Finite Group Databasebeta

← C2×#22578146 #31794910

Subgroups — C2×#22578146 #31794910

Order 17536.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#22578146C2116yesmaximal, direct product factor
2C2×#17984075C213yesmaximal, complemented
3C2×#17984074C216yesmaximal, complemented
4C2×#17984057C212yesmaximal, complemented
5C2×#17984056C213yesmaximal, complemented
6C2×#120×C137C211yesmaximal, characteristic, complemented
7C24×D137C2211yescharacteristic, complemented
8C22×Dic274C2213yescomplemented
9C22×C22⋊D137C2216yes—
10#17984075C22124yescomplemented
11#17984074C22148yescomplemented
12#17984057C22116yescomplemented
13#17984056C22124yescomplemented
14C2×C22⋊Dic137C2213yescomplemented
15C2×C22⋊Dic137C2216yescomplemented
16C2×#14555487C2216yescomplemented
17C2×C4×Dic137C2213yescomplemented
18C25×C137C2211yescharacteristic, complemented
19C22×D4×C137C2213yes—
20#120×C137C2218yescomplemented
21C2×C22⋊C4×C137C2213yescomplemented
22C23×D137—46no—
23C23×D137C2312yes—
24C23×D137C2314yescomplemented
25C2×Dic274—46no—
26C2×Dic274C23112yescomplemented
27C2×C22⋊D137—412no—
28C2×C22⋊D137C23124yes—
29C22×Dic137C2316yes—
30C22⋊Dic137C23112yescomplemented
31C22⋊Dic137C23124yescomplemented
32#14555487C23124yescomplemented
33C4×Dic137C23112yescomplemented
34C24×C137—46no—
35C24×C137C2311yescharacteristic
36C24×C137C2313yes—
37C24×C137C2314yescomplemented
38C2×D4×C137—46no—
39C2×D4×C137C23112yes—
40C22×C4×C137C2313yes—
41C22⋊C4×C137C23112yescomplemented
42C22×D137—212no—
43C22×D137—412no—
44C22×D137—416no—
45C22×D137C2414yes—
46Dic274—412no—
47C22⋊D137—424no—
48C2×Dic137—212no—
49C2×Dic137C24112yes—
50C23×C137—26no—
51C23×C137—26no—
52C23×C137—212no—
53C23×C137—412no—
54C23×C137—416no—
55C23×C137C2416yes—
56C23×C137C2412yes—
57C23×C137C2411yescharacteristic
58D4×C137—412no—
59C2×C4×C137—26no—
60C2×C4×C137C2416yes—
61C22×C137—26no—
62C22×C137—26no—
63C22×C137—212no—
64C22×C137—416no—
65C22×C137—412no—
66C22×C137—44no—
67C22×C137H(5,𝔽2)16yes—
68C22×C137C2511yescharacteristic
69C2×D137—212no—
70C2×D137—416no—
71C2×D137—48no—
72C4×C137—26no—
73Dic137—212no—
74C2×C137—42no—
75C2×C137—46no—
76C2×C137H(3,𝔽4)14yescomplemented
77C2×C137C2×H(5,𝔽2)13yes—
78D137—44no—
79C137C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
80C2×H(3,𝔽4)—1371nomaximal, Sylow
81H(3,𝔽4)—13716no—
82C2×#124—1373no—
83C2×#124—1376no—
84C2×#120—1372no—
85C2×#120—1373no—
86C2×#120D13711yescharacteristic, complemented
87C25—1371no—
88C25C2×D13711yescharacteristic, complemented
89C22×Q8—1373no—
90C22×D4—1376no—
91C22×D4C2×D13713yes—
92#124—13724no—
93#124—13748no—
94#120—13716no—
95#120—13724no—
96#120C2×D13718yescomplemented
97C2×C22⋊C4—1373no—
98C2×C22⋊C4—1376no—
99C2×C22⋊C4—1376no—
100C2×C22⋊C4C2×D13713yescomplemented
101C2×C42—1373no—
102C24—46no—
103C24—5486no—
104C24—1372no—
105C24—1374no—
106C24C22×D13711yescharacteristic
107C24C22×D13713yes—
108C24C22×D13714yescomplemented
109C2×Q8—5486no—
110C2×Q8—13712no—
111C2×D4—46no—
112C2×D4—54812no—
113C2×D4—13724no—
114C2×D4C22×D137112yes—
115C22×C4—1376no—
116C22×C4C22×D13713yes—
117C22⋊C4—13712no—
118C22⋊C4—13724no—
119C22⋊C4—13724no—
120C22⋊C4C22×D137112yescomplemented
121C42—13712no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—27412no—
128C23—54812no—
129C23—54816no—
130C23—1374no—
131C23C23×D13712yes—
132C23C23×D13716yes—
133C23C23×D13711yescharacteristic
134Q8—54812no—
135D4—412no—
136D4—54824no—
137C2×C4—26no—
138C2×C4—27412no—
139C2×C4—13712no—
140C2×C4C23×D13716yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—27412no—
148C22—54816no—
149C22—5488no—
150C22#1798413316yes—
151C22C24×D13711yescharacteristic
152C4—26no—
153C4—27412no—
154C2—46no—
155C2—42no—
156C2—5484no—
157C2#2257814614yesdirect product factor
158C2C2×#1798413313yes—
subgroupG/H
centerC23C23×D137
commutator, cosocleC22×C137C25
FrattiniC22C24×D137
FittingC2×#120×C137C2
socleC23×C137C24
hypercenterC2×#120D137
hypocenterC137C2×H(3,𝔽4)
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #22578146C2direct (N × H)
2. C2×#17984075C2semidirect
3. C2×#17984074C2semidirect
4. C2×#17984057C2semidirect
5. C2×#17984056C2semidirect
6. C2×#120×C137C2semidirect
7. C24×D137C22semidirect
8. C22×Dic274C22semidirect
10. #17984075C22semidirect
11. #17984074C22semidirect
12. #17984057C22semidirect
13. #17984056C22semidirect
14. C2×C22⋊Dic137C22semidirect
15. C2×C22⋊Dic137C22semidirect
16. C2×#14555487C22semidirect
17. C2×C4×Dic137C22semidirect
18. C25×C137C22semidirect
20. #120×C137C22semidirect
21. C2×C22⋊C4×C137C22semidirect
24. C23×D137C23semidirect
26. C2×Dic274C23semidirect
30. C22⋊Dic137C23semidirect
31. C22⋊Dic137C23semidirect
32. #14555487C23semidirect
33. C4×Dic137C23semidirect
37. C24×C137C23semidirect
41. C22⋊C4×C137C23semidirect
76. C2×C137H(3,𝔽4)semidirect
79. C137C2×H(3,𝔽4)semidirect
86. C2×#120D137semidirect
88. C25C2×D137semidirect
96. #120C2×D137semidirect
100. C2×C22⋊C4C2×D137semidirect
108. C24C22×D137semidirect
120. C22⋊C4C22×D137semidirect
157. C2#22578146direct (N × H)

Maximal subgroups

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

subgroupindex
#225781462
C2×#179840752
C2×#179840742
C2×#179840572
C2×#179840562
C2×#120×C1372
C2×H(3,𝔽4)137