Finite Group Databasebeta

← C22×#19149647 #36366778

Subgroups — C22×#19149647 #36366778

Order 22144.

4544 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 725 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
1C25×D173C211yesmaximal, characteristic, complemented
2C23×C22⋊D173C212yesmaximal, complemented
3C2×#19149647C2124yesmaximal, direct product factor
4C22×C22⋊Dic173C211yesmaximal, characteristic, complemented
5C22×#14583803C212yesmaximal, complemented
6C23×D4×C173C211yesmaximal, characteristic, complemented
7C24×D173—28no—
8C24×D173—216no—
9C24×D173C2212yes—
10C24×D173C22112yescomplemented
11C22×C22⋊D173—216no—
12C22×C22⋊D173C22124yescomplemented
13C23×Dic173C2212yescomplemented
14#19149647C22164yesdirect product factor
15C2×C22⋊Dic173C22112yescomplemented
16C2×#14583803C22124yescomplemented
17C25×C173C2211yescharacteristic
18C25×C173C2211yescharacteristic, complemented
19C22×D4×C173—28no—
20C22×D4×C173C22112yescomplemented
21C23×C4×C173C2211yescharacteristic, complemented
22C23×D173—2128no—
23C23×D173—296no—
24C23×D173—28no—
25C23×D173—28no—
26C23×D173—248no—
27C23×D173D4116yescomplemented
28C23×D173C23112yes—
29C23×D173C23116yescomplemented
30C2×C22⋊D173—296no—
31C2×C22⋊D173C23132yescomplemented
32C22×Dic173—28no—
33C22×Dic173C23112yescomplemented
34C22⋊Dic173C23116yescomplemented
35#14583803C23132yescomplemented
36C24×C173—28no—
37C24×C173—24no—
38C24×C173—28no—
39C24×C173D418yescomplemented
40C24×C173C2316yes—
41C24×C173C2311yescharacteristic
42C24×C173C2316yescomplemented
43C2×D4×C173—248no—
44C2×D4×C173C23116yescomplemented
45C22×C4×C173—24no—
46C22×C4×C173C2316yescomplemented
47C22×D173—2384no—
48C22×D173—264no—
49C22×D173—224no—
50C22×D173—224no—
51C22×D173—232no—
52C22×D173—264no—
53C22×D173C2×D4148yescomplemented
54C22×D173C2418yes—
55C22⋊D173—264no—
56C2×Dic173—224no—
57C2×Dic173C2418yescomplemented
58C23×C173—232no—
59C23×C173—224no—
60C23×C173—224no—
61C23×C173—212no—
62C23×C173—416no—
63C23×C173C2×D418yes—
64C23×C173C2×D414yes—
65C23×C173C2×D4124yescomplemented
66C23×C173C2414yes—
67C23×C173C2413yes—
68C23×C173C2414yescomplemented
69D4×C173—232no—
70C2×C4×C173—212no—
71C2×C4×C173C2414yescomplemented
72C22×C173—248no—
73C22×C173—28no—
74C22×C173—24no—
75C22×C173—28no—
76C22×C173—424no—
77C22×C173#120116yescomplemented
78C22×C173C22×D4112yes—
79C22×C173C22×D416yes—
80C22×C173C22×D418yescomplemented
81C22×C173C2511yescharacteristic
82C2×D173—2128no—
83C2×D173—296no—
84C2×D173—28no—
85C2×D173—28no—
86C2×D173C22×D4116yescomplemented
87C4×C173—24no—
88Dic173—28no—
89C2×C173—28no—
90C2×C173—44no—
91C2×C173C2×#120112yescomplemented
92C2×C173C23×D412yes—
93C2×C173C23×D411yescharacteristic
94D173—216no—
95C173C22×#12011yescharacteristic, complemented, Sylow
96C22×#120—1731nomaximal, Sylow
97C26—1731no—
98C23×D4—1732no—
99C23×D4D17311yescharacteristic, complemented
100C2×#120—17324no—
101C22×C22⋊C4—1731no—
102C22×C22⋊C4—1732no—
103C25—34616no—
104C25—3468no—
105C25—17312no—
106C25—1732no—
107C25C2×D17311yescharacteristic
108C25C2×D17311yescharacteristic, complemented
109C22×D4—28no—
110C22×D4—34616no—
111C22×D4—17324no—
112C22×D4C2×D173112yescomplemented
113C23×C4—1732no—
114C23×C4C2×D17311yescharacteristic, complemented
115#120—17364no—
