Finite Group Databasebeta

← #32911353

Subgroups — #32911353

Order 18176.

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#22603153C2128yesmaximal, complemented
2#22603150C2135yesmaximal, complemented
3C2×#17986834C2135yesmaximal, complemented
4#585×D71C2128yesmaximal, complemented
5C71×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#17986836C22156yescomplemented
7#17986835C221420yescomplemented
8#17986834C221280yescomplemented
9#17986834C221560yescomplemented
10C2×#14556175C221210yescomplemented
112−1+4×D71C221168yescomplemented
12C2×#14556173C221315yescomplemented
13H(5,𝔽2)×D71C221280yescomplemented
14C22×#11765864C221105yescomplemented
15C2×#41×D71C221210yescomplemented
16#585×C71C22128yescomplemented
17C2×H(5,𝔽2)×C71C22135yescomplemented
18#14556176C231280yescomplemented
19#14556175C2311680yescomplemented
20#14556174C231560yescomplemented
21#14556173C2312520yescomplemented
22C2×#11765866C23170yescomplemented
23C2×#11765865C231630yescomplemented
24C2×#11765864C231630yescomplemented
25C2×#11765864C2311260yescomplemented
26#41×D71C231560yescomplemented
27#41×D71C2311680yescomplemented
28C22×Dic142C231105yescomplemented
29C2×Q8×D71C231210yescomplemented
30C22×C22⋊D71C231210yescomplemented
31C2×D4×D71C231630yescomplemented
32C23×Dic71C23130yescomplemented
33C22×C4×D71C231105yescomplemented
342−1+4×C71C23156yescomplemented
35H(5,𝔽2)×C71C231280yescomplemented
36C2×#41×C71C231210yescomplemented
37C22×D4×C71C231105yescomplemented
38C23×D71C241105yes—
39#11765866C24156yes—
40#11765866C241280yes—
41#11765865C2411680yes—
42#11765864C241280yes—
43#11765864C2412520yes—
44C2×Dic142C241630yes—
45Q8×D71C241560yes—
46C2×D284C241210yes—
47C2×C22⋊D71C2411260yes—
48D4×D71C2411680yes—
49C22×Dic71C241105yes—
50C22×Dic71C241210yes—
51C2×C4×D71C241210yes—
52C2×C4×D71C241630yes—
53C24×C71C24130yes—
54#41×C71C241560yes—
55C2×Q8×C71C24170yes—
56C2×D4×C71C241630yes—
57C22×C4×C71C241105yes—
58C23×C71—830no—
59C23×C71C251105yes—
60C22×D71—8105no—
61C22×D71C251210yes—
62Q8×C71C25156yes—
63D4×C71C251280yes—
64C2×C4×C71C251210yes—
65C22⋊D71C251560yes—
66C2×Dic71C25135yes—
67C2×Dic71C251315yes—
68D284C251168yes—
69C4×D71C25128yes—
70C4×D71C251420yes—
71Dic142C251280yes—
72C22×C71—4105no—
73C22×C71C26135yes—
74C2×D71—4210no—
75C2×D71C26128yes—
76C4×C71C26128yes—
77Dic71C26135yes—
78Dic71C2611yescharacteristic
79#63103—711nomaximal, Sylow
80C2×C71—235no—
81C2×C71C2711yescharacteristic
82D71—228no—
832−1+6—7128no—
84H(7,𝔽2)—7135no—
85H(7,𝔽2)D7111yescharacteristic, complemented
86C2×#585—7128no—
87C2×#585—7135no—
88C71#6310311yescharacteristic, complemented, Sylow
89#585—7156no—
90#585—71280no—
91#585—71560no—
92#585—71420no—
93#585C2×D71128yescomplemented
94C2×2−1+4—71168no—
95C2×2−1+4—71210no—
96C2×H(5,𝔽2)—71280no—
97C2×H(5,𝔽2)—71315no—
98C2×H(5,𝔽2)C2×D71135yescomplemented
99C22×#41—71210no—
100C22×#41—71105no—
