Finite Group Databasebeta

← C22×D4×#10621924 #36370537

Subgroups — C22×D4×#10621924 #36370537

Order 22144.

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×#10621924C212yesmaximal, complemented
2C2×D4×#10621924C2124yesmaximal, direct product factor
3C22×C4⋊#10621924C211yesmaximal, characteristic, complemented
4C22×C22⋊#10621924C212yesmaximal, complemented
5C22×C4×#10621924C211yesmaximal, characteristic, complemented
6C22×D4×D173C211yesmaximal, characteristic
7C23×#10621924—216no—
8C23×#10621924C2211yescharacteristic
9C23×#10621924C2212yes—
10C23×#10621924C2211yescharacteristic
11C23×#10621924C22124yescomplemented
12D4×#10621924C22164yesdirect product factor
13C2×C4⋊#10621924C22112yescomplemented
14C2×C22⋊#10621924C22124yescomplemented
15C2×C4×#10621924C22112yescomplemented
16C24×D173C2212yes—
17C22×D692C411yescharacteristic, complemented
18C22×C22⋊D173C412yescomplemented
19C2×D4×D173C4112yescomplemented
20C2×D4×D173C22112yes—
21C22×C4×D173C2211yescharacteristic
22C22×D4×C173C411yescharacteristic, complemented
23C22×#10621924—296no—
24C22×#10621924—28no—
25C22×#10621924—24no—
26C22×#10621924D418yesdirect product factor
27C22×#10621924C2316yes—
28C22×#10621924C23112yes—
29C22×#10621924C2316yes—
30C22×#10621924C23132yescomplemented
31C4⋊#10621924C23116yescomplemented
32C22⋊#10621924C23132yescomplemented
33C4×#10621924C23116yescomplemented
34C23×D173—28no—
35C23×D173—28no—
36C23×D173C2×C412yescomplemented
37C23×D173C2×C4112yescomplemented
38C23×D173C2311yescharacteristic
39C23×D173C23112yes—
40C2×D692C2×C4112yescomplemented
41C2×C22⋊D173C2×C4124yescomplemented
42D4×D173C2×C4148yescomplemented
43D4×D173C23116yes—
44C22×Dic173C2×C411yescharacteristic, complemented
45C2×C4×D173C2×C416yescomplemented
46C2×C4×D173C2316yes—
47C24×C173C2×C412yescomplemented
48C2×D4×C173C2×C4112yescomplemented
49C22×C4×C173C2×C411yescharacteristic, complemented
50C2×#10621924—224no—
51C2×#10621924—212no—
52C2×#10621924—264no—
53C2×#10621924C2×D4124yesdirect product factor
54C2×#10621924C2414yes—
55C2×#10621924C2418yes—
56C2×#10621924C2414yes—
57C22×D173—28no—
58C22×D173—248no—
59C22×D173—224no—
60C22×D173—224no—
61C22×D173C22×C413yes—
62C22×D173C22×C4124yescomplemented
63C22×D173C22×C4112yescomplemented
64C22×D173C2×D414yes—
65C22×D173#4114yes—
66C22×D173C2418yes—
67C22×D173C2413yes—
68D692C22×C4116yescomplemented
69C22⋊D173C22×C4132yescomplemented
70C2×Dic173C22×C416yescomplemented
71C4×D173C22×C4112yescomplemented
72C4×D173C2414yes—
73C23×C173—28no—
74C23×C173C22×C411yescharacteristic
75C23×C173C22×C4112yescomplemented
76D4×C173C22×C4116yescomplemented
77C2×C4×C173C22×C416yescomplemented
78C22×C173—224no—
79C22×C173C4×D414yescomplemented
80C22×C173C23×C413yes—
81C22×C173C23×C418yescomplemented
82C2×D173—224no—
83C2×D173—248no—
84C2×D173—28no—
85C2×D173—28no—
86C2×D173C4×D4112yescomplemented
87C2×D173C23×C413yes—
88C2×D173C23×C418yescomplemented
89C2×D173C22×D416yes—
90C2×D173C2×#4116yes—
91C2×D173C2511yescharacteristic
92#10621924—28no—
93#10621924—24no—
94#10621924C22×D418yesdirect product factor
95C4×C173C23×C414yescomplemented
96Dic173C23×C414yescomplemented
97C2×C173—28no—
98C2×C173C2×C4×D416yescomplemented
99C2×C173C24×C411yescharacteristic
100D173—28no—
101D173C2×C4×D416yescomplemented
102D173C23×D411yescharacteristic
103D173C22×#4111yescharacteristic
104C173C22×C4×D411yescharacteristic, complemented, Sylow
105C22×C4×D4—1731nomaximal, Sylow
106C23×D4—1731no—
107C24×C4—1732no—
108C2×C4×D4—17324no—
109C22×C4⋊C4—1731no—
110C22×C22⋊C4—1732no—
111C22×C42—1731no—
112C25—1732no—
113C22×D4—1732no—
114C22×D4—17312no—
115C22×D4—17312no—
116C22×D4—1731no—
117C22×D4#1062192411yescharacteristic, direct product factor
118C23×C4—34616no—
119C23×C4—1731no—
120C23×C4—1732no—
121C23×C4—17324no—
122C23×C4—1731no—
123C23×C4—1731no—
124C4×D4—17364no—
125C2×C4⋊C4—17312no—
126C2×C22⋊C4—17324no—
127C2×C42—17312no—
128C24—3468no—
