Finite Group Databasebeta

← H(7,𝔽2)×D71 #32911347

Subgroups — H(7,𝔽2)×D71 #32911347

Order 18176.

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#22603151C2128yesmaximal, complemented
2#22603150C2135yesmaximal, complemented
3#585×D71C2128yesmaximal, complemented
4C2×H(5,𝔽2)×D71C2135yesmaximal, complemented
5C71×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#17986836C221168yescomplemented
7#17986835C221420yescomplemented
8#17986834C221280yescomplemented
92−1+4×D71C22156yescomplemented
10C2×#14556174C221210yescomplemented
11C2×#14556173C221315yescomplemented
12H(5,𝔽2)×D71C221280yescomplemented
13H(5,𝔽2)×D71C221560yescomplemented
14C2×#41×D71C221210yescomplemented
15C22×D4×D71C221105yescomplemented
16#585×C71C22128yescomplemented
17C2×H(5,𝔽2)×C71C22135yescomplemented
18C24×D71C23130yescomplemented
19#14556176C231280yescomplemented
20#14556175C231560yescomplemented
21#14556174C2311680yescomplemented
22#14556173C2312520yescomplemented
23C2×#11765866C231210yescomplemented
24C2×#11765865C231630yescomplemented
25C2×#11765864C231630yescomplemented
26#41×D71C231560yescomplemented
27#41×D71C2311680yescomplemented
28C2×Q8×D71C23170yescomplemented
29C22×D284C231105yescomplemented
30C22×C22⋊D71C231210yescomplemented
31C2×D4×D71C231630yescomplemented
32C2×D4×D71C2311260yescomplemented
33C22×C4×D71C231105yescomplemented
342−1+4×C71C23156yescomplemented
35H(5,𝔽2)×C71C231280yescomplemented
36C2×#41×C71C231210yescomplemented
37C22×D4×C71C231105yescomplemented
38C23×D71—860no—
39C23×D71C241105yes—
40C23×D71C241210yescomplemented
41#11765866C241560yescomplemented
42#11765865C2411680yescomplemented
43#11765864C2411680yescomplemented
44C2×Dic142C241210yescomplemented
45Q8×D71C24156yes—
46Q8×D71C241280yescomplemented
47C2×D284C241630yescomplemented
48C2×C22⋊D71C2411260yescomplemented
49D4×D71C241280yes—
50D4×D71C2412520yescomplemented
51C22×Dic71C241105yescomplemented
52C2×C4×D71C241210yes—
53C2×C4×D71C241630yescomplemented
54C24×C71C24130yescomplemented
55#41×C71C241560yescomplemented
56C2×Q8×C71C24170yescomplemented
57C2×D4×C71C241630yescomplemented
58C22×C4×C71C241105yescomplemented
59C23×C71—830no—
60C23×C71C251105yes—
61C22×D71—4210no—
62C22×D71—8210no—
63C22×D71C25135yes—
64C22×D71C251315yes—
65Q8×C71C25156yes—
66D4×C71C251280yes—
67C2×C4×C71C251210yes—
68C22⋊D71C251560yes—
69C2×Dic71C251210yes—
70D284C251280yes—
71C4×D71C25128yes—
72C4×D71C251420yes—
73Dic142C251168yes—
74C22×C71—4105no—
75C22×C71C26135yes—
76C2×D71—270no—
77C2×D71—4315no—
78C2×D71C26135yes—
79C2×D71C2611yescharacteristic
80C4×C71C26128yes—
81Dic71C26128yes—
82C2×H(7,𝔽2)—711nomaximal, Sylow
83C2×C71—235no—
84C2×C71C2711yescharacteristic
85D71—235no—
86D71H(7,𝔽2)12yesdirect product factor
87H(7,𝔽2)—7128no—
88H(7,𝔽2)—7135no—
89H(7,𝔽2)D7111yescharacteristic, direct product factor
90C2×#585—7128no—
91C22×H(5,𝔽2)—7135no—
92C71C2×H(7,𝔽2)11yescharacteristic, complemented, Sylow
93#585—71168no—
94#585—71280no—
95#585—71420no—
96#585C2×D71128yescomplemented
97C2×2−1+4—7156no—
98C2×H(5,𝔽2)—71280no—
99C2×H(5,𝔽2)—71560no—
100C2×H(5,𝔽2)—71210no—
101C2×H(5,𝔽2)—71315no—
102C2×H(5,𝔽2)C2×D71135yescomplemented
103C22×#41—71210no—
104C23×D4—71105no—
105C25—7130no—
1062−1+4—71560no—
1072−1+4—71280no—
1082−1+4C22×D71156yescomplemented
109H(5,𝔽2)—711680no—
110H(5,𝔽2)—712520no—
111H(5,𝔽2)C22×D711280yescomplemented
112C2×#41—71560no—
113C2×#41—711680no—
