Finite Group Databasebeta

← H(5,𝔽2)×#595 #14545765

Subgroups — H(5,𝔽2)×#595 #14545765

Order 2176.

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#11763878C216yesmaximal, complemented
2#11763876C219yesmaximal, complemented
3#41×#595C216yesmaximal, complemented
4C2×D4×#595C219yesmaximal, complemented
5H(5,𝔽2)×D17C211yesmaximal, characteristic
6C23×#595C2216yescomplemented
7C17×H(5,𝔽2)C411yescharacteristic, complemented
8#10587696C416yescomplemented
9#41×D17C2216yes—
10#10587691C419yescomplemented
11C2×D4×D17C2219yes—
12C2×C22⋊#595C22118yescomplemented
13Q8×#595C2212yescomplemented
14Q8×#595C2216yescomplemented
15D4×#595C22118yescomplemented
16D4×#595C22136yescomplemented
17D4×#595C22118yescomplemented
18#10587677C22118yescomplemented
19C2×C4⋊#595C2219yescomplemented
20C2×C4×#595C2219yescomplemented
21C23×D17C2316yes—
22C22×#595—412no—
23C22×#595C2319yes—
24C22×#595C2319yes—
25C22×#595C23118yes—
26#41×C17C2×C416yescomplemented
27C2×D4×C17C2×C419yescomplemented
28C2×C22⋊D17C2×C4118yescomplemented
29#63256C2×C416yescomplemented
30Q8×D17C2312yes—
31#63254C2×C4118yescomplemented
32D4×D17C2×C4136yescomplemented
33D4×D17C23118yes—
34#63252C2×C4118yescomplemented
35C2×D68C2×C419yescomplemented
36C2×C4×D17C2319yes—
37C22⋊#595C23136yes—
38C4⋊#595C23118yes—
39C4⋊#595C2316yes—
40C4×#595C2316yes—
41C4×#595C23118yes—
42C23×C17C22×C416yescomplemented
43C22×D17—46no—
44C22×D17—46no—
45C22×D17C22×C4118yescomplemented
46C22×D17C2419yes—
47C2×#595—218no—
48C2×#595—418no—
49C2×#595—49no—
50C2×#595C2416yes—
51C2×#595C2419yes—
52C2×#595C2411yescharacteristic
53Q8×C17C22×C412yescomplemented
54D4×C17C22×C4118yescomplemented
55C2×C4×C17C22×C419yescomplemented
56C22⋊D17C22×C4136yescomplemented
57C2×Dic17C22×C419yescomplemented
58D68C22×C4118yescomplemented
59C4×D17C22×C4118yescomplemented
60C4×D17C2416yes—
61Dic34C22×C416yescomplemented
62C4×H(5,𝔽2)—171nomaximal, Sylow
63C22×C17—46no—
64C22×C17C23×C419yes—
65C2×D17—29no—
66C2×D17—29no—
67C2×D17—418no—
68C2×D17C23×C419yes—
69C2×D17C2511yescharacteristic
70#595—29no—
71#595—26no—
72#595H(5,𝔽2)12yesdirect product factor
73C4×C17C23×C416yes—
74Dic17C23×C416yes—
75C2×H(5,𝔽2)—171no—
76#520—176no—
77#518—179no—
78C4×#41—176no—
79C2×C4×D4—179no—
80C2×C17—29no—
81C2×C17C24×C411yescharacteristic
82D17—29no—
83D17C2×H(5,𝔽2)11yescharacteristic
84D17#58511yescharacteristic
85H(5,𝔽2)—176no—
86H(5,𝔽2)—179no—
87H(5,𝔽2)#59511yescharacteristic, direct product factor
88C2×#41—176no—
89C22×D4—179no—
90C23×C4—176no—
91C4×Q8—172no—
92C4×Q8—176no—
93C4×D4—1718no—
94C4×D4—1736no—
95C4×D4—1718no—
96#117—1718no—
97C2×C4⋊C4—179no—
98C2×C22⋊C4—1718no—
99C2×C42—179no—
