Finite Group Databasebeta

← H(7,𝔽2)×D67 #31769726

Subgroups — H(7,𝔽2)×D67 #31769726

Order 17152.

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#22575020C2128yesmaximal, complemented
2#22575019C2135yesmaximal, complemented
3#585×D67C2128yesmaximal, complemented
4C2×H(5,𝔽2)×D67C2135yesmaximal, complemented
5C67×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#16864233C221168yescomplemented
7#16864232C221420yescomplemented
8#16864231C221280yescomplemented
92−1+4×D67C22156yescomplemented
10C2×#14524089C221210yescomplemented
11C2×#14524088C221315yescomplemented
12H(5,𝔽2)×D67C221280yescomplemented
13H(5,𝔽2)×D67C221560yescomplemented
14C2×#41×D67C221210yescomplemented
15C22×D4×D67C221105yescomplemented
16#585×C67C22128yescomplemented
17C2×H(5,𝔽2)×C67C22135yescomplemented
18C24×D67C23130yescomplemented
19#14524091C231280yescomplemented
20#14524090C231560yescomplemented
21#14524089C2311680yescomplemented
22#14524088C2312520yescomplemented
23C2×#11761629C231210yescomplemented
24C2×#11761628C231630yescomplemented
25C2×#11761627C231630yescomplemented
26#41×D67C231560yescomplemented
27#41×D67C2311680yescomplemented
28C2×Q8×D67C23170yescomplemented
29C22×D268C231105yescomplemented
30C22×C22⋊D67C231210yescomplemented
31C2×D4×D67C231630yescomplemented
32C2×D4×D67C2311260yescomplemented
33C22×C4×D67C231105yescomplemented
342−1+4×C67C23156yescomplemented
35H(5,𝔽2)×C67C231280yescomplemented
36C2×#41×C67C231210yescomplemented
37C22×D4×C67C231105yescomplemented
38C23×D67—860no—
39C23×D67C241105yes—
40C23×D67C241210yescomplemented
41#11761629C241560yescomplemented
42#11761628C2411680yescomplemented
43#11761627C2411680yescomplemented
44C2×Dic134C241210yescomplemented
45Q8×D67C24156yes—
46Q8×D67C241280yescomplemented
47C2×D268C241630yescomplemented
48C2×C22⋊D67C2411260yescomplemented
49D4×D67C241280yes—
50D4×D67C2412520yescomplemented
51C22×Dic67C241105yescomplemented
52C2×C4×D67C241210yes—
53C2×C4×D67C241630yescomplemented
54C24×C67C24130yescomplemented
55#41×C67C241560yescomplemented
56C2×Q8×C67C24170yescomplemented
57C2×D4×C67C241630yescomplemented
58C22×C4×C67C241105yescomplemented
59C23×C67—830no—
60C23×C67C251105yes—
61C22×D67—4210no—
62C22×D67—8210no—
63C22×D67C25135yes—
64C22×D67C251315yes—
65Q8×C67C25156yes—
66D4×C67C251280yes—
67C2×C4×C67C251210yes—
68C22⋊D67C251560yes—
69C2×Dic67C251210yes—
70D268C251280yes—
71C4×D67C25128yes—
72C4×D67C251420yes—
73Dic134C251168yes—
74C22×C67—4105no—
75C22×C67C26135yes—
76C2×D67—270no—
77C2×D67—4315no—
78C2×D67C26135yes—
79C2×D67C2611yescharacteristic
80C4×C67C26128yes—
81Dic67C26128yes—
82C2×H(7,𝔽2)—671nomaximal, Sylow
83C2×C67—235no—
84C2×C67C2711yescharacteristic
85D67—235no—
86D67H(7,𝔽2)12yesdirect product factor
87H(7,𝔽2)—6728no—
88H(7,𝔽2)—6735no—
89H(7,𝔽2)D6711yescharacteristic, direct product factor
90C2×#585—6728no—
91C22×H(5,𝔽2)—6735no—
92C67C2×H(7,𝔽2)11yescharacteristic, complemented, Sylow
93#585—67168no—
94#585—67280no—
95#585—67420no—
96#585C2×D67128yescomplemented
97C2×2−1+4—6756no—
98C2×H(5,𝔽2)—67280no—
99C2×H(5,𝔽2)—67560no—
100C2×H(5,𝔽2)—67210no—
101C2×H(5,𝔽2)—67315no—
102C2×H(5,𝔽2)C2×D67135yescomplemented
103C22×#41—67210no—
104C23×D4—67105no—
105C25—6730no—
1062−1+4—67560no—
1072−1+4—67280no—
1082−1+4C22×D67156yescomplemented
109H(5,𝔽2)—671680no—
110H(5,𝔽2)—672520no—
111H(5,𝔽2)C22×D671280yescomplemented
112C2×#41—67560no—
113C2×#41—671680no—
