Finite Group Databasebeta

← C22×D4×D21 #13062372

Subgroups — C22×D4×D21 #13062372

Order 1344.

2724 conjugacy classes of subgroups; only the 659 normal subgroups are shown.

normal subgroups only

This group has 2724 conjugacy classes of subgroups — too many to draw. Only its 659 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×D21C212yesmaximal, complemented
2C22×C3×C7×D4C211yesmaximal, characteristic, complemented
3C22×#4268C212yesmaximal, complemented
4C2×D4×D21C2124yesmaximal, direct product factor
5C22×D84C211yesmaximal, characteristic, complemented
6C22×C4×D21C211yesmaximal, characteristic, complemented
7C22×D4×D7—31nomaximal, Hall
8C24×C3×C7C2212yescomplemented
9C23×D21—216no—
10C23×D21C2211yescharacteristic
11C23×D21C22124yescomplemented
12C23×D21C2212yescomplemented
13C2×C3×C7×D4C22112yescomplemented
14C22×C3×C4×C7C2211yescharacteristic, complemented
15C2×#4268C22124yescomplemented
16C22×Dic21C2211yescharacteristic, complemented
17D4×D21C22164yesdirect product factor
18C2×D84C22112yescomplemented
19C2×C4×D21C22112yescomplemented
20C24×D7—32no—
21C22×C7×D4D311yescharacteristic, complemented
22C22×C22⋊D7—32no—
23C2×D4×D7—324no—
24C22×D28—31no—
25C22×C4×D7—31no—
26C22×D3×D4—71nomaximal, Hall
27C23×C3×C7—28no—
28C23×C3×C7C2311yescharacteristic
29C23×C3×C7C23112yescomplemented
30C22×D21—28no—
31C22×D21—296no—
32C22×D21D418yesdirect product factor
33C22×D21C2316yes—
34C22×D21C23132yescomplemented
35C22×D21C23112yescomplemented
36C3×C7×D4C23116yescomplemented
37C2×C3×C4×C7C2316yescomplemented
38#4268C23132yescomplemented
39C2×Dic21C2316yescomplemented
40D84C23116yescomplemented
41C4×D21C23116yescomplemented
42C24×C7C2×D312yescomplemented
43C23×D7—31no—
44C23×D7—32no—
45C23×D7—324no—
46C23×D7—616no—
47C2×C7×D4C2×D3112yescomplemented
48C22×C4×C7C2×D311yescharacteristic, complemented
49C2×C22⋊D7—324no—
50C22×Dic7—31no—
51D4×D7—364no—
52C2×D28—312no—
53C2×C4×D7—312no—
54C24×D3—72no—
55C22×C3×D4D711yescharacteristic, complemented
56C22×#67—72no—
57C2×D3×D4—724no—
58C22×D12—71no—
59C22×C4×D3—71no—
60C22×C3×C7—224no—
61C22×C3×C7C2×D414yescomplemented
62C22×C3×C7C2413yes—
63C22×C3×C7C2418yescomplemented
64C2×D21—224no—
65C2×D21—264no—
66C2×D21C2×D4124yesdirect product factor
67C2×D21C2414yes—
68C2×D21C2418yescomplemented
69C3×C4×C7C2414yescomplemented
70Dic21C2414yescomplemented
71C23×D4—211noSylow
72C23×C7—28no—
73C23×C7C22×D311yescharacteristic
74C23×C7C22×D3112yescomplemented
75C22×D7—38no—
76C22×D7—36no—
77C22×D7—312no—
78C22×D7—332no—
79C22×D7—696no—
80C22×D7—68no—
81C7×D4C22×D3116yescomplemented
82C2×C4×C7C22×D316yescomplemented
83C22⋊D7—332no—
84C2×Dic7—36no—
85D28—316no—
86C4×D7—316no—
87C24×C3C2×D712yescomplemented
88C23×D3—71no—
89C23×D3—72no—
90C23×D3—724no—
91C23×D3—1416no—
92C2×C3×D4C2×D7112yescomplemented
93C22×C3×C4C2×D711yescharacteristic, complemented
94C2×#67—724no—
95C22×Dic3—71no—
96D3×D4—764no—
97C2×D12—712no—
98C2×C4×D3—712no—
99C2×C3×C7—28no—
100C2×C3×C7C22×D416yescomplemented
101C2×C3×C7C2511yescharacteristic
102D21—28no—
103D21C22×D418yesdirect product factor
104C25—212no—
105C22×D4—212no—
106C22×D4—2124no—
107C22×D4—211no—
108C22×D4D2111yescharacteristic, direct product factor
109C23×C4—211no—
110C22×C7—224no—
111C22×C7D3×D414yescomplemented
112C22×C7C23×D313yes—
113C22×C7C23×D318yescomplemented
114C2×D7—324no—
115C2×D7—34no—
116C2×D7—38no—
117C2×D7—664no—
118C2×D7—624no—
119C4×C7C23×D314yescomplemented
120Dic7—34no—
121C23×C3—28no—
122C23×C3C22×D711yescharacteristic
123C23×C3C22×D7112yescomplemented
124C22×D3—78no—
125C22×D3—76no—
126C22×D3—712no—
127C22×D3—732no—
128C22×D3—1496no—
129C22×D3—148no—
130C3×D4C22×D7116yescomplemented
131C2×C3×C4C22×D716yescomplemented
132#67—732no—
133C2×Dic3—76no—
134D12—716no—
135C4×D3—716no—
