Finite Group Databasebeta

← C22×D4×#261 #13131810

Subgroups — C22×D4×#261 #13131810

Order 1664.

2528 conjugacy classes of subgroups; only the 850 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 850 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×#261C212yesmaximal, complemented
2C22×D4×D13C211yesmaximal, characteristic
3C22×C22⋊#261C212yesmaximal, complemented
4C2×D4×#261C2124yesmaximal, direct product factor
5C22×C4⋊#261C211yesmaximal, characteristic, complemented
6C22×C4×#261C211yesmaximal, characteristic, complemented
7C24×D13C2212yes—
8C23×#261—216no—
9C23×#261C2211yescharacteristic
10C23×#261C2212yes—
11C23×#261C2211yescharacteristic
12C23×#261C22124yescomplemented
13C22×D4×C13C411yescharacteristic, complemented
14C22×C22⋊D13C412yescomplemented
15C2×D4×D13C4112yescomplemented
16C2×D4×D13C22112yes—
17C22×D52C411yescharacteristic, complemented
18C22×C4×D13C2211yescharacteristic
19C2×C22⋊#261C22124yescomplemented
20D4×#261C22164yesdirect product factor
21C2×C4⋊#261C22112yescomplemented
22C2×C4×#261C22112yescomplemented
23C24×C13C2×C412yescomplemented
24C23×D13—28no—
25C23×D13—28no—
26C23×D13C2×C4112yescomplemented
27C23×D13C2×C412yescomplemented
28C23×D13C23112yes—
29C23×D13C2311yescharacteristic
30C22×#261—28no—
31C22×#261—24no—
32C22×#261—296no—
33C22×#261D418yesdirect product factor
34C22×#261C2316yes—
35C22×#261C23112yes—
36C22×#261C2316yes—
37C22×#261C23132yescomplemented
38C2×D4×C13C2×C4112yescomplemented
39C22×C4×C13C2×C411yescharacteristic, complemented
40C2×C22⋊D13C2×C4124yescomplemented
41C22×Dic13C2×C411yescharacteristic, complemented
42D4×D13C2×C4148yescomplemented
43D4×D13C23116yes—
44C2×D52C2×C4112yescomplemented
45C2×C4×D13C2×C416yescomplemented
46C2×C4×D13C2316yes—
47C22⋊#261C23132yescomplemented
48C4⋊#261C23116yescomplemented
49C4×#261C23116yescomplemented
50C22×C4×D4—131nomaximal, Sylow
51C23×C13—28no—
52C23×C13C22×C411yescharacteristic
53C23×C13C22×C4112yescomplemented
54C22×D13—28no—
55C22×D13—248no—
56C22×D13—224no—
57C22×D13—224no—
58C22×D13C22×C413yes—
59C22×D13C22×C4124yescomplemented
60C22×D13C22×C4112yescomplemented
61C22×D13C2×D414yes—
62C22×D13#4114yes—
63C22×D13C2418yes—
64C22×D13C2413yes—
65C2×#261—224no—
66C2×#261—212no—
67C2×#261—264no—
68C2×#261C2×D4124yesdirect product factor
69C2×#261C2414yes—
70C2×#261C2418yes—
71C2×#261C2414yes—
72D4×C13C22×C4116yescomplemented
73C2×C4×C13C22×C416yescomplemented
74C22⋊D13C22×C4132yescomplemented
75C2×Dic13C22×C416yescomplemented
76D52C22×C4116yescomplemented
77C4×D13C22×C4112yescomplemented
78C4×D13C2414yes—
79C23×D4—131no—
80C24×C4—132no—
81C2×C4×D4—1324no—
82C22×C4⋊C4—131no—
83C22×C22⋊C4—132no—
84C22×C42—131no—
85C22×C13—224no—
86C22×C13C4×D414yescomplemented
87C22×C13C23×C413yes—
88C22×C13C23×C418yescomplemented
89C2×D13—224no—
90C2×D13—248no—
91C2×D13—28no—
92C2×D13—28no—
93C2×D13C4×D4112yescomplemented
94C2×D13C23×C413yes—
95C2×D13C23×C418yescomplemented
96C2×D13C22×D416yes—
97C2×D13C2×#4116yes—
98C2×D13C2511yescharacteristic
99#261—28no—
100#261—24no—
101#261C22×D418yesdirect product factor
102C4×C13C23×C414yescomplemented
103Dic13C23×C414yescomplemented
104C25—132no—
105C22×D4—132no—
106C22×D4—1312no—
107C22×D4—1312no—
108C22×D4—131no—
109C22×D4#26111yescharacteristic, direct product factor
110C23×C4—131no—
111C23×C4—131no—
112C23×C4—131no—
113C23×C4—132no—
114C23×C4—1324no—
115C23×C4—2616no—
116C4×D4—1364no—
117C2×C4⋊C4—1312no—
118C2×C22⋊C4—1324no—
119C2×C42—1312no—
120C2×C13—28no—
121C2×C13C2×C4×D416yescomplemented
122C2×C13C24×C411yescharacteristic
123D13—28no—
124D13C2×C4×D416yescomplemented
125D13C23×D411yescharacteristic
126D13C22×#4111yescharacteristic
