Finite Group Databasebeta

← H(5,𝔽2)×#1181 #16836473

Subgroups — H(5,𝔽2)×#1181 #16836473

Order 3712.

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#14272232C216yesmaximal, complemented
2#14272230C219yesmaximal, complemented
3#41×#1181C216yesmaximal, complemented
4C2×D4×#1181C219yesmaximal, complemented
5H(5,𝔽2)×D29C211yesmaximal, characteristic
6C23×#1181C2216yescomplemented
7C29×H(5,𝔽2)C411yescharacteristic, complemented
8#11745364C416yescomplemented
9#41×D29C2216yes—
10#11745359C419yescomplemented
11C2×D4×D29C2219yes—
12C2×C22⋊#1181C22118yescomplemented
13Q8×#1181C2212yescomplemented
14Q8×#1181C2216yescomplemented
15D4×#1181C22118yescomplemented
16D4×#1181C22136yescomplemented
17D4×#1181C22118yescomplemented
18#11745345C22118yescomplemented
19C2×C4⋊#1181C2219yescomplemented
20C2×C4×#1181C2219yescomplemented
21C23×D29C2316yes—
22C22×#1181—412no—
23C22×#1181C2319yes—
24C22×#1181C2319yes—
25C22×#1181C23118yes—
26#41×C29C2×C416yescomplemented
27C2×D4×C29C2×C419yescomplemented
28C2×C22⋊D29C2×C4118yescomplemented
29#90809C2×C416yescomplemented
30Q8×D29C2312yes—
31#90807C2×C4118yescomplemented
32D4×D29C2×C4136yescomplemented
33D4×D29C23118yes—
34#90805C2×C4118yescomplemented
35C2×D116C2×C419yescomplemented
36C2×C4×D29C2319yes—
37C22⋊#1181C23136yes—
38C4⋊#1181C23118yes—
39C4⋊#1181C2316yes—
40C4×#1181C2316yes—
41C4×#1181C23118yes—
42C23×C29C22×C416yescomplemented
43C22×D29—46no—
44C22×D29—46no—
45C22×D29C22×C4118yescomplemented
46C22×D29C2419yes—
47C2×#1181—218no—
48C2×#1181—418no—
49C2×#1181—49no—
50C2×#1181C2416yes—
51C2×#1181C2419yes—
52C2×#1181C2411yescharacteristic
53Q8×C29C22×C412yescomplemented
54D4×C29C22×C4118yescomplemented
55C2×C4×C29C22×C419yescomplemented
56C22⋊D29C22×C4136yescomplemented
57C2×Dic29C22×C419yescomplemented
58D116C22×C4118yescomplemented
59C4×D29C22×C4118yescomplemented
60C4×D29C2416yes—
61Dic58C22×C416yescomplemented
62C4×H(5,𝔽2)—291nomaximal, Sylow
63C22×C29—46no—
64C22×C29C23×C419yes—
65C2×D29—29no—
66C2×D29—29no—
67C2×D29—418no—
68C2×D29C23×C419yes—
69C2×D29C2511yescharacteristic
70#1181—29no—
71#1181—26no—
72#1181H(5,𝔽2)12yesdirect product factor
73C4×C29C23×C416yes—
74Dic29C23×C416yes—
75C2×H(5,𝔽2)—291no—
76#520—296no—
77#518—299no—
78C4×#41—296no—
79C2×C4×D4—299no—
80C2×C29—29no—
81C2×C29C24×C411yescharacteristic
82D29—29no—
83D29C2×H(5,𝔽2)11yescharacteristic
84D29#58511yescharacteristic
85H(5,𝔽2)—296no—
86H(5,𝔽2)—299no—
87H(5,𝔽2)#118111yescharacteristic, direct product factor
88C2×#41—296no—
89C22×D4—299no—
90C23×C4—296no—
91C4×Q8—292no—
92C4×Q8—296no—
93C4×D4—2918no—
94C4×D4—2936no—
95C4×D4—2918no—
96#117—2918no—
97C2×C4⋊C4—299no—
98C2×C22⋊C4—2918no—
99C2×C42—299no—
