Finite Group Databasebeta

← C22×#19128805 #36337359

Subgroups — C22×#19128805 #36337359

Order 21376.

2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 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
1C23×D668C211yesmaximal, characteristic, complemented
2C23×C22⋊D167C211yesmaximal, characteristic, complemented
3C23×C22⋊D167C211yesmaximal, characteristic, complemented
4C23×C4×D167C211yesmaximal, characteristic, complemented
5C2×#19128805C2124yesmaximal, direct product factor
6C22×Dic167⋊C4C211yesmaximal, characteristic, complemented
7C22×(C2×D167)⋊C4C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C167C211yesmaximal, characteristic, complemented
9C24×D167C2211yescharacteristic
10C24×D167C2211yescharacteristic
11C22×D668—28no—
12C22×D668C22112yescomplemented
13C22×C22⋊D167—28no—
14C22×C22⋊D167—28no—
15C22×C22⋊D167C22112yescomplemented
16C22×C22⋊D167C22112yescomplemented
17C23×Dic167C2211yescharacteristic
18C23×Dic167C2211yescharacteristic
19C22×C4×D167—28no—
20C22×C4×D167C22112yescomplemented
21#19128805C22164yesdirect product factor
22C2×Dic167⋊C4C22112yescomplemented
23C2×(C2×D167)⋊C4C22112yescomplemented
24C25×C167C2211yescharacteristic
25C23×C4×C167C2211yescharacteristic
26C23×C4×C167C2211yescharacteristic
27C2×C22⋊C4×C167C22112yescomplemented
28C23×D167—24no—
29C23×D167—24no—
30C23×D167—24no—
31C23×D167—24no—
32C23×D167—24no—
33C23×D167D418yescomplemented
34C23×D167C2316yes—
35C23×D167C2316yes—
36C2×D668—248no—
37C2×D668C23116yescomplemented
38C2×C22⋊D167—248no—
39C2×C22⋊D167—248no—
40C2×C22⋊D167C23116yescomplemented
41C2×C22⋊D167C23116yescomplemented
42C22×Dic167—24no—
43C22×Dic167D418yescomplemented
44C22×Dic167C2316yes—
45C22×Dic167C2316yes—
46C2×C4×D167—248no—
47C2×C4×D167C23116yescomplemented
48Dic167⋊C4C23116yescomplemented
49(C2×D167)⋊C4C23116yescomplemented
50C24×C167—24no—
51C24×C167—24no—
52C24×C167—24no—
53C24×C167C2316yes—
54C24×C167C2311yescharacteristic
55C22×C4×C167—24no—
56C22×C4×C167—24no—
57C22×C4×C167C2316yes—
58C22×C4×C167C2316yes—
59C22⋊C4×C167C23116yescomplemented
60C22×D167—232no—
61C22×D167—212no—
62C22×D167—212no—
63C22×D167—212no—
64C22×D167—212no—
65C22×D167—212no—
66C22×D167—416no—
67C22×D167C2×D4124yescomplemented
68C22×D167C2414yes—
69C22×D167C2414yes—
70D668—232no—
71C22⋊D167—232no—
72C22⋊D167—232no—
73C2×Dic167—212no—
74C2×Dic167C2×D4124yescomplemented
75C2×Dic167C2414yes—
76C2×Dic167C2414yes—
77C4×D167—232no—
78C23×C167—212no—
79C23×C167—212no—
80C23×C167—212no—
81C23×C167—416no—
82C23×C167C2×D414yes—
83C23×C167C2×D414yes—
84C23×C167#4114yes—
85C23×C167C2414yes—
86C23×C167C2413yes—
87C2×C4×C167—212no—
88C2×C4×C167—212no—
89C2×C4×C167C2414yes—
90C2×C4×C167C2414yes—
91C22×C167—24no—
92C22×C167—24no—
93C22×C167—24no—
94C22×C167—424no—
95C22×C167#121116yescomplemented
96C22×C167C22×D416yes—
97C22×C167C22×D416yes—
98C22×C167C2×#4116yes—
99C22×C167C2511yescharacteristic
100C2×D167—248no—
101C2×D167—24no—
102C2×D167—24no—
103C2×D167—24no—
104C2×D167—24no—
105C2×D167—24no—
106C2×D167—424no—
107C2×D167C22×D418yescomplemented
108C4×C167—24no—
109C4×C167—24no—
110Dic167—24no—
111Dic167C22×D418yescomplemented
112C2×C167—44no—
113C2×C167C2×#121112yescomplemented
114C2×C167C23×D411yescharacteristic
115C2×C167C23×D411yescharacteristic
116C2×C167C22×#4111yescharacteristic
117D167—28no—
118D167—44no—
119C167C22×#12111yescharacteristic, complemented, Sylow
120C22×#121—1671nomaximal, Sylow
121C23×D4—1671no—
122C23×D4—1671no—
123C23×D4—1671no—
124C24×C4—1671no—
125C2×#121—16724no—
126C22×C4⋊C4—1671no—
127C22×C22⋊C4—1671no—
128C22×C22⋊C4D16711yescharacteristic, complemented
