Finite Group Databasebeta

← C2×#126×D17 #14542038

Subgroups — C2×#126×D17 #14542038

Order 2176.

1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.

normal subgroups only

This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 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
1C2×#10587584C213yesmaximal, complemented
2C2×#10587562C211yesmaximal, characteristic, complemented
3C2×#10587565C213yesmaximal, complemented
4#126×D17C2116yesmaximal, direct product factor
5C2×C4⋊C4×D17C213yesmaximal, complemented
6C2×C22⋊C4×D17C213yesmaximal
7C2×C42×D17C211yesmaximal, characteristic, complemented
8C2×C17×#126C211yesmaximal, characteristic, complemented
9C24×D17C2211yescharacteristic
10C22×C4×D17C2213yes—
11C22×C4×D17C2213yes—
12C17×#126C2218yescomplemented
13C2×C4⋊C4×C17C2213yescomplemented
14C2×C22⋊C4×C17C2213yes—
15C2×C42×C17C2211yescharacteristic, complemented
16C2×C22⋊Dic17C2213yes—
17C2×#63227C2213yes—
18C2×#63227C2213yes—
19C2×C4⋊Dic17C2213yescomplemented
20C2×#63225C2213yescomplemented
21C2×#63225C2213yescomplemented
22C2×C4×Dic17C2213yescomplemented
23#10587584C22124yescomplemented
24C4⋊C4×D17C22124yescomplemented
25C22⋊C4×D17C22124yes—
26#10587565C22124yescomplemented
27#10587562C2218yescomplemented
28C42×D17C2218yescomplemented
29C24×C17C2311yescharacteristic
30C23×D17—212no—
31C23×D17C2314yes—
32C23×D17C2311yescharacteristic
33C23×D17C2311yescharacteristic
34C22×C4×C17C2313yes—
35C22×C4×C17C2313yes—
36C22×Dic17C2313yes—
37C22×Dic17C2313yes—
38C2×C4×D17—212no—
39C2×C4×D17—212no—
40C2×C4×D17C23112yes—
41C2×C4×D17C23112yes—
42C4⋊C4×C17C23112yescomplemented
43C22⋊C4×C17C23112yes—
44C42×C17C2314yescomplemented
45C22⋊Dic17C23112yes—
46#63227C23112yes—
47#63227C23112yes—
48C4⋊Dic17C23112yescomplemented
49#63225C23112yescomplemented
50#63225C23112yescomplemented
51C4×Dic17C23112yescomplemented
52C23×C17—26no—
53C23×C17C2412yes—
54C23×C17C2411yescharacteristic
55C22×D17—26no—
56C22×D17—224no—
57C22×D17—416no—
58C22×D17#41112yes—
59C22×D17C2412yes—
60C22×D17C2412yes—
61C2×C4×C17—26no—
62C2×C4×C17—26no—
63C2×C4×C17C2416yes—
64C2×C4×C17C2416yes—
65C2×Dic17—26no—
66C2×Dic17—26no—
67C2×Dic17C2416yes—
68C2×Dic17C2416yes—
69C4×D17—224no—
70C4×D17—224no—
71C22×#126—171nomaximal, Sylow
72C22×C17—26no—
73C22×C17—44no—
74C22×C17C2×#4116yes—
75C22×C17C2511yescharacteristic
76C2×D17—26no—
77C2×D17—44no—
78C2×D17—416no—
79C2×D17#126116yesdirect product factor
80C2×D17C2×#41112yes—
81C4×C17—26no—
82C4×C17—26no—
83Dic17—26no—
84Dic17—26no—
85C2×#126—1716no—
86C2×#126—173no—
87C2×#126—173no—
88C2×#126—171no—
89C2×#126D1711yescharacteristic, direct product factor
90C22×C4⋊C4—173no—
91C22×C22⋊C4—173no—
92C22×C42—171no—
93C2×C17—42no—
94C2×C17C2×#12614yescomplemented
95C2×C17C22×#4113yes—
96D17—42no—
97D17C2×#12618yesdirect product factor
98C25—171no—
99C23×C4—173no—
100C23×C4—173no—
101#126—1724no—
102#126—1724no—
