Finite Group Databasebeta

← #30593916

Subgroups — #30593916

Order 15616.

59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.

normal subgroups only

This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 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#21460631C2127yesmaximal, complemented
2#21460630C2136yesmaximal, complemented
3C2×#16840599C2127yesmaximal, complemented
4#585×D61C2136yesmaximal, complemented
5C61×2−1+6C211yesmaximal, characteristic, complemented
6#16840599C221216yescomplemented
7#16840599C221432yescomplemented
8#16840598C221540yescomplemented
9#16840597C221120yescomplemented
10C2×#14519825C221135yescomplemented
112−1+4×D61C221216yescomplemented
12C2×#14519823C221270yescomplemented
13H(5,𝔽2)×D61C221360yescomplemented
14C22×#11758226C22145yescomplemented
15C2×#41×D61C221270yescomplemented
16C61×#585C22136yescomplemented
17C2×2−1+4×C61C22127yescomplemented
18#14519825C2311080yescomplemented
19#14519824C231720yescomplemented
20#14519823C2312160yescomplemented
21#14519822C2311080yescomplemented
22C2×#11758226C231270yescomplemented
23C2×#11758226C231540yescomplemented
24C2×#11758222C231810yescomplemented
25C2×#11758224C231270yescomplemented
26#41×D61C231720yescomplemented
27#41×D61C2312160yescomplemented
28C2×Q8×D61C231270yescomplemented
29C22×D244C231135yescomplemented
30C2×D4×D61C231810yescomplemented
31C22×C4×D61C231135yescomplemented
322−1+4×C61C231216yescomplemented
33H(5,𝔽2)×C61C231120yescomplemented
34C2×#41×C61C231270yescomplemented
35C22×Q8×C61C23145yescomplemented
36C23×D61C241135yes—
37#41×C61C241720yes—
38C2×Q8×C61C241270yes—
39C2×D4×C61C241270yes—
40C22×C4×C61C241135yes—
41C2×C22⋊D61C241540yes—
42C22×Dic61C24145yes—
43#11758226C241120yes—
44#11758226C2411080yes—
45Q8×D61C241720yes—
46#11758224C241216yes—
47#11758224C2411080yes—
48D4×D61C2412160yes—
49#11758222C2412160yes—
50C2×D244C241810yes—
51C2×C4×D61C241270yes—
52C2×C4×D61C241810yes—
53C2×Dic122C241270yes—
54C23×C61C25145yes—
55C22×D61—8135no—
56C22×D61C251270yes—
57Q8×C61C251120yes—
58D4×C61C251216yes—
59C2×C4×C61C251270yes—
60C22⋊D61C251432yes—
61C2×Dic61C251135yes—
62C2×Dic61C25127yes—
63D244C251360yes—
64C4×D61C25136yes—
65C4×D61C251540yes—
66Dic122C251216yes—
67#63103—611nomaximal, Sylow
68C22×C61—445no—
69C22×C61C26127yes—
70C2×D61—4270no—
71C2×D61C26136yes—
72C4×C61C26136yes—
73Dic61C26127yes—
74Dic61C2611yescharacteristic
752−1+6—6127no—
762−1+6D6111yescharacteristic, complemented
77H(7,𝔽2)—6136no—
78C2×#585—6136no—
79C2×#585—6127no—
80C2×C61—227no—
81C2×C61C2711yescharacteristic
82D61—236no—
83#585—61216no—
84#585—61432no—
85#585—61120no—
86#585—61540no—
87#585C2×D61136yescomplemented
88C2×2−1+4—61216no—
89C2×2−1+4—61135no—
90C2×2−1+4C2×D61127yescomplemented
91C2×H(5,𝔽2)—61360no—
92C2×H(5,𝔽2)—61270no—
