Finite Group Databasebeta

← #120×#10621924 #36370026

Subgroups — #120×#10621924 #36370026

Order 22144.

1224 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 210 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×#10621924C211yesmaximal, characteristic, complemented
2C2×D4×#10621924C213yesmaximal, complemented
3#26012439C213yesmaximal, complemented
4#26012438C213yesmaximal, complemented
5#26012432C211yesmaximal, characteristic, complemented
6C22⋊C4×#10621924C213yesmaximal, complemented
7#120×D173C211yesmaximal, characteristic
8C23×#10621924—212no—
9C23×#10621924C2213yes—
10C23×#10621924C2213yes—
11C23×#10621924C2211yescharacteristic, complemented
12D4×#10621924—212no—
13C2×C4⋊#10621924C2213yescomplemented
14C2×C22⋊#10621924C2213yes—
15C2×C22⋊#10621924C2213yes—
16C2×C22⋊#10621924C2213yescomplemented
17C2×C4×#10621924C2213yescomplemented
18(C2×#10621924)⋊C4C2216yescomplemented
19C24×D173C2211yescharacteristic
20C2×D4×D173C2213yes—
21#19149649C413yescomplemented
22#19149648C411yescharacteristic, complemented
23#19149647C413yescomplemented
24C22⋊C4×D173C2213yes—
25#120×C173C411yescharacteristic, complemented
26C22×#10621924—232no—
27C22×#10621924—212no—
28C22×#10621924—212no—
29C22×#10621924—26no—
30C22×#10621924—26no—
31C22×#10621924D4112yescomplemented
32C22×#10621924C2313yes—
33C22×#10621924C2311yescharacteristic
34C22×#10621924C2313yes—
35C22×#10621924C2311yescharacteristic
36C4⋊#10621924—26no—
37C22⋊#10621924—26no—
38C22⋊#10621924—26no—
39C22⋊#10621924—26no—
40C4×#10621924—26no—
41C23×D173—26no—
42C23×D173—26no—
43C23×D173C2×C413yescomplemented
44C23×D173C2313yes—
45C23×D173C2311yescharacteristic
46C2×D692C2×C413yescomplemented
47C2×C22⋊D173C2×C413yescomplemented
48C2×C22⋊D173C2×C413yescomplemented
49D4×D173—26no—
50D4×D173—26no—
51C2×C4×D173C2313yes—
52C22⋊Dic173C2×C413yescomplemented
53#14583803C2×C416yescomplemented
54C24×C173C2×C411yescharacteristic, complemented
55C2×D4×C173C2×C413yescomplemented
56C22⋊C4×C173C2×C413yescomplemented
57C2×#10621924—224no—
58C2×#10621924—26no—
59C2×#10621924—224no—
60C2×#10621924—23no—
61C2×#10621924—23no—
62C2×#10621924—26no—
63C2×#10621924—44no—
64C2×#10621924C2×D416yes—
65C2×#10621924C2×D416yes—
66C22×D173—224no—
67C22×D173—28no—
68C22×D173—26no—
69C22×D173—212no—
70C22×D173—26no—
71C22×D173—26no—
72C22×D173—23no—
73C22×D173—23no—
74C22×D173C22×C413yes—
75C22×D173C22×C411yescharacteristic, complemented
76C22×D173C2×D416yes—
77C22×D173#4116yes—
78C22×D173C2411yescharacteristic
79D692—26no—
80C22⋊D173—26no—
81C22⋊D173—26no—
82C2×Dic173C22×C413yescomplemented
83C4×D173—23no—
84C4×D173—23no—
85C23×C173—26no—
86C23×C173C22×C413yes—
87C23×C173C22×C411yescharacteristic, complemented
88D4×C173—26no—
89C2×C4×C173C22×C413yescomplemented
90C22×C173—28no—
91C22×C173—26no—
92C22×C173—23no—
93C22×C173C4×D416yescomplemented
94C22×C173C23×C411yescharacteristic
95C2×D173—224no—
96C2×D173—26no—
97C2×D173—26no—
98C2×D173—26no—
99C2×D173—212no—
100C2×D173—23no—
101C2×D173—41no—
102C2×D173—43no—
103C2×D173C4×D416yescomplemented
104C2×D173C22×D413yes—
105C2×D173C2×#4113yes—
106#10621924—26no—
107#10621924—41no—
108#10621924—43no—
109#10621924#12014yesdirect product factor
110C4×C173—23no—
111Dic173—23no—
112C2×C173—26no—
113C2×C173—41no—
114C2×C173C2×C4×D413yes—
115D173—26no—
116D173—41no—
117D173C2×#12011yescharacteristic
118D173#52513yes—
119C173C4×#12011yescharacteristic, complemented, Sylow
120C4×#120—1731nomaximal, Sylow
121C24×C4—1731no—
122C2×#120—1731no—
123C2×C4×D4—1733no—
124#386—1733no—
125#385—1733no—
126#379—1731no—
127C4×C22⋊C4—1733no—
128C25—1731no—
