Finite Group Databasebeta

← H(5,𝔽2)×#10587739 #31797770

Subgroups — H(5,𝔽2)×#10587739 #31797770

Order 17536.

1660 conjugacy classes of subgroups; only the 754 normal subgroups are shown.

normal subgroups only

This group has 1660 conjugacy classes of subgroups — too many to draw. Only its 754 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#22578472C216yesmaximal, complemented
2#22578470C219yesmaximal, complemented
3#41×#10587739C216yesmaximal, complemented
4C2×D4×#10587739C219yesmaximal, complemented
5H(5,𝔽2)×D137C211yesmaximal, characteristic
6C23×#10587739C2216yescomplemented
7Q8×#10587739C2212yescomplemented
8Q8×#10587739C2216yescomplemented
9D4×#10587739C22118yescomplemented
10D4×#10587739C22136yescomplemented
11D4×#10587739C22118yescomplemented
12#17984167C22118yescomplemented
13C2×C4⋊#10587739C2219yescomplemented
14C2×C22⋊#10587739C22118yescomplemented
15C2×C4×#10587739C2219yescomplemented
16#17984134C416yescomplemented
17#17984133C419yescomplemented
18#41×D137C2216yes—
19C2×D4×D137C2219yes—
20H(5,𝔽2)×C137C411yescharacteristic, complemented
21C22×#10587739—412no—
22C22×#10587739C2319yes—
23C22×#10587739C2319yes—
24C22×#10587739C23118yes—
25C4⋊#10587739C23118yes—
26C4⋊#10587739C2316yes—
27C22⋊#10587739C23136yes—
28C4×#10587739C2316yes—
29C4×#10587739C23118yes—
30C23×D137C2316yes—
31#14555511C2×C416yescomplemented
32#14555510C2×C4118yescomplemented
33#14555509C2×C4118yescomplemented
34Q8×D137C2312yes—
35C2×D548C2×C419yescomplemented
36C2×C22⋊D137C2×C4118yescomplemented
37D4×D137C2×C4136yescomplemented
38D4×D137C23118yes—
39C2×C4×D137C2319yes—
40#41×C137C2×C416yescomplemented
41C2×D4×C137C2×C419yescomplemented
42C2×#10587739—218no—
43C2×#10587739—418no—
44C2×#10587739—49no—
45C2×#10587739C2416yes—
46C2×#10587739C2419yes—
47C2×#10587739C2411yescharacteristic
48C22×D137—46no—
49C22×D137—46no—
50C22×D137C22×C4118yescomplemented
51C22×D137C2419yes—
52Dic274C22×C416yescomplemented
53D548C22×C4118yescomplemented
54C22⋊D137C22×C4136yescomplemented
55C2×Dic137C22×C419yescomplemented
56C4×D137C22×C4118yescomplemented
57C4×D137C2416yes—
58C23×C137C22×C416yescomplemented
59Q8×C137C22×C412yescomplemented
60D4×C137C22×C4118yescomplemented
61C2×C4×C137C22×C419yescomplemented
62C22×C137—46no—
63C22×C137C23×C419yes—
64C2×D137—29no—
65C2×D137—29no—
66C2×D137—418no—
67C2×D137C23×C419yes—
68C2×D137C2511yescharacteristic
69#10587739—29no—
70#10587739—26no—
71#10587739H(5,𝔽2)12yesdirect product factor
72C4×C137C23×C416yes—
73Dic137C23×C416yes—
74C2×C137—29no—
75C2×C137C24×C411yescharacteristic
76D137—29no—
77D137C2×H(5,𝔽2)11yescharacteristic
78D137#58511yescharacteristic
79C137C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
80C4×H(5,𝔽2)—1371nomaximal, Sylow
81C2×H(5,𝔽2)—1371no—
82#520—1376no—
83#518—1379no—
84C4×#41—1376no—
85C2×C4×D4—1379no—
86H(5,𝔽2)—1376no—
87H(5,𝔽2)—1379no—
88H(5,𝔽2)#1058773911yescharacteristic, direct product factor
89C2×#41—1376no—
90C22×D4—1379no—
91C23×C4—1376no—
