Finite Group Databasebeta

← C2×#22578240 #31795598

Subgroups — C2×#22578240 #31795598

Order 17536.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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#22578240C2132yesmaximal, direct product factor
2C2×#17984136C215yesmaximal, complemented
3C2×#17984134C2110yesmaximal, complemented
4C22×#14555511C215yesmaximal, complemented
5C2×#41×D137C2110yesmaximal, complemented
6C2×2−1+4×C137C211yesmaximal, characteristic, complemented
7#17984136C22180yescomplemented
8#17984134C221160yescomplemented
9C2×#14555511C22110yescomplemented
10C2×#14555511C22120yescomplemented
11C2×#14555511C22180yescomplemented
12C2×#14555510C22130yescomplemented
13C2×#14555509C22110yescomplemented
14#41×D137C221160yescomplemented
15C2×Q8×D137C22110yescomplemented
16C22×D548C22115yescomplemented
17C2×D4×D137C22130yescomplemented
18C22×C4×D137C22115yescomplemented
192−1+4×C137C22116yescomplemented
20C2×#41×C137C22110yescomplemented
21C22×Q8×C137C2215yescomplemented
22C23×D137C23115yes—
23#14555511C23180yescomplemented
24#14555511C231160yescomplemented
25#14555510C231240yescomplemented
26#14555509C23180yescomplemented
27C2×Dic274C23110yes—
28Q8×D137C23180yescomplemented
29C2×D548C23130yes—
30C2×D548C231120yescomplemented
31C2×C22⋊D137C23120yes—
32D4×D137C231240yescomplemented
33C22×Dic137C2315yes—
34C2×C4×D137C23110yes—
35C2×C4×D137C23130yes—
36C2×C4×D137C231120yescomplemented
37#41×C137C23180yescomplemented
38C2×Q8×C137C23110yes—
39C2×Q8×C137C23140yescomplemented
40C2×D4×C137C23110yes—
41C22×C4×C137C23115yes—
42C22×D137—430no—
43C22×D137C24110yes—
44C22×D137C24160yes—
45Dic274C24140yes—
46D548C241120yes—
47C22⋊D137C24180yes—
48C2×Dic137C2411yescharacteristic
49C2×Dic137C2415yes—
50C2×Dic137C24120yes—
51C4×D137C24140yes—
52C4×D137C241120yes—
53C23×C137C2415yes—
54Q8×C137C24140yes—
55D4×C137C24140yes—
56C2×C4×C137C24110yes—
57C2×C4×C137C24160yes—
58C22×C137—210no—
59C22×C137C25110yes—
60C22×C137C2511yescharacteristic
61C2×D137—220no—
62C2×D137—460no—
63C2×D137C25120yes—
64C4×C137C25120yes—
65Dic137C2512yes—
66Dic137C25110yes—
67C2×C137—210no—
68C2×C137#58512yescomplemented
69C2×C137C2611yescharacteristic
70D137—220no—
71C137C2×#58511yescharacteristic, complemented, Sylow
72C2×#585—1371nomaximal, Sylow
73#585—13732no—
74C2×2−1+4—1375no—
75C2×2−1+4D13711yescharacteristic, complemented
76C2×H(5,𝔽2)—13710no—
77C22×#41—13710no—
78C22×#41—1375no—
792−1+4—13780no—
802−1+4C2×D137116yescomplemented
81H(5,𝔽2)—137160no—
82C2×#41—137160no—
83C2×#41—13710no—
84C2×#41—13720no—
85C2×#41—13780no—
86C2×#41—13710no—
87C2×#41—13730no—
88C2×#41C2×D137110yescomplemented
89C22×Q8—13710no—
90C22×Q8C2×D13715yescomplemented
91C22×D4—13730no—
92C22×D4—13715no—
93C23×C4—13715no—
94C24—13715no—
95#41—13780no—
96#41—137160no—
97#41—13780no—
98#41—137240no—
99#41C22×D137180yescomplemented
100C2×Q8—13780no—
101C2×Q8—13710no—
102C2×Q8C22×D137110yes—
103C2×Q8C22×D137140yescomplemented
104C2×D4—13720no—
105C2×D4—137240no—
106C2×D4—13730no—
107C2×D4—137120no—
108C2×D4C22×D137110yes—
109C22×C4—1375no—
110C22×C4—13710no—
111C22×C4—13730no—
112C22×C4—137120no—
113C22×C4C22×D137115yes—
114C23—54830no—
115C23—13710no—
116C23—13760no—
117C23C23×D13715yes—
118Q8—13740no—
119Q8C23×D137140yes—
120D4—13780no—
121D4—137120no—
122D4C23×D137140yes—
123C2×C4—13720no—
124C2×C4—1371no—
125C2×C4—1375no—
126C2×C4—13740no—
127C2×C4—137120no—
128C2×C4C23×D137160yes—
129C2×C4C23×D137110yes—
130C22—210no—
131C22—27420no—
132C22—54860no—
133C22—13720no—
134C22C24×D137110yes—
135C22C24×D13711yescharacteristic
136C4—1372no—
137C4—13710no—
138C4C24×D137120yes—
139C2—210no—
140C2—27420no—
141C2#2257824012yesdirect product factor
142C2C25×D13711yescharacteristic
subgroupG/H
centerC22C24×D137
commutator, cosocleC2×C137C26
FrattiniC2C25×D137
FittingC2×2−1+4×C137C2
socleC22×C137C25
hypercenterC2×2−1+4D137
hypocenterC137C2×#585
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #22578240C2direct (N × H)
2. C2×#17984136C2semidirect
3. C2×#17984134C2semidirect
4. C22×#14555511C2semidirect
5. C2×#41×D137C2semidirect
6. C2×2−1+4×C137C2semidirect
7. #17984136C22semidirect
8. #17984134C22semidirect
9. C2×#14555511C22semidirect
10. C2×#14555511C22semidirect
11. C2×#14555511C22semidirect
12. C2×#14555510C22semidirect
13. C2×#14555509C22semidirect
14. #41×D137C22semidirect
15. C2×Q8×D137C22semidirect
16. C22×D548C22semidirect
17. C2×D4×D137C22semidirect
18. C22×C4×D137C22semidirect
19. 2−1+4×C137C22semidirect
20. C2×#41×C137C22semidirect
21. C22×Q8×C137C22semidirect
23. #14555511C23semidirect
24. #14555511C23semidirect
25. #14555510C23semidirect
26. #14555509C23semidirect
28. Q8×D137C23semidirect
30. C2×D548C23semidirect
32. D4×D137C23semidirect
36. C2×C4×D137C23semidirect
37. #41×C137C23semidirect
39. C2×Q8×C137C23semidirect
68. C2×C137#585semidirect
71. C137C2×#585semidirect
75. C2×2−1+4D137semidirect
80. 2−1+4C2×D137semidirect
88. C2×#41C2×D137semidirect
90. C22×Q8C2×D137semidirect
99. #41C22×D137semidirect
103. C2×Q8C22×D137semidirect
141. C2#22578240direct (N × H)

Maximal subgroups

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

subgroupindex
#225782402
C2×#179841362
C2×#179841342
C22×#145555112
C2×#41×D1372
C2×2−1+4×C1372
C2×#585137