Finite Group Databasebeta

← #2842×D67 #31062240

Subgroups — #2842×D67 #31062240

Order 17152.

3294 conjugacy classes of subgroups; only the 579 normal subgroups are shown.

normal subgroups only

This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 579 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#22568474C214yesmaximal, complemented
2#22568473C213yesmaximal, complemented
3#22568472C218yesmaximal, complemented
4#399×D67C214yesmaximal, complemented
5#396×D67C218yesmaximal, complemented
6#379×D67C213yesmaximal
7C67×#2842C211yesmaximal, characteristic, complemented
8C25×D67C2211yescharacteristic
9C2×C4⋊C4×D67C22112yescomplemented
10C2×C22⋊C4×D67C22112yes—
11C2×C22⋊C4×D67C2216yes—
12#16863333C2214yescomplemented
13#16863332C22112yescomplemented
14#16863331C22112yescomplemented
15#16863320C2218yescomplemented
16#16863319C22124yescomplemented
17#16863318C22124yescomplemented
18#16863252C2213yes—
19#16863251C22112yes—
20#16863251C2216yes—
21#95×D67C2214yescomplemented
22#399×C67C2214yescomplemented
23#396×C67C2218yescomplemented
24#379×C67C2213yes—
25C24×D67—412no—
26C24×D67Q818yes—
27C24×D67C2312yes—
28C24×D67C2311yescharacteristic
29C24×D67C2312yes—
30C24×D67C2311yescharacteristic
31C22×C4×D67C23112yes—
32C2×C4⋊Dic67C23112yescomplemented
33C2×Dic67⋊C4C23124yescomplemented
34C4⋊C4×D67—424no—
35C2×C22⋊Dic67C23112yes—
36C2×C22⋊Dic67C2316yes—
37C2×(C2×D67)⋊C4C23124yes—
38C2×(C2×D67)⋊C4C23112yes—
39C22⋊C4×D67—424no—
40C22⋊C4×D67—412no—
41C67⋊#95C23112yescomplemented
42C25×C67C2311yescharacteristic
43C2×C4⋊C4×C67C23112yescomplemented
44C2×C22⋊C4×C67C23112yes—
45C2×C22⋊C4×C67C2316yes—
46#95×C67C2314yescomplemented
47C23×D67—224no—
48C23×D67—212no—
49C23×D67—448no—
50C23×D67—412no—
51C23×D67—46no—
52C23×D67—464no—
53C23×D67C2×Q818yes—
54C23×D67C2×Q814yes—
55C23×D67C2×Q818yes—
56C23×D67C2×Q814yes—
57C23×D67C2412yes—
58C23×D67C2411yescharacteristic
59C23×D67C2411yescharacteristic
60C22×Dic67C24112yes—
61C2×C4×D67—224no—
62C2×C4×D67—248no—
63C4⋊Dic67—412no—
64Dic67⋊C4—424no—
65C22⋊Dic67—46no—
66C22⋊Dic67—412no—
67(C2×D67)⋊C4—412no—
68(C2×D67)⋊C4—424no—
69C24×C67—46no—
70C24×C67C2×Q814yes—
71C24×C67C2412yes—
72C24×C67C2411yescharacteristic
73C22×C4×C67C24112yes—
74C4⋊C4×C67—412no—
75C22⋊C4×C67—412no—
76C22⋊C4×C67—46no—
77C23×C67—212no—
78C23×C67—26no—
79C23×C67—412no—
80C23×C67—416no—
81C23×C67C22×Q814yes—
82C23×C67C22×Q812yes—
83C23×C67C2511yescharacteristic
84C22×D67—26no—
85C22×D67—212no—
86C22×D67—248no—
87C22×D67—224no—
88C22×D67—424no—
89C22×D67—412no—
90C22×D67—432no—
91C22×D67—416no—
92C22×D67—4128no—
93C22×D67—432no—
94C22×D67—416no—
95C22×D67C22×Q814yes—
96C22×D67C22×Q812yes—
97C22×D67C22×Q812yes—
98C22×D67H(5,𝔽2)112yes—
99C2×C4×C67—224no—
100C2×C4×C67—212no—
101C2×Dic67—212no—
102C2×Dic67—224no—
103C4×D67—448no—
104C22×C67—212no—
105C22×C67—26no—
106C22×C67—416no—
107C22×C67—48no—
108C22×C67—44no—
109C22×C67C23×Q811yescharacteristic
110C22×C67C2×H(5,𝔽2)16yes—
111C2×D67—26no—
112C2×D67—212no—
113C2×D67—432no—
114C2×D67—416no—
115C2×D67—44no—
116C2×D67—48no—
117C2×D67—432no—
118C2×D67—416no—
119C2×D67#543112yes—
120C2×D67#560116yes—
121C4×C67—412no—
122Dic67—412no—
123C2×#2842—671nomaximal, Sylow
124C2×C67—44no—
125C2×C67—42no—
126C2×C67C2×#54313yes—
127C2×C67C2×#56014yes—
128D67—42no—
129D67—44no—
130D67#284218yesdirect product factor
131#2842—674no—
132#2842—678no—
133#2842—673no—
134#2842D6711yescharacteristic, direct product factor
135C2×#399—674no—
136C2×#396—678no—
