Finite Group Databasebeta

← #28377843

Subgroups — #28377843

Order 13952.

5992 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.

normal subgroups only

This group has 5992 conjugacy classes of subgroups — too many to draw. Only its 3201 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#20295863C2112yesmaximal, complemented
2#20295862C2115yesmaximal, complemented
3C2×#16812745C2115yesmaximal, complemented
4H(5,𝔽2)×D109C2120yesmaximal, complemented
5#585×C109C211yesmaximal, characteristic, complemented
6#16812747C22130yescomplemented
7#16812746C22120yescomplemented
8#16812745C221120yescomplemented
9#16812745C22160yescomplemented
10#16812744C22190yescomplemented
11C2×#13182030C22130yescomplemented
12C2×#13182029C22145yescomplemented
13#41×D109C221120yescomplemented
14C22×D436C22115yescomplemented
15C2×D4×D109C22190yescomplemented
162−1+4×C109C2216yescomplemented
17H(5,𝔽2)×C109C22110yescomplemented
18C2×#41×C109C22115yescomplemented
19C23×D109C23130yescomplemented
20#13182030C231120yescomplemented
21#13182029C231180yescomplemented
22#13182029C23160yescomplemented
23#13182028C231120yescomplemented
24C2×Dic218C23115yescomplemented
25Q8×D109C23140yescomplemented
26C2×D436C23190yescomplemented
27C2×D436C23145yescomplemented
28C2×C22⋊D109C23190yescomplemented
29D4×D109C231360yescomplemented
30C2×C4×D109C23190yescomplemented
31#41×C109C23120yescomplemented
32#41×C109C23160yescomplemented
33C2×Q8×C109C23115yescomplemented
34C2×D4×C109C23145yescomplemented
35C22×C4×C109C23115yescomplemented
36C23×C109C24115yes—
37C22×D109—830no—
38C22×D109C24190yes—
39Q8×C109C24120yes—
40D4×C109C24160yes—
41C2×C4×C109C24115yes—
42C2×C4×C109C24145yes—
43C22⋊D109C241120yes—
44C2×Dic109C24130yes—
45D436C24190yes—
46D436C24110yes—
47C4×D109C241120yes—
48Dic218C24130yes—
49Dic218C2416yes—
50C22×C109—415no—
51C22×C109C25115yes—
52C2×D109—490no—
53C2×D109C25120yes—
54C4×C109C2511yescharacteristic
55C4×C109C25115yes—
56Dic109C25112yes—
57C2×C109—215no—
58C2×C109C2611yescharacteristic
59D109—220no—
60H(7,𝔽2)—1091nomaximal, Sylow
61C109H(7,𝔽2)11yescharacteristic, complemented, Sylow
62#585—10912no—
63#585—10915no—
64#585D10911yescharacteristic, complemented
65C2×H(5,𝔽2)—10920no—
66C2×H(5,𝔽2)—10915no—
672−1+4—10920no—
682−1+4—10930no—
692−1+4C2×D10916yescomplemented
70H(5,𝔽2)—109120no—
71H(5,𝔽2)—10960no—
72H(5,𝔽2)—10990no—
73H(5,𝔽2)C2×D109110yescomplemented
74C2×#41—109120no—
75C2×#41—10930no—
76C2×#41—10945no—
77C2×#41C2×D109115yescomplemented
78C22×D4—10990no—
79C22×D4—10915no—
80C24—10930no—
81#41—109120no—
82#41—109120no—
83#41—109180no—
84#41—10960no—
85#41C22×D109160yescomplemented
86#41C22×D109120yescomplemented
87C2×Q8—10940no—
88C2×Q8—10915no—
89C2×Q8C22×D109115yescomplemented
90C2×D4—10990no—
91C2×D4—109360no—
92C2×D4—10990no—
93C2×D4—10945no—
94C2×D4C22×D109145yescomplemented
95C22×C4—10990no—
96C22×C4C22×D109115yescomplemented
97C23—87230no—
98C23—10990no—
99C23C23×D109115yes—
100Q8—10930no—
101Q8—1096no—
102Q8C23×D109120yes—
103D4—109120no—
104D4—10990no—
105D4—10910no—
106D4C23×D109160yes—
107C2×C4—10930no—
108C2×C4—109120no—
109C2×C4C23×D109145yes—
110C2×C4C23×D109115yes—
111C22—415no—
112C22—10920no—
113C22—43690no—
114C22C24×D109115yes—
115C4—10912no—
116C4C24×D109115yes—
117C4C24×D10911yescharacteristic
118C2—215no—
119C2—21820no—
120C2C25×D10911yescharacteristic
subgroupG/H
center, FrattiniC2C25×D109
commutator, socle, cosocleC2×C109C26
Fitting#585×C109C2
hypercenter#585D109
hypocenterC109H(7,𝔽2)
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20295863C2semidirect
2. #20295862C2semidirect
3. C2×#16812745C2semidirect
4. H(5,𝔽2)×D109C2semidirect
5. #585×C109C2semidirect
6. #16812747C22semidirect
7. #16812746C22semidirect
8. #16812745C22semidirect
9. #16812745C22semidirect
10. #16812744C22semidirect
11. C2×#13182030C22semidirect
12. C2×#13182029C22semidirect
13. #41×D109C22semidirect
14. C22×D436C22semidirect
15. C2×D4×D109C22semidirect
16. 2−1+4×C109C22semidirect
17. H(5,𝔽2)×C109C22semidirect
18. C2×#41×C109C22semidirect
19. C23×D109C23semidirect
20. #13182030C23semidirect
21. #13182029C23semidirect
22. #13182029C23semidirect
23. #13182028C23semidirect
24. C2×Dic218C23semidirect
25. Q8×D109C23semidirect
26. C2×D436C23semidirect
27. C2×D436C23semidirect
28. C2×C22⋊D109C23semidirect
29. D4×D109C23semidirect
30. C2×C4×D109C23semidirect
31. #41×C109C23semidirect
32. #41×C109C23semidirect
33. C2×Q8×C109C23semidirect
34. C2×D4×C109C23semidirect
35. C22×C4×C109C23semidirect
61. C109H(7,𝔽2)semidirect
64. #585D109semidirect
69. 2−1+4C2×D109semidirect
73. H(5,𝔽2)C2×D109semidirect
77. C2×#41C2×D109semidirect
85. #41C22×D109semidirect
86. #41C22×D109semidirect
89. C2×Q8C22×D109semidirect
94. C2×D4C22×D109semidirect
96. C22×C4C22×D109semidirect

Maximal subgroups

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

subgroupindex
#202958632
#202958622
C2×#168127452
H(5,𝔽2)×D1092
#585×C1092
H(7,𝔽2)109