Finite Group Databasebeta

← #535×C173 #26011158

Subgroups — #535×C173 #26011158

Order 11072.

334 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C2×#41×C173C211yesmaximal, characteristic, complemented
2C22×D4×C173C211yesmaximal, characteristic, complemented
3#127×C173C212yesmaximal, complemented
4#124×C173C212yesmaximal, complemented
5#121×C173C214yesmaximal, complemented
6#120×C173C214yesmaximal, complemented
7#117×C173C211yesmaximal, characteristic, complemented
8C24×C173C2212yes—
9#41×C173—24no—
10C2×Q8×C173C2211yescharacteristic, complemented
11C2×D4×C173—24no—
12C2×D4×C173C2212yes—
13C2×D4×C173C2212yes—
14C2×D4×C173C2212yes—
15C2×D4×C173C2211yescharacteristic, complemented
16C2×D4×C173C2218yescomplemented
17C22×C4×C173C2211yescharacteristic
18C22×C4×C173C2212yescomplemented
19C4⋊C4×C173C2212yescomplemented
20C22⋊C4×C173C2212yes—
21C22⋊C4×C173C2218yescomplemented
22C42×C173C2212yescomplemented
23C23×C173—24no—
24C23×C173—44no—
25C23×C173C2314yes—
26C23×C173C2312yes—
27C23×C173C2311yescharacteristic
28Q8×C173—22no—
29D4×C173—24no—
30D4×C173—24no—
31D4×C173—24no—
32D4×C173—22no—
33D4×C173—48no—
34C2×C4×C173—24no—
35C2×C4×C173D414yescomplemented
36C2×C4×C173C2314yes—
37C2×C4×C173C2312yes—
38C2×C4×C173C2311yescharacteristic
39C2×C4×C173C2311yescharacteristic
40C22×C173—22no—
41C22×C173—22no—
42C22×C173—24no—
43C22×C173—24no—
44C22×C173—28no—
45C22×C173—48no—
46C22×C173C2×D412yes—
47C22×C173C2411yescharacteristic
48C4×C173—22no—
49C4×C173—24no—
50C4×C173C2×D412yes—
51C4×C173C2×D412yes—
52C2×C173—22no—
53C2×C173—42no—
54C2×C173—44no—
55C2×C173C22×D411yescharacteristic
56C2×C173H(5,𝔽2)12yes—
57C173#53511yescharacteristic, direct product factor, Sylow
58#535C17311yesmaximal, characteristic, direct product factor, Sylow
59C2×#41C2×C17311yescharacteristic, complemented
60C22×D4C2×C17311yescharacteristic, complemented
61#127C2×C17312yescomplemented
62#124C2×C17312yescomplemented
63#121C2×C17314yescomplemented
64#120C2×C17314yescomplemented
65#117C2×C17311yescharacteristic, complemented
66C24C22×C17312yes—
67#41—24no—
68C2×Q8C22×C17311yescharacteristic, complemented
69C2×D4—24no—
70C2×D4C22×C17312yes—
71C2×D4C22×C17312yes—
72C2×D4C22×C17312yes—
73C2×D4C22×C17311yescharacteristic, complemented
74C2×D4C22×C17318yescomplemented
75C22×C4C22×C17311yescharacteristic
76C22×C4C22×C17312yescomplemented
77C4⋊C4C22×C17312yescomplemented
78C22⋊C4C22×C17312yes—
79C22⋊C4C22×C17318yescomplemented
80C42C22×C17312yescomplemented
81C23—24no—
82C23—44no—
83C23C23×C17314yes—
84C23C23×C17312yes—
85C23C23×C17311yescharacteristic
86Q8—22no—
87D4—24no—
88D4—24no—
89D4—24no—
90D4—22no—
91D4—48no—
92C2×C4—24no—
93C2×C4D4×C17314yescomplemented
94C2×C4C23×C17314yes—
95C2×C4C23×C17312yes—
96C2×C4C23×C17311yescharacteristic
97C2×C4C23×C17311yescharacteristic
98C22—22no—
99C22—22no—
100C22—24no—
101C22—24no—
102C22—28no—
103C22—48no—
104C22C2×D4×C17312yes—
105C22C24×C17311yescharacteristic
106C4—22no—
107C4—24no—
108C4C2×D4×C17312yes—
109C4C2×D4×C17312yes—
110C2—22no—
111C2—42no—
112C2—44no—
113C2C22×D4×C17311yescharacteristic
114C2H(5,𝔽2)×C17312yes—
subgroupG/H
center, socleC22×C173C24
commutator, Frattini, cosocleC22C24×C173
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C173 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#41×C173C2semidirect
2. C22×D4×C173C2semidirect
3. #127×C173C2semidirect
4. #124×C173C2semidirect
5. #121×C173C2semidirect
6. #120×C173C2semidirect
7. #117×C173C2semidirect
10. C2×Q8×C173C22semidirect
15. C2×D4×C173C22semidirect
16. C2×D4×C173C22semidirect
18. C22×C4×C173C22semidirect
19. C4⋊C4×C173C22semidirect
21. C22⋊C4×C173C22semidirect
22. C42×C173C22semidirect
35. C2×C4×C173D4semidirect
57. C173#535direct (N × H)
58. #535C173direct (N × H)
59. C2×#41C2×C173semidirect
60. C22×D4C2×C173semidirect
61. #127C2×C173semidirect
62. #124C2×C173semidirect
63. #121C2×C173semidirect
64. #120C2×C173semidirect
65. #117C2×C173semidirect
68. C2×Q8C22×C173semidirect
73. C2×D4C22×C173semidirect
74. C2×D4C22×C173semidirect
76. C22×C4C22×C173semidirect
77. C4⋊C4C22×C173semidirect
79. C22⋊C4C22×C173semidirect
80. C42C22×C173semidirect
93. C2×C4D4×C173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×C1732
C22×D4×C1732
#127×C1732
#124×C1732
#121×C1732
#120×C1732
#117×C1732
#535173