82C4×D4—101no—
83#117—101no—
84C2×C42—51no—
85C4⋊C8—101no—
86C4⋊C8—101no—
87C22⋊C8—101no—
88C22⋊C8—101no—
89#97—52no—
90#97—52no—
91C22×C5C2×C22⋊C411yescharacteristic
92C2×D5—21no—
93C2×D5—21no—
94C2×D5—21no—
95C2×D5—41no—
96C2×D5—41no—
97C2×D5—41no—
98C4×C5—21no—
99C4×C5—21no—
100C4×C5—21no—
101C4×C5—41no—
102C4×C5—41no—
103Dic5—21no—
104Dic5—21no—
105Dic5—21no—
106Dic5—41no—
107C2×Q8C2×AGL(1,𝔽5)11yescharacteristic
108C2×D4—51no—
109C2×D4—101no—
110C22×C4—51no—
111C22×C4—51no—
112C22×C4—51no—
113C22×C4—101no—
114C2×C8—102no—
115C2×C8—102no—
116C4⋊C4—21no—
117C4⋊C4—51no—
118C4⋊C4—101no—
119C4⋊C4—101no—
120C4⋊C4—101no—
121C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
122C22⋊C4—101no—
123C22⋊C4—101no—
124C22⋊C4—101no—
125C22⋊C4—101no—
126C22⋊C4—101no—
127C42—21no—
128C42—51no—
129C42—51no—
130C42—51no—
131C42C2×AGL(1,𝔽5)11yescharacteristic
132C2×C5#41311yescharacteristic
133C2×C5#42111yescharacteristic
134C2×C5#42111yescharacteristic
135D5—41no—
136D5—81no—
137C23—51no—
138C23—101no—
139Q8—41no—
140D4—201no—
141D4—202no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—51no—
145C2×C4—51no—
146C2×C4—51no—
147C2×C4—101no—
148C2×C4—201no—
149C2×C4—201no—
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#153511yescharacteristic, complemented, Sylow
158C22—101no—
159C22—101no—
160C22—101no—
161C22—201no—
162C22—201no—
163C22—201no—
164C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—101no—
171C4—101no—
172C4—101no—
173C4—201no—
174C2—201no—
175C2—401no—
176C2#6576111yescharacteristic
177C2#6578411yescharacteristic
178C2#6579511yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC4×C5×Q8C4
socleC22×C5C2×C22⋊C4
prime radical#4041C4
cosocleC4×Dic5C23
hypercenterC4×Q8AGL(1,𝔽5)
hypocenterC5#1535
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. #65684C2semidirect
3. #65683C2semidirect
4. #64927C2semidirect
5. #64870C2semidirect
157. C5#1535semidirect

Maximal subgroups

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

subgroupindex
#659142
#656842
#656832
#649272
#648702
#15355