137C2×#379—673no—
138C67C2×#284211yescharacteristic, complemented, Sylow
139C26—671no—
140C22×C4⋊C4—6712no—
141C22×C22⋊C4—6712no—
142C22×C22⋊C4—676no—
143#399—6712no—
144#399—6712no—
145#399—674no—
146#399C2×D6714yescomplemented
147#396—6724no—
148#396—6724no—
149#396—678no—
150#396C2×D6718yescomplemented
151#379—673no—
152#379—6712no—
153#379—676no—
154#379C2×D6713yes—
155C2×#95—674no—
156C25—26812no—
157C25—678no—
158C25—671no—
159C25—672no—
160C25—672no—
161C25—671no—
162C25C22×D6711yescharacteristic
163C23×C4—6712no—
164C2×C4⋊C4—26824no—
165C2×C4⋊C4—6724no—
166C2×C4⋊C4—6712no—
167C2×C4⋊C4C22×D67112yescomplemented
168C2×C22⋊C4—26824no—
169C2×C22⋊C4—26812no—
170C2×C22⋊C4—6712no—
171C2×C22⋊C4—676no—
172C2×C22⋊C4—6724no—
173C2×C22⋊C4—6712no—
174C2×C22⋊C4C22×D67112yes—
175C2×C22⋊C4C22×D6716yes—
176#95—6712no—
177#95C22×D6714yescomplemented
178C24—46no—
179C24—26812no—
180C24—2686no—
181C24—26864no—
182C24—26848no—
183C24—674no—
184C24—678no—
185C24—671no—
186C24—678no—
187C24—674no—
188C24—672no—
189C24—671no—
190C24—13424no—
191C24—13412no—
192C24Q8×D6714yes—
193C24C23×D6712yes—
194C24C23×D6711yescharacteristic
195C22×C4—6712no—
196C22×C4—13448no—
197C22×C4—13424no—
198C22×C4C23×D67112yes—
199C4⋊C4—412no—
200C4⋊C4—26824no—
201C4⋊C4—26812no—
202C22⋊C4—412no—
203C22⋊C4—46no—
204C22⋊C4—26824no—
205C22⋊C4—26812no—
206C22⋊C4—26812no—
207C22⋊C4—2686no—
208C23—212no—
209C23—26no—
210C23—412no—
211C23—416no—
212C23—26824no—
213C23—26812no—
214C23—268128no—
215C23—26832no—
216C23—26816no—
217C23—26832no—
218C23—26816no—
219C23—672no—
220C23—6712no—
221C23—674no—
222C23—672no—
223C23—13448no—
224C23—13424no—
225C23—13412no—
226C23—1346no—
227C23C2×Q8×D6714yes—
228C23C2×Q8×D6712yes—
229C23C24×D6711yescharacteristic
230C2×C4—224no—
231C2×C4—212no—
232C2×C4—26848no—
233C2×C4—13412no—
234C2×C4—13424no—
235C22—212no—
236C22—26no—
237C22—416no—
238C22—48no—
239C22—44no—
240C22—26832no—
241C22—26816no—
242C22—26832no—
243C22—26816no—
244C22—2688no—
245C22—2684no—
246C22—6712no—
247C22—6716no—
248C22—13412no—
249C22—1346no—
250C22C22×Q8×D6711yescharacteristic
251C22H(5,𝔽2)×D6716yes—
252C4—412no—
253C4—26812no—
254C2—44no—
255C2—42no—
256C2—2684no—
257C2—2682no—
258C2—678no—
259C2#543×D6713yes—
260C2#560×D6714yes—
subgroupG/H
center, FrattiniC23C24×D67
commutator, socle, cosocleC23×C67C25
FittingC67×#2842C2
prime radicalC25×D67C22
hypercenter#2842D67
hypocenterC67C2×#2842
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #2842

Complements

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

NH ≅ G/Ntype
1. #22568474C2semidirect
2. #22568473C2semidirect
3. #22568472C2semidirect
4. #399×D67C2semidirect
5. #396×D67C2semidirect
7. C67×#2842C2semidirect
9. C2×C4⋊C4×D67C22semidirect
12. #16863333C22semidirect
13. #16863332C22semidirect
14. #16863331C22semidirect
15. #16863320C22semidirect
16. #16863319C22semidirect
17. #16863318C22semidirect
21. #95×D67C22semidirect
22. #399×C67C22semidirect
23. #396×C67C22semidirect
32. C2×C4⋊Dic67C23semidirect
33. C2×Dic67⋊C4C23semidirect
41. C67⋊#95C23semidirect
43. C2×C4⋊C4×C67C23semidirect
46. #95×C67C23semidirect
130. D67#2842direct (N × H)
134. #2842D67direct (N × H)
138. C67C2×#2842semidirect
146. #399C2×D67semidirect
150. #396C2×D67semidirect
167. C2×C4⋊C4C22×D67semidirect
177. #95C22×D67semidirect

Maximal subgroups

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

subgroupindex
#225684742
#225684732
#225684722
#399×D672
#396×D672
#379×D672
C67×#28422
C2×#284267