Finite Group Databasebeta

← C2×#22553220 #30669156

Subgroups — C2×#22553220 #30669156

Order 16768.

1812 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 285 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×C22⋊D131C212yesmaximal, complemented
2C22×C22⋊Dic131C212yesmaximal, complemented
3C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
4#22553220C218yesmaximal, direct product factor
5C2×C131⋊#95C211yesmaximal, characteristic, complemented
6C23×D4×C131C211yesmaximal, characteristic, complemented
7C24×D131C2211yescharacteristic
8C22×C22⋊D131—28no—
9C22×C22⋊D131—28no—
10C22×C22⋊D131—28no—
11C22×C22⋊D131C2218yescomplemented
12C23×Dic131C2211yescharacteristic
13C23×Dic131C2212yes—
14C2×C22⋊Dic131—28no—
15C2×C22⋊Dic131C2218yescomplemented
16C2×(C2×D131)⋊C4—24no—
17C2×(C2×D131)⋊C4C2214yescomplemented
18C131⋊#95C2214yescomplemented
19C25×C131C2212yes—
20C22×D4×C131—28no—
21C22×D4×C131—24no—
22C22×D4×C131C2214yescomplemented
23C23×C4×C131C2211yescharacteristic
24C23×D131—24no—
25C23×D131—24no—
26C23×D131—22no—
27C23×D131—22no—
28C23×D131D414yescomplemented
29C23×D131C2312yes—
30C2×C22⋊D131—264no—
31C2×C22⋊D131—216no—
32C2×C22⋊D131—216no—
33C2×C22⋊D131—216no—
34C22×Dic131—24no—
35C22×Dic131—24no—
36C22×Dic131—22no—
37C22×Dic131—24no—
38C22×Dic131D418yescomplemented
39C22×Dic131C2312yes—
40C22×Dic131C2314yes—
41C22⋊Dic131—216no—
42(C2×D131)⋊C4—28no—
43C24×C131—24no—
44C24×C131—24no—
45C24×C131—24no—
46C24×C131—24no—
47C24×C131—24no—
48C24×C131—24no—
49C24×C131D418yescomplemented
50C24×C131C2314yes—
51C24×C131C2311yescharacteristic
52C2×D4×C131—232no—
53C2×D4×C131—216no—
54C2×D4×C131—28no—
55C22×C4×C131—24no—
56C22×C4×C131D414yescomplemented
57C22×C4×C131C2312yes—
58C22×D131—216no—
59C22×D131—28no—
60C22×D131—22no—
61C22×D131—24no—
62C22×D131—24no—
63C22×D131—22no—
64C22×D131—44no—
65C22×D131—48no—
66C22×D131—44no—
67C22×D131C2×D414yescomplemented
68C22⋊D131—264no—
69C2×Dic131—216no—
70C2×Dic131—24no—
71C2×Dic131—24no—
72C2×Dic131—22no—
73C2×Dic131—24no—
74C2×Dic131—44no—
75C2×Dic131C2×D418yescomplemented
76C23×C131—216no—
77C23×C131—216no—
78C23×C131—216no—
79C23×C131—24no—
80C23×C131—24no—
81C23×C131—24no—
82C23×C131—24no—
83C23×C131—24no—
84C23×C131—24no—
85C23×C131—48no—
86C23×C131—48no—
87C23×C131—48no—
88C23×C131—48no—
89C23×C131C2×D412yes—
90C23×C131C2×D414yes—
91C23×C131C2×D414yes—
92C23×C131C2×D412yes—
93C23×C131C2×D418yescomplemented
94C23×C131#4112yes—
95C23×C131C2411yescharacteristic
96D4×C131—232no—
97C2×C4×C131—28no—
98C2×C4×C131—24no—
99C2×C4×C131C2×D414yescomplemented
100C22×C131—28no—
101C22×C131—28no—
102C22×C131—28no—
103C22×C131—44no—
104C22×C131—44no—
105C22×C131—44no—
106C22×C131—44no—
107C22×C131—432no—
108C22×C131#12018yes—
109C22×C131#12014yes—
110C22×C131#12114yes—
111C22×C131#12118yes—
112C22×C131#12714yes—
113C22×C131C22×D411yescharacteristic
114C22×C131C22×D412yes—
115C22×C131C22×D412yes—
116C22×C131C22×D411yescharacteristic
117C22×C131C2×#4111yescharacteristic
118C2×D131—28no—
119C2×D131—24no—
120C2×D131—42no—
121C2×D131—44no—
122C2×D131—42no—
123C2×D131—416no—
124C4×C131—24no—
125Dic131—28no—
126Dic131—42no—
127C2×C131—48no—
128C2×C131#39218yescomplemented
129C2×C131C2×#12012yes—
130C2×C131C2×#12011yescharacteristic
131C2×C131C2×#12111yescharacteristic
132C2×C131C2×#12112yes—
133C2×C131C2×#12711yescharacteristic
134D131—44no—
135C131C2×#39211yescharacteristic, complemented, Sylow
136C2×#392—1311nomaximal, Sylow
137C23×D4—1312no—
138C23×D4D13111yescharacteristic, complemented
139C22×C22⋊C4—1312no—
140C22×C22⋊C4—1311no—
141#392—1318no—
