Finite Group Databasebeta

← #41×#1042 #13107795

Subgroups — #41×#1042 #13107795

Order 1600.

640 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C5×D5×#41C211yesmaximal, characteristic
2Q8×#1042C212yesmaximal, complemented
3D4×#1042C216yesmaximal, complemented
4#11717425C213yesmaximal, complemented
5C2×C4×#1042C213yesmaximal, complemented
6C22×#1042C2216yes—
7C52×#41C411yescharacteristic, complemented
8C5×#713C411yescharacteristic, complemented
9C5×Q8×D5C2211yescharacteristic
10C5×#711C413yescomplemented
11C5×D4×D5C2213yes—
12C5×#709C413yescomplemented
13C2×C4×C5×D5C2213yes—
14C22⋊#1042C2216yes—
15C4⋊#1042C2216yes—
16C4×#1042C2211yescharacteristic
17C4×#1042C2213yes—
18C4×#1042C2216yes—
19#41×AGL(1,𝔽5)—51nomaximal
20#41×Dic5—51nomaximal
21C22×C5×D5C2313yes—
22C2×#1042—212no—
23C2×#1042C2316yes—
24C2×#1042C2312yes—
25C52×Q8C2×C411yescharacteristic, complemented
26C52×D4C2×C413yescomplemented
27C2×C4×C52C2×C413yescomplemented
28C5×C22⋊D5C2×C416yescomplemented
29C2×C5×Dic5C2×C413yescomplemented
30C5×D20C2×C413yescomplemented
31C4×C5×D5C2×C413yescomplemented
32C4×C5×D5C2×C413yescomplemented
33C4×C5×D5C2311yescharacteristic
34C4×C5×D5C2313yes—
35C5×Dic10C2×C413yescomplemented
36C2×C5×#41—51no—
37D5×#41D511yescharacteristic
38Q8×AGL(1,𝔽5)—52no—
39D4×AGL(1,𝔽5)—56no—
40#4130—53no—
41C2×C4×AGL(1,𝔽5)—53no—
42Q8×Dic5—52no—
43D4×Dic5—56no—
44#4072—53no—
45C2×C4×Dic5—53no—
46C22×C52C22×C413yes—
47C2×C5×D5—26no—
48C2×C5×D5C22×C413yes—
49C2×C5×D5C2411yescharacteristic
50#1042—26no—
51#1042#4114yesdirect product factor
52C4×C52C22×C413yes—
53C4×C52C22×C411yescharacteristic
54C5×Dic5C22×C413yes—
55C5×Dic5C22×C411yescharacteristic
56C22×AGL(1,𝔽5)—56no—
57C5×#41—41no—
58C5×#41—51no—
59C5×#41—53no—
60C5×#41—53no—
61C5×#41Dic511yescharacteristic, complemented
62C5×#41AGL(1,𝔽5)11yescharacteristic, complemented
63C2×C5×Q8—51no—
64C2×C5×D4—53no—
65C22×C4×C5—53no—
66C22×Dic5—56no—
67#713Dic511yescharacteristic, complemented
68Q8×D5C2×D511yescharacteristic
69#711Dic513yescomplemented
70D4×D5C2×D513yes—
71#709Dic513yescomplemented
72C2×C4×D5C2×D513yes—
73C22⋊AGL(1,𝔽5)—56no—
74C4⋊AGL(1,𝔽5)—56no—
75C4×AGL(1,𝔽5)—51no—
76C4×AGL(1,𝔽5)—53no—
77C4×AGL(1,𝔽5)—56no—
78C22⋊Dic5—56no—
79C4⋊Dic5—56no—
80C4×Dic5—51no—
81C4×Dic5—53no—
82C4×Dic5—56no—
83C4×#41—251noSylow
84C2×C52—23no—
85C2×C52C23×C411yescharacteristic
86C5×D5—23no—
87C5×D5C2×#4112yes—
88C23×C5—53no—
89C22×D5C22×D513yes—
90C2×AGL(1,𝔽5)—52no—
91C2×AGL(1,𝔽5)—56no—
92C2×AGL(1,𝔽5)—1012no—
93C5×Q8—41no—
94C5×Q8—53no—
95C5×Q8C2×Dic511yescharacteristic, complemented
96C5×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
97C5×D4—43no—
98C5×D4—56no—
99C5×D4—53no—
100C5×D4C2×Dic513yescomplemented
101C5×D4C2×AGL(1,𝔽5)13yescomplemented
102C2×C4×C5—43no—
103C2×C4×C5—53no—
104C2×C4×C5—53no—
105C2×C4×C5—53no—
106C2×C4×C5—51no—
107C2×C4×C5—53no—
108C2×C4×C5C2×Dic513yescomplemented
109C2×C4×C5C2×AGL(1,𝔽5)13yescomplemented
110C22⋊D5C2×Dic516yescomplemented
111C2×Dic5—52no—
112C2×Dic5—56no—
113C2×Dic5—1012no—
114C2×Dic5C2×Dic513yescomplemented
115D20C2×Dic513yescomplemented
116C4×D5C2×Dic513yescomplemented
117C4×D5C2×Dic513yescomplemented
118C4×D5C22×D511yescharacteristic
119C4×D5C22×D513yes—
120Dic10C2×Dic513yescomplemented
