Finite Group Databasebeta

← C23×#87830 #72221626

Subgroups — C23×#87830 #72221626

Order 3200.

7362 conjugacy classes of subgroups; only the 1803 normal subgroups are shown.

normal subgroups only

This group has 7362 conjugacy classes of subgroups — too many to draw. Only its 1803 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×#1044C211yesmaximal, characteristic
2C24×#1043C211yesmaximal, characteristic
3C23×#6036C211yesmaximal, characteristic
4C22×#87830C2128yesmaximal, direct product factor
5C24×C5⋊D5C2211yescharacteristic
6C23×#1044C4116yescomplemented
7C23×#1044C22114yes—
8C23×#1043C4116yescomplemented
9C23×#1043C22114yes—
10C23×C5×Dic5C412yescomplemented
11C22×#6036C4114yescomplemented
12C22×#6036C22114yes—
13C2×#87830C221112yesdirect product factor
14C23×C4×AGL(1,𝔽5)—52nomaximal
15C24×C52C2×C411yescharacteristic
16C23×C5⋊D5C2×C417yes—
17C23×C5⋊D5C2×C418yes—
18C23×C5⋊D5C2×C418yes—
19C23×C5⋊D5C2317yes—
20C22×#1044C2×C41112yescomplemented
21C22×#1044C23128yes—
22C22×#1043C2×C41112yescomplemented
23C22×#1043C23128yes—
24C22×C5×Dic5C2×C4128yescomplemented
25C2×#6036C2×C4184yescomplemented
26C2×#6036C23128yes—
27#87830C23164yesdirect product factor
28C24×AGL(1,𝔽5)—52no—
29C24×AGL(1,𝔽5)—52no—
30C24×AGL(1,𝔽5)—102no—
31C23×C4×D5—52no—
32C22×C4×AGL(1,𝔽5)—556no—
33C23×C52C4218yescomplemented
34C23×C52C22×C417yes—
35C22×C5⋊D5C42156yescomplemented
36C22×C5⋊D5C22×C4121yes—
37C22×C5⋊D5C22×C4128yes—
38C22×C5⋊D5C22×C4128yes—
39C22×C5⋊D5C2417yes—
40C2×#1044C22×C41112yescomplemented
41C2×#1044C2418yes—
42C2×#1043C22×C41112yescomplemented
43C2×#1043C2418yes—
44C2×C5×Dic5C22×C4156yescomplemented
45#6036C22×C4156yescomplemented
46#6036C2418yes—
47C24×D5—52no—
48C24×D5—102no—
49C23×AGL(1,𝔽5)—532no—
50C23×AGL(1,𝔽5)—528no—
51C23×AGL(1,𝔽5)—532no—
52C23×AGL(1,𝔽5)—528no—
53C23×AGL(1,𝔽5)—1032no—
54C23×AGL(1,𝔽5)—1028no—
55C23×C4×C5—52no—
56C23×Dic5AGL(1,𝔽5)12yescomplemented
57C22×C4×D5—528no—
58C22×C4×D5—528no—
59C2×C4×AGL(1,𝔽5)—5224no—
60C23×C42—251noSylow
61C22×C52C2×C42128yescomplemented
62C22×C52C23×C417yes—
63C2×C5⋊D5C2×C42184yescomplemented
64C2×C5⋊D5C23×C4114yes—
65C2×C5⋊D5C23×C4114yes—
66C2×C5⋊D5C23×C417yes—
67C2×C5⋊D5C2511yescharacteristic
68#1044C23×C4116yescomplemented
69#1043C23×C4116yescomplemented
70C5×Dic5C23×C4116yescomplemented
71C24×C5—22no—
72C24×C5C2×AGL(1,𝔽5)12yes—
73C23×D5—516no—
74C23×D5—516no—
75C23×D5—514no—
76C23×D5—514no—
77C23×D5—1016no—
78C23×D5—1014no—
79C23×D5—1016no—
80C23×D5—1014no—
81C22×AGL(1,𝔽5)—556no—
82C22×AGL(1,𝔽5)—556no—
83C22×AGL(1,𝔽5)—5224no—
84C22×AGL(1,𝔽5)—5224no—
85C22×AGL(1,𝔽5)—10224no—
86C22×AGL(1,𝔽5)—1056no—
87C22×C4×C5—528no—
88C22×Dic5C2×AGL(1,𝔽5)128yescomplemented
89C2×C4×D5—556no—
90C2×C4×D5—5168no—
91C4×AGL(1,𝔽5)—5128no—
92C24×C4—251no—
93C24×C4—251no—
94C24×C4—251no—
95C22×C42—2528no—
96C2×C52C22×C42114yescomplemented
97C2×C52C24×C411yescharacteristic
98C5⋊D5C22×C42114yescomplemented
99C5⋊D5C24×C411yescharacteristic
100C5⋊D5C24×C411yescharacteristic
101C23×C5—216no—
102C23×C5—214no—
103C23×C5C4×AGL(1,𝔽5)116yescomplemented
104C23×C5C22×AGL(1,𝔽5)114yes—
105C22×D5—5112no—
106C22×D5—556no—
107C22×D5—556no—
108C22×D5—542no—
109C22×D5—514no—
110C22×D5—10112no—
111C22×D5—1056no—
112C22×D5—1042no—
113C22×D5—1056no—
114C22×D5—1014no—
115C2×AGL(1,𝔽5)—5224no—
