Finite Group Databasebeta

← H(5,𝔽2)×#10624382 #37550702

Subgroups — H(5,𝔽2)×#10624382 #37550702

Order 23168.

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#26042348C216yesmaximal, complemented
2#26042346C219yesmaximal, complemented
3#41×#10624382C216yesmaximal, complemented
4C2×D4×#10624382C219yesmaximal, complemented
5H(5,𝔽2)×D181C211yesmaximal, characteristic
6C23×#10624382C2216yescomplemented
7Q8×#10624382C2212yescomplemented
8Q8×#10624382C2216yescomplemented
9D4×#10624382C22118yescomplemented
10D4×#10624382C22136yescomplemented
11D4×#10624382C22118yescomplemented
12#19153895C22118yescomplemented
13C2×C4⋊#10624382C2219yescomplemented
14C2×C22⋊#10624382C22118yescomplemented
15C2×C4×#10624382C2219yescomplemented
16#19153862C416yescomplemented
17#19153861C419yescomplemented
18#41×D181C2216yes—
19C2×D4×D181C2219yes—
20H(5,𝔽2)×C181C411yescharacteristic, complemented
21C22×#10624382—412no—
22C22×#10624382C2319yes—
23C22×#10624382C2319yes—
24C22×#10624382C23118yes—
25C4⋊#10624382C23118yes—
26C4⋊#10624382C2316yes—
27C22⋊#10624382C23136yes—
28C4×#10624382C2316yes—
29C4×#10624382C23118yes—
30C23×D181C2316yes—
31#15668450C2×C416yescomplemented
32#15668449C2×C4118yescomplemented
33#15668448C2×C4118yescomplemented
34Q8×D181C2312yes—
35C2×D724C2×C419yescomplemented
36C2×C22⋊D181C2×C4118yescomplemented
37D4×D181C2×C4136yescomplemented
38D4×D181C23118yes—
39C2×C4×D181C2319yes—
40#41×C181C2×C416yescomplemented
41C2×D4×C181C2×C419yescomplemented
42C2×#10624382—218no—
43C2×#10624382—418no—
44C2×#10624382—49no—
45C2×#10624382C2416yes—
46C2×#10624382C2419yes—
47C2×#10624382C2411yescharacteristic
48C22×D181—46no—
49C22×D181—46no—
50C22×D181C22×C4118yescomplemented
51C22×D181C2419yes—
52Dic362C22×C416yescomplemented
53D724C22×C4118yescomplemented
54C22⋊D181C22×C4136yescomplemented
55C2×Dic181C22×C419yescomplemented
56C4×D181C22×C4118yescomplemented
57C4×D181C2416yes—
58C23×C181C22×C416yescomplemented
59Q8×C181C22×C412yescomplemented
60D4×C181C22×C4118yescomplemented
61C2×C4×C181C22×C419yescomplemented
62C22×C181—46no—
63C22×C181C23×C419yes—
64C2×D181—29no—
65C2×D181—29no—
66C2×D181—418no—
67C2×D181C23×C419yes—
68C2×D181C2511yescharacteristic
69#10624382—29no—
70#10624382—26no—
71#10624382H(5,𝔽2)12yesdirect product factor
72C4×C181C23×C416yes—
73Dic181C23×C416yes—
74C2×C181—29no—
75C2×C181C24×C411yescharacteristic
76D181—29no—
77D181C2×H(5,𝔽2)11yescharacteristic
78D181#58511yescharacteristic
79C181C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
80C4×H(5,𝔽2)—1811nomaximal, Sylow
81C2×H(5,𝔽2)—1811no—
82#520—1816no—
83#518—1819no—
84C4×#41—1816no—
85C2×C4×D4—1819no—
86H(5,𝔽2)—1816no—
87H(5,𝔽2)—1819no—
88H(5,𝔽2)#1062438211yescharacteristic, direct product factor
89C2×#41—1816no—
90C22×D4—1819no—
91C23×C4—1816no—
92C4×Q8—1812no—
93C4×Q8—1816no—
94C4×D4—18118no—
95C4×D4—18136no—
96C4×D4—18118no—