116C2×C22⋊C4—17312no—
117C2×C22⋊C4—17324no—
118C24—28no—
119C24—28no—
120C24—24no—
121C24—346128no—
122C24—34696no—
123C24—3468no—
124C24—3468no—
125C24—34648no—
126C24—17316no—
127C24—17316no—
128C24—17312no—
129C24C22⋊D17318yescomplemented
130C24C22×D17316yes—
131C24C22×D17311yescharacteristic
132C24C22×D17316yescomplemented
133C2×D4—248no—
134C2×D4—34696no—
135C2×D4—17332no—
136C2×D4C22×D173116yescomplemented
137C22×C4—24no—
138C22×C4—3468no—
139C22×C4—17312no—
140C22×C4C22×D17316yescomplemented
141C22⋊C4—17316no—
142C22⋊C4—17332no—
143C23—232no—
144C23—224no—
145C23—224no—
146C23—212no—
147C23—416no—
148C23—346384no—
149C23—34664no—
150C23—34664no—
151C23—34624no—
152C23—34624no—
153C23—34632no—
154C23—17348no—
155C23—1738no—
156C23D4×D17318yes—
157C23C2×C22⋊D17314yes—
158C23C2×C22⋊D173124yescomplemented
159C23C23×D17314yes—
160C23C23×D17313yes—
161C23C23×D17314yescomplemented
162D4—232no—
163D4—34664no—
164C2×C4—212no—
165C2×C4—34624no—
166C2×C4—1738no—
167C2×C4C23×D17314yescomplemented
168C22—248no—
169C22—28no—
170C22—28no—
171C22—24no—
172C22—424no—
173C22—346128no—
174C22—34696no—
175C22—3468no—
176C22—3468no—
177C22—17316no—
178C22#19149647116yesdirect product factor
179C22C2×D4×D173112yes—
180C22C22×C22⋊D17316yes—
181C22C22×C22⋊D17318yescomplemented
182C22C24×D17311yescharacteristic
183C4—24no—
184C4—3468no—
185C2—28no—
186C2—44no—
187C2—34616no—
188C2C2×#19149647112yesdirect product factor
189C2C22×D4×D17312yes—
190C2C23×C22⋊D17311yescharacteristic
subgroupG/H
centerC24C22×D173
commutator, cosocleC22×C173C25
FrattiniC22C24×D173
FittingC23×D4×C173C2
socleC24×C173C23
hypercenterC23×D4D173
hypocenterC173C22×#120
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C25×D173C2semidirect
2. C23×C22⋊D173C2semidirect
3. C2×#19149647C2direct (N × H)
4. C22×C22⋊Dic173C2semidirect
5. C22×#14583803C2semidirect
6. C23×D4×C173C2semidirect
10. C24×D173C22semidirect
12. C22×C22⋊D173C22semidirect
13. C23×Dic173C22semidirect
14. #19149647C22direct (N × H)
15. C2×C22⋊Dic173C22semidirect
16. C2×#14583803C22semidirect
18. C25×C173C22semidirect
20. C22×D4×C173C22semidirect
21. C23×C4×C173C22semidirect
27. C23×D173D4semidirect
29. C23×D173C23semidirect
31. C2×C22⋊D173C23semidirect
33. C22×Dic173C23semidirect
34. C22⋊Dic173C23semidirect
35. #14583803C23semidirect
39. C24×C173D4semidirect
42. C24×C173C23semidirect
44. C2×D4×C173C23semidirect
46. C22×C4×C173C23semidirect
53. C22×D173C2×D4semidirect
57. C2×Dic173C24semidirect
65. C23×C173C2×D4semidirect
68. C23×C173C24semidirect
71. C2×C4×C173C24semidirect
77. C22×C173#120semidirect
80. C22×C173C22×D4semidirect
86. C2×D173C22×D4semidirect
91. C2×C173C2×#120semidirect
95. C173C22×#120semidirect
99. C23×D4D173semidirect
108. C25C2×D173semidirect
112. C22×D4C2×D173semidirect
114. C23×C4C2×D173semidirect
129. C24C22⋊D173semidirect
132. C24C22×D173semidirect
136. C2×D4C22×D173semidirect
140. C22×C4C22×D173semidirect
158. C23C2×C22⋊D173semidirect
161. C23C23×D173semidirect
167. C2×C4C23×D173semidirect
178. C22#19149647direct (N × H)
181. C22C22×C22⋊D173semidirect
188. C2C2×#19149647direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D1732
C23×C22⋊D1732
C2×#191496472
C22×C22⋊Dic1732
C22×#145838032
C23×D4×C1732
C22×#120173