Finite Group Databasebeta

← C2×#22568206 #31061949

Subgroups — C2×#22568206 #31061949

Order 17152.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 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×#14524012C212yesmaximal, complemented
2#22568206C2116yesmaximal, direct product factor
3C2×#16863321C213yesmaximal, complemented
4C2×#16863312C216yesmaximal, complemented
5C2×#16863305C213yesmaximal, complemented
6C2×#379×C67C211yesmaximal, characteristic, complemented
7C23×D268C2211yescharacteristic, complemented
8C23×C22⋊D67C2216yes—
9C2×#14524012—216no—
10C2×#14524012C22116yescomplemented
11C22×C4⋊Dic67C2213yescomplemented
12C22×C22⋊Dic67C2216yes—
13C22×(C2×D67)⋊C4C2216yescomplemented
14#16863321C22124yescomplemented
15#16863312C22148yescomplemented
16#16863305C22124yescomplemented
17C2×C67⋊#95C2213yescomplemented
18C2×C67⋊#95C2213yescomplemented
19C26×C67C2211yescharacteristic
20C22×C22⋊C4×C67C2213yes—
21C22×C22⋊C4×C67C2213yes—
22#379×C67C2218yescomplemented
23C24×D67C2312yes—
24C22×D268—46no—
25C22×D268C2314yescomplemented
26C22×C22⋊D67—224no—
27C22×C22⋊D67—412no—
28C22×C22⋊D67—412no—
29C22×C22⋊D67C23124yes—
30C23×Dic67C2316yes—
31#14524012—264no—
32C2×C4⋊Dic67—46no—
33C2×C4⋊Dic67C23112yescomplemented
34C2×C22⋊Dic67—224no—
35C2×C22⋊Dic67C23124yes—
36C2×(C2×D67)⋊C4—412no—
37C2×(C2×D67)⋊C4C23124yescomplemented
38C67⋊#95C23112yescomplemented
39C67⋊#95C23112yescomplemented
40C25×C67—46no—
41C25×C67—46no—
42C25×C67D418yescomplemented
43C25×C67C2313yes—
44C25×C67C2314yes—
45C23×C4×C67C2311yescharacteristic
46C23×C4×C67C2313yes—
47C2×C22⋊C4×C67—46no—
48C2×C22⋊C4×C67—46no—
49C2×C22⋊C4×C67C23112yes—
50C2×C22⋊C4×C67C23112yes—
51C23×D67—24no—
52C23×D67—212no—
53C23×D67—212no—
54C23×D67C2414yes—
55C2×D268—412no—
56C2×D268—416no—
57C2×C22⋊D67—248no—
58C2×C22⋊D67—424no—
59C2×C22⋊D67—424no—
60C2×C22⋊D67—496no—
61C22×Dic67—212no—
62C22×Dic67—212no—
63C22×Dic67—212no—
64C22×Dic67C24112yes—
65C4⋊Dic67—412no—
66C22⋊Dic67—248no—
67(C2×D67)⋊C4—424no—
68C24×C67—26no—
69C24×C67—26no—
70C24×C67—212no—
71C24×C67—212no—
72C24×C67—448no—
73C24×C67—412no—
74C24×C67—412no—
75C24×C67—448no—
76C24×C67—416no—
77C24×C67C2×D4112yes—
78C24×C67C2×D4116yescomplemented
79C24×C67C2416yes—
80C24×C67C2411yescharacteristic
81C22×C4×C67—212no—
82C22×C4×C67—26no—
83C22×C4×C67—26no—
84C22×C4×C67C2412yes—
85C22×C4×C67C2416yes—
86C22⋊C4×C67—412no—
87C22⋊C4×C67—412no—
88C23×C67—224no—
89C23×C67—248no—
90C23×C67—26no—
91C23×C67—26no—
92C23×C67—212no—
93C23×C67—212no—
94C23×C67—448no—
95C23×C67—448no—
96C23×C67—416no—
97C23×C67—4128no—
98C23×C67—424no—
99C23×C67—412no—
100C23×C67—412no—
101C23×C67C22×D4112yes—
102C23×C67C22×D412yes—
103C23×C67H(5,𝔽2)16yes—
104C23×C67H(5,𝔽2)16yes—
105C23×C67C2511yescharacteristic
106C22×D67—24no—
107C22×D67—212no—
108C22×D67—212no—
109C22×D67—424no—
110C22×D67—424no—
111C22×D67—48no—
112C2×C4×C67—212no—
113C2×C4×C67—26no—
114C2×C4×C67—26no—
115C2×C4×C67—412no—
116C2×C4×C67—44no—
117C22⋊D67—496no—
118C2×Dic67—212no—
119C2×Dic67—212no—
120C2×Dic67—212no—
121C2×Dic67—424no—
122D268—416no—
123C22×C67—224no—
124C22×C67—212no—
125C22×C67—464no—
126C22×C67—412no—
127C22×C67—46no—
128C22×C67—46no—
129C22×C67—448no—
130C22×C67#534112yes—
131C22×C67#560112yes—
132C22×C67H(3,𝔽4)14yes—
133C22×C67C23×D411yescharacteristic
134C22×C67C2×H(5,𝔽2)13yes—
135C22×C67C2×H(5,𝔽2)13yes—
136C2×D67—412no—
137C2×D67—412no—
138C2×D67—44no—
139C2×D67—816no—
140C4×C67—46no—
141C4×C67—42no—
142Dic67—412no—
143C2×#2812—671nomaximal, Sylow
144C2×C67—412no—
145C2×C67#281218yescomplemented
146C2×C67C2×#53413yes—
147C2×C67C2×#56013yes—
148C2×C67C2×H(3,𝔽4)11yescharacteristic
149D67—84no—
150C22×#120—672no—
151#2812—6716no—
152C2×#397—673no—
153C2×#394—676no—
154C2×#392—673no—
155C2×#379D6711yescharacteristic, complemented
156C67C2×#281211yescharacteristic, complemented, Sylow
