Finite Group Databasebeta

← #120×D25 #13098912

Subgroups — #120×D25 #13098912

Order 1600.

993 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C24×D25C211yesmaximal, characteristic, complemented
2C2×D4×D25C213yesmaximal, complemented
3C25×#120C211yesmaximal, characteristic, complemented
4#11716515C211yesmaximal, characteristic, complemented
5#11716499C213yesmaximal, complemented
6#11716444C213yesmaximal, complemented
7C22⋊C4×D25C213yesmaximal, complemented
8C24×C25C2211yescharacteristic, complemented
9C23×D25—212no—
10C23×D25C2213yes—
11C23×D25C2213yescomplemented
12C23×D25C2211yescharacteristic, complemented
13C2×D4×C25C2213yescomplemented
14C2×C22⋊D25C2213yescomplemented
15C2×C22⋊D25C2213yescomplemented
16D4×D25—212no—
17C2×D100C2213yescomplemented
18C2×C4×D25C2213yescomplemented
19C22⋊C4×C25C2213yescomplemented
20C22⋊Dic25C2213yescomplemented
21#87720C2216yescomplemented
22D5×#120—51nomaximal
23C23×C25—26no—
24C23×C25C2313yes—
25C23×C25C2311yescharacteristic, complemented
26C22×D25—232no—
27C22×D25—26no—
28C22×D25—212no—
29C22×D25—212no—
30C22×D25—26no—
31C22×D25D4112yescomplemented
32C22×D25C2313yes—
33C22×D25C2311yescharacteristic
34C22×D25C2311yescharacteristic, complemented
35D4×C25—26no—
36C2×C4×C25C2313yescomplemented
37C22⋊D25—26no—
38C22⋊D25—26no—
39C2×Dic25C2313yescomplemented
40D100—26no—
41C4×D25—26no—
42C24×D5—51no—
43C2×D4×D5—53no—
44C5×#120D511yescharacteristic
45#4099—51no—
46#4083—53no—
47#4028—53no—
48D5×C22⋊C4—53no—
49C22×C25—28no—
50C22×C25—26no—
51C22×C25—23no—
52C22×C25C2×D416yescomplemented
53C22×C25C2411yescharacteristic
54C2×D25—224no—
55C2×D25—26no—
56C2×D25—224no—
57C2×D25—23no—
58C2×D25—44no—
59C2×D25C2×D416yes—
60C2×D25C2×D416yescomplemented
61C4×C25—23no—
62Dic25—23no—
63C24×C5C2×D511yescharacteristic
64C23×D5—53no—
65C23×D5—51no—
66C23×D5—53no—
67C23×D5—1012no—
68C2×C5×D4C2×D513yes—
69C2×C22⋊D5—53no—
70C2×C22⋊D5—53no—
71D4×D5—1012no—
72C2×D20—53no—
73C2×C4×D5—53no—
74C5×C22⋊C4C2×D513yes—
75C22⋊Dic5—53no—
76#685—56no—
77C2×#120—251noSylow
78C2×C25—26no—
79C2×C25—41no—
80C2×C25C22×D413yes—
81D25—26no—
82D25—41no—
83D25#12014yesdirect product factor
84C23×C5—26no—
85C23×C5C22×D513yes—
86C23×C5C22×D511yescharacteristic
87C22×D5—51no—
88C22×D5—512no—
89C22×D5—53no—
90C22×D5—51no—
91C22×D5—1012no—
92C22×D5—1012no—
93C22×D5—106no—
94C22×D5—1032no—
95C22×D5—106no—
96C5×D4—26no—
97C2×C4×C5C22×D513yes—
98C22⋊D5—106no—
99C22⋊D5—106no—
100C2×Dic5—53no—
101D20—106no—
102C4×D5—106no—
103C25—251no—
104C22×D4—253no—
105#120—251no—
106#120—253no—
107#120—253no—
108#120D2511yescharacteristic, direct product factor
109C2×C22⋊C4—253no—
