Finite Group Databasebeta

← C22×#16862219 #30679182

Subgroups — C22×#16862219 #30679182

Order 16768.

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×D524C211yesmaximal, characteristic, complemented
2C23×C22⋊D131C211yesmaximal, characteristic, complemented
3C23×C22⋊D131C211yesmaximal, characteristic, complemented
4C23×C4×D131C211yesmaximal, characteristic, complemented
5C2×#16862219C2124yesmaximal, direct product factor
6C22×Dic131⋊C4C211yesmaximal, characteristic, complemented
7C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C131C211yesmaximal, characteristic, complemented
9C24×D131C2211yescharacteristic
10C24×D131C2211yescharacteristic
11C22×D524—28no—
12C22×D524C22112yescomplemented
13C22×C22⋊D131—28no—
14C22×C22⋊D131—28no—
15C22×C22⋊D131C22112yescomplemented
16C22×C22⋊D131C22112yescomplemented
17C23×Dic131C2211yescharacteristic
18C23×Dic131C2211yescharacteristic
19C22×C4×D131—28no—
20C22×C4×D131C22112yescomplemented
21#16862219C22164yesdirect product factor
22C2×Dic131⋊C4C22112yescomplemented
23C2×(C2×D131)⋊C4C22112yescomplemented
24C25×C131C2211yescharacteristic
25C23×C4×C131C2211yescharacteristic
26C23×C4×C131C2211yescharacteristic
27C2×C22⋊C4×C131C22112yescomplemented
28C23×D131—24no—
29C23×D131—24no—
30C23×D131—24no—
31C23×D131—24no—
32C23×D131—24no—
33C23×D131D418yescomplemented
34C23×D131C2316yes—
35C23×D131C2316yes—
36C2×D524—248no—
37C2×D524C23116yescomplemented
38C2×C22⋊D131—248no—
39C2×C22⋊D131—248no—
40C2×C22⋊D131C23116yescomplemented
41C2×C22⋊D131C23116yescomplemented
42C22×Dic131—24no—
43C22×Dic131D418yescomplemented
44C22×Dic131C2316yes—
45C22×Dic131C2316yes—
46C2×C4×D131—248no—
47C2×C4×D131C23116yescomplemented
48Dic131⋊C4C23116yescomplemented
49(C2×D131)⋊C4C23116yescomplemented
50C24×C131—24no—
51C24×C131—24no—
52C24×C131—24no—
53C24×C131C2316yes—
54C24×C131C2311yescharacteristic
55C22×C4×C131—24no—
56C22×C4×C131—24no—
57C22×C4×C131C2316yes—
58C22×C4×C131C2316yes—
59C22⋊C4×C131C23116yescomplemented
60C22×D131—232no—
61C22×D131—212no—
62C22×D131—212no—
63C22×D131—212no—
64C22×D131—212no—
65C22×D131—212no—
66C22×D131—416no—
67C22×D131C2×D4124yescomplemented
68C22×D131C2414yes—
69C22×D131C2414yes—
70D524—232no—
71C22⋊D131—232no—
72C22⋊D131—232no—
73C2×Dic131—212no—
74C2×Dic131C2×D4124yescomplemented
75C2×Dic131C2414yes—
76C2×Dic131C2414yes—
77C4×D131—232no—
78C23×C131—212no—
79C23×C131—212no—
80C23×C131—212no—
81C23×C131—416no—
82C23×C131C2×D414yes—
83C23×C131C2×D414yes—
84C23×C131#4114yes—
85C23×C131C2414yes—
86C23×C131C2413yes—
87C2×C4×C131—212no—
88C2×C4×C131—212no—
89C2×C4×C131C2414yes—
90C2×C4×C131C2414yes—
91C22×C131—24no—
92C22×C131—24no—
93C22×C131—24no—
94C22×C131—424no—
95C22×C131#121116yescomplemented
96C22×C131C22×D416yes—
97C22×C131C22×D416yes—
98C22×C131C2×#4116yes—
99C22×C131C2511yescharacteristic
100C2×D131—248no—
101C2×D131—24no—
102C2×D131—24no—
103C2×D131—24no—
104C2×D131—24no—
105C2×D131—24no—
106C2×D131—424no—
107C2×D131C22×D418yescomplemented
108C4×C131—24no—
109C4×C131—24no—
110Dic131—24no—
111Dic131C22×D418yescomplemented
112C2×C131—44no—
113C2×C131C2×#121112yescomplemented
114C2×C131C23×D411yescharacteristic
115C2×C131C23×D411yescharacteristic
116C2×C131C22×#4111yescharacteristic
117D131—28no—
118D131—44no—
119C131C22×#12111yescharacteristic, complemented, Sylow
120C22×#121—1311nomaximal, Sylow
121C23×D4—1311no—
122C23×D4—1311no—
123C23×D4—1311no—
124C24×C4—1311no—
125C2×#121—13124no—
126C22×C4⋊C4—1311no—
127C22×C22⋊C4—1311no—
128C22×C22⋊C4D13111yescharacteristic, complemented
129C25—1311no—
