Finite Group Databasebeta

← C22×#17984888 #31821206

Subgroups — C22×#17984888 #31821206

Order 17792.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#17984888C2148yesmaximal, direct product factor
2C22×#14555786C212yesmaximal, complemented
3C22×#14555785C213yesmaximal, complemented
4C23×D556C213yesmaximal, complemented
5C22×D4×D139C216yesmaximal, complemented
6C22×#41×C139C211yesmaximal, characteristic, complemented
7C24×D139C2216yescomplemented
8#17984888C221256yesdirect product factor
9C2×#14555786C22148yescomplemented
10C2×#14555785C22172yescomplemented
11C22×Dic278C2211yescharacteristic, complemented
12C22×D556C2216yescomplemented
13C22×D556C2213yescomplemented
14C22×D556C22172yescomplemented
15C22×C22⋊D139C2216yescomplemented
16C2×D4×D139C221144yescomplemented
17C22×C4×D139C2216yescomplemented
18C2×#41×C139C22124yescomplemented
19C22×Q8×C139C2211yescharacteristic, complemented
20C22×D4×C139C2213yescomplemented
21C23×C4×C139C2213yescomplemented
22C23×D139—424no—
23C23×D139C2316yes—
24C23×D139C23172yescomplemented
25#14555786C231128yescomplemented
26#14555785C231192yescomplemented
27C2×Dic278C23112yescomplemented
28C2×D556C23172yescomplemented
29C2×D556C23136yescomplemented
30C2×D556C231192yescomplemented
31C2×C22⋊D139C23172yescomplemented
32D4×D139C231384yescomplemented
33C22×Dic139C2312yes—
34C2×C4×D139C23172yescomplemented
35C24×C139C2313yes—
36#41×C139C23164yescomplemented
37C2×Q8×C139C23112yescomplemented
38C2×D4×C139C23136yescomplemented
39C22×C4×C139C2313yes—
40C22×C4×C139C2311yescharacteristic
41C22×C4×C139C23136yescomplemented
42C22×D139—224no—
43C22×D139—4144no—
44C22×D139C24136yes—
45C22×D139C24196yescomplemented
46Dic278C24116yescomplemented
47D556C24196yescomplemented
48D556C24148yescomplemented
49C22⋊D139C24196yescomplemented
50C2×Dic139C24112yes—
51C4×D139C24196yescomplemented
52C23×C139—212no—
53C23×C139C24118yes—
54C23×C139C2411yescharacteristic
55Q8×C139C24116yescomplemented
56D4×C139C24148yescomplemented
57C2×C4×C139C2416yes—
58C2×C4×C139C24118yes—
59C2×C4×C139C24148yescomplemented
60C22×C139—236no—
61C22×C139H(5,𝔽2)14yescomplemented
62C22×C139C25112yes—
63C22×C139C2513yes—
64C2×D139—272no—
65C2×D139—496no—
66C2×D139C25124yes—
67C4×C139C2514yes—
68C4×C139C25112yes—
69Dic139C2518yes—
70C2×C139—212no—
71C2×C139C2×H(5,𝔽2)16yescomplemented
72C2×C139C2611yescharacteristic
73D139—224no—
74C139C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1391nomaximal, Sylow
76C2×H(5,𝔽2)—13948no—
77C22×#41—1392no—
78C22×#41—1393no—
79C22×#41D13911yescharacteristic, complemented
80C23×D4—1396no—
81C23×D4—1393no—
82C25—1396no—
83H(5,𝔽2)—139256no—
84C2×#41—13948no—
85C2×#41—13972no—
86C2×#41C2×D139124yescomplemented
87C22×Q8—1391no—
88C22×Q8C2×D13911yescharacteristic, complemented
89C22×D4—1396no—
90C22×D4—139144no—
91C22×D4—1396no—
92C22×D4—1393no—
93C22×D4—13972no—
94C22×D4C2×D13913yescomplemented
95C23×C4—1396no—
96C23×C4C2×D13913yescomplemented
97C24—55624no—
98C24—1396no—
99C24—13972no—
100C24C22×D13913yes—
101#41—139128no—
102#41—139192no—
103#41C22×D139164yescomplemented
