Finite Group Databasebeta

← D61×H(7,𝔽2) #30593908

Subgroups — D61×H(7,𝔽2) #30593908

Order 15616.

60570 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.

normal subgroups only

This group has 60570 conjugacy classes of subgroups — too many to draw. Only its 32041 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#21460630C2128yesmaximal, complemented
2#21460629C2135yesmaximal, complemented
3#585×D61C2128yesmaximal, complemented
4C2×H(5,𝔽2)×D61C2135yesmaximal, complemented
5C61×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#16840599C221168yescomplemented
7#16840598C221420yescomplemented
8#16840597C221280yescomplemented
92−1+4×D61C22156yescomplemented
10C2×#14519823C221210yescomplemented
11C2×#14519822C221315yescomplemented
12H(5,𝔽2)×D61C221280yescomplemented
13H(5,𝔽2)×D61C221560yescomplemented
14C2×#41×D61C221210yescomplemented
15C22×D4×D61C221105yescomplemented
16C61×#585C22128yescomplemented
17C2×H(5,𝔽2)×C61C22135yescomplemented
18C24×D61C23130yescomplemented
19#14519825C231280yescomplemented
20#14519824C231560yescomplemented
21#14519823C2311680yescomplemented
22#14519822C2312520yescomplemented
23C2×#11758226C231210yescomplemented
24C2×#11758222C231630yescomplemented
25C2×#11758224C231630yescomplemented
26#41×D61C231560yescomplemented
27#41×D61C2311680yescomplemented
28C2×Q8×D61C23170yescomplemented
29C22×D244C231105yescomplemented
30C22×C22⋊D61C231210yescomplemented
31C2×D4×D61C231630yescomplemented
32C2×D4×D61C2311260yescomplemented
33C22×C4×D61C231105yescomplemented
342−1+4×C61C23156yescomplemented
35H(5,𝔽2)×C61C231280yescomplemented
36C2×#41×C61C231210yescomplemented
37C22×D4×C61C231105yescomplemented
38C24×C61C24130yescomplemented
39C23×D61—860no—
40C23×D61C241105yes—
41C23×D61C241210yescomplemented
42#41×C61C241560yescomplemented
43C2×Q8×C61C24170yescomplemented
44C2×D4×C61C241630yescomplemented
45C22×C4×C61C241105yescomplemented
46C2×C22⋊D61C2411260yescomplemented
47C22×Dic61C241105yescomplemented
48#11758226C241560yescomplemented
49Q8×D61C24156yes—
50Q8×D61C241280yescomplemented
51#11758224C2411680yescomplemented
52D4×D61C241280yes—
53D4×D61C2412520yescomplemented
54#11758222C2411680yescomplemented
55C2×D244C241630yescomplemented
56C2×C4×D61C241210yes—
57C2×C4×D61C241630yescomplemented
58C2×Dic122C241210yescomplemented
59C23×C61—830no—
60C23×C61C251105yes—
61C22×D61—4210no—
62C22×D61—8210no—
63C22×D61C25135yes—
64C22×D61C251315yes—
65Q8×C61C25156yes—
66D4×C61C251280yes—
67C2×C4×C61C251210yes—
68C22⋊D61C251560yes—
69C2×Dic61C251210yes—
70D244C251280yes—
71C4×D61C25128yes—
72C4×D61C251420yes—
73Dic122C251168yes—
74C2×H(7,𝔽2)—611nomaximal, Sylow
75C22×C61—4105no—
76C22×C61C26135yes—
77C2×D61—270no—
78C2×D61—4315no—
79C2×D61C26135yes—
80C2×D61C2611yescharacteristic
81C4×C61C26128yes—
82Dic61C26128yes—
83H(7,𝔽2)—6128no—
84H(7,𝔽2)—6135no—
85H(7,𝔽2)D6111yescharacteristic, direct product factor
86C2×#585—6128no—
87C22×H(5,𝔽2)—6135no—
88C2×C61—235no—
89C2×C61C2711yescharacteristic
90D61—235no—
91D61H(7,𝔽2)12yesdirect product factor
92#585—61168no—
93#585—61280no—
94#585—61420no—
95#585C2×D61128yescomplemented
96C2×2−1+4—6156no—
97C2×H(5,𝔽2)—61280no—
98C2×H(5,𝔽2)—61560no—
99C2×H(5,𝔽2)—61210no—
100C2×H(5,𝔽2)—61315no—
101C2×H(5,𝔽2)C2×D61135yescomplemented
102C22×#41—61210no—
103C23×D4—61105no—
104C61C2×H(7,𝔽2)11yescharacteristic, complemented, Sylow
105C25—6130no—
1062−1+4—61560no—
1072−1+4—61280no—
1082−1+4C22×D61156yescomplemented
109H(5,𝔽2)—611680no—
110H(5,𝔽2)—612520no—
