Finite Group Databasebeta

← C22×#17984061 #31794010

Subgroups — C22×#17984061 #31794010

Order 17536.

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×D548C211yesmaximal, characteristic, complemented
2C23×C22⋊D137C211yesmaximal, characteristic, complemented
3C23×C22⋊D137C211yesmaximal, characteristic, complemented
4C23×C4×D137C211yesmaximal, characteristic, complemented
5C2×#17984061C2124yesmaximal, direct product factor
6C22×#14555489C211yesmaximal, characteristic, complemented
7C22×#14555487C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C137C211yesmaximal, characteristic, complemented
9C24×D137C2211yescharacteristic
10C24×D137C2211yescharacteristic
11C22×D548—28no—
12C22×D548C22112yescomplemented
13C22×C22⋊D137—28no—
14C22×C22⋊D137—28no—
15C22×C22⋊D137C22112yescomplemented
16C22×C22⋊D137C22112yescomplemented
17C23×Dic137C2211yescharacteristic
18C23×Dic137C2211yescharacteristic
19C22×C4×D137—28no—
20C22×C4×D137C22112yescomplemented
21#17984061C22164yesdirect product factor
22C2×#14555489C22112yescomplemented
23C2×#14555487C22112yescomplemented
24C25×C137C2211yescharacteristic
25C23×C4×C137C2211yescharacteristic
26C23×C4×C137C2211yescharacteristic
27C2×C22⋊C4×C137C22112yescomplemented
28C23×D137—24no—
29C23×D137—24no—
30C23×D137—24no—
31C23×D137—24no—
32C23×D137—24no—
33C23×D137D418yescomplemented
34C23×D137C2316yes—
35C23×D137C2316yes—
36C2×D548—248no—
37C2×D548C23116yescomplemented
38C2×C22⋊D137—248no—
39C2×C22⋊D137—248no—
40C2×C22⋊D137C23116yescomplemented
41C2×C22⋊D137C23116yescomplemented
42C22×Dic137—24no—
43C22×Dic137D418yescomplemented
44C22×Dic137C2316yes—
45C22×Dic137C2316yes—
46C2×C4×D137—248no—
47C2×C4×D137C23116yescomplemented
48#14555489C23116yescomplemented
49#14555487C23116yescomplemented
50C24×C137—24no—
51C24×C137—24no—
52C24×C137—24no—
53C24×C137C2316yes—
54C24×C137C2311yescharacteristic
55C22×C4×C137—24no—
56C22×C4×C137—24no—
57C22×C4×C137C2316yes—
58C22×C4×C137C2316yes—
59C22⋊C4×C137C23116yescomplemented
60C22×D137—232no—
61C22×D137—212no—
62C22×D137—212no—
63C22×D137—212no—
64C22×D137—212no—
65C22×D137—212no—
66C22×D137—416no—
67C22×D137C2×D4124yescomplemented
68C22×D137C2414yes—
69C22×D137C2414yes—
70D548—232no—
71C22⋊D137—232no—
72C22⋊D137—232no—
73C2×Dic137—212no—
74C2×Dic137C2×D4124yescomplemented
75C2×Dic137C2414yes—
76C2×Dic137C2414yes—
77C4×D137—232no—
78C23×C137—212no—
79C23×C137—212no—
80C23×C137—212no—
81C23×C137—416no—
82C23×C137C2×D414yes—
83C23×C137C2×D414yes—
84C23×C137#4114yes—
85C23×C137C2414yes—
86C23×C137C2413yes—
87C2×C4×C137—212no—
88C2×C4×C137—212no—
89C2×C4×C137C2414yes—
90C2×C4×C137C2414yes—
91C22×C137—24no—
92C22×C137—24no—
93C22×C137—24no—
94C22×C137—424no—
95C22×C137#121116yescomplemented
96C22×C137C22×D416yes—
97C22×C137C22×D416yes—
98C22×C137C2×#4116yes—
99C22×C137C2511yescharacteristic
100C2×D137—248no—
101C2×D137—24no—
102C2×D137—24no—
103C2×D137—24no—
104C2×D137—24no—
105C2×D137—24no—
106C2×D137—424no—
107C2×D137C22×D418yescomplemented
108C4×C137—24no—
109C4×C137—24no—
110Dic137—24no—
111Dic137C22×D418yescomplemented
112C2×C137—44no—
113C2×C137C2×#121112yescomplemented
114C2×C137C23×D411yescharacteristic
115C2×C137C23×D411yescharacteristic
116C2×C137C22×#4111yescharacteristic
117D137—28no—
118D137—44no—
119C137C22×#12111yescharacteristic, complemented, Sylow
120C22×#121—1371nomaximal, Sylow
121C23×D4—1371no—
122C23×D4—1371no—
123C23×D4—1371no—
124C24×C4—1371no—
125C2×#121—13724no—
126C22×C4⋊C4—1371no—
127C22×C22⋊C4—1371no—
128C22×C22⋊C4D13711yescharacteristic, complemented
