Finite Group Databasebeta

← C2×#22565928 #31022744

Subgroups — C2×#22565928 #31022744

Order 17152.

5194 conjugacy classes of subgroups; only the 907 normal subgroups are shown.

normal subgroups only

This group has 5194 conjugacy classes of subgroups — too many to draw. Only its 907 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×#14524026C213yesmaximal
2#22565928C2116yesmaximal, direct product factor
3C2×#16863315C214yesmaximal, complemented
4C2×#16863307C214yesmaximal, complemented
5C2×#16863259C213yesmaximal, complemented
6C24×Q8×C67C211yesmaximal, characteristic, complemented
7C23×C22⋊D67C2211yescharacteristic
8C2×#14524026—224no—
9C2×#14524026C22124yes—
10C22×Dic67⋊C4C2216yes—
11C22×C22⋊Dic67C2213yes—
12C22×(C2×D67)⋊C4C2216yes—
13#16863315C22132yescomplemented
14#16863307C22132yescomplemented
15#16863259C22124yescomplemented
16C2×C67⋊#95C22112yes—
17C23×Q8×C67—224no—
18C23×Q8×C67C2214yes—
19C23×Q8×C67C2218yescomplemented
20C24×C4×C67C2213yes—
21C24×D67C2311yescharacteristic
22C22×C22⋊D67—212no—
23C22×C22⋊D67C2314yes—
24C23×Dic67C2316yes—
25C23×Dic67C2311yescharacteristic
26#14524026—296no—
27C2×Dic67⋊C4—224no—
28C2×Dic67⋊C4C23124yes—
29C2×C22⋊Dic67—212no—
30C2×C22⋊Dic67C23112yes—
31C2×(C2×D67)⋊C4—224no—
32C2×(C2×D67)⋊C4C23124yes—
33C67⋊#95C23148yes—
34C25×C67C2311yescharacteristic
35C22×Q8×C67—2128no—
36C22×Q8×C67—248no—
37C22×Q8×C67—296no—
38C22×Q8×C67C23116yes—
39C23×C4×C67—224no—
40C23×C4×C67D4124yes—
41C23×C4×C67C2316yes—
42C23×C4×C67C23112yes—
43C23×D67—26no—
44C23×D67—28no—
45C23×D67C2412yes—
46C2×C22⋊D67—224no—
47C2×C22⋊D67—416no—
48C22×Dic67—212no—
49C22×Dic67—26no—
50C22×Dic67—224no—
51C22×Dic67C24112yes—
52C22×Dic67C2412yes—
53Dic67⋊C4—248no—
54C22⋊Dic67—224no—
55(C2×D67)⋊C4—248no—
56C24×C67—28no—
57C24×C67C2×D4112yes—
58C24×C67C2412yes—
59C24×C67C2411yescharacteristic
60C2×Q8×C67—2128no—
61C2×Q8×C67—2256no—
62C2×Q8×C67—296no—
63C22×C4×C67—296no—
64C22×C4×C67—248no—
65C22×C4×C67—224no—
66C22×C4×C67C2×D4124yes—
67C22×C4×C67C2×D4148yes—
68C22×C4×C67C24112yes—
69C23×C67—248no—
70C23×C67—28no—
71C23×C67#120116yes—
72C23×C67C22×D4112yes—
73C23×C67C22×D416yes—
74C23×C672−1+418yes—
75C23×C67C2511yescharacteristic
76C22×D67—26no—
77C22×D67—28no—
78C22×D67—44no—
79C22×D67—424no—
80Q8×C67—2128no—
81C2×C4×C67—296no—
82C2×C4×C67—248no—
83C2×C4×C67—224no—
84C2×C4×C67C22×D4124yes—
85C22⋊D67—416no—
86C2×Dic67—212no—
87C2×Dic67—26no—
88C2×Dic67—224no—
89C2×Dic67—44no—
90C2×Dic67—424no—
91C22×C67—232no—
92C22×C67—224no—
93C22×C67C2×#12018yes—
94C22×C67C2×#12014yes—
95C22×C67#536124yes—
96C22×C67C23×D413yes—
97C22×C67C2×2−1+414yes—
98C2×D67—42no—
99C2×D67—412no—
100C2×D67—88no—
101C4×C67—224no—
102Dic67—42no—
103Dic67—412no—
104C2×#2614—671nomaximal, Sylow
105C2×C67—28no—
106C2×C67#261418yescomplemented
107C2×C67C22×#12011yescharacteristic
108C2×C67C2×#53616yes—
109D67—82no—
110C24×Q8D6711yescharacteristic, complemented
