Finite Group Databasebeta

← #13098307

Subgroups — #13098307

Order 1600.

297 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 51 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#11716563C211yesmaximal, characteristic, complemented
2C25×#120C211yesmaximal, characteristic, complemented
3#11716515C212yesmaximal, complemented
4#11716382C211yesmaximal, characteristic, complemented
5#11716354C212yesmaximal, complemented
6C24×C25C2211yescharacteristic, complemented
7C2×D4×C25—21no—
8C2×D4×C25C2211yescharacteristic
9C2×C22⋊D25—21no—
10C2×C22⋊D25—22no—
11C2×C22⋊D25C2212yes—
12#87746—22no—
13D4×D25—22no—
14#87744—21no—
15C22⋊C4×C25—21no—
16C22⋊C4×C25C2211yescharacteristic, complemented
17C22⋊Dic25—22no—
18C22⋊Dic25C2212yescomplemented
19#65321—51nomaximal
20C23×C25—22no—
21C23×C25—21no—
22C23×C25—42no—
23C23×C25D411yescharacteristic, complemented
24C23×C25C2311yescharacteristic
25C22×D25—21no—
26C22×D25D412yescomplemented
27D4×C25—22no—
28D4×C25—42no—
29C2×C4×C25—21no—
30C2×C4×C25D411yescharacteristic, complemented
31C22⋊D25—22no—
32C22⋊D25—22no—
33C22⋊D25—22no—
34C22⋊D25—44no—
35C2×Dic25—21no—
36C2×Dic25—22no—
37C2×Dic25D412yescomplemented
38D100—21no—
39C4×D25—22no—
40Dic50—21no—
41#4144—51no—
42C5×#120D511yescharacteristic
43#4099—52no—
44#3966—51no—
45#3938—52no—
46C22×C25—24no—
47C22×C25—22no—
48C22×C25—21no—
49C22×C25—41no—
50C22×C25—41no—
51C22×C25—42no—
52C22×C25—41no—
53C22×C25—41no—
54C22×C25C2×D412yes—
55C22×C25C2×D411yescharacteristic
56C2×D25—22no—
57C2×D25—42no—
58C2×D25—81no—
59C4×C25—21no—
60C4×C25—41no—
61Dic25—22no—
62Dic25—42no—
63C24×C5C2×D511yescharacteristic
64C2×C5×D4—21no—
65C2×C5×D4C2×D511yescharacteristic
66C2×C22⋊D5—52no—
67C2×C22⋊D5—101no—
68C2×C22⋊D5—102no—
69#711—102no—
70D4×D5—102no—
71#709—101no—
72C5×C22⋊C4—21no—
73C5×C22⋊C4C2×D511yescharacteristic
74C22⋊Dic5—52no—
75C22⋊Dic5—102no—
76B(4,𝔽2)—251noSylow
77C2×C25—22no—
78C2×C25—21no—
79C2×C25—41no—
80C2×C25—42no—
81C2×C25#12011yescharacteristic
82D25—42no—
83C23×C5—22no—
84C23×C5—21no—
85C23×C5—42no—
86C23×C5C22⋊D511yescharacteristic
87C23×C5C22×D511yescharacteristic
88C22×D5—52no—
89C22×D5—101no—
90C5×D4—22no—
91C5×D4—42no—
92C2×C4×C5—21no—
93C2×C4×C5C22⋊D511yescharacteristic
94C22⋊D5—102no—
95C22⋊D5—102no—
96C22⋊D5—102no—
97C22⋊D5—204no—
98C2×Dic5—52no—
99C2×Dic5—101no—
100C2×Dic5—102no—
101D20—101no—
102C4×D5—102no—
103Dic10—101no—
104H(5,𝔽2)—251no—
105#120—252no—
106#120D2511yescharacteristic, complemented
107#99—251no—
108#99—252no—
109C25B(4,𝔽2)11yescharacteristic, complemented, Sylow
110C22×C5—24no—
111C22×C5—22no—
112C22×C5—21no—
113C22×C5—41no—
114C22×C5—41no—
115C22×C5—42no—
116C22×C5—41no—
117C22×C5—41no—
118C22×C5D4×D512yes—
119C22×C5C2×C22⋊D511yescharacteristic
120C2×D5—102no—
121C2×D5—202no—
122C2×D5—401no—
123C4×C5—21no—
124C4×C5—41no—
125Dic5—102no—
126Dic5—202no—
127C24C2×D2511yescharacteristic, complemented
128#41—502no—
129#41—501no—
130C2×D4—21no—
131C2×D4—252no—
132C2×D4—502no—
133C2×D4—501no—
134C2×D4—502no—
135C2×D4C2×D2511yescharacteristic
136C22⋊C4—21no—
137C22⋊C4—252no—
138C22⋊C4—502no—
139C22⋊C4C2×D2511yescharacteristic, complemented
140C2×C5—22no—
141C2×C5—21no—
142C2×C5—41no—
143C2×C5—42no—
144C2×C5#408311yescharacteristic
145D5—202no—
146C23—22no—
147C23—21no—
148C23—42no—
149C23—252no—
150C23—501no—
151C23C22⋊D2511yescharacteristic, complemented
152C23C22×D2511yescharacteristic
153Q8—501no—
154D4—22no—
155D4—42no—
156D4—501no—
157D4—502no—
158D4—502no—
159D4—502no—
160D4—1004no—
161C2×C4—21no—
162C2×C4—252no—
163C2×C4—502no—
164C2×C4—501no—
165C2×C4—502no—
166C2×C4C22⋊D2511yescharacteristic, complemented
167C5#6532111yescharacteristic
168C22—22no—
169C22—24no—
170C22—21no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22—502no—
177C22—1002no—
178C22—2001no—
179C22D4×D2512yes—
180C22C2×C22⋊D2511yescharacteristic
181C4—21no—
182C4—41no—
183C4—502no—
184C4—1002no—
185C2—21no—
186C2—22no—
187C2—42no—
188C2—41no—
189C2—1002no—
190C2#1171649911yescharacteristic
subgroupG/H
centerC2#11716499
commutator, cosocleC23×C25C23
FrattiniC23×C5C22×D5
FittingC25×#120C2
socleC2×C5#4083
hypercenter#120D25
hypocenterC25B(4,𝔽2)
LCSG ▷ C23×C25 ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #11716563C2semidirect
2. C25×#120C2semidirect
3. #11716515C2semidirect
4. #11716382C2semidirect
5. #11716354C2semidirect
6. C24×C25C22semidirect
16. C22⋊C4×C25C22semidirect
18. C22⋊Dic25C22semidirect
23. C23×C25D4semidirect
26. C22×D25D4semidirect
30. C2×C4×C25D4semidirect
37. C2×Dic25D4semidirect
106. #120D25semidirect
109. C25B(4,𝔽2)semidirect
127. C24C2×D25semidirect
139. C22⋊C4C2×D25semidirect
151. C23C22⋊D25semidirect
166. C2×C4C22⋊D25semidirect

Maximal subgroups

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

subgroupindex
#117165632
C25×#1202
#117165152
#117163822
#117163542
#653215