Finite Group Databasebeta

← #10604980

Subgroups — #10604980

Order 640.

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

normal subgroups only

This group has 234 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#66150C211yesmaximal, characteristic
2#65761C211yesmaximal, characteristic
3#65745C211yesmaximal, characteristic
4#64916C214yesmaximal, complemented
5C22×C5×Q8C411yescharacteristic, complemented
6C2×#709C2211yescharacteristic
7#4130—21no—
8D5⋊M16—21no—
9#4093—21no—
10#4092—21no—
11#4092—21no—
12#4090—21no—
13#4075C412yescomplemented
14#4075C2212yes—
15#4072C411yescharacteristic, complemented
16#4013C2212yes—
17#4011C2212yes—
18#4005—21no—
19#3999—21no—
20#2123—51nomaximal, Sylow
21C2×C5×Q8—21no—
22C2×C5×Q8—21no—
23C2×C5×Q8—41no—
24C2×C5×Q8—41no—
25C22×C4×C5C2×C412yes—
26C22×C4×C5C2×C411yescharacteristic
27C2×C22⋊D5C2×C411yescharacteristic
28C2×C22⋊D5C2311yescharacteristic
29#709—41no—
30#709—41no—
31C2×D20D411yescharacteristic
32C2×C4×D5D411yescharacteristic
33C2×C4×D5D411yescharacteristic
34C2×Dic10D411yescharacteristic
35C22⋊AGL(1,𝔽5)—22no—
36#704—22no—
37C4⋊AGL(1,𝔽5)—41no—
38C4×AGL(1,𝔽5)—41no—
39#700—41no—
40D5⋊C8—41no—
41C22⋊Dic5C2×C412yes—
42#685—22no—
43#685—22no—
44C4⋊Dic5—21no—
45#683—22no—
46#683—22no—
47C4×Dic5—21no—
48#536—51no—
49#413—51no—
50#410—51no—
51#352—54no—
52C23×C5C22×C411yescharacteristic
53C22×D5C22⋊C411yescharacteristic
54C22×D5C2×D411yescharacteristic
55C2×AGL(1,𝔽5)—42no—
56C5×Q8—42no—
57C5×Q8—41no—
58C5×Q8—41no—
59C2×C4×C5—21no—
60C2×C4×C5—21no—
61C2×C4×C5—22no—
62C2×C4×C5—42no—
63C2×C4×C5C22⋊C411yescharacteristic
64C2×C4×C5C22⋊C411yescharacteristic
65C22⋊D5—41no—
66C22⋊D5—41no—
67C2×Dic5—22no—
68C2×Dic5C22⋊C411yescharacteristic
69C2×Dic5C2×D411yescharacteristic
70D20—41no—
71C4×D5—21no—
72C4×D5—21no—
73C4×D5—21no—
74C4×D5—21no—
75Dic10—41no—
76#170—42no—
77C2×#41—51no—
78C22×Q8AGL(1,𝔽5)11yescharacteristic, complemented
79C2×M16—101no—
80C4⋊Q8—101no—
81#124—101no—
82#123—52no—
83#123—52no—
84C22⋊Q8—101no—
85C22⋊Q8—101no—
86#117—51no—
87#117—101no—
88#101—101no—
89#100—52no—
90#99—52no—
91#99—101no—
92C22×C5—21no—
93C22×C5—41no—
94C22×C5C2×C22⋊C411yescharacteristic
95C2×D5—21no—
96C2×D5—21no—
97C2×D5—41no—
98C2×D5—41no—
99AGL(1,𝔽5)—82no—
100C4×C5—21no—
101C4×C5—42no—
102C4×C5#9911yescharacteristic
103C4×C5#9911yescharacteristic
104Dic5—21no—
105Dic5—21no—
106Dic5—42no—
107#41—201no—
108#41—201no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8—51no—
114C2×D4—51no—
115C2×D4—51no—
116C2×D4—51no—
117C22×C4—51no—
118C22×C4—51no—
119C22×C4C2×AGL(1,𝔽5)12yes—
120C22×C4C2×AGL(1,𝔽5)11yescharacteristic
121M16—102no—
122M16—201no—
123C2×C8—201no—
124C4⋊C4—102no—
125C4⋊C4—102no—
126C4⋊C4—101no—
127C4⋊C4—201no—
128C22⋊C4—52no—
129C22⋊C4—102no—
130C22⋊C4—102no—
131C22⋊C4—102no—
132C42—101no—
133C42—201no—
134C2×C5—21no—
135C2×C5—41no—
136C2×C5C2×#9911yescharacteristic
137D5—41no—
138D5—41no—
139C23—51no—
140C23—51no—
141C23C22×AGL(1,𝔽5)11yescharacteristic
142Q8—42no—
143Q8—41no—
144Q8—41no—
145Q8—201no—
146D4—201no—
147D4—201no—
148D4—201no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—51no—
154C2×C4—51no—
155C2×C4—101no—
156C2×C4—101no—
157C2×C4—101no—
158C2×C4—101no—
159C2×C4—102no—
160C2×C4—202no—
161C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
162C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
163C8—202no—
164C5#212311yescharacteristic, complemented, Sylow
165C22—21no—
166C22—41no—
167C22—101no—
168C22—101no—
169C22—201no—
170C22—201no—
171C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
172C4—21no—
173C4—42no—
174C4—101no—
175C4—101no—
176C4—202no—
177C4—402no—
178C4#401111yescharacteristic
179C4#401111yescharacteristic
180C2—21no—
181C2—41no—
182C2—201no—
183C2—201no—
184C2C2×#401111yescharacteristic
subgroupG/H
centerC2C2×#4011
commutatorC23×C5C22×C4
FrattiniC23C22×AGL(1,𝔽5)
FittingC22×C5×Q8C4
socleC2×C5C2×#99
prime radicalC2×#709C22
cosocleC2×C22⋊D5C23
hypercenterC22×Q8AGL(1,𝔽5)
hypocenterC5#2123
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #64916C2semidirect
5. C22×C5×Q8C4semidirect
13. #4075C4semidirect
15. #4072C4semidirect
78. C22×Q8AGL(1,𝔽5)semidirect
164. C5#2123semidirect

Maximal subgroups

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

subgroupindex
#661502
#657612
#657452
#649162
#21235