Finite Group Databasebeta

← C2×#31772673 #46831005

Subgroups — C2×#31772673 #46831005

Order 34624.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 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#31772673C2116yesmaximal, direct product factor
2C2×#22575836C212yesmaximal, complemented
3C2×#22575835C214yesmaximal, complemented
4C22×C22⋊D541C214yesmaximal, complemented
5C2×D4×D541C214yesmaximal, complemented
6C22×D4×C541C211yesmaximal, characteristic, complemented
7C23×D541C2214yescomplemented
8#22575836C22116yescomplemented
9#22575835C22132yescomplemented
10C2×Dic1082C2212yescomplemented
11C2×D2164C2212yescomplemented
12C2×C22⋊D541C2218yescomplemented
13C2×C22⋊D541C2214yescomplemented
14C2×C22⋊D541C22132yescomplemented
15D4×D541C22132yescomplemented
16C22×Dic541C2214yescomplemented
17C2×C4×D541C2214yescomplemented
18C24×C541C2212yescomplemented
19C2×D4×C541C2214yescomplemented
20C2×D4×C541C2218yescomplemented
21C22×C4×C541C2211yescharacteristic, complemented
22C22×D541—48no—
23C22×D541C2314yes—
24C22×D541C23116yescomplemented
25Dic1082C2318yescomplemented
26D2164C2318yescomplemented
27C22⋊D541C23132yescomplemented
28C22⋊D541C23116yescomplemented
29C2×Dic541C2314yes—
30C2×Dic541C23116yescomplemented
31C4×D541C23116yescomplemented
32C23×C541—44no—
33C23×C541C2311yescharacteristic
34C23×C541C2314yes—
35C23×C541C2318yescomplemented
36D4×C541C23116yescomplemented
37C2×C4×C541C2312yes—
38C2×C4×C541C2314yescomplemented
39C22×C541—22no—
40C22×C541—28no—
41C22×C541—48no—
42C22×C541C2412yes—
43C22×C541C2418yes—
44C22×C541C2411yescharacteristic
45C2×D541—28no—
46C2×D541—416no—
47C2×D541C2418yes—
48C4×C541C2414yes—
49Dic541C2418yes—
50C2×C541—28no—
51C2×C541—22no—
52C2×C541H(5,𝔽2)12yescomplemented
53C2×C541C2511yescharacteristic
54D541—28no—
55C541C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
56C2×H(5,𝔽2)—5411nomaximal, Sylow
57H(5,𝔽2)—54116no—
58C2×#41—5414no—
59C2×#41—5412no—
60C22×D4—5414no—
61C22×D4—5414no—
62C22×D4D54111yescharacteristic, complemented
63C24—5414no—
64C24C2×D54112yescomplemented
65#41—54132no—
66#41—54116no—
67C2×Q8—5412no—
68C2×D4—5418no—
69C2×D4—5414no—
70C2×D4—54132no—
71C2×D4—54132no—
72C2×D4—5412no—
73C2×D4C2×D54114yescomplemented
74C2×D4C2×D54118yescomplemented
75C22×C4—5414no—
76C22×C4—5414no—
77C22×C4C2×D54111yescharacteristic, complemented
78C23—44no—
79C23—5414no—
80C23—54116no—
81C23—21648no—
82C23C22×D54114yes—
83C23C22×D54111yescharacteristic
84C23C22×D54118yescomplemented
85Q8—5418no—
86D4—54132no—
87D4—54116no—
88D4—5418no—
89D4C22×D541116yescomplemented
90C2×C4—54116no—
91C2×C4—5414no—
92C2×C4—54116no—
93C2×C4C22×D54112yes—
94C2×C4C22×D54114yescomplemented
95C22—22no—
96C22—28no—
97C22—48no—
98C22—5418no—
99C22—10828no—
100C22—216416no—
101C22C23×D54118yes—
102C22C23×D54112yes—
103C22C23×D54111yescharacteristic
104C4—5418no—
105C4C23×D54114yes—
106C2—22no—
107C2—28no—
108C2—10828no—
109C2#3177267312yesdirect product factor
110C2C24×D54111yescharacteristic
subgroupG/H
centerC22C23×D541
commutator, cosocleC2×C541C25
FrattiniC2C24×D541
FittingC22×D4×C541C2
socleC22×C541C24
hypercenterC22×D4D541
hypocenterC541C2×H(5,𝔽2)
LCSG ▷ C2×C541 ▷ C541
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #31772673C2direct (N × H)
2. C2×#22575836C2semidirect
3. C2×#22575835C2semidirect
4. C22×C22⋊D541C2semidirect
5. C2×D4×D541C2semidirect
6. C22×D4×C541C2semidirect
7. C23×D541C22semidirect
8. #22575836C22semidirect
9. #22575835C22semidirect
10. C2×Dic1082C22semidirect
11. C2×D2164C22semidirect
12. C2×C22⋊D541C22semidirect
13. C2×C22⋊D541C22semidirect
14. C2×C22⋊D541C22semidirect
15. D4×D541C22semidirect
16. C22×Dic541C22semidirect
17. C2×C4×D541C22semidirect
18. C24×C541C22semidirect
19. C2×D4×C541C22semidirect
20. C2×D4×C541C22semidirect
21. C22×C4×C541C22semidirect
24. C22×D541C23semidirect
25. Dic1082C23semidirect
26. D2164C23semidirect
27. C22⋊D541C23semidirect
28. C22⋊D541C23semidirect
30. C2×Dic541C23semidirect
31. C4×D541C23semidirect
35. C23×C541C23semidirect
36. D4×C541C23semidirect
38. C2×C4×C541C23semidirect
52. C2×C541H(5,𝔽2)semidirect
55. C541C2×H(5,𝔽2)semidirect
62. C22×D4D541semidirect
64. C24C2×D541semidirect
73. C2×D4C2×D541semidirect
74. C2×D4C2×D541semidirect
77. C22×C4C2×D541semidirect
84. C23C22×D541semidirect
89. D4C22×D541semidirect
94. C2×C4C22×D541semidirect
109. C2#31772673direct (N × H)

Maximal subgroups

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

subgroupindex
#317726732
C2×#225758362
C2×#225758352
C22×C22⋊D5412
C2×D4×D5412
C22×D4×C5412
C2×H(5,𝔽2)541