116C2×AGL(1,𝔽5)—516no—
117C2×AGL(1,𝔽5)—5224no—
118C2×AGL(1,𝔽5)—516no—
119C2×AGL(1,𝔽5)—10224no—
120C2×AGL(1,𝔽5)—1016no—
121C2×C4×C5—556no—
122C2×Dic5C22×AGL(1,𝔽5)156yescomplemented
123C4×D5—5112no—
124C4×D5—516no—
125C25—251no—
126C23×C4—52no—
127C23×C4—2514no—
128C23×C4—2514no—
129C23×C4—2516no—
130C23×C4—2514no—
131C23×C4—2516no—
132C23×C4—2514no—
133C2×C42—25112no—
134C52C23×C4211yescharacteristic, complemented, Sylow
135C22×C5—214no—
136C22×C5—256no—
137C22×C5C2×C4×AGL(1,𝔽5)156yescomplemented
138C22×C5C23×AGL(1,𝔽5)114yes—
139C2×D5—5168no—
140C2×D5—528no—
141C2×D5—528no—
142C2×D5—514no—
143C2×D5—52no—
144C2×D5—10168no—
145C2×D5—1028no—
146C2×D5—1028no—
147C2×D5—1014no—
148C2×D5—102no—
149AGL(1,𝔽5)—532no—
150AGL(1,𝔽5)—532no—
151AGL(1,𝔽5)—1032no—
152C4×C5—516no—
153Dic5C23×AGL(1,𝔽5)116yescomplemented
154C24—258no—
155C24—258no—
156C24—257no—
157C24—257no—
158C24D5⋊AGL(1,𝔽5)11yescharacteristic
159C22×C4—528no—
160C22×C4—2528no—
161C22×C4—2584no—
162C22×C4—2528no—
163C22×C4—2528no—
164C22×C4—25112no—
165C22×C4—25112no—
166C42—2564no—
167C2×C5—228no—
168C2×C5—22no—
169C2×C5C22×C4×AGL(1,𝔽5)128yescomplemented
170C2×C5C24×AGL(1,𝔽5)12yes—
171D5—52no—
172D5—52no—
173D5—528no—
174D5—102no—
175D5—102no—
176D5—1028no—
177C23—257no—
178C23—2521no—
179C23—2528no—
180C23—2528no—
181C23—2556no—
182C23#8783018yesdirect product factor
183C23C2×D5⋊AGL(1,𝔽5)17yes—
184C2×C4—556no—
185C2×C4—258no—
186C2×C4—2556no—
187C2×C4—258no—
188C2×C4—258no—
189C2×C4—25112no—
190C2×C4—25112no—
191C5—22no—
192C5C23×C4×AGL(1,𝔽5)12yescomplemented
193C22—2514no—
194C22—2514no—
195C22—2584no—
196C22—251no—
197C22—257no—
198C22C2×#87830128yesdirect product factor
199C22C22×D5⋊AGL(1,𝔽5)17yes—
200C4—516no—
201C4—2516no—
202C4—2516no—
203C2—251no—
204C2—251no—
205C2—2514no—
206C2C22×#87830114yesdirect product factor
207C2C23×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC24D5⋊AGL(1,𝔽5)
commutator, hypocenterC52C23×C42
FrattiniC2C23×D5⋊AGL(1,𝔽5)
Fitting, socleC24×C52C2×C4
prime radicalC24×C5⋊D5C22
cosocleC2×C5⋊D5C25
LCSG ▷ C52
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C22×#87830C2direct (N × H)
6. C23×#1044C4semidirect
8. C23×#1043C4semidirect
10. C23×C5×Dic5C4semidirect
11. C22×#6036C4semidirect
13. C2×#87830C22direct (N × H)
20. C22×#1044C2×C4semidirect
22. C22×#1043C2×C4semidirect
24. C22×C5×Dic5C2×C4semidirect
25. C2×#6036C2×C4semidirect
27. #87830C23direct (N × H)
33. C23×C52C42semidirect
35. C22×C5⋊D5C42semidirect
40. C2×#1044C22×C4semidirect
42. C2×#1043C22×C4semidirect
44. C2×C5×Dic5C22×C4semidirect
45. #6036C22×C4semidirect
56. C23×Dic5AGL(1,𝔽5)semidirect
61. C22×C52C2×C42semidirect
63. C2×C5⋊D5C2×C42semidirect
68. #1044C23×C4semidirect
69. #1043C23×C4semidirect
70. C5×Dic5C23×C4semidirect
88. C22×Dic5C2×AGL(1,𝔽5)semidirect
96. C2×C52C22×C42semidirect
98. C5⋊D5C22×C42semidirect
103. C23×C5C4×AGL(1,𝔽5)semidirect
122. C2×Dic5C22×AGL(1,𝔽5)semidirect
134. C52C23×C42semidirect
137. C22×C5C2×C4×AGL(1,𝔽5)semidirect
153. Dic5C23×AGL(1,𝔽5)semidirect
169. C2×C5C22×C4×AGL(1,𝔽5)semidirect
182. C23#87830direct (N × H)
192. C5C23×C4×AGL(1,𝔽5)semidirect
198. C22C2×#87830direct (N × H)
206. C2C22×#87830direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#10442
C24×#10432
C23×#60362
C22×#878302
C23×C4×AGL(1,𝔽5)5