129C24—3468no—
130C24—1732no—
131C24—17312no—
132C24—17312no—
133C24—1731no—
134C24C2×#1062192412yescomplemented
135C2×D4—17324no—
136C2×D4—17348no—
137C2×D4—17316no—
138C2×D4—17312no—
139C2×D4C2×#10621924112yesdirect product factor
140C22×C4—3468no—
141C22×C4—3464no—
142C22×C4—34696no—
143C22×C4—1736no—
144C22×C4—17312no—
145C22×C4—17332no—
146C22×C4—1738no—
147C22×C4—1736no—
148C22×C4—1731no—
149C22×C4—1736no—
150C22×C4—1736no—
151C22×C4C2×#1062192411yescharacteristic, complemented
152C4⋊C4—17316no—
153C22⋊C4—17332no—
154C42—17316no—
155C23—28no—
156C23—34624no—
157C23—34624no—
158C23—34648no—
159C23—3468no—
160C23—17312no—
161C23—17324no—
162C23—1738no—
163C23—1734no—
164C23—1734no—
165C23—1733no—
166C23—1733no—
167C23C22×#1062192411yescharacteristic
168C23C22×#10621924112yescomplemented
169D4—17332no—
170D4—17316no—
171D4C22×#10621924116yesdirect product factor
172C2×C4—34624no—
173C2×C4—34612no—
174C2×C4—34664no—
175C2×C4—1734no—
176C2×C4—1738no—
177C2×C4—17324no—
178C2×C4—1734no—
179C2×C4—1736no—
180C2×C4—17312no—
181C2×C4—1734no—
182C2×C4C22×#1062192416yescomplemented
183C22—224no—
184C22—3468no—
185C22—34648no—
186C22—3468no—
187C22—34624no—
188C22—1738no—
189C22—1736no—
190C22—1736no—
191C22—17312no—
192C22—1731no—
193C22—1733no—
194C22D4×#1062192414yesdirect product factor
195C22C23×#1062192413yes—
196C22C23×#1062192418yescomplemented
197C4—3468no—
198C4—3464no—
199C4—1738no—
200C4—1734no—
201C4C23×#1062192414yescomplemented
202C2—28no—
203C2—3468no—
204C2—1736no—
205C2—1731no—
206C2—1731no—
207C2C2×D4×#1062192416yesdirect product factor
208C2C24×#1062192411yescharacteristic
subgroupG/H
centerC23C22×#10621924
commutatorC2×C173C24×C4
FrattiniC2C24×#10621924
FittingC22×D4×C173C4
socleC23×C173C22×C4
prime radicalC22×D4×D173C2
cosocleC2×D173C25
hypercenterC22×D4#10621924
hypocenterC173C22×C4×D4
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×#10621924C2semidirect
2. C2×D4×#10621924C2direct (N × H)
3. C22×C4⋊#10621924C2semidirect
4. C22×C22⋊#10621924C2semidirect
5. C22×C4×#10621924C2semidirect
11. C23×#10621924C22semidirect
12. D4×#10621924C22direct (N × H)
13. C2×C4⋊#10621924C22semidirect
14. C2×C22⋊#10621924C22semidirect
15. C2×C4×#10621924C22semidirect
17. C22×D692C4semidirect
18. C22×C22⋊D173C4semidirect
19. C2×D4×D173C4semidirect
22. C22×D4×C173C4semidirect
26. C22×#10621924D4direct (N × H)
30. C22×#10621924C23semidirect
31. C4⋊#10621924C23semidirect
32. C22⋊#10621924C23semidirect
33. C4×#10621924C23semidirect
36. C23×D173C2×C4semidirect
37. C23×D173C2×C4semidirect
40. C2×D692C2×C4semidirect
41. C2×C22⋊D173C2×C4semidirect
42. D4×D173C2×C4semidirect
44. C22×Dic173C2×C4semidirect
45. C2×C4×D173C2×C4semidirect
47. C24×C173C2×C4semidirect
48. C2×D4×C173C2×C4semidirect
49. C22×C4×C173C2×C4semidirect
53. C2×#10621924C2×D4direct (N × H)
62. C22×D173C22×C4semidirect
63. C22×D173C22×C4semidirect
68. D692C22×C4semidirect
69. C22⋊D173C22×C4semidirect
70. C2×Dic173C22×C4semidirect
71. C4×D173C22×C4semidirect
75. C23×C173C22×C4semidirect
76. D4×C173C22×C4semidirect
77. C2×C4×C173C22×C4semidirect
79. C22×C173C4×D4semidirect
81. C22×C173C23×C4semidirect
86. C2×D173C4×D4semidirect
88. C2×D173C23×C4semidirect
94. #10621924C22×D4direct (N × H)
95. C4×C173C23×C4semidirect
96. Dic173C23×C4semidirect
98. C2×C173C2×C4×D4semidirect
101. D173C2×C4×D4semidirect
104. C173C22×C4×D4semidirect
117. C22×D4#10621924direct (N × H)
134. C24C2×#10621924semidirect
139. C2×D4C2×#10621924direct (N × H)
151. C22×C4C2×#10621924semidirect
168. C23C22×#10621924semidirect
171. D4C22×#10621924direct (N × H)
182. C2×C4C22×#10621924semidirect
194. C22D4×#10621924direct (N × H)
196. C22C23×#10621924semidirect
201. C4C23×#10621924semidirect
207. C2C2×D4×#10621924direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#106219242
C2×D4×#106219242
C22×C4⋊#106219242
C22×C22⋊#106219242
C22×C4×#106219242
C22×D4×D1732
C22×C4×D4173