127C24—131no—
128C24—132no—
129C24—1312no—
130C24—1312no—
131C24—268no—
132C24—268no—
133C24C2×#26112yescomplemented
134C2×D4—1324no—
135C2×D4—1316no—
136C2×D4—1348no—
137C2×D4—1312no—
138C2×D4C2×#261112yesdirect product factor
139C22×C4—136no—
140C22×C4—138no—
141C22×C4—131no—
142C22×C4—136no—
143C22×C4—136no—
144C22×C4—136no—
145C22×C4—1312no—
146C22×C4—1332no—
147C22×C4—2696no—
148C22×C4—268no—
149C22×C4—264no—
150C22×C4C2×#26111yescharacteristic, complemented
151C4⋊C4—1316no—
152C22⋊C4—1332no—
153C42—1316no—
154C13C22×C4×D411yescharacteristic, complemented, Sylow
155C23—28no—
156C23—134no—
157C23—134no—
158C23—133no—
159C23—133no—
160C23—1312no—
161C23—138no—
162C23—1324no—
163C23—2624no—
164C23—2648no—
165C23—2624no—
166C23—268no—
167C23C22×#26111yescharacteristic
168C23C22×#261112yescomplemented
169D4—1332no—
170D4—1316no—
171D4C22×#261116yesdirect product factor
172C2×C4—134no—
173C2×C4—1324no—
174C2×C4—136no—
175C2×C4—134no—
176C2×C4—1312no—
177C2×C4—134no—
178C2×C4—138no—
179C2×C4—2664no—
180C2×C4—2624no—
181C2×C4—2612no—
182C2×C4C22×#26116yescomplemented
183C22—224no—
184C22—1312no—
185C22—136no—
186C22—136no—
187C22—133no—
188C22—131no—
189C22—138no—
190C22—268no—
191C22—268no—
192C22—2648no—
193C22—2624no—
194C22D4×#26114yesdirect product factor
195C22C23×#26113yes—
196C22C23×#26118yescomplemented
197C4—138no—
198C4—134no—
199C4—264no—
200C4—268no—
201C4C23×#26114yescomplemented
202C2—28no—
203C2—131no—
204C2—131no—
205C2—136no—
206C2—268no—
207C2C2×D4×#26116yesdirect product factor
208C2C24×#26111yescharacteristic
subgroupG/H
centerC23C22×#261
commutatorC2×C13C24×C4
FrattiniC2C24×#261
FittingC22×D4×C13C4
socleC23×C13C22×C4
prime radicalC22×D4×D13C2
cosocleC2×D13C25
hypercenterC22×D4#261
hypocenterC13C22×C4×D4
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×#261C2semidirect
3. C22×C22⋊#261C2semidirect
4. C2×D4×#261C2direct (N × H)
5. C22×C4⋊#261C2semidirect
6. C22×C4×#261C2semidirect
12. C23×#261C22semidirect
13. C22×D4×C13C4semidirect
14. C22×C22⋊D13C4semidirect
15. C2×D4×D13C4semidirect
17. C22×D52C4semidirect
19. C2×C22⋊#261C22semidirect
20. D4×#261C22direct (N × H)
21. C2×C4⋊#261C22semidirect
22. C2×C4×#261C22semidirect
23. C24×C13C2×C4semidirect
26. C23×D13C2×C4semidirect
27. C23×D13C2×C4semidirect
33. C22×#261D4direct (N × H)
37. C22×#261C23semidirect
38. C2×D4×C13C2×C4semidirect
39. C22×C4×C13C2×C4semidirect
40. C2×C22⋊D13C2×C4semidirect
41. C22×Dic13C2×C4semidirect
42. D4×D13C2×C4semidirect
44. C2×D52C2×C4semidirect
45. C2×C4×D13C2×C4semidirect
47. C22⋊#261C23semidirect
48. C4⋊#261C23semidirect
49. C4×#261C23semidirect
53. C23×C13C22×C4semidirect
59. C22×D13C22×C4semidirect
60. C22×D13C22×C4semidirect
68. C2×#261C2×D4direct (N × H)
72. D4×C13C22×C4semidirect
73. C2×C4×C13C22×C4semidirect
74. C22⋊D13C22×C4semidirect
75. C2×Dic13C22×C4semidirect
76. D52C22×C4semidirect
77. C4×D13C22×C4semidirect
86. C22×C13C4×D4semidirect
88. C22×C13C23×C4semidirect
93. C2×D13C4×D4semidirect
95. C2×D13C23×C4semidirect
101. #261C22×D4direct (N × H)
102. C4×C13C23×C4semidirect
103. Dic13C23×C4semidirect
109. C22×D4#261direct (N × H)
121. C2×C13C2×C4×D4semidirect
124. D13C2×C4×D4semidirect
133. C24C2×#261semidirect
138. C2×D4C2×#261direct (N × H)
150. C22×C4C2×#261semidirect
154. C13C22×C4×D4semidirect
168. C23C22×#261semidirect
171. D4C22×#261direct (N × H)
182. C2×C4C22×#261semidirect
194. C22D4×#261direct (N × H)
196. C22C23×#261semidirect
201. C4C23×#261semidirect
207. C2C2×D4×#261direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#2612
C22×D4×D132
C22×C22⋊#2612
C2×D4×#2612
C22×C4⋊#2612
C22×C4×#2612
C22×C4×D413