111C22×#123—673no—
112#2614—6716no—
113C2×#395—674no—
114C2×#393—674no—
115C2×#381—673no—
116C67C2×#261411yescharacteristic, complemented, Sylow
117C23×Q8—224no—
118C23×Q8C2×D6714yes—
119C23×Q8C2×D6718yescomplemented
120C23×D4—671no—
121C24×C4C2×D6713yes—
122C2×#123—6724no—
123C2×#123—13424no—
124C22×C4⋊C4—676no—
125C22×C22⋊C4—673no—
126C22×C22⋊C4—676no—
127#395—6732no—
128#393—6732no—
129#381—6724no—
130C2×#95—6712no—
131C25—671no—
132C25C22×D6711yescharacteristic
133C22×Q8—2128no—
134C22×Q8—248no—
135C22×Q8—296no—
136C22×Q8C22×D67116yes—
137C22×D4—674no—
138C22×D4—13412no—
139C23×C4—224no—
140C23×C4—676no—
141C23×C4—671no—
142C23×C4C22⋊D67124yes—
143C23×C4C22×D67112yes—
144C23×C4C22×D6716yes—
145#123—13496no—
146C2×C4⋊C4—6724no—
147C2×C4⋊C4—13424no—
148C2×C22⋊C4—6712no—
149C2×C22⋊C4—6724no—
150C2×C22⋊C4—13412no—
151C2×C22⋊C4—13424no—
152#95—6748no—
153C24—28no—
154C24—672no—
155C24—1346no—
156C24—1348no—
157C24C2×C22⋊D67112yes—
158C24C23×D6712yes—
159C24C23×D6711yescharacteristic
160C2×Q8—2128no—
161C2×Q8—2256no—
162C2×Q8—296no—
163C2×D4—26816no—
164C2×D4—13424no—
165C22×C4—296no—
166C22×C4—248no—
167C22×C4—224no—
168C22×C4—6712no—
169C22×C4—672no—
170C22×C4—13412no—
171C22×C4—1346no—
172C22×C4—13424no—
173C22×C4C2×C22⋊D67124yes—
174C22×C4C2×C22⋊D67148yes—
175C22×C4C23×D67112yes—
176C4⋊C4—13448no—
177C22⋊C4—13424no—
178C22⋊C4—13448no—
179C23—248no—
180C23—28no—
181C23—2684no—
182C23—26824no—
183C23—1346no—
184C23—1348no—
185C23#14524012116yes—
186C23C22×C22⋊D67112yes—
187C23C22×C22⋊D6716yes—
188C23#1452409118yes—
189C23C24×D6711yescharacteristic
190Q8—2128no—
191D4—26816no—
192C2×C4—296no—
193C2×C4—248no—
194C2×C4—224no—
195C2×C4—2684no—
196C2×C4—26824no—
197C2×C4—13412no—
198C2×C4—1346no—
199C2×C4—13424no—
200C2×C4C22×C22⋊D67124yes—
201C22—232no—
202C22—224no—
203C22—2682no—
204C22—26812no—
205C22—5368no—
206C22C2×#1452401218yes—
207C22C2×#1452401214yes—
208C22#16863970124yes—
209C22C23×C22⋊D6713yes—
210C22C2×#1452409114yes—
211C4—224no—
212C4—2682no—
213C4—26812no—
214C2—28no—
215C2—5362no—
216C2#2256592818yesdirect product factor
217C2C22×#1452401211yescharacteristic
218C2C2×#1686397016yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC24×Q8×C67C2
socleC24×C67C24
prime radicalC23×C22⋊D67C22
hypercenterC24×Q8D67
hypocenterC67C2×#2614
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
2. #22565928C2direct (N × H)
3. C2×#16863315C2semidirect
4. C2×#16863307C2semidirect
5. C2×#16863259C2semidirect
6. C24×Q8×C67C2semidirect
13. #16863315C22semidirect
14. #16863307C22semidirect
15. #16863259C22semidirect
19. C23×Q8×C67C22semidirect
106. C2×C67#2614semidirect
110. C24×Q8D67semidirect
116. C67C2×#2614semidirect
119. C23×Q8C2×D67semidirect
216. C2#22565928direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240262
#225659282
C2×#168633152
C2×#168633072
C2×#168632592
C24×Q8×C672
C2×#261467