Finite Group Databasebeta

← C2×#22568474 #31062243

Subgroups — C2×#22568474 #31062243

Order 17152.

3294 conjugacy classes of subgroups; only the 627 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 627 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
1#22568474C2116yesmaximal, direct product factor
2C2×#16863332C213yesmaximal, complemented
3C2×#16863320C212yesmaximal, complemented
4C2×#16863318C216yesmaximal, complemented
5C2×#16863251C213yesmaximal
6C2×#399×C67C211yesmaximal, characteristic, complemented
7C25×D67C2211yescharacteristic
8C22×C4⋊Dic67C2213yescomplemented
9C22×Dic67⋊C4C2216yescomplemented
10C22×C22⋊Dic67C2213yes—
11C22×(C2×D67)⋊C4C2216yes—
12C22×(C2×D67)⋊C4C2216yes—
13#16863332C22124yescomplemented
14#16863320C22116yescomplemented
15#16863318C22148yescomplemented
16#16863251C22124yes—
17C2×C67⋊#95C2213yescomplemented
18C22×C4⋊C4×C67C2213yescomplemented
19C22×C22⋊C4×C67C2213yes—
20#399×C67C2218yescomplemented
21C2×#95×C67C2211yescharacteristic, complemented
22C24×D67—46no—
23C24×D67—46no—
24C24×D67Q818yes—
25C24×D67C2312yes—
26C24×D67C2314yes—
27C23×Dic67C2316yes—
28C2×C4⋊Dic67—46no—
29C2×C4⋊Dic67C23112yescomplemented
30C2×Dic67⋊C4—412no—
31C2×Dic67⋊C4C23124yescomplemented
32C2×C22⋊Dic67—46no—
33C2×C22⋊Dic67C23112yes—
34C2×(C2×D67)⋊C4—412no—
35C2×(C2×D67)⋊C4—412no—
36C2×(C2×D67)⋊C4C23124yes—
37C2×(C2×D67)⋊C4C23124yes—
38C67⋊#95C23112yescomplemented
39C25×C67C2311yescharacteristic
40C23×C4×C67C2313yes—
41C23×C4×C67C2313yes—
42C2×C4⋊C4×C67—46no—
43C2×C4⋊C4×C67C23112yescomplemented
44C2×C22⋊C4×C67—46no—
45C2×C22⋊C4×C67C23112yes—
46#95×C67C2314yescomplemented
47C23×D67—212no—
48C23×D67—212no—
49C23×D67—448no—
50C23×D67—412no—
51C23×D67—412no—
52C23×D67—416no—
53C23×D67—448no—
54C23×D67C2×Q818yes—
55C23×D67C2×Q8116yes—
56C23×D67C2414yes—
57C22×Dic67—212no—
58C22×Dic67—212no—
59C22×Dic67—212no—
60C22×Dic67C24112yes—
61C4⋊Dic67—412no—
62Dic67⋊C4—424no—
63C22⋊Dic67—412no—
64(C2×D67)⋊C4—424no—
65(C2×D67)⋊C4—424no—
66C24×C67—26no—
67C24×C67—26no—
68C24×C67C2×Q814yes—
69C24×C67C2412yes—
70C24×C67C2411yescharacteristic
71C22×C4×C67—26no—
72C22×C4×C67—26no—
73C22×C4×C67—212no—
74C22×C4×C67—212no—
75C22×C4×C67C2416yes—
76C22×C4×C67C2416yes—
77C4⋊C4×C67—412no—
78C22⋊C4×C67—412no—
79C23×C67—224no—
80C23×C67—26no—
81C23×C67—26no—
82C23×C67—44no—
83C23×C67—412no—
84C23×C67C22×Q814yes—
85C23×C67C22×Q812yes—
86C23×C67H(5,𝔽2)16yes—
87C23×C67H(5,𝔽2)16yes—
88C23×C67C2511yescharacteristic
89C22×D67—248no—
90C22×D67—212no—
91C22×D67—212no—
92C22×D67—448no—
93C22×D67—4128no—
94C22×D67—416no—
95C22×D67—448no—
96C22×D67—48no—
97C22×D67—424no—
98C22×D67C22×Q818yes—
99C2×C4×C67—26no—
100C2×C4×C67—26no—
101C2×C4×C67—212no—
102C2×C4×C67—212no—
103C2×C4×C67—412no—
104C2×C4×C67—412no—
105C2×Dic67—212no—
106C2×Dic67—212no—
107C2×Dic67—212no—
108C2×Dic67—424no—
109C22×C67—212no—
110C22×C67—416no—
111C22×C67—42no—
112C22×C67—46no—
113C22×C67#543112yes—
114C22×C67#56014yes—
115C22×C67#560112yes—
116C22×C67C23×Q811yescharacteristic
117C22×C67C2×H(5,𝔽2)13yes—
118C22×C67C2×H(5,𝔽2)13yes—
119C2×D67—224no—
120C2×D67—464no—
121C2×D67—432no—
122C2×D67—44no—
123C2×D67—412no—
124C4×C67—46no—
125C4×C67—46no—
126Dic67—412no—
127C2×#2842—671nomaximal, Sylow
128C2×C67—44no—
129C2×C67#284218yescomplemented
130C2×C67C2×#54313yes—
131C2×C67C2×#56011yescharacteristic
132C2×C67C2×#56013yes—
133D67—48no—
134#2842—6716no—
135C2×#399—673no—
136C2×#399D6711yescharacteristic, complemented
137C2×#396—672no—
138C2×#396—676no—
139C2×#379—673no—
140C67C2×#284211yescharacteristic, complemented, Sylow
141C26—671no—
142C22×C4⋊C4—676no—
