Finite Group Databasebeta

← C2×#22553459 #30675164

Subgroups — C2×#22553459 #30675164

Order 16768.

812 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#14523513C211yesmaximal, characteristic, complemented
2C22×#14523502C211yesmaximal, characteristic, complemented
3C22×D1048C211yesmaximal, characteristic, complemented
4C2×#16862220C211yesmaximal, characteristic, complemented
5#22553459C218yesmaximal, direct product factor
6C2×#16862171C211yesmaximal, characteristic, complemented
7C2×#16862168C211yesmaximal, characteristic, complemented
8C2×C22⋊C8×C131C211yesmaximal, characteristic, complemented
9C2×#14523513—24no—
10C2×#14523513—24no—
11C2×#14523513—22no—
12C2×#14523513C2214yescomplemented
13C22×Dic262C2211yescharacteristic
14C22×D524C2211yescharacteristic
15C22×D524C2211yescharacteristic
16C22×C22⋊D131—21no—
17C22×C22⋊D131—21no—
18C22×C4×D131—21no—
19C2×#14523502—24no—
20C2×#14523502C2214yescomplemented
21C2×D1048—24no—
22C2×D1048C2214yescomplemented
23#16862220—22no—
24#16862220C2214yescomplemented
25C2×C4⋊Dic131C2211yescharacteristic
26C2×(C2×D131)⋊C4—21no—
27#16862171C2214yescomplemented
28#16862168C2214yescomplemented
29C23×C4×C131C2211yescharacteristic
30C22×C8×C131C2211yescharacteristic
31C22×C8×C131C2211yescharacteristic
32C22⋊C8×C131C2214yescomplemented
33C23×D131—21no—
34C23×D131—21no—
35#14523513—216no—
36#14523513—28no—
37#14523513—28no—
38C2×Dic262—22no—
39C2×Dic262—21no—
40C2×Dic262D414yescomplemented
41C2×Dic262C2312yes—
42C2×D524—22no—
43C2×D524—21no—
44C2×D524—22no—
45C2×D524—22no—
46C2×D524—21no—
47C2×D524D414yescomplemented
48C2×D524C2312yes—
49C2×D524C2312yes—
50C2×C22⋊D131—28no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131—24no—
53C2×C22⋊D131—44no—
54C22×Dic131—21no—
55C22×Dic131—21no—
56C2×C4×D131—28no—
57C2×C4×D131—24no—
58#14523502—28no—
59D1048—28no—
60C4⋊Dic131—21no—
61C4⋊Dic131C2312yes—
62(C2×D131)⋊C4—24no—
63C24×C131D411yescharacteristic
64C22×C4×C131—22no—
65C22×C4×C131—22no—
66C22×C4×C131D412yes—
67C22×C4×C131D411yescharacteristic
68C22×C4×C131C2312yes—
69C22×C4×C131C2311yescharacteristic
70C2×C8×C131—22no—
71C2×C8×C131—22no—
72C2×C8×C131C2312yes—
73C2×C8×C131C2312yes—
74C22×D131—24no—
75C22×D131—22no—
76C22×D131—22no—
77C22×D131—42no—
78C22×D131—42no—
79C22×D131—42no—
80C22×D131—42no—
81C22×D131—42no—
82Dic262—24no—
83Dic262—22no—
84Dic262C2×D414yescomplemented
85D524—24no—
86D524—22no—
87D524—22no—
88D524—22no—
89D524—42no—
90D524C2×D414yescomplemented
91C22⋊D131—216no—
92C22⋊D131—48no—
93C2×Dic131—24no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131—42no—
97C4×D131—216no—
98C23×C131—22no—
99C23×C131—22no—
100C23×C131—22no—
101C23×C131C2×D412yes—
102C23×C131C2×D411yescharacteristic
103C2×C4×C131—22no—
104C2×C4×C131—24no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131C2×D412yes—
108C2×C4×C131C2×D412yes—
109C2×C4×C131C2×D412yes—
110C2×C4×C131C2×D411yescharacteristic
111C2×C4×C131C2411yescharacteristic
112C8×C131—22no—
113C8×C131—22no—
114C22×C131—22no—
115C22×C131—22no—
116C22×C131—22no—
117C22×C131—44no—
118C22×C131#12012yes—
119C22×C131#13512yes—
120C22×C131AGL(1,ℤ/8)12yes—
121C22×C131C22×D411yescharacteristic
122C2×D131—24no—
123C2×D131—42no—
124C2×D131—42no—
125C2×D131—48no—
126C2×D131—42no—
127C2×D131—42no—
128C2×D131—42no—
129C2×D131—84no—
130C4×C131—22no—
131C4×C131#12012yes—
132C4×C131C22×D411yescharacteristic
133C4×C131C22×D411yescharacteristic
134Dic131—24no—
135Dic131—42no—
136C2×C131—42no—
137C2×C131#44914yescomplemented
138C2×C131C2×#12011yescharacteristic
139C2×C131C2×#13511yescharacteristic
140C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
141D131—44no—
142D131—82no—
143C131C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1311nomaximal, Sylow