93C22×#41—61270no—
94C22×#41—6145no—
95C61#6310311yescharacteristic, complemented, Sylow
962−1+4—61720no—
972−1+4—611080no—
982−1+4C22×D611216yescomplemented
99H(5,𝔽2)—612160no—
100H(5,𝔽2)—611080no—
101H(5,𝔽2)C22×D611120yescomplemented
102C2×#41—61720no—
103C2×#41—612160no—
104C2×#41—61270no—
105C2×#41—61540no—
106C2×#41—61270no—
107C2×#41—61810no—
108C2×#41C22×D611270yescomplemented
109C22×Q8—61270no—
110C22×Q8C22×D61145yescomplemented
111C22×D4—61810no—
112C22×D4—61135no—
113C23×C4—61135no—
114C24—61135no—
115#41—61120no—
116#41—611080no—
117#41—61216no—
118#41—611080no—
119#41—612160no—
120#41C23×D611720yes—
121C2×Q8—61720no—
122C2×Q8—61270no—
123C2×Q8C23×D611270yes—
124C2×D4—61540no—
125C2×D4—612160no—
126C2×D4—61810no—
127C2×D4C23×D611270yes—
128C22×C4—6145no—
129C22×C4—61270no—
130C22×C4—61810no—
131C22×C4C23×D611135yes—
132C23—61270no—
133C23—488135no—
134C23C24×D61145yes—
135Q8—61216no—
136Q8C24×D611120yes—
137D4—61432no—
138D4—61360no—
139D4C24×D611216yes—
140C2×C4—6127no—
141C2×C4—61135no—
142C2×C4—6136no—
143C2×C4—61540no—
144C2×C4C24×D611270yes—
145C22—445no—
146C22—6136no—
147C22—244270no—
148C22C25×D61127yes—
149C4—611no—
150C4—6127no—
151C4C25×D61136yes—
152C2—227no—
153C2—12236no—
154C2C26×D6111yescharacteristic
subgroupG/H
center, FrattiniC2C26×D61
commutator, socle, cosocleC2×C61C27
FittingC61×2−1+6C2
hypercenter2−1+6D61
hypocenterC61#63103
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #21460631C2semidirect
2. #21460630C2semidirect
3. C2×#16840599C2semidirect
4. #585×D61C2semidirect
5. C61×2−1+6C2semidirect
6. #16840599C22semidirect
7. #16840599C22semidirect
8. #16840598C22semidirect
9. #16840597C22semidirect
10. C2×#14519825C22semidirect
11. 2−1+4×D61C22semidirect
12. C2×#14519823C22semidirect
13. H(5,𝔽2)×D61C22semidirect
14. C22×#11758226C22semidirect
15. C2×#41×D61C22semidirect
16. C61×#585C22semidirect
17. C2×2−1+4×C61C22semidirect
18. #14519825C23semidirect
19. #14519824C23semidirect
20. #14519823C23semidirect
21. #14519822C23semidirect
22. C2×#11758226C23semidirect
23. C2×#11758226C23semidirect
24. C2×#11758222C23semidirect
25. C2×#11758224C23semidirect
26. #41×D61C23semidirect
27. #41×D61C23semidirect
28. C2×Q8×D61C23semidirect
29. C22×D244C23semidirect
30. C2×D4×D61C23semidirect
31. C22×C4×D61C23semidirect
32. 2−1+4×C61C23semidirect
33. H(5,𝔽2)×C61C23semidirect
34. C2×#41×C61C23semidirect
35. C22×Q8×C61C23semidirect
76. 2−1+6D61semidirect
87. #585C2×D61semidirect
90. C2×2−1+4C2×D61semidirect
95. C61#63103semidirect
98. 2−1+4C22×D61semidirect
101. H(5,𝔽2)C22×D61semidirect
108. C2×#41C22×D61semidirect
110. C22×Q8C22×D61semidirect

Maximal subgroups

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

subgroupindex
#214606312
#214606302
C2×#168405992
#585×D612
C61×2−1+62
#6310361