Finite Group Databasebeta

← #72244792

Subgroups — #72244792

Order 6800.

248 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 49 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#72244791C212yesmaximal, complemented
2#72244790C212yesmaximal, complemented
3#72244779C211yesmaximal, characteristic, complemented
4D52×D17C211yesmaximal, characteristic
5#72244730C211yesmaximal, characteristic, complemented
6(C5⋊D5)⋊D17C411yescharacteristic, complemented
7D5×D85C2212yes—
8D52×C17C411yescharacteristic, complemented
9D17×C5⋊D5C2211yescharacteristic
10C5×D5×D17C412yescomplemented
11#13133347C2212yescomplemented
12#13133344C2211yescharacteristic, complemented
13#13133343C2211yescharacteristic, complemented
14C2×D5⋊#595—52nomaximal
15C5⋊D85C2311yescharacteristic
16C17×C5⋊D5C2×C411yescharacteristic, complemented
17C5×D85C2×C412yescomplemented
18C5×D5×C17C2×C412yescomplemented
19C52×D17C2×C411yescharacteristic, complemented
20C2×D5×D17—52no—
21C2×#66776—52no—
22C2×#66775—52no—
23D5⋊#595—52no—
24D5⋊#595—52no—
25D5⋊#595—52no—
26D5⋊#595—52no—
27D5⋊#595—102no—
28C52×C17C22×C411yescharacteristic, complemented, Hall
29C2×D5⋊AGL(1,𝔽5)—171nomaximal, Hall
30C2×D85—52no—
31C2×D5×C17—52no—
32C2×C5×D17—52no—
33D5×D17—52no—
34D5×D17—52no—
35D5×D17—52no—
36D5×D17—102no—
37D5×D17AGL(1,𝔽5)12yescomplemented
38#66776—52no—
39#66776—52no—
40#66776—102no—
41#66775—52no—
42#66775—52no—
43#66775—102no—
44C22×#595—251noHall
45C2×D52—171no—
46C2×#1044—171no—
47C2×#1043—171no—
48D5⋊AGL(1,𝔽5)—172no—
49D5⋊AGL(1,𝔽5)—172no—
50C2×C5×C17—52no—
51D85—52no—
52D85—102no—
53D85C2×AGL(1,𝔽5)12yescomplemented
54C5×D17—22no—
55C5×D17—52no—
56C5×D17C2×AGL(1,𝔽5)12yescomplemented
57D5×C17—52no—
58D5×C17—102no—
59D5×C17C2×AGL(1,𝔽5)12yescomplemented
60C22×D17—251no—
61C2×#595—251no—
62C2×#595—251no—
63C2×#595—252no—
64C2×#595—252no—
65C2×C5⋊D5—171no—
66C2×C5×D5—172no—
67D52—172no—
68D52—171no—
69D52#59511yescharacteristic, complemented
70#1044—172no—
71#1043—171no—
72#1043—171no—
73C5×C17—22no—
74C5×C17C22×AGL(1,𝔽5)12yescomplemented
75C22×AGL(1,𝔽5)—852no—
76C22×C17—251no—
77C2×D17—52no—
78C2×D17—251no—
79C2×D17—252no—
80C2×D17—251no—
81#595—251no—
82#595—252no—
83#595—251no—
84C2×C52—171no—
85C5⋊D5—171no—
86C5⋊D5C2×#59511yescharacteristic, complemented
87C5×D5—172no—
88C5×D5C2×#59512yescomplemented
89C22×D5—852no—
90C2×AGL(1,𝔽5)—852no—
91C2×AGL(1,𝔽5)—852no—
92C2×AGL(1,𝔽5)—852no—
93C2×AGL(1,𝔽5)—852no—
94C2×AGL(1,𝔽5)—852no—
95C2×AGL(1,𝔽5)—852no—
96C2×AGL(1,𝔽5)—1702no—
97C2×C17—52no—
98C2×C17—251no—
99D17—52no—
100D17—251no—
101D17D5⋊AGL(1,𝔽5)11yescharacteristic, complemented
102C52C22×#59511yescharacteristic, complemented, Sylow
103C22×C5—852no—
104C2×D5—52no—
105C2×D5—172no—
106C2×D5—852no—
107C2×D5—852no—
108C2×D5—852no—
109C2×D5—852no—
110C2×D5—1702no—
111AGL(1,𝔽5)—852no—
112AGL(1,𝔽5)—852no—
113AGL(1,𝔽5)—852no—
114AGL(1,𝔽5)—852no—
115AGL(1,𝔽5)—1702no—
116AGL(1,𝔽5)—1702no—
117C17C2×D5⋊AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
118C22×C4—4251noSylow
119C2×C5—52no—
120C2×C5—172no—
121C2×C5—342no—
122C2×C5—852no—
123D5—52no—
124D5—102no—
125D5—172no—
126D5—852no—
127D5—1702no—
128D5D5⋊#59512yescomplemented
129C23—4251no—
130C2×C4—4252no—
131C2×C4—4252no—
132C2×C4—4251no—
133C2×C4—4251no—
134C5—22no—
135C5C2×D5⋊#59512yescomplemented
136C22—251no—
137C22—852no—
138C22—4252no—
139C22—4251no—
140C22—4251no—
141C4—4251no—
142C4—4251no—
143C4—4252no—
144C2—52no—
145C2—171no—
146C2—251no—
147C2—852no—
148C2—4251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC52×C17C22×C4
prime radicalD52×D17C2
cosocleC5⋊D85C23
LCSG ▷ C52×C17
UCSC1

Complements

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

NH ≅ G/Ntype
1. #72244791C2semidirect
2. #72244790C2semidirect
3. #72244779C2semidirect
5. #72244730C2semidirect
6. (C5⋊D5)⋊D17C4semidirect
8. D52×C17C4semidirect
10. C5×D5×D17C4semidirect
11. #13133347C22semidirect
12. #13133344C22semidirect
13. #13133343C22semidirect
16. C17×C5⋊D5C2×C4semidirect
17. C5×D85C2×C4semidirect
18. C5×D5×C17C2×C4semidirect
19. C52×D17C2×C4semidirect
28. C52×C17C22×C4semidirect
37. D5×D17AGL(1,𝔽5)semidirect
53. D85C2×AGL(1,𝔽5)semidirect
56. C5×D17C2×AGL(1,𝔽5)semidirect
59. D5×C17C2×AGL(1,𝔽5)semidirect
69. D52#595semidirect
74. C5×C17C22×AGL(1,𝔽5)semidirect
86. C5⋊D5C2×#595semidirect
88. C5×D5C2×#595semidirect
101. D17D5⋊AGL(1,𝔽5)semidirect
102. C52C22×#595semidirect
117. C17C2×D5⋊AGL(1,𝔽5)semidirect
128. D5D5⋊#595semidirect
135. C5C2×D5⋊#595semidirect

Maximal subgroups

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

subgroupindex
#722447912
#722447902
#722447792
D52×D172
#722447302
C2×D5⋊#5955
C2×D5⋊AGL(1,𝔽5)17