114C2×#41—71210no—
115C2×#41—71630no—
116C2×#41—71630no—
117C2×#41C22×D711210yescomplemented
118C22×Q8—7170no—
119C22×D4—71210no—
120C22×D4—71630no—
121C22×D4—711260no—
122C22×D4—71105no—
123C22×D4C22×D711105yescomplemented
124C23×C4—71105no—
125C24—56860no—
126C24—71105no—
127C24—71210no—
128C24C23×D71130yescomplemented
129#41—71560no—
130#41—711680no—
131#41—711680no—
132#41C23×D711560yescomplemented
133C2×Q8—7156no—
134C2×Q8—71280no—
135C2×Q8—71210no—
136C2×Q8C23×D71170yescomplemented
137C2×D4—711260no—
138C2×D4—71280no—
139C2×D4—712520no—
140C2×D4—71630no—
141C2×D4C23×D711630yescomplemented
142C22×C4—71105no—
143C22×C4—71210no—
144C22×C4—71630no—
145C22×C4C23×D711105yescomplemented
146C23—830no—
147C23—284210no—
148C23—568210no—
149C23—71315no—
150C23—7135no—
151C23C24×D711105yes—
152Q8—71168no—
153Q8C24×D71156yes—
154D4—71560no—
155D4—71280no—
156D4C24×D711280yes—
157C2×C4—71210no—
158C2×C4—7128no—
159C2×C4—71420no—
160C2×C4C24×D711210yes—
161C22—4105no—
162C22—284315no—
163C22—711no—
164C22—7135no—
165C22—14270no—
166C22C25×D71135yes—
167C4—7128no—
168C4C25×D71128yes—
169C2—235no—
170C2—712no—
171C2—14235no—
172C2C26×D7111yescharacteristic
subgroupG/H
center, FrattiniC2C26×D71
commutator, socle, cosocleC2×C71C27
FittingC71×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D71
hypocenterC71C2×H(7,𝔽2)
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22603151C2semidirect
2. #22603150C2semidirect
3. #585×D71C2semidirect
4. C2×H(5,𝔽2)×D71C2semidirect
5. C71×H(7,𝔽2)C2semidirect
6. #17986836C22semidirect
7. #17986835C22semidirect
8. #17986834C22semidirect
9. 2−1+4×D71C22semidirect
10. C2×#14556174C22semidirect
11. C2×#14556173C22semidirect
12. H(5,𝔽2)×D71C22semidirect
13. H(5,𝔽2)×D71C22semidirect
14. C2×#41×D71C22semidirect
15. C22×D4×D71C22semidirect
16. #585×C71C22semidirect
17. C2×H(5,𝔽2)×C71C22semidirect
18. C24×D71C23semidirect
19. #14556176C23semidirect
20. #14556175C23semidirect
21. #14556174C23semidirect
22. #14556173C23semidirect
23. C2×#11765866C23semidirect
24. C2×#11765865C23semidirect
25. C2×#11765864C23semidirect
26. #41×D71C23semidirect
27. #41×D71C23semidirect
28. C2×Q8×D71C23semidirect
29. C22×D284C23semidirect
30. C22×C22⋊D71C23semidirect
31. C2×D4×D71C23semidirect
32. C2×D4×D71C23semidirect
33. C22×C4×D71C23semidirect
34. 2−1+4×C71C23semidirect
35. H(5,𝔽2)×C71C23semidirect
36. C2×#41×C71C23semidirect
37. C22×D4×C71C23semidirect
40. C23×D71C24semidirect
41. #11765866C24semidirect
42. #11765865C24semidirect
43. #11765864C24semidirect
44. C2×Dic142C24semidirect
46. Q8×D71C24semidirect
47. C2×D284C24semidirect
48. C2×C22⋊D71C24semidirect
50. D4×D71C24semidirect
51. C22×Dic71C24semidirect
53. C2×C4×D71C24semidirect
54. C24×C71C24semidirect
55. #41×C71C24semidirect
56. C2×Q8×C71C24semidirect
57. C2×D4×C71C24semidirect
58. C22×C4×C71C24semidirect
86. D71H(7,𝔽2)direct (N × H)
89. H(7,𝔽2)D71direct (N × H)
92. C71C2×H(7,𝔽2)semidirect
96. #585C2×D71semidirect
102. C2×H(5,𝔽2)C2×D71semidirect
108. 2−1+4C22×D71semidirect
111. H(5,𝔽2)C22×D71semidirect
117. C2×#41C22×D71semidirect
123. C22×D4C22×D71semidirect
128. C24C23×D71semidirect
132. #41C23×D71semidirect
136. C2×Q8C23×D71semidirect
141. C2×D4C23×D71semidirect
145. C22×C4C23×D71semidirect

Maximal subgroups

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

subgroupindex
#226031512
#226031502
#585×D712
C2×H(5,𝔽2)×D712
C71×H(7,𝔽2)2
C2×H(7,𝔽2)71