103#126—178no—
104#126C2×D1718yesdirect product factor
105C2×C4⋊C4—173no—
106C2×C4⋊C4—173no—
107C2×C4⋊C4—173no—
108C2×C4⋊C4—1724no—
109C2×C4⋊C4C2×D1713yescomplemented
110C2×C22⋊C4—173no—
111C2×C22⋊C4—173no—
112C2×C22⋊C4—173no—
113C2×C22⋊C4—1724no—
114C2×C22⋊C4C2×D1713yes—
115C2×C42—173no—
116C2×C42—178no—
117C2×C42C2×D1711yescharacteristic, complemented
118C17C22×#12611yescharacteristic, complemented, Sylow
119C24—171no—
120C24—174no—
121C24—171no—
122C24—3412no—
123C24C22×D1711yescharacteristic
124C22×C4—173no—
125C22×C4—173no—
126C22×C4—1712no—
127C22×C4—1712no—
128C22×C4—3412no—
129C22×C4—3412no—
130C22×C4C22×D1713yes—
131C22×C4C22×D1713yes—
132C4⋊C4—1712no—
133C4⋊C4—1712no—
134C4⋊C4—1712no—
135C4⋊C4C22×D17112yescomplemented
136C22⋊C4—1712no—
137C22⋊C4—1712no—
138C22⋊C4—1712no—
139C22⋊C4C22×D17112yes—
140C42—1712no—
141C42C22×D1714yescomplemented
142C23—26no—
143C23—1712no—
144C23—172no—
145C23—172no—
146C23—3424no—
147C23—346no—
148C23—6816no—
149C23C23×D1712yes—
150C23C23×D1711yescharacteristic
151C2×C4—26no—
152C2×C4—26no—
153C2×C4—176no—
154C2×C4—176no—
155C2×C4—3424no—
156C2×C4—3424no—
157C2×C4—346no—
158C2×C4—346no—
159C2×C4C23×D1716yes—
160C2×C4C23×D1716yes—
161C22—26no—
162C22—44no—
163C22—1716no—
164C22—1712no—
165C22—346no—
166C22—6816no—
167C22—684no—
168C22#41×D1716yes—
169C22C24×D1711yescharacteristic
170C4—26no—
171C4—26no—
172C4—346no—
173C4—346no—
174C2—42no—
175C2—178no—
176C2—682no—
177C2#126×D1714yesdirect product factor
178C2C2×#41×D1713yes—
subgroupG/H
centerC23C23×D17
commutator, cosocleC22×C17C25
FrattiniC22C24×D17
FittingC2×C17×#126C2
socleC23×C17C24
prime radicalC24×D17C22
hypercenterC2×#126D17
hypocenterC17C22×#126
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
2. C2×#10587562C2semidirect
3. C2×#10587565C2semidirect
4. #126×D17C2direct (N × H)
5. C2×C4⋊C4×D17C2semidirect
7. C2×C42×D17C2semidirect
8. C2×C17×#126C2semidirect
12. C17×#126C22semidirect
13. C2×C4⋊C4×C17C22semidirect
15. C2×C42×C17C22semidirect
19. C2×C4⋊Dic17C22semidirect
20. C2×#63225C22semidirect
21. C2×#63225C22semidirect
22. C2×C4×Dic17C22semidirect
23. #10587584C22semidirect
24. C4⋊C4×D17C22semidirect
26. #10587565C22semidirect
27. #10587562C22semidirect
28. C42×D17C22semidirect
42. C4⋊C4×C17C23semidirect
44. C42×C17C23semidirect
48. C4⋊Dic17C23semidirect
49. #63225C23semidirect
50. #63225C23semidirect
51. C4×Dic17C23semidirect
79. C2×D17#126direct (N × H)
89. C2×#126D17direct (N × H)
94. C2×C17C2×#126semidirect
97. D17C2×#126direct (N × H)
104. #126C2×D17direct (N × H)
109. C2×C4⋊C4C2×D17semidirect
117. C2×C42C2×D17semidirect
118. C17C22×#126semidirect
135. C4⋊C4C22×D17semidirect
141. C42C22×D17semidirect
177. C2#126×D17direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#105875842
C2×#105875622
C2×#105875652
#126×D172
C2×C4⋊C4×D172
C2×C22⋊C4×D172
C2×C42×D172
C2×C17×#1262
C22×#12617