111H(5,𝔽2)C22×D611280yescomplemented
112C2×#41—61560no—
113C2×#41—611680no—
114C2×#41—61210no—
115C2×#41—61630no—
116C2×#41—61630no—
117C2×#41C22×D611210yescomplemented
118C22×Q8—6170no—
119C22×D4—61210no—
120C22×D4—61630no—
121C22×D4—611260no—
122C22×D4—61105no—
123C22×D4C22×D611105yescomplemented
124C23×C4—61105no—
125C24—61210no—
126C24—61105no—
127C24—48860no—
128C24C23×D61130yescomplemented
129#41—61560no—
130#41—611680no—
131#41—611680no—
132#41C23×D611560yescomplemented
133C2×Q8—6156no—
134C2×Q8—61280no—
135C2×Q8—61210no—
136C2×Q8C23×D61170yescomplemented
137C2×D4—611260no—
138C2×D4—61280no—
139C2×D4—612520no—
140C2×D4—61630no—
141C2×D4C23×D611630yescomplemented
142C22×C4—61105no—
143C22×C4—61210no—
144C22×C4—61630no—
145C22×C4C23×D611105yescomplemented
146C23—830no—
147C23—6135no—
148C23—61315no—
149C23—488210no—
150C23—244210no—
151C23C24×D611105yes—
152Q8—61168no—
153Q8C24×D61156yes—
154D4—61560no—
155D4—61280no—
156D4C24×D611280yes—
157C2×C4—61210no—
158C2×C4—6128no—
159C2×C4—61420no—
160C2×C4C24×D611210yes—
161C22—4105no—
162C22—611no—
163C22—6135no—
164C22—12270no—
165C22—244315no—
166C22C25×D61135yes—
167C4—6128no—
168C4C25×D61128yes—
169C2—235no—
170C2—612no—
171C2—12235no—
172C2C26×D6111yescharacteristic
subgroupG/H
center, FrattiniC2C26×D61
commutator, socle, cosocleC2×C61C27
FittingC61×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D61
hypocenterC61C2×H(7,𝔽2)
LCSG ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #21460630C2semidirect
2. #21460629C2semidirect
3. #585×D61C2semidirect
4. C2×H(5,𝔽2)×D61C2semidirect
5. C61×H(7,𝔽2)C2semidirect
6. #16840599C22semidirect
7. #16840598C22semidirect
8. #16840597C22semidirect
9. 2−1+4×D61C22semidirect
10. C2×#14519823C22semidirect
11. C2×#14519822C22semidirect
12. H(5,𝔽2)×D61C22semidirect
13. H(5,𝔽2)×D61C22semidirect
14. C2×#41×D61C22semidirect
15. C22×D4×D61C22semidirect
16. C61×#585C22semidirect
17. C2×H(5,𝔽2)×C61C22semidirect
18. C24×D61C23semidirect
19. #14519825C23semidirect
20. #14519824C23semidirect
21. #14519823C23semidirect
22. #14519822C23semidirect
23. C2×#11758226C23semidirect
24. C2×#11758222C23semidirect
25. C2×#11758224C23semidirect
26. #41×D61C23semidirect
27. #41×D61C23semidirect
28. C2×Q8×D61C23semidirect
29. C22×D244C23semidirect
30. C22×C22⋊D61C23semidirect
31. C2×D4×D61C23semidirect
32. C2×D4×D61C23semidirect
33. C22×C4×D61C23semidirect
34. 2−1+4×C61C23semidirect
35. H(5,𝔽2)×C61C23semidirect
36. C2×#41×C61C23semidirect
37. C22×D4×C61C23semidirect
38. C24×C61C24semidirect
41. C23×D61C24semidirect
42. #41×C61C24semidirect
43. C2×Q8×C61C24semidirect
44. C2×D4×C61C24semidirect
45. C22×C4×C61C24semidirect
46. C2×C22⋊D61C24semidirect
47. C22×Dic61C24semidirect
48. #11758226C24semidirect
50. Q8×D61C24semidirect
51. #11758224C24semidirect
53. D4×D61C24semidirect
54. #11758222C24semidirect
55. C2×D244C24semidirect
57. C2×C4×D61C24semidirect
58. C2×Dic122C24semidirect
85. H(7,𝔽2)D61direct (N × H)
91. D61H(7,𝔽2)direct (N × H)
95. #585C2×D61semidirect
101. C2×H(5,𝔽2)C2×D61semidirect
104. C61C2×H(7,𝔽2)semidirect
108. 2−1+4C22×D61semidirect
111. H(5,𝔽2)C22×D61semidirect
117. C2×#41C22×D61semidirect
123. C22×D4C22×D61semidirect
128. C24C23×D61semidirect
132. #41C23×D61semidirect
136. C2×Q8C23×D61semidirect
141. C2×D4C23×D61semidirect
145. C22×C4C23×D61semidirect

Maximal subgroups

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

subgroupindex
#214606302
#214606292
#585×D612
C2×H(5,𝔽2)×D612
C61×H(7,𝔽2)2
C2×H(7,𝔽2)61