114C2×#41—67210no—
115C2×#41—67630no—
116C2×#41—67630no—
117C2×#41C22×D671210yescomplemented
118C22×Q8—6770no—
119C22×D4—67210no—
120C22×D4—67630no—
121C22×D4—671260no—
122C22×D4—67105no—
123C22×D4C22×D671105yescomplemented
124C23×C4—67105no—
125C24—53660no—
126C24—67105no—
127C24—67210no—
128C24C23×D67130yescomplemented
129#41—67560no—
130#41—671680no—
131#41—671680no—
132#41C23×D671560yescomplemented
133C2×Q8—6756no—
134C2×Q8—67280no—
135C2×Q8—67210no—
136C2×Q8C23×D67170yescomplemented
137C2×D4—671260no—
138C2×D4—67280no—
139C2×D4—672520no—
140C2×D4—67630no—
141C2×D4C23×D671630yescomplemented
142C22×C4—67105no—
143C22×C4—67210no—
144C22×C4—67630no—
145C22×C4C23×D671105yescomplemented
146C23—830no—
147C23—268210no—
148C23—536210no—
149C23—67315no—
150C23—6735no—
151C23C24×D671105yes—
152Q8—67168no—
153Q8C24×D67156yes—
154D4—67560no—
155D4—67280no—
156D4C24×D671280yes—
157C2×C4—67210no—
158C2×C4—6728no—
159C2×C4—67420no—
160C2×C4C24×D671210yes—
161C22—4105no—
162C22—268315no—
163C22—671no—
164C22—6735no—
165C22—13470no—
166C22C25×D67135yes—
167C4—6728no—
168C4C25×D67128yes—
169C2—235no—
170C2—672no—
171C2—13435no—
172C2C26×D6711yescharacteristic
subgroupG/H
center, FrattiniC2C26×D67
commutator, socle, cosocleC2×C67C27
FittingC67×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D67
hypocenterC67C2×H(7,𝔽2)
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22575020C2semidirect
2. #22575019C2semidirect
3. #585×D67C2semidirect
4. C2×H(5,𝔽2)×D67C2semidirect
5. C67×H(7,𝔽2)C2semidirect
6. #16864233C22semidirect
7. #16864232C22semidirect
8. #16864231C22semidirect
9. 2−1+4×D67C22semidirect
10. C2×#14524089C22semidirect
11. C2×#14524088C22semidirect
12. H(5,𝔽2)×D67C22semidirect
13. H(5,𝔽2)×D67C22semidirect
14. C2×#41×D67C22semidirect
15. C22×D4×D67C22semidirect
16. #585×C67C22semidirect
17. C2×H(5,𝔽2)×C67C22semidirect
18. C24×D67C23semidirect
19. #14524091C23semidirect
20. #14524090C23semidirect
21. #14524089C23semidirect
22. #14524088C23semidirect
23. C2×#11761629C23semidirect
24. C2×#11761628C23semidirect
25. C2×#11761627C23semidirect
26. #41×D67C23semidirect
27. #41×D67C23semidirect
28. C2×Q8×D67C23semidirect
29. C22×D268C23semidirect
30. C22×C22⋊D67C23semidirect
31. C2×D4×D67C23semidirect
32. C2×D4×D67C23semidirect
33. C22×C4×D67C23semidirect
34. 2−1+4×C67C23semidirect
35. H(5,𝔽2)×C67C23semidirect
36. C2×#41×C67C23semidirect
37. C22×D4×C67C23semidirect
40. C23×D67C24semidirect
41. #11761629C24semidirect
42. #11761628C24semidirect
43. #11761627C24semidirect
44. C2×Dic134C24semidirect
46. Q8×D67C24semidirect
47. C2×D268C24semidirect
48. C2×C22⋊D67C24semidirect
50. D4×D67C24semidirect
51. C22×Dic67C24semidirect
53. C2×C4×D67C24semidirect
54. C24×C67C24semidirect
55. #41×C67C24semidirect
56. C2×Q8×C67C24semidirect
57. C2×D4×C67C24semidirect
58. C22×C4×C67C24semidirect
86. D67H(7,𝔽2)direct (N × H)
89. H(7,𝔽2)D67direct (N × H)
92. C67C2×H(7,𝔽2)semidirect
96. #585C2×D67semidirect
102. C2×H(5,𝔽2)C2×D67semidirect
108. 2−1+4C22×D67semidirect
111. H(5,𝔽2)C22×D67semidirect
117. C2×#41C22×D67semidirect
123. C22×D4C22×D67semidirect
128. C24C23×D67semidirect
132. #41C23×D67semidirect
136. C2×Q8C23×D67semidirect
141. C2×D4C23×D67semidirect
145. C22×C4C23×D67semidirect

Maximal subgroups

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

subgroupindex
#225750202
#225750192
#585×D672
C2×H(5,𝔽2)×D672
C67×H(7,𝔽2)2
C2×H(7,𝔽2)67