100C17C4×H(5,𝔽2)11yescharacteristic, complemented, Sylow
101C24—176no—
102#41—176no—
103#41—1718no—
104#41—1718no—
105#41C2×#59516yescomplemented
106C2×Q8—172no—
107C2×D4—1718no—
108C2×D4—1736no—
109C2×D4—1718no—
110C2×D4—179no—
111C2×D4C2×#59519yescomplemented
112C22×C4—179no—
113C22×C4—179no—
114C22×C4—179no—
115C22×C4—1718no—
116C22×C4—6812no—
117C4⋊C4—1718no—
118C4⋊C4—176no—
119C22⋊C4—1736no—
120C42—176no—
121C42—1718no—
122C23—179no—
123C23—1718no—
124C23—686no—
125C23—686no—
126C23C22×#59516yescomplemented
127Q8—176no—
128Q8C22×#59512yescomplemented
129D4—1736no—
130D4—1718no—
131D4C22×#595118yescomplemented
132C2×C4—171no—
133C2×C4—179no—
134C2×C4—176no—
135C2×C4—1718no—
136C2×C4—176no—
137C2×C4—179no—
138C2×C4—3418no—
139C2×C4—6818no—
140C2×C4—689no—
141C2×C4C22×#59519yescomplemented
142C22—46no—
143C22—171no—
144C22—179no—
145C22—349no—
146C22—349no—
147C22—6818no—
148C22C23×#59519yes—
149C4—172no—
150C4—176no—
151C4—349no—
152C4—346no—
153C4C23×#59516yes—
154C2—29no—
155C2—171no—
156C2—171no—
157C2—349no—
158C2C24×#59511yescharacteristic
subgroupG/H
center, FrattiniC2C24×#595
commutator, socleC2×C17C24×C4
FittingC17×H(5,𝔽2)C4
prime radicalH(5,𝔽2)×D17C2
cosocleC2×D17C25
hypercenterH(5,𝔽2)#595
hypocenterC17C4×H(5,𝔽2)
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #11763878C2semidirect
2. #11763876C2semidirect
3. #41×#595C2semidirect
4. C2×D4×#595C2semidirect
6. C23×#595C22semidirect
7. C17×H(5,𝔽2)C4semidirect
8. #10587696C4semidirect
10. #10587691C4semidirect
12. C2×C22⋊#595C22semidirect
13. Q8×#595C22semidirect
14. Q8×#595C22semidirect
15. D4×#595C22semidirect
16. D4×#595C22semidirect
17. D4×#595C22semidirect
18. #10587677C22semidirect
19. C2×C4⋊#595C22semidirect
20. C2×C4×#595C22semidirect
26. #41×C17C2×C4semidirect
27. C2×D4×C17C2×C4semidirect
28. C2×C22⋊D17C2×C4semidirect
29. #63256C2×C4semidirect
31. #63254C2×C4semidirect
32. D4×D17C2×C4semidirect
34. #63252C2×C4semidirect
35. C2×D68C2×C4semidirect
42. C23×C17C22×C4semidirect
45. C22×D17C22×C4semidirect
53. Q8×C17C22×C4semidirect
54. D4×C17C22×C4semidirect
55. C2×C4×C17C22×C4semidirect
56. C22⋊D17C22×C4semidirect
57. C2×Dic17C22×C4semidirect
58. D68C22×C4semidirect
59. C4×D17C22×C4semidirect
61. Dic34C22×C4semidirect
72. #595H(5,𝔽2)direct (N × H)
87. H(5,𝔽2)#595direct (N × H)
100. C17C4×H(5,𝔽2)semidirect
105. #41C2×#595semidirect
111. C2×D4C2×#595semidirect
126. C23C22×#595semidirect
128. Q8C22×#595semidirect
131. D4C22×#595semidirect
141. C2×C4C22×#595semidirect

Maximal subgroups

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

subgroupindex
#117638782
#117638762
#41×#5952
C2×D4×#5952
H(5,𝔽2)×D172
C4×H(5,𝔽2)17