130C25—1311no—
131C25C2×D13111yescharacteristic
132C22×D4—2628no—
133C22×D4—2628no—
134C22×D4—2628no—
135C22×D4—13112no—
136C22×D4—13112no—
137C22×D4—13112no—
138C23×C4—2628no—
139C23×C4—1311no—
140C23×C4—13112no—
141C23×C4—1311no—
142C23×C4C2×D13111yescharacteristic
143C23×C4C2×D13111yescharacteristic
144#121—13164no—
145C2×C4⋊C4—13112no—
146C2×C22⋊C4—13112no—
147C2×C22⋊C4C2×D131112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—2624no—
152C24—2624no—
153C24—2624no—
154C24—2624no—
155C24—2624no—
156C24—1316no—
157C24—1318no—
158C24—1316no—
159C24C22×D13116yes—
160C24C22×D13111yescharacteristic
161C2×D4—26248no—
162C2×D4—26248no—
163C2×D4—26248no—
164C2×D4—13116no—
165C2×D4—13116no—
166C2×D4—13116no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—26248no—
170C22×C4—2624no—
171C22×C4—1316no—
172C22×C4—13116no—
173C22×C4—1318no—
174C22×C4—1316no—
175C22×C4C22×D13116yes—
176C22×C4C22×D13116yes—
177C4⋊C4—13116no—
178C22⋊C4—13116no—
179C22⋊C4C22×D131116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—26232no—
185C23—26212no—
186C23—26212no—
187C23—26212no—
188C23—26212no—
189C23—26212no—
190C23—52416no—
191C23—1314no—
192C23—13124no—
193C23—1314no—
194C23D4×D13114yes—
195C23D4×D13114yes—
196C23#1452351314yes—
197C23C23×D13114yes—
198C23C23×D13113yes—
199D4—26232no—
200D4—26232no—
201D4—26232no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—26232no—
205C2×C4—26212no—
206C2×C4—1314no—
207C2×C4—13124no—
208C2×C4—1314no—
209C2×C4C23×D13114yes—
210C2×C4C23×D13114yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—2624no—
216C22—2624no—
217C22—26248no—
218C22—2624no—
219C22—2624no—
220C22—2624no—
221C22—52424no—
222C22—1318no—
223C22#16862219116yesdirect product factor
224C22C2×D4×D13116yes—
225C22C2×D4×D13116yes—
226C22C2×#1452351316yes—
227C22C24×D13111yescharacteristic
228C4—24no—
229C4—24no—
230C4—2624no—
231C4—1318no—
232C2—44no—
233C2—2628no—
234C2—5244no—
235C2C2×#16862219112yesdirect product factor
236C2C22×D4×D13111yescharacteristic
237C2C22×D4×D13111yescharacteristic
238C2C22×#1452351311yescharacteristic
subgroupG/H
centerC24C22×D131
commutator, cosocleC22×C131C25
FrattiniC22C24×D131
FittingC22×C22⋊C4×C131C2
socleC24×C131C23
hypercenterC22×C22⋊C4D131
hypocenterC131C22×#121
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D524C2semidirect
2. C23×C22⋊D131C2semidirect
3. C23×C22⋊D131C2semidirect
4. C23×C4×D131C2semidirect
5. C2×#16862219C2direct (N × H)
6. C22×Dic131⋊C4C2semidirect
7. C22×(C2×D131)⋊C4C2semidirect
8. C22×C22⋊C4×C131C2semidirect
12. C22×D524C22semidirect
15. C22×C22⋊D131C22semidirect
16. C22×C22⋊D131C22semidirect
20. C22×C4×D131C22semidirect
21. #16862219C22direct (N × H)
22. C2×Dic131⋊C4C22semidirect
23. C2×(C2×D131)⋊C4C22semidirect
27. C2×C22⋊C4×C131C22semidirect
33. C23×D131D4semidirect
37. C2×D524C23semidirect
40. C2×C22⋊D131C23semidirect
41. C2×C22⋊D131C23semidirect
43. C22×Dic131D4semidirect
47. C2×C4×D131C23semidirect
48. Dic131⋊C4C23semidirect
49. (C2×D131)⋊C4C23semidirect
59. C22⋊C4×C131C23semidirect
67. C22×D131C2×D4semidirect
74. C2×Dic131C2×D4semidirect
95. C22×C131#121semidirect
107. C2×D131C22×D4semidirect
111. Dic131C22×D4semidirect
113. C2×C131C2×#121semidirect
119. C131C22×#121semidirect
128. C22×C22⋊C4D131semidirect
147. C2×C22⋊C4C2×D131semidirect
179. C22⋊C4C22×D131semidirect
223. C22#16862219direct (N × H)
235. C2C2×#16862219direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5242
C23×C22⋊D1312
C23×C4×D1312
C2×#168622192
C22×Dic131⋊C42
C22×(C2×D131)⋊C42
C22×C22⋊C4×C1312
C22×#121131