92C4×Q8—1372no—
93C4×Q8—1376no—
94C4×D4—13718no—
95C4×D4—13736no—
96C4×D4—13718no—
97#117—13718no—
98C2×C4⋊C4—1379no—
99C2×C22⋊C4—13718no—
100C2×C42—1379no—
101C24—1376no—
102#41—1376no—
103#41—13718no—
104#41—13718no—
105#41C2×#1058773916yescomplemented
106C2×Q8—1372no—
107C2×D4—13718no—
108C2×D4—13736no—
109C2×D4—13718no—
110C2×D4—1379no—
111C2×D4C2×#1058773919yescomplemented
112C22×C4—54812no—
113C22×C4—1379no—
114C22×C4—1379no—
115C22×C4—13718no—
116C22×C4—1379no—
117C4⋊C4—13718no—
118C4⋊C4—1376no—
119C22⋊C4—13736no—
120C42—1376no—
121C42—13718no—
122C23—5486no—
123C23—5486no—
124C23—1379no—
125C23—13718no—
126C23C22×#1058773916yescomplemented
127Q8—1376no—
128Q8C22×#1058773912yescomplemented
129D4—13736no—
130D4—13718no—
131D4C22×#10587739118yescomplemented
132C2×C4—27418no—
133C2×C4—54818no—
134C2×C4—5489no—
135C2×C4—1376no—
136C2×C4—1379no—
137C2×C4—1371no—
138C2×C4—1379no—
139C2×C4—1376no—
140C2×C4—13718no—
141C2×C4C22×#1058773919yescomplemented
142C22—46no—
143C22—2749no—
144C22—2749no—
145C22—54818no—
146C22—1379no—
147C22—1371no—
148C22C23×#1058773919yes—
149C4—2749no—
150C4—2746no—
151C4—1372no—
152C4—1376no—
153C4C23×#1058773916yes—
154C2—29no—
155C2—2749no—
156C2—1371no—
157C2—1371no—
158C2C24×#1058773911yescharacteristic
subgroupG/H
center, FrattiniC2C24×#10587739
commutator, socleC2×C137C24×C4
FittingH(5,𝔽2)×C137C4
prime radicalH(5,𝔽2)×D137C2
cosocleC2×D137C25
hypercenterH(5,𝔽2)#10587739
hypocenterC137C4×H(5,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22578472C2semidirect
2. #22578470C2semidirect
3. #41×#10587739C2semidirect
4. C2×D4×#10587739C2semidirect
6. C23×#10587739C22semidirect
7. Q8×#10587739C22semidirect
8. Q8×#10587739C22semidirect
9. D4×#10587739C22semidirect
10. D4×#10587739C22semidirect
11. D4×#10587739C22semidirect
12. #17984167C22semidirect
13. C2×C4⋊#10587739C22semidirect
14. C2×C22⋊#10587739C22semidirect
15. C2×C4×#10587739C22semidirect
16. #17984134C4semidirect
17. #17984133C4semidirect
20. H(5,𝔽2)×C137C4semidirect
31. #14555511C2×C4semidirect
32. #14555510C2×C4semidirect
33. #14555509C2×C4semidirect
35. C2×D548C2×C4semidirect
36. C2×C22⋊D137C2×C4semidirect
37. D4×D137C2×C4semidirect
40. #41×C137C2×C4semidirect
41. C2×D4×C137C2×C4semidirect
50. C22×D137C22×C4semidirect
52. Dic274C22×C4semidirect
53. D548C22×C4semidirect
54. C22⋊D137C22×C4semidirect
55. C2×Dic137C22×C4semidirect
56. C4×D137C22×C4semidirect
58. C23×C137C22×C4semidirect
59. Q8×C137C22×C4semidirect
60. D4×C137C22×C4semidirect
61. C2×C4×C137C22×C4semidirect
71. #10587739H(5,𝔽2)direct (N × H)
79. C137C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#10587739direct (N × H)
105. #41C2×#10587739semidirect
111. C2×D4C2×#10587739semidirect
126. C23C22×#10587739semidirect
128. Q8C22×#10587739semidirect
131. D4C22×#10587739semidirect
141. C2×C4C22×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784722
#225784702
#41×#105877392
C2×D4×#105877392
H(5,𝔽2)×D1372
C4×H(5,𝔽2)137