129C22×D4—1733no—
130C23×C4—34612no—
131C23×C4—1731no—
132C23×C4—1733no—
133C23×C4—1733no—
134#120—1731no—
135#120—1733no—
136#120—1733no—
137#120#1062192411yescharacteristic, direct product factor
138C4×D4—34612no—
139C2×C4⋊C4—1733no—
140C2×C22⋊C4—1733no—
141C2×C22⋊C4—1733no—
142C2×C22⋊C4—1733no—
143C2×C22⋊C4—1733no—
144C2×C42—1733no—
145#95—1736no—
146C24—3466no—
147C24—3466no—
148C24—1733no—
149C24—1731no—
150C24—1733no—
151C24C2×#1062192411yescharacteristic, complemented
152C2×D4—3466no—
153C2×D4—3466no—
154C2×D4—1733no—
155C2×D4—1733no—
156C2×D4—1733no—
157C2×D4C2×#1062192413yescomplemented
158C22×C4—34632no—
159C22×C4—3466no—
160C22×C4—34612no—
161C22×C4—34612no—
162C22×C4—3466no—
163C22×C4—1733no—
164C22×C4—1731no—
165C22×C4—17312no—
166C22×C4—1733no—
167C22×C4—1733no—
168C22×C4—1731no—
169C4⋊C4—3466no—
170C22⋊C4—3466no—
171C22⋊C4—3466no—
172C22⋊C4—3466no—
173C22⋊C4—1733no—
174C22⋊C4—1736no—
175C22⋊C4C2×#1062192413yescomplemented
176C42—3466no—
177C23—26no—
178C23—3468no—
179C23—34624no—
180C23—3466no—
181C23—34612no—
182C23—3466no—
183C23—3463no—
184C23—3463no—
185C23—3466no—
186C23—1731no—
187C23—1736no—
188C23—1736no—
189C23—1733no—
190C23—1731no—
191C23C22×#1062192413yes—
192C23C22×#1062192411yescharacteristic, complemented
193D4—26no—
194D4—3466no—
195D4—3466no—
196D4—3466no—
197C2×C4—3463no—
198C2×C4—3463no—
199C2×C4—34624no—
200C2×C4—3466no—
201C2×C4—34624no—
202C2×C4—3463no—
203C2×C4—3463no—
204C2×C4—3466no—
205C2×C4—1736no—
206C2×C4—1733no—
207C2×C4—1736no—
208C2×C4—6924no—
209C2×C4C22×#1062192413yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—34624no—
214C22—3466no—
215C22—34612no—
216C22—3466no—
217C22—3466no—
218C22—3463no—
219C22—1736no—
220C22—1733no—
221C22—1733no—
222C22—6921no—
223C22—6923no—
224C22D4×#1062192416yescomplemented
225C22C23×#1062192411yescharacteristic
226C4—23no—
227C4—3466no—
228C4—3463no—
229C4—1734no—
230C4—6921no—
231C4—6923no—
232C2—26no—
233C2—41no—
234C2—3466no—
235C2—1731no—
236C2—1733no—
237C2—6921no—
238C2C2×D4×#1062192413yes—
subgroupG/H
center, FrattiniC22C23×#10621924
commutator, socleC22×C173C23×C4
Fitting#120×C173C4
prime radical#120×D173C2
cosocleC22×D173C24
hypercenter#120#10621924
hypocenterC173C4×#120
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#10621924C2semidirect
2. C2×D4×#10621924C2semidirect
3. #26012439C2semidirect
4. #26012438C2semidirect
5. #26012432C2semidirect
6. C22⋊C4×#10621924C2semidirect
11. C23×#10621924C22semidirect
13. C2×C4⋊#10621924C22semidirect
16. C2×C22⋊#10621924C22semidirect
17. C2×C4×#10621924C22semidirect
18. (C2×#10621924)⋊C4C22semidirect
21. #19149649C4semidirect
22. #19149648C4semidirect
23. #19149647C4semidirect
25. #120×C173C4semidirect
31. C22×#10621924D4semidirect
43. C23×D173C2×C4semidirect
46. C2×D692C2×C4semidirect
47. C2×C22⋊D173C2×C4semidirect
48. C2×C22⋊D173C2×C4semidirect
52. C22⋊Dic173C2×C4semidirect
53. #14583803C2×C4semidirect
54. C24×C173C2×C4semidirect
55. C2×D4×C173C2×C4semidirect
56. C22⋊C4×C173C2×C4semidirect
75. C22×D173C22×C4semidirect
82. C2×Dic173C22×C4semidirect
87. C23×C173C22×C4semidirect
89. C2×C4×C173C22×C4semidirect
93. C22×C173C4×D4semidirect
103. C2×D173C4×D4semidirect
109. #10621924#120direct (N × H)
119. C173C4×#120semidirect
137. #120#10621924direct (N × H)
151. C24C2×#10621924semidirect
157. C2×D4C2×#10621924semidirect
175. C22⋊C4C2×#10621924semidirect
192. C23C22×#10621924semidirect
209. C2×C4C22×#10621924semidirect
224. C22D4×#10621924semidirect

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
#260124392
#260124382
#260124322
C22⋊C4×#106219242
#120×D1732
C4×#120173