Finite Group Databasebeta

← C2×#14536824 #17240881

Subgroups — C2×#14536824 #17240881

Order 4352.

3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3666 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×#10587560C211yesmaximal, characteristic, complemented
2#14536824C2116yesmaximal, direct product factor
3C2×#11762726C216yesmaximal, complemented
4C2×#11762718C216yesmaximal, complemented
5C2×#11762677C211yesmaximal, characteristic, complemented
6C22×C17×#126C211yesmaximal, characteristic, complemented
7C23×D68C2216yes—
8C23×C22⋊D17C2214yes—
9C2×#10587560—28no—
10C2×#10587560C2218yescomplemented
11C22×#63227C22112yes—
12#11762726C22148yescomplemented
13#11762718C22148yescomplemented
14#11762677C2218yescomplemented
15C2×#10587502C2216yes—
16C2×C17×#126—28no—
17C2×C17×#126C2218yescomplemented
18C22×C4⋊C4×C17C2213yes—
19C22×C22⋊C4×C17C2213yes—
20C22×C42×C17C2211yescharacteristic
21C25×C17C2311yescharacteristic
22C24×D17C2314yes—
23C23×C4×C17C2313yes—
24C23×C4×C17C2313yes—
25C22×C22⋊D17—424no—
26C22×C22⋊D17C23116yes—
27C23×Dic17C2314yes—
28C22×D68—224no—
29C22×D68—424no—
30C22×D68C23124yes—
31C17×#126—232no—
32C2×C4⋊C4×C17—212no—
33C2×C4⋊C4×C17C23112yes—
34C2×C22⋊C4×C17—212no—
35C2×C22⋊C4×C17C23112yes—
36C2×C42×C17D418yescomplemented
37C2×C42×C17C2314yes—
38C2×#63227—424no—
39C2×#63227C23148yes—
40#10587560—232no—
41#10587502C23124yes—
42C24×C17—22no—
43C24×C17—212no—
44C24×C17C2412yes—
45C24×C17C2411yescharacteristic
46C23×D17—28no—
47C23×D17—224no—
48C23×D17—224no—
49C23×D17C2418yes—
50C22×C4×C17—212no—
51C22×C4×C17—26no—
52C22×C4×C17—212no—
53C22×C4×C17C2×D4112yes—
54C22×C4×C17C2416yes—
55C22×C4×C17C2416yes—
56C2×C22⋊D17—448no—
57C2×C22⋊D17—464no—
58C22×Dic17—224no—
59C22×Dic17C2418yes—
60C2×D68—248no—
61C2×D68—448no—
62C2×D68—496no—
63C4⋊C4×C17—224no—
64C22⋊C4×C17—224no—
65C42×C17C2×D4116yescomplemented
66#63227—448no—
67C2×#2818—171nomaximal, Sylow
68C23×C17—22no—
69C23×C17—212no—
70C23×C17—412no—
71C23×C17—416no—
72C23×C17C22×D412yes—
73C23×C17H(5,𝔽2)112yes—
74C23×C17C2511yescharacteristic
75C22×D17—28no—
76C22×D17—224no—
77C22×D17—224no—
78C22×D17—448no—
79C22×D17—416no—
80C22×D17—448no—
81C2×C4×C17—224no—
82C2×C4×C17—212no—
83C2×C4×C17—26no—
84C2×C4×C17—212no—
85C2×C4×C17—412no—
86C2×C4×C17C22×D4112yes—
87C22⋊D17—464no—
88C2×Dic17—224no—
89C2×Dic17—416no—
90D68—496no—
91C22×#127—171no—
92C22×#126D1711yescharacteristic, complemented
93#2818—1716no—
94C2×#394—176no—
95C2×#392—176no—
96C2×#383—171no—
97C22×C17—46no—
98C22×C17—48no—
99C22×C17—88no—
100C22×C17#535112yes—
101C22×C17#560116yes—
102C22×C17C23×D411yescharacteristic
103C22×C17C2×H(5,𝔽2)16yes—
104C2×D17—424no—
105C2×D17—48no—
106C2×D17—424no—
107C2×D17—832no—
108C4×C17—212no—
109C4×C17—46no—
110Dic17—48no—
111C23×D4—174no—
112C23×D4—176no—
113C2×#127—178no—
114C2×#127—348no—
115C2×#126—28no—
116C2×#126C2×D1718yescomplemented
117C22×C4⋊C4C2×D1713yes—