104C2×Q8—13912no—
105C2×Q8C22×D139112yescomplemented
106C2×D4—13972no—
107C2×D4—139384no—
108C2×D4—13972no—
109C2×D4—13936no—
110C2×D4—139192no—
111C2×D4C22×D139136yescomplemented
112C22×C4—1392no—
113C22×C4—13972no—
114C22×C4C22×D13913yes—
115C22×C4C22×D13911yescharacteristic
116C22×C4C22×D139136yescomplemented
117C23—212no—
118C23—27824no—
119C23—556144no—
120C23—13936no—
121C23—13996no—
122C23C23×D139118yes—
123C23C23×D13911yescharacteristic
124Q8—13916no—
125Q8C23×D139116yescomplemented
126D4—13996no—
127D4—13996no—
128D4—13948no—
129D4C23×D139148yescomplemented
130C2×C4—13912no—
131C2×C4—13996no—
132C2×C4C23×D139118yes—
133C2×C4C23×D13916yes—
134C2×C4C23×D139148yescomplemented
135C22—236no—
136C22—27872no—
137C22—55696no—
138C22—13924no—
139C22#1798488814yesdirect product factor
140C22C24×D139112yes—
141C22C24×D13913yes—
142C4—1398no—
143C4C24×D139112yes—
144C4C24×D13914yes—
145C2—212no—
146C2—27824no—
147C2C2×#1798488816yesdirect product factor
148C2C25×D13911yescharacteristic
subgroupG/H
centerC23C23×D139
commutator, cosocleC2×C139C26
FrattiniC2C25×D139
FittingC22×#41×C139C2
socleC23×C139C24
hypercenterC22×#41D139
hypocenterC139C22×H(5,𝔽2)
LCSG ▷ C2×C139 ▷ C139
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#17984888C2direct (N × H)
2. C22×#14555786C2semidirect
3. C22×#14555785C2semidirect
4. C23×D556C2semidirect
5. C22×D4×D139C2semidirect
6. C22×#41×C139C2semidirect
7. C24×D139C22semidirect
8. #17984888C22direct (N × H)
9. C2×#14555786C22semidirect
10. C2×#14555785C22semidirect
11. C22×Dic278C22semidirect
12. C22×D556C22semidirect
13. C22×D556C22semidirect
14. C22×D556C22semidirect
15. C22×C22⋊D139C22semidirect
16. C2×D4×D139C22semidirect
17. C22×C4×D139C22semidirect
18. C2×#41×C139C22semidirect
19. C22×Q8×C139C22semidirect
20. C22×D4×C139C22semidirect
21. C23×C4×C139C22semidirect
24. C23×D139C23semidirect
25. #14555786C23semidirect
26. #14555785C23semidirect
27. C2×Dic278C23semidirect
28. C2×D556C23semidirect
29. C2×D556C23semidirect
30. C2×D556C23semidirect
31. C2×C22⋊D139C23semidirect
32. D4×D139C23semidirect
34. C2×C4×D139C23semidirect
36. #41×C139C23semidirect
37. C2×Q8×C139C23semidirect
38. C2×D4×C139C23semidirect
41. C22×C4×C139C23semidirect
45. C22×D139C24semidirect
46. Dic278C24semidirect
47. D556C24semidirect
48. D556C24semidirect
49. C22⋊D139C24semidirect
51. C4×D139C24semidirect
55. Q8×C139C24semidirect
56. D4×C139C24semidirect
59. C2×C4×C139C24semidirect
61. C22×C139H(5,𝔽2)semidirect
71. C2×C139C2×H(5,𝔽2)semidirect
74. C139C22×H(5,𝔽2)semidirect
79. C22×#41D139semidirect
86. C2×#41C2×D139semidirect
88. C22×Q8C2×D139semidirect
94. C22×D4C2×D139semidirect
96. C23×C4C2×D139semidirect
103. #41C22×D139semidirect
105. C2×Q8C22×D139semidirect
111. C2×D4C22×D139semidirect
116. C22×C4C22×D139semidirect
125. Q8C23×D139semidirect
129. D4C23×D139semidirect
134. C2×C4C23×D139semidirect
139. C22#17984888direct (N × H)
147. C2C2×#17984888direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179848882
C22×#145557862
C22×#145557852
C23×D5562
C22×D4×D1392
C22×#41×C1392
C22×H(5,𝔽2)139