97#117—18118no—
98C2×C4⋊C4—1819no—
99C2×C22⋊C4—18118no—
100C2×C42—1819no—
101C24—1816no—
102#41—1816no—
103#41—18118no—
104#41—18118no—
105#41C2×#1062438216yescomplemented
106C2×Q8—1812no—
107C2×D4—18118no—
108C2×D4—18118no—
109C2×D4—18136no—
110C2×D4—1819no—
111C2×D4C2×#1062438219yescomplemented
112C22×C4—1819no—
113C22×C4—1819no—
114C22×C4—1819no—
115C22×C4—18118no—
116C22×C4—72412no—
117C4⋊C4—18118no—
118C4⋊C4—1816no—
119C22⋊C4—18136no—
120C42—1816no—
121C42—18118no—
122C23—1819no—
123C23—18118no—
124C23—7246no—
125C23—7246no—
126C23C22×#1062438216yescomplemented
127Q8—1816no—
128Q8C22×#1062438212yescomplemented
129D4—18136no—
130D4—18118no—
131D4C22×#10624382118yescomplemented
132C2×C4—36218no—
133C2×C4—1816no—
134C2×C4—1819no—
135C2×C4—1811no—
136C2×C4—1819no—
137C2×C4—1816no—
138C2×C4—18118no—
139C2×C4—72418no—
140C2×C4—7249no—
141C2×C4C22×#1062438219yescomplemented
142C22—46no—
143C22—3629no—
144C22—3629no—
145C22—1819no—
146C22—1811no—
147C22—72418no—
148C22C23×#1062438219yes—
149C4—3629no—
150C4—3626no—
151C4—1812no—
152C4—1816no—
153C4C23×#1062438216yes—
154C2—29no—
155C2—3629no—
156C2—1811no—
157C2—1811no—
158C2C24×#1062438211yescharacteristic
subgroupG/H
center, FrattiniC2C24×#10624382
commutator, socleC2×C181C24×C4
FittingH(5,𝔽2)×C181C4
prime radicalH(5,𝔽2)×D181C2
cosocleC2×D181C25
hypercenterH(5,𝔽2)#10624382
hypocenterC181C4×H(5,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26042348C2semidirect
2. #26042346C2semidirect
3. #41×#10624382C2semidirect
4. C2×D4×#10624382C2semidirect
6. C23×#10624382C22semidirect
7. Q8×#10624382C22semidirect
8. Q8×#10624382C22semidirect
9. D4×#10624382C22semidirect
10. D4×#10624382C22semidirect
11. D4×#10624382C22semidirect
12. #19153895C22semidirect
13. C2×C4⋊#10624382C22semidirect
14. C2×C22⋊#10624382C22semidirect
15. C2×C4×#10624382C22semidirect
16. #19153862C4semidirect
17. #19153861C4semidirect
20. H(5,𝔽2)×C181C4semidirect
31. #15668450C2×C4semidirect
32. #15668449C2×C4semidirect
33. #15668448C2×C4semidirect
35. C2×D724C2×C4semidirect
36. C2×C22⋊D181C2×C4semidirect
37. D4×D181C2×C4semidirect
40. #41×C181C2×C4semidirect
41. C2×D4×C181C2×C4semidirect
50. C22×D181C22×C4semidirect
52. Dic362C22×C4semidirect
53. D724C22×C4semidirect
54. C22⋊D181C22×C4semidirect
55. C2×Dic181C22×C4semidirect
56. C4×D181C22×C4semidirect
58. C23×C181C22×C4semidirect
59. Q8×C181C22×C4semidirect
60. D4×C181C22×C4semidirect
61. C2×C4×C181C22×C4semidirect
71. #10624382H(5,𝔽2)direct (N × H)
79. C181C4×H(5,𝔽2)semidirect
88. H(5,𝔽2)#10624382direct (N × H)
105. #41C2×#10624382semidirect
111. C2×D4C2×#10624382semidirect
126. C23C22×#10624382semidirect
128. Q8C22×#10624382semidirect
131. D4C22×#10624382semidirect
141. C2×C4C22×#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423482
#260423462
#41×#106243822
C2×D4×#106243822
H(5,𝔽2)×D1812
C4×H(5,𝔽2)181