143C22×C4⋊C4—673no—
144C22×C4⋊C4C2×D6713yescomplemented
145C22×C22⋊C4—673no—
146C22×C22⋊C4—676no—
147C22×C22⋊C4—676no—
148C22×C22⋊C4C2×D6713yes—
149#399—6724no—
150#399C2×D6718yescomplemented
151#396—6748no—
152#396—6716no—
153#379—6724no—
154C2×#95—673no—
155C2×#95C2×D6711yescharacteristic, complemented
156C25—2686no—
157C25—2686no—
158C25—678no—
159C25—674no—
160C25—672no—
161C25C22×D6711yescharacteristic
162C23×C4—676no—
163C23×C4C22×D6713yes—
164C23×C4C22×D6713yes—
165C2×C4⋊C4—46no—
166C2×C4⋊C4—26812no—
167C2×C4⋊C4—2686no—
168C2×C4⋊C4—6724no—
169C2×C4⋊C4—6712no—
170C2×C4⋊C4C22×D67112yescomplemented
171C2×C22⋊C4—46no—
172C2×C22⋊C4—26812no—
173C2×C22⋊C4—26812no—
174C2×C22⋊C4—2686no—
175C2×C22⋊C4—6712no—
176C2×C22⋊C4—6724no—
177C2×C22⋊C4—6724no—
178C2×C22⋊C4C22×D67112yes—
179#95—6712no—
180#95C22×D6714yescomplemented
181C24—26no—
182C24—26no—
183C24—26848no—
184C24—26812no—
185C24—26812no—
186C24—26816no—
187C24—26848no—
188C24—6716no—
189C24—678no—
190C24—674no—
191C24—13412no—
192C24—13412no—
193C24Q8×D6714yes—
194C24C23×D6712yes—
195C24C23×D6711yescharacteristic
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—212no—
199C22×C4—212no—
200C22×C4—6712no—
201C22×C4—13412no—
202C22×C4—13412no—
203C22×C4—13412no—
204C22×C4C23×D6716yes—
205C22×C4C23×D6716yes—
206C4⋊C4—412no—
207C4⋊C4—26824no—
208C4⋊C4—26812no—
209C22⋊C4—412no—
210C22⋊C4—26824no—
211C22⋊C4—26824no—
212C22⋊C4—26812no—
213C23—224no—
214C23—26no—
215C23—26no—
216C23—44no—
217C23—412no—
218C23—26848no—
219C23—268128no—
220C23—26816no—
221C23—26848no—
222C23—2688no—
223C23—26824no—
224C23—678no—
225C23—13448no—
226C23—13412no—
227C23—13412no—
228C23C2×Q8×D6714yes—
229C23C2×Q8×D6712yes—
230C23#1452408816yes—
231C23#1452408916yes—
232C23C24×D6711yescharacteristic
233C2×C4—26no—
234C2×C4—26no—
235C2×C4—212no—
236C2×C4—212no—
237C2×C4—412no—
238C2×C4—412no—
239C2×C4—26824no—
240C2×C4—13412no—
241C2×C4—13412no—
242C2×C4—13412no—
243C22—212no—
244C22—416no—
245C22—42no—
246C22—46no—
247C22—26864no—
248C22—26832no—
249C22—2684no—
250C22—26812no—
251C22—13424no—
252C22#16864021112yes—
253C22#1686413314yes—
254C22#16864136112yes—
255C22C22×Q8×D6711yescharacteristic
256C22C2×#1452408813yes—
257C22C2×#1452408913yes—
258C4—46no—
259C4—46no—
260C4—26812no—
261C2—44no—
262C2—2688no—
263C2#2256847418yesdirect product factor
264C2C2×#1686402113yes—
265C2C2×#1686413311yescharacteristic
266C2C2×#1686413613yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC2×#399×C67C2
socleC24×C67C24
prime radicalC25×D67C22
hypercenterC2×#399D67
hypocenterC67C2×#2842
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. #22568474C2direct (N × H)
2. C2×#16863332C2semidirect
3. C2×#16863320C2semidirect
4. C2×#16863318C2semidirect
6. C2×#399×C67C2semidirect
8. C22×C4⋊Dic67C22semidirect
9. C22×Dic67⋊C4C22semidirect
13. #16863332C22semidirect
14. #16863320C22semidirect
15. #16863318C22semidirect
17. C2×C67⋊#95C22semidirect
18. C22×C4⋊C4×C67C22semidirect
20. #399×C67C22semidirect
21. C2×#95×C67C22semidirect
29. C2×C4⋊Dic67C23semidirect
31. C2×Dic67⋊C4C23semidirect
38. C67⋊#95C23semidirect
43. C2×C4⋊C4×C67C23semidirect
46. #95×C67C23semidirect
129. C2×C67#2842semidirect
136. C2×#399D67semidirect
140. C67C2×#2842semidirect
144. C22×C4⋊C4C2×D67semidirect
150. #399C2×D67semidirect
155. C2×#95C2×D67semidirect
170. C2×C4⋊C4C22×D67semidirect
180. #95C22×D67semidirect
263. C2#22568474direct (N × H)

Maximal subgroups

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

subgroupindex
#225684742
C2×#168633322
C2×#168633202
C2×#168633182
C2×#168632512
C2×#399×C672
C2×#284267