Finite Group Databasebeta

← C2×#31774640 #46850485

Subgroups — C2×#31774640 #46850485

Order 34688.

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#31774640C2116yesmaximal, direct product factor
2C2×#22576147C213yesmaximal, complemented
3C2×#22576146C216yesmaximal, complemented
4C2×#22576129C212yesmaximal, complemented
5C2×#22576128C213yesmaximal, complemented
6C2×#120×C271C211yesmaximal, characteristic, complemented
7C24×D271C2211yescharacteristic, complemented
8C22×Dic542C2213yescomplemented
9C22×C22⋊D271C2216yes—
10#22576147C22124yescomplemented
11#22576146C22148yescomplemented
12#22576129C22116yescomplemented
13#22576128C22124yescomplemented
14C2×C22⋊Dic271C2213yescomplemented
15C2×C22⋊Dic271C2216yescomplemented
16C2×(C2×D271)⋊C4C2216yescomplemented
17C2×C4×Dic271C2213yescomplemented
18C25×C271C2211yescharacteristic, complemented
19C22×D4×C271C2213yes—
20#120×C271C2218yescomplemented
21C2×C22⋊C4×C271C2213yescomplemented
22C23×D271—46no—
23C23×D271C2312yes—
24C23×D271C2314yescomplemented
25C2×Dic542—46no—
26C2×Dic542C23112yescomplemented
27C2×C22⋊D271—412no—
28C2×C22⋊D271C23124yes—
29C22×Dic271C2316yes—
30C22⋊Dic271C23112yescomplemented
31C22⋊Dic271C23124yescomplemented
32(C2×D271)⋊C4C23124yescomplemented
33C4×Dic271C23112yescomplemented
34C24×C271—46no—
35C24×C271C2313yes—
36C24×C271C2311yescharacteristic
37C24×C271C2314yescomplemented
38C2×D4×C271—46no—
39C2×D4×C271C23112yes—
40C22×C4×C271C2313yes—
41C22⋊C4×C271C23112yescomplemented
42C22×D271—212no—
43C22×D271—412no—
44C22×D271—416no—
45C22×D271C2414yes—
46Dic542—412no—
47C22⋊D271—424no—
48C2×Dic271—212no—
49C2×Dic271C24112yes—
50C23×C271—26no—
51C23×C271—26no—
52C23×C271—212no—
53C23×C271—412no—
54C23×C271—416no—
55C23×C271C2416yes—
56C23×C271C2412yes—
57C23×C271C2411yescharacteristic
58D4×C271—412no—
59C2×C4×C271—26no—
60C2×C4×C271C2416yes—
61C22×C271—26no—
62C22×C271—26no—
63C22×C271—212no—
64C22×C271—416no—
65C22×C271—412no—
66C22×C271—44no—
67C22×C271H(5,𝔽2)16yes—
68C22×C271C2511yescharacteristic
69C2×D271—212no—
70C2×D271—416no—
71C2×D271—48no—
72C4×C271—26no—
73Dic271—212no—
74C2×C271—42no—
75C2×C271—46no—
76C2×C271H(3,𝔽4)14yescomplemented
77C2×C271C2×H(5,𝔽2)13yes—
78D271—44no—
79C271C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
80C2×H(3,𝔽4)—2711nomaximal, Sylow
81H(3,𝔽4)—27116no—
82C2×#124—2713no—
83C2×#124—2716no—
84C2×#120—2712no—
85C2×#120—2713no—
86C2×#120D27111yescharacteristic, complemented
87C25—2711no—
88C25C2×D27111yescharacteristic, complemented
89C22×Q8—2713no—
90C22×D4—2716no—
91C22×D4C2×D27113yes—
92#124—27124no—
93#124—27148no—
94#120—27116no—
95#120—27124no—
96#120C2×D27118yescomplemented
97C2×C22⋊C4—2713no—
98C2×C22⋊C4—2716no—
99C2×C22⋊C4—2716no—
100C2×C22⋊C4C2×D27113yescomplemented
101C2×C42—2713no—
102C24—46no—
103C24—2712no—
104C24—2714no—
105C24—10846no—
106C24C22×D27111yescharacteristic
107C24C22×D27113yes—
108C24C22×D27114yescomplemented
109C2×Q8—27112no—
110C2×Q8—10846no—
111C2×D4—46no—
112C2×D4—27124no—
113C2×D4—108412no—
114C2×D4C22×D271112yes—
115C22×C4—2716no—
116C22×C4C22×D27113yes—
117C22⋊C4—27112no—
118C22⋊C4—27124no—
119C22⋊C4—27124no—
120C22⋊C4C22×D271112yescomplemented
121C42—27112no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—2714no—
128C23—54212no—
129C23—108412no—
130C23—108416no—
131C23C23×D27112yes—
132C23C23×D27116yes—
133C23C23×D27111yescharacteristic
134Q8—108412no—
135D4—412no—
136D4—108424no—
137C2×C4—26no—
138C2×C4—27112no—
139C2×C4—54212no—
140C2×C4C23×D27116yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—54212no—
148C22—108416no—
149C22—10848no—
150C22#2257620516yes—
151C22C24×D27111yescharacteristic
152C4—26no—
153C4—54212no—
154C2—46no—
155C2—42no—
156C2—10844no—
157C2#3177464014yesdirect product factor
158C2C2×#2257620513yes—
subgroupG/H
centerC23C23×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC2×#120×C271C2
socleC23×C271C24
hypercenterC2×#120D271
hypocenterC271C2×H(3,𝔽4)
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #31774640C2direct (N × H)
2. C2×#22576147C2semidirect
3. C2×#22576146C2semidirect
4. C2×#22576129C2semidirect
5. C2×#22576128C2semidirect
6. C2×#120×C271C2semidirect
7. C24×D271C22semidirect
8. C22×Dic542C22semidirect
10. #22576147C22semidirect
11. #22576146C22semidirect
12. #22576129C22semidirect
13. #22576128C22semidirect
14. C2×C22⋊Dic271C22semidirect
15. C2×C22⋊Dic271C22semidirect
16. C2×(C2×D271)⋊C4C22semidirect
17. C2×C4×Dic271C22semidirect
18. C25×C271C22semidirect
20. #120×C271C22semidirect
21. C2×C22⋊C4×C271C22semidirect
24. C23×D271C23semidirect
26. C2×Dic542C23semidirect
30. C22⋊Dic271C23semidirect
31. C22⋊Dic271C23semidirect
32. (C2×D271)⋊C4C23semidirect
33. C4×Dic271C23semidirect
37. C24×C271C23semidirect
41. C22⋊C4×C271C23semidirect
76. C2×C271H(3,𝔽4)semidirect
79. C271C2×H(3,𝔽4)semidirect
86. C2×#120D271semidirect
88. C25C2×D271semidirect
96. #120C2×D271semidirect
100. C2×C22⋊C4C2×D271semidirect
108. C24C22×D271semidirect
120. C22⋊C4C22×D271semidirect
157. C2#31774640direct (N × H)

Maximal subgroups

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

subgroupindex
#317746402
C2×#225761472
C2×#225761462
C2×#225761292
C2×#225761282
C2×#120×C2712
C2×H(3,𝔽4)271