110C25C2×#12011yescharacteristic, complemented, Sylow
111C22×C5—28no—
112C22×C5—26no—
113C22×C5—23no—
114C22×C5D4×D516yes—
115C22×C5C23×D511yescharacteristic
116C2×D5—56no—
117C2×D5—56no—
118C2×D5—1024no—
119C2×D5—1024no—
120C2×D5—106no—
121C2×D5—103no—
122C2×D5—204no—
123C4×C5—23no—
124Dic5—103no—
125C24—253no—
126C24—253no—
127C24—251no—
128C24—5012no—
129C24C2×D2511yescharacteristic, complemented
130C2×D4—253no—
131C2×D4—253no—
132C2×D4—253no—
133C2×D4—5012no—
134C2×D4C2×D2513yescomplemented
135C22×C4—253no—
136C22⋊C4—253no—
137C22⋊C4—256no—
138C22⋊C4C2×D2513yescomplemented
139C2×C5—26no—
140C2×C5—41no—
141C2×C5C2×D4×D513yes—
142D5—54no—
143D5—106no—
144D5—201no—
145C23—26no—
146C23—251no—
147C23—2512no—
148C23—253no—
149C23—251no—
150C23—5012no—
151C23—5012no—
152C23—506no—
153C23—5032no—
154C23—506no—
155C23C22×D2513yes—
156C23C22×D2511yescharacteristic, complemented
157D4—26no—
158D4—506no—
159D4—506no—
160D4—506no—
161C2×C4—253no—
162C2×C4—506no—
163C2×C4C22×D2513yescomplemented
164C5D5×#12011yescharacteristic
165C22—28no—
166C22—26no—
167C22—23no—
168C22—256no—
169C22—256no—
170C22—5024no—
171C22—5024no—
172C22—506no—
173C22—503no—
174C22—1004no—
175C22D4×D2516yescomplemented
176C22C23×D2511yescharacteristic
177C4—23no—
178C4—503no—
179C2—26no—
180C2—41no—
181C2—254no—
182C2—506no—
183C2—1001no—
184C2C2×D4×D2513yes—
subgroupG/H
centerC22C23×D25
commutator, cosocleC22×C25C24
Frattini, socleC22×C5C23×D5
FittingC25×#120C2
hypercenter#120D25
hypocenterC25C2×#120
LCSG ▷ C22×C25 ▷ C25
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D25C2semidirect
2. C2×D4×D25C2semidirect
3. C25×#120C2semidirect
4. #11716515C2semidirect
5. #11716499C2semidirect
6. #11716444C2semidirect
7. C22⋊C4×D25C2semidirect
8. C24×C25C22semidirect
11. C23×D25C22semidirect
12. C23×D25C22semidirect
13. C2×D4×C25C22semidirect
14. C2×C22⋊D25C22semidirect
15. C2×C22⋊D25C22semidirect
17. C2×D100C22semidirect
18. C2×C4×D25C22semidirect
19. C22⋊C4×C25C22semidirect
20. C22⋊Dic25C22semidirect
21. #87720C22semidirect
25. C23×C25C23semidirect
31. C22×D25D4semidirect
34. C22×D25C23semidirect
36. C2×C4×C25C23semidirect
39. C2×Dic25C23semidirect
52. C22×C25C2×D4semidirect
60. C2×D25C2×D4semidirect
83. D25#120direct (N × H)
108. #120D25direct (N × H)
110. C25C2×#120semidirect
129. C24C2×D25semidirect
134. C2×D4C2×D25semidirect
138. C22⋊C4C2×D25semidirect
156. C23C22×D25semidirect
163. C2×C4C22×D25semidirect
175. C22D4×D25semidirect

Maximal subgroups

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

subgroupindex
C24×D252
C2×D4×D252
C25×#1202
#117165152
#117164992
#117164442
C22⋊C4×D252
D5×#1205