Finite Group Databasebeta

← C2×#22566311 #31022373

Subgroups — C2×#22566311 #31022373

Order 17152.

4886 conjugacy classes of subgroups; only the 907 normal subgroups are shown.

normal subgroups only

This group has 4886 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×#14524017C216yesmaximal, complemented
2#22566311C2116yesmaximal, direct product factor
3C2×#16863333C214yesmaximal, complemented
4C2×#16863306C214yesmaximal, complemented
5C22×#95×C67C211yesmaximal, characteristic, complemented
6C23×D268C2216yes—
7C23×C22⋊D67C2214yes—
8C2×#14524017—248no—
9C2×#14524017C22148yescomplemented
10C22×C4⋊Dic67C2216yes—
11C22×(C2×D67)⋊C4C22112yes—
12#16863333C22132yescomplemented
13#16863306C22132yescomplemented
14C24×C4×C67C2213yes—
15C2×#95×C67C2214yes—
16C2×#95×C67C2218yescomplemented
17C24×D67C2314yes—
18C22×D268—224no—
19C22×D268—424no—
20C22×D268C23124yes—
21C22×C22⋊D67—248no—
22C22×C22⋊D67C23116yes—
23C23×Dic67C2314yes—
24#14524017—2192no—
25C2×C4⋊Dic67—224no—
26C2×C4⋊Dic67C23124yes—
27C2×(C2×D67)⋊C4—248no—
28C2×(C2×D67)⋊C4C23148yes—
29C25×C67C2311yescharacteristic
30C23×C4×C67—412no—
31C23×C4×C67D4124yescomplemented
32C23×C4×C67C2316yes—
33C23×C4×C67C23112yes—
34#95×C67C23116yes—
35C23×D67—224no—
36C23×D67—224no—
37C23×D67—28no—
38C23×D67C2418yes—
39C2×D268—248no—
40C2×D268—448no—
41C2×D268—496no—
42C2×C22⋊D67—296no—
43C2×C22⋊D67—464no—
44C22×Dic67—224no—
45C22×Dic67C2418yes—
46C4⋊Dic67—248no—
47(C2×D67)⋊C4—296no—
48C24×C67—26no—
49C24×C67—22no—
50C24×C67C2×D4112yes—
51C24×C67C2412yes—
52C24×C67C2411yescharacteristic
53C22×C4×C67—224no—
54C22×C4×C67—448no—
55C22×C4×C67—424no—
56C22×C4×C67C2×D4124yes—
57C22×C4×C67C2×D4148yescomplemented
58C22×C4×C67C24112yes—
59C23×C67—224no—
60C23×C67—224no—
61C23×C67—26no—
62C23×C67—22no—
63C23×C67#127116yes—
64C23×C67C22×D4112yes—
65C23×C67C22×D416yes—
66C23×C67H(5,𝔽2)16yes—
67C23×C672−1+412yes—
68C23×C67C2511yescharacteristic
69C22×D67—224no—
70C22×D67—224no—
71C22×D67—28no—
72C22×D67—416no—
73C22×D67—448no—
74C22×D67—448no—
75C2×C4×C67—248no—
76C2×C4×C67—224no—
77C2×C4×C67—448no—
78C2×C4×C67C22×D4124yes—
79C22⋊D67—464no—
80C2×Dic67—224no—
81C2×Dic67—416no—
82D268—496no—
83C22×C67—232no—
84C22×C67—212no—
85C22×C67—212no—
86C22×C67C2×#12718yes—
87C22×C67C2×#12714yes—
88C22×C67#534112yes—
89C22×C67#537112yes—
90C22×C67C23×D413yes—
91C22×C67C2×H(5,𝔽2)13yes—
92C22×C67C2×2−1+411yescharacteristic
93C2×D67—48no—
94C2×D67—424no—
95C2×D67—424no—
96C2×D67—832no—
97C4×C67—224no—
98Dic67—48no—
99C2×#2656—671nomaximal, Sylow
100C2×C67—28no—
101C2×C67#265618yescomplemented
102C2×C67C22×#12711yescharacteristic
103C2×C67C2×#53413yes—
104C2×C67C2×#53713yes—
105D67—88no—
106C22×#121—676no—
107#2656—6716no—
108C2×#399—674no—
109C2×#392—674no—
110C22×#95D6711yescharacteristic, complemented
111C67C2×#265611yescharacteristic, complemented, Sylow