136C3×C7C23×D411yescharacteristic, complemented, Hall
137C24—211no—
138C24—212no—
139C24—2124no—
140C24—4216no—
141C24C2×D2112yescomplemented
142C2×D4—2124no—
143C2×D4—2164no—
144C2×D4—2112no—
145C2×D4C2×D21112yesdirect product factor
146C22×C4—211no—
147C22×C4—2112no—
148C22×C4C2×D2111yescharacteristic, complemented
149C2×C7—28no—
150C2×C7C2×D3×D416yescomplemented
151C2×C7C24×D311yescharacteristic
152D7—38no—
153D7—68no—
154C22×C3—224no—
155C22×C3D4×D714yescomplemented
156C22×C3C23×D713yes—
157C22×C3C23×D718yescomplemented
158C2×D3—724no—
159C2×D3—74no—
160C2×D3—78no—
161C2×D3—1464no—
162C2×D3—1424no—
163C3×C4C23×D714yescomplemented
164Dic3—74no—
165C23—28no—
166C23—218no—
167C23—216no—
168C23—2112no—
169C23—2132no—
170C23—4296no—
171C23—428no—
172C23C22×D2111yescharacteristic
173C23C22×D21112yescomplemented
174D4—2132no—
175D4—2116no—
176D4C22×D21116yesdirect product factor
177C2×C4—216no—
178C2×C4—2116no—
179C2×C4C22×D2116yescomplemented
180C7C22×D3×D411yescharacteristic, complemented, Sylow
181C2×C3—28no—
182C2×C3C2×D4×D716yescomplemented
183C2×C3C24×D711yescharacteristic
184D3—78no—
185D3—148no—
186C22—224no—
187C22—214no—
188C22—2124no—
189C22—218no—
190C22—4264no—
191C22—4224no—
192C22D4×D2114yesdirect product factor
193C22C23×D2113yes—
194C22C23×D2118yescomplemented
195C4—214no—
196C4C23×D2114yescomplemented
197C3C22×D4×D711yescharacteristic, complemented, Sylow
198C2—28no—
199C2—218no—
200C2—428no—
201C2C2×D4×D2116yesdirect product factor
202C2C24×D2111yescharacteristic
subgroupG/H
centerC23C22×D21
commutator, cosocleC2×C3×C7C25
FrattiniC2C24×D21
FittingC22×C3×C7×D4C2
socleC23×C3×C7C23
hypercenterC22×D4D21
hypocenterC3×C7C23×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×D21C2semidirect
2. C22×C3×C7×D4C2semidirect
3. C22×#4268C2semidirect
4. C2×D4×D21C2direct (N × H)
5. C22×D84C2semidirect
6. C22×C4×D21C2semidirect
8. C24×C3×C7C22semidirect
11. C23×D21C22semidirect
12. C23×D21C22semidirect
13. C2×C3×C7×D4C22semidirect
14. C22×C3×C4×C7C22semidirect
15. C2×#4268C22semidirect
16. C22×Dic21C22semidirect
17. D4×D21C22direct (N × H)
18. C2×D84C22semidirect
19. C2×C4×D21C22semidirect
21. C22×C7×D4D3semidirect
29. C23×C3×C7C23semidirect
32. C22×D21D4direct (N × H)
34. C22×D21C23semidirect
35. C22×D21C23semidirect
36. C3×C7×D4C23semidirect
37. C2×C3×C4×C7C23semidirect
38. #4268C23semidirect
39. C2×Dic21C23semidirect
40. D84C23semidirect
41. C4×D21C23semidirect
42. C24×C7C2×D3semidirect
47. C2×C7×D4C2×D3semidirect
48. C22×C4×C7C2×D3semidirect
55. C22×C3×D4D7semidirect
61. C22×C3×C7C2×D4semidirect
63. C22×C3×C7C24semidirect
66. C2×D21C2×D4direct (N × H)
68. C2×D21C24semidirect
69. C3×C4×C7C24semidirect
70. Dic21C24semidirect
74. C23×C7C22×D3semidirect
81. C7×D4C22×D3semidirect
82. C2×C4×C7C22×D3semidirect
87. C24×C3C2×D7semidirect
92. C2×C3×D4C2×D7semidirect
93. C22×C3×C4C2×D7semidirect
100. C2×C3×C7C22×D4semidirect
103. D21C22×D4direct (N × H)
108. C22×D4D21direct (N × H)
111. C22×C7D3×D4semidirect
113. C22×C7C23×D3semidirect
119. C4×C7C23×D3semidirect
123. C23×C3C22×D7semidirect
130. C3×D4C22×D7semidirect
131. C2×C3×C4C22×D7semidirect
136. C3×C7C23×D4semidirect
141. C24C2×D21semidirect
145. C2×D4C2×D21direct (N × H)
148. C22×C4C2×D21semidirect
150. C2×C7C2×D3×D4semidirect
155. C22×C3D4×D7semidirect
157. C22×C3C23×D7semidirect
163. C3×C4C23×D7semidirect
173. C23C22×D21semidirect
176. D4C22×D21direct (N × H)
179. C2×C4C22×D21semidirect
180. C7C22×D3×D4semidirect
182. C2×C3C2×D4×D7semidirect
192. C22D4×D21direct (N × H)
194. C22C23×D21semidirect
196. C4C23×D21semidirect
197. C3C22×D4×D7semidirect
201. C2C2×D4×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D212
C22×C3×C7×D42
C22×#42682
C2×D4×D212
C22×D842
C22×C4×D212
C22×D4×D73
C22×D3×D47