Finite Group Databasebeta

← #13107796

Subgroups — #13107796

Order 1600.

608 conjugacy classes of subgroups; only the 225 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 225 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⋊#1042C2213yes—
16C4⋊#1042C2216yes—
17C4⋊#1042C2211yescharacteristic
18C4×#1042C2216yes—
19#66266—51nomaximal
20#66161—51nomaximal
21C22×C5×D5C2313yes—
22C2×#1042—46no—
23C2×#1042C2312yes—
24C2×#1042C2316yes—
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×D5C2313yes—
34C4×C5×D5C2311yescharacteristic
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—23no—
48C2×C5×D5—23no—
49C2×C5×D5C22×C413yes—
50C2×C5×D5C2411yescharacteristic
51#1042—26no—
52#1042—22no—
53C4×C52C22×C413yes—
54C4×C52C22×C411yescharacteristic
55C5×Dic5C22×C413yes—
56C5×Dic5C22×C411yescharacteristic
57C22×AGL(1,𝔽5)—56no—
58C5×#41—41no—
59C5×#41—51no—
60C5×#41—53no—
61C5×#41—53no—
62C5×#41Dic511yescharacteristic, complemented
63C5×#41AGL(1,𝔽5)11yescharacteristic, complemented
64C2×C5×Q8—51no—
65C2×C5×D4—53no—
66C22×C4×C5—53no—
67C22×Dic5—56no—
68#713Dic511yescharacteristic, complemented
69Q8×D5C2×D511yescharacteristic
70#711Dic513yescomplemented
71D4×D5C2×D513yes—
72#709Dic513yescomplemented
73C2×C4×D5C2×D513yes—
74C22⋊AGL(1,𝔽5)—56no—
75C4⋊AGL(1,𝔽5)—53no—
76C4⋊AGL(1,𝔽5)—56no—
77C4⋊AGL(1,𝔽5)—51no—
78C4×AGL(1,𝔽5)—56no—
79C22⋊Dic5—56no—
80C4⋊Dic5—53no—
81C4⋊Dic5—56no—
82C4⋊Dic5—51no—
83C4×Dic5—56no—
84#520—251noSylow
85C2×C52—23no—
86C2×C52C23×C411yescharacteristic
87C5×D5—23no—
88C5×D5H(5,𝔽2)11yescharacteristic
89C5×D52−1+411yescharacteristic
90C23×C5—53no—
91C22×D5C22×D513yes—
92C2×AGL(1,𝔽5)—52no—
93C2×AGL(1,𝔽5)—56no—
94C2×AGL(1,𝔽5)—206no—
95C5×Q8—41no—
96C5×Q8—53no—
97C5×Q8C2×Dic511yescharacteristic, complemented
98C5×Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
99C5×D4—43no—
100C5×D4—56no—
101C5×D4—53no—
102C5×D4C2×Dic513yescomplemented
103C5×D4C2×AGL(1,𝔽5)13yescomplemented
104C2×C4×C5—43no—
105C2×C4×C5—53no—
106C2×C4×C5—53no—
107C2×C4×C5—53no—
108C2×C4×C5—53no—
109C2×C4×C5—51no—
110C2×C4×C5C2×Dic513yescomplemented
111C2×C4×C5C2×AGL(1,𝔽5)13yescomplemented
112C22⋊D5C2×Dic516yescomplemented
113C2×Dic5—52no—
114C2×Dic5—56no—
115C2×Dic5—206no—
116C2×Dic5C2×Dic513yescomplemented
117D20C2×Dic513yescomplemented
118C4×D5C2×Dic513yescomplemented
119C4×D5C2×Dic513yescomplemented
120C4×D5C22×D513yes—
121C4×D5C22×D511yescharacteristic
122Dic10C2×Dic513yescomplemented
123C2×#41—51no—
124C4×Q8—252no—
125C4×D4—256no—
126#117—253no—
127C2×C4⋊C4—253no—
128C52#52011yescharacteristic, complemented, Sylow
129C22×C5—43no—
130C22×C5—51no—
131C22×C5—53no—
132C22×C5—103no—