145C22×#41—1311no—
146C22×SD16—1311no—
147C22×D8—1311no—
148C2×#121—1311no—
149#449—1318no—
150C2×#103—1311no—
151C2×#102—1311no—
152C2×C22⋊C8D13111yescharacteristic, complemented
153C2×#41—2624no—
154C2×#41—2624no—
155C2×#41—2622no—
156C2×#41—1314no—
157C22×Q8—1311no—
158C22×D4—2621no—
159C22×D4—2621no—
160C22×D4—1311no—
161C22×D4—1311no—
162C23×C4—2621no—
163C23×C4C2×D13111yescharacteristic
164C2×SD16—2624no—
165C2×SD16—1314no—
166C2×D8—2624no—
167C2×D8—1314no—
168C22×C8C2×D13111yescharacteristic
169C22×C8C2×D13111yescharacteristic
170#121—2622no—
171#121—1314no—
172C2×C4⋊C4—1311no—
173C2×C22⋊C4—2621no—
174#103—1314no—
175#102—1314no—
176C22⋊C8C2×D13114yescomplemented
177C24—2621no—
178C24—2621no—
179C24D52411yescharacteristic
180#41—26216no—
181#41—2628no—
182#41—2628no—
183C2×Q8—2622no—
184C2×Q8—2621no—
185C2×Q8—1314no—
186C2×Q8—1312no—
187C2×D4—2628no—
188C2×D4—2624no—
189C2×D4—2624no—
190C2×D4—2622no—
191C2×D4—2622no—
192C2×D4—2622no—
193C2×D4—2621no—
194C2×D4—2621no—
195C2×D4—5244no—
196C2×D4—1314no—
197C2×D4—1312no—
198C2×D4—1312no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—2628no—
202C22×C4—2621no—
203C22×C4—2624no—
204C22×C4—2621no—
205C22×C4D52412yes—
206C22×C4D52411yescharacteristic
207C22×C4C22×D13112yes—
208C22×C4C22×D13111yescharacteristic
209SD16—2628no—
210D8—2628no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8C22×D13112yes—
214C2×C8C22×D13112yes—
215C4⋊C4—2621no—
216C4⋊C4—1312no—
217C22⋊C4—2624no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—2624no—
222C23—2622no—
223C23—2622no—
224C23—5242no—
225C23—5242no—
226C23—5242no—
227C23—5242no—
228C23—5242no—
229C23C2×D52412yes—
230C23C2×D52411yescharacteristic
231Q8—2624no—
232Q8—2622no—
233Q8—1314no—
234D4—26216no—
235D4—2624no—
236D4—2622no—
237D4—2622no—
238D4—2622no—
239D4—5248no—
240D4—5242no—
241D4—1314no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—26216no—
247C2×C4—2624no—
248C2×C4—2622no—
249C2×C4—2622no—
250C2×C4—5242no—
251C2×C4D4×D13112yes—
252C2×C4D4×D13112yes—
253C2×C4C2×D52412yes—
254C2×C4C2×D52411yescharacteristic
255C2×C4C23×D13111yescharacteristic
256C8—22no—
257C8—22no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—2624no—
263C22—5248no—
264C22—5242no—
265C22—5242no—
266C22—5242no—
267C22—5242no—
268C22—5242no—
269C22—10484no—
270C22#1686221612yes—
271C22#1686226812yes—
272C22#1686227412yes—
273C22C22×D52411yescharacteristic
274C4—22no—
275C4—2624no—
276C4—5242no—
277C4#1686221612yes—
278C4C2×D4×D13111yescharacteristic
279C4C2×D4×D13111yescharacteristic
280C2—42no—
281C2—5244no—
282C2—10482no—
283C2#2255345914yesdirect product factor
284C2C2×#1686221611yescharacteristic
285C2C2×#1686226811yescharacteristic
286C2C2×#1686227411yescharacteristic
subgroupG/H
centerC23C2×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×C22⋊C8×C131C2
socleC23×C131C2×D4
hypercenterC2×C22⋊C8D131
hypocenterC131C2×#449
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C22×#14523502C2semidirect
3. C22×D1048C2semidirect
4. C2×#16862220C2semidirect
5. #22553459C2direct (N × H)
6. C2×#16862171C2semidirect
7. C2×#16862168C2semidirect
8. C2×C22⋊C8×C131C2semidirect
12. C2×#14523513C22semidirect
20. C2×#14523502C22semidirect
22. C2×D1048C22semidirect
24. #16862220C22semidirect
27. #16862171C22semidirect
28. #16862168C22semidirect
32. C22⋊C8×C131C22semidirect
40. C2×Dic262D4semidirect
47. C2×D524D4semidirect
84. Dic262C2×D4semidirect
90. D524C2×D4semidirect
137. C2×C131#449semidirect
143. C131C2×#449semidirect
152. C2×C22⋊C8D131semidirect
176. C22⋊C8C2×D131semidirect
283. C2#22553459direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145235132
C22×#145235022
C22×D10482
C2×#168622202
#225534592
C2×#168621712
C2×#168621682
C2×C22⋊C8×C1312
C2×#449131