112C23×D4—674no—
113C23×D4—676no—
114C24×C4C2×D6713yes—
115C2×#121—6748no—
116C2×#121—13448no—
117C22×C4⋊C4—676no—
118C22×C22⋊C4—6712no—
119#399—6732no—
120#392—6732no—
121C2×#95C2×D6714yes—
122C2×#95C2×D6718yescomplemented
123C25—674no—
124C25C22×D6711yescharacteristic
125C22×D4—26824no—
126C22×D4—6716no—
127C22×D4—6724no—
128C22×D4—13448no—
129C22×D4—13424no—
130C23×C4—412no—
131C23×C4—674no—
132C23×C4D268124yescomplemented
133C23×C4C22×D6716yes—
134C23×C4C22×D67112yes—
135#121—134192no—
136C2×C4⋊C4—6724no—
137C2×C4⋊C4—13424no—
138C2×C22⋊C4—6748no—
139C2×C22⋊C4—13448no—
140#95C22×D67116yes—
141C24—26no—
142C24—22no—
143C24—678no—
144C24—13424no—
145C24—13424no—
146C24—1348no—
147C24C2×D268112yes—
148C24C23×D6712yes—
149C24C23×D6711yescharacteristic
150C2×D4—26896no—
151C2×D4—26848no—
152C2×D4—26864no—
153C2×D4—13496no—
154C2×D4—13448no—
155C22×C4—224no—
156C22×C4—448no—
157C22×C4—424no—
158C22×C4—678no—
159C22×C4—13424no—
160C22×C4C2×D268124yes—
161C22×C4C2×D268148yescomplemented
162C22×C4C23×D67112yes—
163C4⋊C4—13448no—
164C22⋊C4—13496no—
165C23—224no—
166C23—26no—
167C23—22no—
168C23—224no—
169C23—26816no—
170C23—26848no—
171C23—26848no—
172C23—13424no—
173C23—13424no—
174C23—1348no—
175C23#14524040116yes—
176C23C22×D268112yes—
177C23C22×D26816yes—
178C23#1452408816yes—
179C23#1452409112yes—
180C23C24×D6711yescharacteristic
181D4—26896no—
182D4—26864no—
183C2×C4—248no—
184C2×C4—224no—
185C2×C4—448no—
186C2×C4—26816no—
187C2×C4—13424no—
188C2×C4C22×D268124yes—
189C22—232no—
190C22—212no—
191C22—212no—
192C22—2688no—
193C22—26824no—
194C22—26824no—
195C22—53632no—
196C22C2×#1452404018yes—
197C22C2×#1452404014yes—
198C22#16863958112yes—
199C22#16863977112yes—
200C22C23×D26813yes—
201C22C2×#1452408813yes—
202C22C2×#1452409111yescharacteristic
203C4—224no—
204C4—2688no—
205C2—28no—
206C2—5368no—
207C2#2256631118yesdirect product factor
208C2C22×#1452404011yescharacteristic
209C2C2×#1686395813yes—
210C2C2×#1686397713yes—
subgroupG/H
centerC24C23×D67
commutator, cosocleC23×C67C25
FrattiniC23C24×D67
FittingC22×#95×C67C2
socleC24×C67C24
hypercenterC22×#95D67
hypocenterC67C2×#2656
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C22×#14524017C2semidirect
2. #22566311C2direct (N × H)
3. C2×#16863333C2semidirect
4. C2×#16863306C2semidirect
5. C22×#95×C67C2semidirect
9. C2×#14524017C22semidirect
12. #16863333C22semidirect
13. #16863306C22semidirect
16. C2×#95×C67C22semidirect
31. C23×C4×C67D4semidirect
57. C22×C4×C67C2×D4semidirect
101. C2×C67#2656semidirect
110. C22×#95D67semidirect
111. C67C2×#2656semidirect
122. C2×#95C2×D67semidirect
132. C23×C4D268semidirect
161. C22×C4C2×D268semidirect
207. C2#22566311direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240172
#225663112
C2×#168633332
C2×#168633062
C22×#95×C672
C2×#265667