133C22×C5—103no—
134C22×C5C22×Dic513yes—
135C22×C5C22×AGL(1,𝔽5)13yes—
136C2×D5—23no—
137C2×D5—23no—
138C2×D5C22×Dic513yes—
139C2×D5C23×D511yescharacteristic
140AGL(1,𝔽5)—106no—
141AGL(1,𝔽5)—102no—
142C4×C5—41no—
143C4×C5—43no—
144C4×C5—51no—
145C4×C5—53no—
146C4×C5C22×Dic513yes—
147C4×C5C22×Dic511yescharacteristic
148C4×C5C22×AGL(1,𝔽5)11yescharacteristic
149C4×C5C22×AGL(1,𝔽5)13yes—
150Dic5—102no—
151Dic5—106no—
152Dic5C22×Dic513yes—
153Dic5C22×Dic511yescharacteristic
154#41—51no—
155#41—53no—
156#41—53no—
157#41#104211yescharacteristic, complemented
158C2×Q8—51no—
159C2×D4—53no—
160C22×C4—53no—
161C22×C4—256no—
162C4⋊C4—253no—
163C4⋊C4—256no—
164C4⋊C4—251no—
165C22⋊C4—256no—
166C42—256no—
167C2×C5—23no—
168C2×C5—23no—
169C2×C5—41no—
170C2×C5—51no—
171C2×C5—51no—
172C2×C5—83no—
173C2×C5—103no—
174C2×C5C23×Dic511yescharacteristic
175C2×C5C23×AGL(1,𝔽5)11yescharacteristic
176D5—23no—
177D5#414911yescharacteristic
178D5#415011yescharacteristic
179C23—53no—
180Q8—53no—
181Q8C2×#104211yescharacteristic, complemented
182D4—56no—
183D4—53no—
184D4C2×#104213yescomplemented
185C2×C4—53no—
186C2×C4—53no—
187C2×C4—51no—
188C2×C4—53no—
189C2×C4—53no—
190C2×C4—252no—
191C2×C4—256no—
192C2×C4—1006no—
193C2×C4C2×#104213yescomplemented
194C5—41no—
195C5#6616111yescharacteristic, complemented
196C5#6626611yescharacteristic, complemented
197C22—51no—
198C22—53no—
199C22—103no—
200C22—103no—
201C22C22×#104213yes—
202C4—51no—
203C4—53no—
204C4—506no—
205C4—502no—
206C4C22×#104211yescharacteristic
207C4C22×#104213yes—
208C2—23no—
209C2—51no—
210C2—51no—
211C2—103no—
212C2C23×#104211yescharacteristic
subgroupG/H
center, FrattiniC2C23×#1042
commutator, socleC2×C52C23×C4
FittingC52×#41C4
prime radicalC5×D5×#41C2
cosocleC2×C5×D5C24
hypercenter#41#1042
hypocenterC52#520
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2 ◁ #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
62. C5×#41Dic5semidirect
63. C5×#41AGL(1,𝔽5)semidirect
68. #713Dic5semidirect
70. #711Dic5semidirect
72. #709Dic5semidirect
97. C5×Q8C2×Dic5semidirect
98. C5×Q8C2×AGL(1,𝔽5)semidirect
102. C5×D4C2×Dic5semidirect
103. C5×D4C2×AGL(1,𝔽5)semidirect
110. C2×C4×C5C2×Dic5semidirect
111. C2×C4×C5C2×AGL(1,𝔽5)semidirect
112. C22⋊D5C2×Dic5semidirect
116. C2×Dic5C2×Dic5semidirect
117. D20C2×Dic5semidirect
118. C4×D5C2×Dic5semidirect
119. C4×D5C2×Dic5semidirect
122. Dic10C2×Dic5semidirect
128. C52#520semidirect
157. #41#1042semidirect
181. Q8C2×#1042semidirect
184. D4C2×#1042semidirect
193. C2×C4C2×#1042semidirect
195. C5#66161semidirect
196. C5#66266semidirect

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
#662665
#661615