100C29C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
101C24—296no—
102#41—296no—
103#41—2918no—
104#41—2918no—
105#41C2×#118116yescomplemented
106C2×Q8—292no—
107C2×D4—2918no—
108C2×D4—2918no—
109C2×D4—2936no—
110C2×D4—299no—
111C2×D4C2×#118119yescomplemented
112C22×C4—299no—
113C22×C4—299no—
114C22×C4—299no—
115C22×C4—2918no—
116C22×C4—11612no—
117C4⋊C4—2918no—
118C4⋊C4—296no—
119C22⋊C4—2936no—
120C42—296no—
121C42—2918no—
122C23—299no—
123C23—2918no—
124C23—1166no—
125C23—1166no—
126C23C22×#118116yescomplemented
127Q8—296no—
128Q8C22×#118112yescomplemented
129D4—2936no—
130D4—2918no—
131D4C22×#1181118yescomplemented
132C2×C4—291no—
133C2×C4—299no—
134C2×C4—296no—
135C2×C4—2918no—
136C2×C4—296no—
137C2×C4—299no—
138C2×C4—5818no—
139C2×C4—11618no—
140C2×C4—1169no—
141C2×C4C22×#118119yescomplemented
142C22—46no—
143C22—291no—
144C22—299no—
145C22—589no—
146C22—589no—
147C22—11618no—
148C22C23×#118119yes—
149C4—292no—
150C4—296no—
151C4—589no—
152C4—586no—
153C4C23×#118116yes—
154C2—29no—
155C2—291no—
156C2—291no—
157C2—589no—
158C2C24×#118111yescharacteristic
subgroupG/H
center, FrattiniC2C24×#1181
commutator, socleC2×C29C24×C4
FittingC29×H(5,𝔽2)C4
prime radicalH(5,𝔽2)×D29C2
cosocleC2×D29C25
hypercenterH(5,𝔽2)#1181
hypocenterC29C4×H(5,𝔽2)
LCSG ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14272232C2semidirect
2. #14272230C2semidirect
3. #41×#1181C2semidirect
4. C2×D4×#1181C2semidirect
6. C23×#1181C22semidirect
7. C29×H(5,𝔽2)C4semidirect
8. #11745364C4semidirect
10. #11745359C4semidirect
12. C2×C22⋊#1181C22semidirect
13. Q8×#1181C22semidirect
14. Q8×#1181C22semidirect
15. D4×#1181C22semidirect
16. D4×#1181C22semidirect
17. D4×#1181C22semidirect
18. #11745345C22semidirect
19. C2×C4⋊#1181C22semidirect
20. C2×C4×#1181C22semidirect
26. #41×C29C2×C4semidirect
27. C2×D4×C29C2×C4semidirect
28. C2×C22⋊D29C2×C4semidirect
29. #90809C2×C4semidirect
31. #90807C2×C4semidirect
32. D4×D29C2×C4semidirect
34. #90805C2×C4semidirect
35. C2×D116C2×C4semidirect
42. C23×C29C22×C4semidirect
45. C22×D29C22×C4semidirect
53. Q8×C29C22×C4semidirect
54. D4×C29C22×C4semidirect
55. C2×C4×C29C22×C4semidirect
56. C22⋊D29C22×C4semidirect
57. C2×Dic29C22×C4semidirect
58. D116C22×C4semidirect
59. C4×D29C22×C4semidirect
61. Dic58C22×C4semidirect
72. #1181H(5,𝔽2)direct (N × H)
87. H(5,𝔽2)#1181direct (N × H)
100. C29C4×H(5,𝔽2)semidirect
105. #41C2×#1181semidirect
111. C2×D4C2×#1181semidirect
126. C23C22×#1181semidirect
128. Q8C22×#1181semidirect
131. D4C22×#1181semidirect
141. C2×C4C22×#1181semidirect

Maximal subgroups

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

subgroupindex
#142722322
#142722302
#41×#11812
C2×D4×#11812
H(5,𝔽2)×D292
C4×H(5,𝔽2)29