1012−1+4—711680no—
1022−1+4—71280no—
1032−1+4C22×D71156yescomplemented
104H(5,𝔽2)—71560no—
105H(5,𝔽2)—712520no—
106H(5,𝔽2)C22×D711280yescomplemented
107C2×#41—71560no—
108C2×#41—711680no—
109C2×#41—7170no—
110C2×#41—71630no—
111C2×#41—711260no—
112C2×#41—71630no—
113C2×#41C22×D711210yescomplemented
114C22×Q8—71210no—
115C22×Q8—71105no—
116C22×D4—71210no—
117C22×D4—71630no—
118C22×D4C22×D711105yescomplemented
119C23×C4—7130no—
120C23×C4—71105no—
121C24—71105no—
122C24C23×D71130yes—
123#41—7156no—
124#41—71280no—
125#41—71280no—
126#41—712520no—
127#41—711680no—
128#41C23×D711560yes—
129C2×Q8—71560no—
130C2×Q8—71630no—
131C2×Q8C23×D71170yes—
132C2×D4—711260no—
133C2×D4—711680no—
134C2×D4—71210no—
135C2×D4C23×D711630yes—
136C22×C4—71105no—
137C22×C4—71210no—
138C22×C4—71210no—
139C22×C4—71630no—
140C22×C4C23×D711105yes—
141C23—830no—
142C23—568105no—
143C23—71210no—
144C23C24×D711105yes—
145Q8—71280no—
146Q8C24×D71156yes—
147D4—71560no—
148D4—71168no—
149D4C24×D711280yes—
150C2×C4—7135no—
151C2×C4—71315no—
152C2×C4—7128no—
153C2×C4—71420no—
154C2×C4C24×D711210yes—
155C22—4105no—
156C22—284210no—
157C22—7128no—
158C22C25×D71135yes—
159C4—711no—
160C4—7135no—
161C4C25×D71128yes—
162C2—235no—
163C2—14228no—
164C2C26×D7111yescharacteristic
subgroupG/H
center, FrattiniC2C26×D71
commutator, socle, cosocleC2×C71C27
FittingC71×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D71
hypocenterC71#63103
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22603153C2semidirect
2. #22603150C2semidirect
3. C2×#17986834C2semidirect
4. #585×D71C2semidirect
5. C71×H(7,𝔽2)C2semidirect
6. #17986836C22semidirect
7. #17986835C22semidirect
8. #17986834C22semidirect
9. #17986834C22semidirect
10. C2×#14556175C22semidirect
11. 2−1+4×D71C22semidirect
12. C2×#14556173C22semidirect
13. H(5,𝔽2)×D71C22semidirect
14. C22×#11765864C22semidirect
15. C2×#41×D71C22semidirect
16. #585×C71C22semidirect
17. C2×H(5,𝔽2)×C71C22semidirect
18. #14556176C23semidirect
19. #14556175C23semidirect
20. #14556174C23semidirect
21. #14556173C23semidirect
22. C2×#11765866C23semidirect
23. C2×#11765865C23semidirect
24. C2×#11765864C23semidirect
25. C2×#11765864C23semidirect
26. #41×D71C23semidirect
27. #41×D71C23semidirect
28. C22×Dic142C23semidirect
29. C2×Q8×D71C23semidirect
30. C22×C22⋊D71C23semidirect
31. C2×D4×D71C23semidirect
32. C23×Dic71C23semidirect
33. C22×C4×D71C23semidirect
34. 2−1+4×C71C23semidirect
35. H(5,𝔽2)×C71C23semidirect
36. C2×#41×C71C23semidirect
37. C22×D4×C71C23semidirect
85. H(7,𝔽2)D71semidirect
88. C71#63103semidirect
93. #585C2×D71semidirect
98. C2×H(5,𝔽2)C2×D71semidirect
103. 2−1+4C22×D71semidirect
106. H(5,𝔽2)C22×D71semidirect
113. C2×#41C22×D71semidirect
118. C22×D4C22×D71semidirect

Maximal subgroups

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

subgroupindex
#226031532
#226031502
C2×#179868342
#585×D712
C71×H(7,𝔽2)2
#6310371