142C2×#95—1311no—
143C25—1311no—
144C25C2×D13112yes—
145C22×D4—28no—
146C22×D4—24no—
147C22×D4—2628no—
148C22×D4—2628no—
149C22×D4—2628no—
150C22×D4—1318no—
151C22×D4C2×D13114yescomplemented
152C23×C4—1311no—
153C23×C4—1312no—
154C23×C4C2×D13111yescharacteristic
155C2×C22⋊C4—2628no—
156C2×C22⋊C4—2624no—
157C2×C22⋊C4—1318no—
158C2×C22⋊C4—1314no—
159#95—1314no—
160C24—24no—
161C24—24no—
162C24—24no—
163C24—24no—
164C24—24no—
165C24—24no—
166C24—2624no—
167C24—2622no—
168C24—2624no—
169C24—2622no—
170C24—1314no—
171C24—1312no—
172C24C22⋊D13118yescomplemented
173C24C22×D13114yes—
174C24C22×D13111yescharacteristic
175C2×D4—232no—
176C2×D4—216no—
177C2×D4—28no—
178C2×D4—26264no—
179C2×D4—26216no—
180C2×D4—26216no—
181C2×D4—26216no—
182C22×C4—24no—
183C22×C4—2624no—
184C22×C4—2624no—
185C22×C4—2624no—
186C22×C4—2622no—
187C22×C4—1312no—
188C22×C4—1318no—
189C22×C4—1314no—
190C22×C4C22⋊D13114yescomplemented
191C22×C4C22×D13112yes—
192C22⋊C4—26216no—
193C22⋊C4—2628no—
194C23—216no—
195C23—216no—
196C23—216no—
197C23—24no—
198C23—24no—
199C23—24no—
200C23—24no—
201C23—24no—
202C23—24no—
203C23—48no—
204C23—48no—
205C23—48no—
206C23—48no—
207C23—26216no—
208C23—2628no—
209C23—2624no—
210C23—2622no—
211C23—2624no—
212C23—2622no—
213C23—5244no—
214C23—5248no—
215C23—5244no—
216C23—1314no—
217C23D4×D13112yes—
218C23D4×D13114yes—
219C23C2×C22⋊D13114yes—
220C23C2×C22⋊D13112yes—
221C23C2×C22⋊D13118yescomplemented
222C23#1452351212yes—
223C23C23×D13111yescharacteristic
224D4—232no—
225D4—26264no—
226C2×C4—28no—
227C2×C4—24no—
228C2×C4—26216no—
229C2×C4—2624no—
230C2×C4—2624no—
231C2×C4—2624no—
232C2×C4—2622no—
233C2×C4—5244no—
234C2×C4—1318no—
235C2×C4C2×C22⋊D13114yescomplemented
236C22—28no—
237C22—28no—
238C22—28no—
239C22—432no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—2628no—
245C22—2624no—
246C22—52416no—
247C22—5242no—
248C22—5244no—
249C22—5242no—
250C22#1686221418yes—
251C22#1686221514yes—
252C22#1686221714yes—
253C22#1686221818yes—
254C22#1686224214yes—
255C22C2×D4×D13111yescharacteristic
256C22C2×D4×D13112yes—
257C22C22×C22⋊D13112yes—
258C22C22×C22⋊D13111yescharacteristic
259C22C2×#1452351211yescharacteristic
260C4—24no—
261C4—2628no—
262C4—5242no—
263C2—48no—
264C2—5244no—
265C2#2255322018yesdirect product factor
266C2C2×#1686221412yes—
267C2C2×#1686221511yescharacteristic
268C2C2×#1686221711yescharacteristic
269C2C2×#1686221812yes—
270C2C2×#1686224211yescharacteristic
subgroupG/H
centerC24C22×D131
commutator, cosocleC23×C131C24
FrattiniC23C23×D131
FittingC23×D4×C131C2
socleC24×C131C23
hypercenterC23×D4D131
hypocenterC131C2×#392
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D131C2semidirect
2. C22×C22⋊Dic131C2semidirect
3. C22×(C2×D131)⋊C4C2semidirect
4. #22553220C2direct (N × H)
5. C2×C131⋊#95C2semidirect
6. C23×D4×C131C2semidirect
11. C22×C22⋊D131C22semidirect
15. C2×C22⋊Dic131C22semidirect
17. C2×(C2×D131)⋊C4C22semidirect
18. C131⋊#95C22semidirect
22. C22×D4×C131C22semidirect
28. C23×D131D4semidirect
38. C22×Dic131D4semidirect
49. C24×C131D4semidirect
56. C22×C4×C131D4semidirect
67. C22×D131C2×D4semidirect
75. C2×Dic131C2×D4semidirect
93. C23×C131C2×D4semidirect
99. C2×C4×C131C2×D4semidirect
128. C2×C131#392semidirect
135. C131C2×#392semidirect
138. C23×D4D131semidirect
151. C22×D4C2×D131semidirect
172. C24C22⋊D131semidirect
190. C22×C4C22⋊D131semidirect
221. C23C2×C22⋊D131semidirect
235. C2×C4C2×C22⋊D131semidirect
265. C2#22553220direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1312
C22×C22⋊Dic1312
C22×(C2×D131)⋊C42
#225532202
C2×C131⋊#952
C23×D4×C1312
C2×#392131