157C26C2×D6711yescharacteristic
158C23×D4—676no—
159C23×D4—671no—
160C2×#120—6716no—
161C2×#120—13416no—
162C22×C4⋊C4—673no—
163C22×C22⋊C4—676no—
164C22×C22⋊C4—676no—
165C22×C22⋊C4C2×D6713yes—
166C22×C22⋊C4C2×D6713yes—
167#397—6724no—
168#394—6748no—
169#392—6724no—
170#379C2×D6718yescomplemented
171C2×#95—673no—
172C2×#95—673no—
173C25—46no—
174C25—46no—
175C25—672no—
176C25D26818yescomplemented
177C25C22×D6713yes—
178C25C22×D6714yes—
179C22×D4—2686no—
180C22×D4—26812no—
181C22×D4—26812no—
182C22×D4—6724no—
183C22×D4—674no—
184C22×D4—13424no—
185C23×C4—676no—
186C23×C4C22×D6713yes—
187C23×C4C22×D6711yescharacteristic
188#120—13464no—
189C2×C4⋊C4—2686no—
190C2×C4⋊C4—6712no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—26812no—
194C2×C22⋊C4—6724no—
195C2×C22⋊C4—6724no—
196C2×C22⋊C4—13424no—
197C2×C22⋊C4C22×D67112yes—
198C2×C22⋊C4C22×D67112yes—
199#95—6712no—
200#95—6712no—
201C24—26no—
202C24—26no—
203C24—212no—
204C24—212no—
205C24—448no—
206C24—412no—
207C24—412no—
208C24—448no—
209C24—416no—
210C24—674no—
211C24—1344no—
212C24—13412no—
213C24—13412no—
214C24C2×D268112yes—
215C24C2×D268116yescomplemented
216C24C23×D6716yes—
217C24C23×D6711yescharacteristic
218C2×D4—26812no—
219C2×D4—26816no—
220C2×D4—26896no—
221C2×D4—26824no—
222C2×D4—26824no—
223C2×D4—13448no—
224C22×C4—212no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—6712no—
228C22×C4—13412no—
229C22×C4—13412no—
230C22×C4—13412no—
231C22×C4C23×D6712yes—
232C22×C4C23×D6716yes—
233C4⋊C4—26812no—
234C22⋊C4—412no—
235C22⋊C4—412no—
236C22⋊C4—26824no—
237C22⋊C4—13448no—
238C23—248no—
239C23—224no—
240C23—26no—
241C23—26no—
242C23—212no—
243C23—212no—
244C23—448no—
245C23—4128no—
246C23—448no—
247C23—416no—
248C23—424no—
249C23—412no—
250C23—412no—
251C23—26824no—
252C23—2688no—
253C23—26824no—
254C23—1344no—
255C23—13412no—
256C23—13412no—
257C23C22×D268112yes—
258C23C22×D26812yes—
259C23#1452408816yes—
260C23#1452408816yes—
261C23C24×D6711yescharacteristic
262D4—26816no—
263D4—26896no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—412no—
268C2×C4—44no—
269C2×C4—26824no—
270C2×C4—13412no—
271C2×C4—13412no—
272C2×C4—13412no—
273C22—224no—
274C22—212no—
275C22—464no—
276C22—448no—
277C22—412no—
278C22—46no—
279C22—46no—
280C22—26812no—
281C22—26812no—
282C22—2684no—
283C22—53616no—
284C22#16863958112yes—
285C22#16864135112yes—
286C22#1686413914yes—
287C22C23×D26811yescharacteristic
288C22C2×#1452408813yes—
289C22C2×#1452408813yes—
290C4—46no—
291C4—42no—
292C4—26812no—
293C2—412no—
294C2—5364no—
295C2#2256820618yesdirect product factor
296C2C2×#1686395813yes—
297C2C2×#1686413513yes—
298C2C2×#1686413911yescharacteristic
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC2×#379×C67C2
socleC24×C67C24
hypercenterC2×#379D67
hypocenterC67C2×#2812
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#14524012C2semidirect
2. #22568206C2direct (N × H)
3. C2×#16863321C2semidirect
4. C2×#16863312C2semidirect
5. C2×#16863305C2semidirect
6. C2×#379×C67C2semidirect
7. C23×D268C22semidirect
10. C2×#14524012C22semidirect
11. C22×C4⋊Dic67C22semidirect
13. C22×(C2×D67)⋊C4C22semidirect
14. #16863321C22semidirect
15. #16863312C22semidirect
16. #16863305C22semidirect
17. C2×C67⋊#95C22semidirect
18. C2×C67⋊#95C22semidirect
22. #379×C67C22semidirect
25. C22×D268C23semidirect
33. C2×C4⋊Dic67C23semidirect
37. C2×(C2×D67)⋊C4C23semidirect
38. C67⋊#95C23semidirect
39. C67⋊#95C23semidirect
42. C25×C67D4semidirect
78. C24×C67C2×D4semidirect
145. C2×C67#2812semidirect
155. C2×#379D67semidirect
156. C67C2×#2812semidirect
170. #379C2×D67semidirect
176. C25D268semidirect
215. C24C2×D268semidirect
295. C2#22568206direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240122
#225682062
C2×#168633212
C2×#168633122
C2×#168633052
C2×#379×C672
C2×#281267