129C25—1371no—
130C25—1371no—
131C25C2×D13711yescharacteristic
132C22×D4—2748no—
133C22×D4—2748no—
134C22×D4—2748no—
135C22×D4—13712no—
136C22×D4—13712no—
137C22×D4—13712no—
138C23×C4—2748no—
139C23×C4—1371no—
140C23×C4—13712no—
141C23×C4—1371no—
142C23×C4C2×D13711yescharacteristic
143C23×C4C2×D13711yescharacteristic
144#121—13764no—
145C2×C4⋊C4—13712no—
146C2×C22⋊C4—13712no—
147C2×C22⋊C4C2×D137112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—2744no—
152C24—2744no—
153C24—2744no—
154C24—2744no—
155C24—2744no—
156C24—1376no—
157C24—1378no—
158C24—1376no—
159C24C22×D13716yes—
160C24C22×D13711yescharacteristic
161C2×D4—27448no—
162C2×D4—27448no—
163C2×D4—27448no—
164C2×D4—13716no—
165C2×D4—13716no—
166C2×D4—13716no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—27448no—
170C22×C4—2744no—
171C22×C4—1376no—
172C22×C4—13716no—
173C22×C4—1378no—
174C22×C4—1376no—
175C22×C4C22×D13716yes—
176C22×C4C22×D13716yes—
177C4⋊C4—13716no—
178C22⋊C4—13716no—
179C22⋊C4C22×D137116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—27432no—
185C23—27412no—
186C23—27412no—
187C23—27412no—
188C23—27412no—
189C23—27412no—
190C23—54816no—
191C23—1374no—
192C23—13724no—
193C23—1374no—
194C23D4×D13714yes—
195C23D4×D13714yes—
196C23#1455551014yes—
197C23C23×D13714yes—
198C23C23×D13713yes—
199D4—27432no—
200D4—27432no—
201D4—27432no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—27432no—
205C2×C4—27412no—
206C2×C4—1374no—
207C2×C4—13724no—
208C2×C4—1374no—
209C2×C4C23×D13714yes—
210C2×C4C23×D13714yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—2744no—
216C22—2744no—
217C22—27448no—
218C22—2744no—
219C22—2744no—
220C22—2744no—
221C22—54824no—
222C22—1378no—
223C22#17984061116yesdirect product factor
224C22C2×D4×D13716yes—
225C22C2×D4×D13716yes—
226C22C2×#1455551016yes—
227C22C24×D13711yescharacteristic
228C4—24no—
229C4—24no—
230C4—2744no—
231C4—1378no—
232C2—44no—
233C2—2748no—
234C2—5484no—
235C2C2×#17984061112yesdirect product factor
236C2C22×D4×D13711yescharacteristic
237C2C22×D4×D13711yescharacteristic
238C2C22×#1455551011yescharacteristic
subgroupG/H
centerC24C22×D137
commutator, cosocleC22×C137C25
FrattiniC22C24×D137
FittingC22×C22⋊C4×C137C2
socleC24×C137C23
hypercenterC22×C22⋊C4D137
hypocenterC137C22×#121
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D548C2semidirect
2. C23×C22⋊D137C2semidirect
3. C23×C22⋊D137C2semidirect
4. C23×C4×D137C2semidirect
5. C2×#17984061C2direct (N × H)
6. C22×#14555489C2semidirect
7. C22×#14555487C2semidirect
8. C22×C22⋊C4×C137C2semidirect
12. C22×D548C22semidirect
15. C22×C22⋊D137C22semidirect
16. C22×C22⋊D137C22semidirect
20. C22×C4×D137C22semidirect
21. #17984061C22direct (N × H)
22. C2×#14555489C22semidirect
23. C2×#14555487C22semidirect
27. C2×C22⋊C4×C137C22semidirect
33. C23×D137D4semidirect
37. C2×D548C23semidirect
40. C2×C22⋊D137C23semidirect
41. C2×C22⋊D137C23semidirect
43. C22×Dic137D4semidirect
47. C2×C4×D137C23semidirect
48. #14555489C23semidirect
49. #14555487C23semidirect
59. C22⋊C4×C137C23semidirect
67. C22×D137C2×D4semidirect
74. C2×Dic137C2×D4semidirect
95. C22×C137#121semidirect
107. C2×D137C22×D4semidirect
111. Dic137C22×D4semidirect
113. C2×C137C2×#121semidirect
119. C137C22×#121semidirect
128. C22×C22⋊C4D137semidirect
147. C2×C22⋊C4C2×D137semidirect
179. C22⋊C4C22×D137semidirect
223. C22#17984061direct (N × H)
235. C2C2×#17984061direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5482
C23×C22⋊D1372
C23×C4×D1372
C2×#179840612
C22×#145554892
C22×#145554872
C22×C22⋊C4×C1372
C22×#121137