129C25—1671no—
130C25—1671no—
131C25C2×D16711yescharacteristic
132C22×D4—3348no—
133C22×D4—3348no—
134C22×D4—3348no—
135C22×D4—16712no—
136C22×D4—16712no—
137C22×D4—16712no—
138C23×C4—3348no—
139C23×C4—1671no—
140C23×C4—16712no—
141C23×C4—1671no—
142C23×C4C2×D16711yescharacteristic
143C23×C4C2×D16711yescharacteristic
144#121—16764no—
145C2×C4⋊C4—16712no—
146C2×C22⋊C4—16712no—
147C2×C22⋊C4C2×D167112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—3344no—
152C24—3344no—
153C24—3344no—
154C24—3344no—
155C24—3344no—
156C24—1676no—
157C24—1678no—
158C24—1676no—
159C24C22×D16716yes—
160C24C22×D16711yescharacteristic
161C2×D4—33448no—
162C2×D4—33448no—
163C2×D4—33448no—
164C2×D4—16716no—
165C2×D4—16716no—
166C2×D4—16716no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—33448no—
170C22×C4—3344no—
171C22×C4—1676no—
172C22×C4—16716no—
173C22×C4—1678no—
174C22×C4—1676no—
175C22×C4C22×D16716yes—
176C22×C4C22×D16716yes—
177C4⋊C4—16716no—
178C22⋊C4—16716no—
179C22⋊C4C22×D167116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—33412no—
185C23—33412no—
186C23—33432no—
187C23—33412no—
188C23—33412no—
189C23—33412no—
190C23—66816no—
191C23—1674no—
192C23—16724no—
193C23—1674no—
194C23D4×D16714yes—
195C23D4×D16714yes—
196C23#1458195014yes—
197C23C23×D16714yes—
198C23C23×D16713yes—
199D4—33432no—
200D4—33432no—
201D4—33432no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—33432no—
205C2×C4—33412no—
206C2×C4—1674no—
207C2×C4—16724no—
208C2×C4—1674no—
209C2×C4C23×D16714yes—
210C2×C4C23×D16714yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—33448no—
216C22—3344no—
217C22—3344no—
218C22—3344no—
219C22—3344no—
220C22—3344no—
221C22—66824no—
222C22—1678no—
223C22#19128805116yesdirect product factor
224C22C2×D4×D16716yes—
225C22C2×D4×D16716yes—
226C22C2×#1458195016yes—
227C22C24×D16711yescharacteristic
228C4—24no—
229C4—24no—
230C4—3344no—
231C4—1678no—
232C2—44no—
233C2—3348no—
234C2—6684no—
235C2C2×#19128805112yesdirect product factor
236C2C22×D4×D16711yescharacteristic
237C2C22×D4×D16711yescharacteristic
238C2C22×#1458195011yescharacteristic
subgroupG/H
centerC24C22×D167
commutator, cosocleC22×C167C25
FrattiniC22C24×D167
FittingC22×C22⋊C4×C167C2
socleC24×C167C23
hypercenterC22×C22⋊C4D167
hypocenterC167C22×#121
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D668C2semidirect
2. C23×C22⋊D167C2semidirect
3. C23×C22⋊D167C2semidirect
4. C23×C4×D167C2semidirect
5. C2×#19128805C2direct (N × H)
6. C22×Dic167⋊C4C2semidirect
7. C22×(C2×D167)⋊C4C2semidirect
8. C22×C22⋊C4×C167C2semidirect
12. C22×D668C22semidirect
15. C22×C22⋊D167C22semidirect
16. C22×C22⋊D167C22semidirect
20. C22×C4×D167C22semidirect
21. #19128805C22direct (N × H)
22. C2×Dic167⋊C4C22semidirect
23. C2×(C2×D167)⋊C4C22semidirect
27. C2×C22⋊C4×C167C22semidirect
33. C23×D167D4semidirect
37. C2×D668C23semidirect
40. C2×C22⋊D167C23semidirect
41. C2×C22⋊D167C23semidirect
43. C22×Dic167D4semidirect
47. C2×C4×D167C23semidirect
48. Dic167⋊C4C23semidirect
49. (C2×D167)⋊C4C23semidirect
59. C22⋊C4×C167C23semidirect
67. C22×D167C2×D4semidirect
74. C2×Dic167C2×D4semidirect
95. C22×C167#121semidirect
107. C2×D167C22×D4semidirect
111. Dic167C22×D4semidirect
113. C2×C167C2×#121semidirect
119. C167C22×#121semidirect
128. C22×C22⋊C4D167semidirect
147. C2×C22⋊C4C2×D167semidirect
179. C22⋊C4C22×D167semidirect
223. C22#19128805direct (N × H)
235. C2C2×#19128805direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D6682
C23×C22⋊D1672
C23×C4×D1672
C2×#191288052
C22×Dic167⋊C42
C22×(C2×D167)⋊C42
C22×C22⋊C4×C1672
C22×#121167