118C22×C22⋊C4—1712no—
119C22×C22⋊C4C2×D1713yes—
120C22×C42C2×D1711yescharacteristic
121#394—1748no—
122#392—1748no—
123#383—178no—
124C2×#95—176no—
125C2×C17—82no—
126C2×C17#281818yescomplemented
127C2×C17C2×#53513yes—
128C2×C17C2×#56014yes—
129D17—88no—
130C25—174no—
131C25C22×D1711yescharacteristic
132C22×D4—1716no—
133C22×D4—1724no—
134C22×D4—3424no—
135C22×D4—6824no—
136C22×D4—6824no—
137C23×C4—174no—
138C23×C4C22×D1713yes—
139C23×C4C22×D1713yes—
140#127—3432no—
141#126—232no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4C22×D17112yes—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—1748no—
146C2×C22⋊C4—6824no—
147C2×C22⋊C4C22×D17112yes—
148C2×C42C22⋊D1718yescomplemented
149C2×C42C22×D1714yes—
150#95—1724no—
151C17C2×#281811yescharacteristic, complemented, Sylow
152C24—212no—
153C24—22no—
154C24—178no—
155C24—3424no—
156C24—3424no—
157C24—348no—
158C24C23×D1712yes—
159C24C23×D1711yescharacteristic
160C2×D4—3448no—
161C2×D4—6848no—
162C2×D4—6896no—
163C2×D4—6848no—
164C2×D4—6864no—
165C22×C4—212no—
166C22×C4—212no—
167C22×C4—26no—
168C22×C4—178no—
169C22×C4—3424no—
170C22×C4C2×C22⋊D17112yes—
171C22×C4C23×D1716yes—
172C22×C4C23×D1716yes—
173C4⋊C4—224no—
174C22⋊C4—224no—
175C22⋊C4—6848no—
176C42C2×C22⋊D17116yescomplemented
177C23—22no—
178C23—212no—
179C23—416no—
180C23—412no—
181C23—348no—
182C23—3424no—
183C23—3424no—
184C23—6816no—
185C23—6848no—
186C23—6848no—
187C23#10587696112yes—
188C23C22×C22⋊D1712yes—
189C23C24×D1711yescharacteristic
190D4—6896no—
191D4—6864no—
192C2×C4—224no—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—412no—
197C2×C4—3424no—
198C2×C4—6816no—
199C2×C4C22×C22⋊D17112yes—
200C22—46no—
201C22—48no—
202C22—88no—
203C22—6824no—
204C22—688no—
205C22—6824no—
206C22—13632no—
207C22#11763373112yes—
208C22#11763550116yes—
209C22C23×C22⋊D1711yescharacteristic
210C22C2×#1058769616yes—
211C4—212no—
212C4—46no—
213C4—688no—
214C2—82no—
215C2—1368no—
216C2#1453682418yesdirect product factor
217C2C2×#1176337313yes—
218C2C2×#1176355014yes—
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC22×C17×#126C2
socleC24×C17C24
hypercenterC22×#126D17
hypocenterC17C2×#2818
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×#10587560C2semidirect
2. #14536824C2direct (N × H)
3. C2×#11762726C2semidirect
4. C2×#11762718C2semidirect
5. C2×#11762677C2semidirect
6. C22×C17×#126C2semidirect
10. C2×#10587560C22semidirect
12. #11762726C22semidirect
13. #11762718C22semidirect
14. #11762677C22semidirect
17. C2×C17×#126C22semidirect
36. C2×C42×C17D4semidirect
65. C42×C17C2×D4semidirect
92. C22×#126D17semidirect
116. C2×#126C2×D17semidirect
126. C2×C17#2818semidirect
148. C2×C42C22⋊D17semidirect
151. C17C2×#2818semidirect
176. C42C2×C22⋊D17semidirect
216. C2#14536824direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875602
#145368242
C2×#117627262
C2×#117627182
C2×#117626772
C22×C17×#1262
C2×#281817