121C2×#41—51no—
122C4×Q8—252no—
123C4×D4—256no—
124#117—253no—
125C2×C42—253no—
126C52C4×#4111yescharacteristic, complemented, Sylow
127C22×C5—43no—
128C22×C5—51no—
129C22×C5—53no—
130C22×C5—106no—
131C22×C5C22×Dic513yes—
132C22×C5C22×AGL(1,𝔽5)13yes—
133C2×D5—26no—
134C2×D5C22×Dic513yes—
135C2×D5C23×D511yescharacteristic
136AGL(1,𝔽5)—54no—
137AGL(1,𝔽5)—106no—
138C4×C5—43no—
139C4×C5—41no—
140C4×C5—51no—
141C4×C5—53no—
142C4×C5C22×Dic513yes—
143C4×C5C22×Dic511yescharacteristic
144C4×C5C22×AGL(1,𝔽5)13yes—
145C4×C5C22×AGL(1,𝔽5)11yescharacteristic
146Dic5—54no—
147Dic5—106no—
148Dic5C22×Dic513yes—
149Dic5C22×Dic511yescharacteristic
150#41—51no—
151#41—53no—
152#41—53no—
153#41#104211yescharacteristic, direct product factor
154C2×Q8—51no—
155C2×D4—53no—
156C22×C4—53no—
157C22×C4—256no—
158C4⋊C4—256no—
159C22⋊C4—256no—
160C42—251no—
161C42—253no—
162C42—256no—
163C2×C5—23no—
164C2×C5—23no—
165C2×C5—41no—
166C2×C5—52no—
167C2×C5—83no—
168C2×C5—103no—
169C2×C5C23×Dic511yescharacteristic
170C2×C5C23×AGL(1,𝔽5)11yescharacteristic
171D5—23no—
172D5D5×#4112yes—
173C23—53no—
174Q8—53no—
175Q8C2×#104211yescharacteristic, complemented
176D4—56no—
177D4—53no—
178D4C2×#104213yescomplemented
179C2×C4—53no—
180C2×C4—53no—
181C2×C4—53no—
182C2×C4—51no—
183C2×C4—53no—
184C2×C4—252no—
185C2×C4—256no—
186C2×C4—5012no—
187C2×C4C2×#104213yescomplemented
188C5—41no—
189C5#41×Dic511yescharacteristic, complemented
190C5#41×AGL(1,𝔽5)11yescharacteristic, complemented
191C22—51no—
192C22—53no—
193C22—106no—
194C22C22×#104213yes—
195C4—51no—
196C4—53no—
197C4—254no—
198C4—506no—
199C4C22×#104213yes—
200C4C22×#104211yescharacteristic
201C2—23no—
202C2—52no—
203C2—103no—
204C2C23×#104211yescharacteristic
subgroupG/H
centerC4C22×#1042
commutator, socleC2×C52C23×C4
FrattiniC2C23×#1042
FittingC52×#41C4
prime radicalC5×D5×#41C2
cosocleC2×C5×D5C24
hypercenter#41#1042
hypocenterC52C4×#41
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. Q8×#1042C2semidirect
3. D4×#1042C2semidirect
4. #11717425C2semidirect
5. C2×C4×#1042C2semidirect
7. C52×#41C4semidirect
8. C5×#713C4semidirect
10. C5×#711C4semidirect
12. C5×#709C4semidirect
25. C52×Q8C2×C4semidirect
26. C52×D4C2×C4semidirect
27. C2×C4×C52C2×C4semidirect
28. C5×C22⋊D5C2×C4semidirect
29. C2×C5×Dic5C2×C4semidirect
30. C5×D20C2×C4semidirect
31. C4×C5×D5C2×C4semidirect
32. C4×C5×D5C2×C4semidirect
35. C5×Dic10C2×C4semidirect
51. #1042#41direct (N × H)
61. C5×#41Dic5semidirect
62. C5×#41AGL(1,𝔽5)semidirect
67. #713Dic5semidirect
69. #711Dic5semidirect
71. #709Dic5semidirect
95. C5×Q8C2×Dic5semidirect
96. C5×Q8C2×AGL(1,𝔽5)semidirect
100. C5×D4C2×Dic5semidirect
101. C5×D4C2×AGL(1,𝔽5)semidirect
108. C2×C4×C5C2×Dic5semidirect
109. C2×C4×C5C2×AGL(1,𝔽5)semidirect
110. C22⋊D5C2×Dic5semidirect
114. C2×Dic5C2×Dic5semidirect
115. D20C2×Dic5semidirect
116. C4×D5C2×Dic5semidirect
117. C4×D5C2×Dic5semidirect
120. Dic10C2×Dic5semidirect
126. C52C4×#41semidirect
153. #41#1042direct (N × H)
175. Q8C2×#1042semidirect
178. D4C2×#1042semidirect
187. C2×C4C2×#1042semidirect
189. C5#41×Dic5semidirect
190. C5#41×AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C5×D5×#412
Q8×#10422
D4×#10422
#117174252
C2×C4×#10422
#41×AGL(1,𝔽5)5
#41×Dic55