Finite Group Databasebeta

← C2×#5720 #86196

Subgroups — C2×#5720 #86196

Order 384.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 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
1C2×C3×#127C211yesmaximal, characteristic, complemented
2C2×#939C216yesmaximal, complemented
3C2×#937C213yesmaximal, complemented
4C2×#935C213yesmaximal, complemented
5#5720C2116yesmaximal, direct product factor
6C2×#876C212yesmaximal, complemented
7C2×#560—31nomaximal, Sylow
8C24×D3C2211yescharacteristic
9C22×C3×D4C2213yes—
10C22×C3×D4C2213yescomplemented
11C22×#67C2216yes—
12C23×Dic3C2213yescomplemented
13C3×#127C2218yescomplemented
14C2×C3×C42C2211yescharacteristic, complemented
15C2×#217C2213yes—
16C2×#217C2213yescomplemented
17#939C22148yescomplemented
18#937C22124yescomplemented
19#935C22124yescomplemented
20C2×(C2×D3)⋊C4C2216yes—
21C2×Dic3⋊C4C2216yescomplemented
22#876C22116yescomplemented
23#560—316no—
24C2×#127D311yescharacteristic, complemented
25C2×#126—32no—
26C2×#123—33no—
27C2×#121—36no—
28C2×#120—33no—
29C24×C3C2311yescharacteristic
30C24×C3C2313yes—
31C23×D3—46no—
32C23×D3C2312yes—
33C23×D3C2314yes—
34C2×C3×D4—46no—
35C2×C3×D4—46no—
36C2×C3×D4C23112yes—
37C2×C3×D4C23112yescomplemented
38C22×C3×C4C2313yes—
39C2×#67—412no—
40C2×#67C23124yes—
41C22×Dic3—46no—
42C22×Dic3C2316yes—
43C22×Dic3C23112yescomplemented
44C3×C42C2314yescomplemented
45#217C23112yes—
46#217C23112yescomplemented
47(C2×D3)⋊C4C23124yes—
48Dic3⋊C4C23124yescomplemented
49C25—31no—
50C22×D4—36no—
51C22×D4C2×D313yes—
52C22×D4C2×D313yescomplemented
53C23×C4—33no—
54#127C2×D318yescomplemented
55#126—316no—
56#123—324no—
57#121—348no—
58#120—324no—
59C2×C4⋊C4—36no—
60C2×C22⋊C4—33no—
61C2×C22⋊C4—33no—
62C2×C22⋊C4—36no—
63C2×C42C2×D311yescharacteristic, complemented
64C23×C3—26no—
65C23×C3—212no—
66C23×C3—26no—
67C23×C3C2412yes—
68C23×C3C2416yes—
69C23×C3C2411yescharacteristic
70C22×D3—212no—
71C22×D3—412no—
72C22×D3—416no—
73C22×D3C2414yes—
74C3×D4—412no—
75C3×D4—412no—
76C2×C3×C4—26no—
77C2×C3×C4C2416yes—
78#67—424no—
79C2×Dic3—212no—
80C2×Dic3—412no—
81C2×Dic3C24112yes—
82C24—32no—
83C24—34no—
84C24—126no—
85C24C22×D311yescharacteristic
86C24C22×D313yes—
87C2×D4—324no—
88C2×D4—46no—
89C2×D4—46no—
90C2×D4—1212no—
91C2×D4C22×D3112yes—
92C2×D4C22×D3112yescomplemented
93C22×C4—36no—
94C22×C4—312no—
95C22×C4—126no—
96C22×C4C22×D313yes—
97C4⋊C4—324no—
98C22⋊C4—312no—
99C22⋊C4—312no—
100C22⋊C4—324no—
101C42C22×D314yescomplemented
102C22×C3—26no—
103C22×C3—212no—
104C22×C3—26no—
105C22×C3—44no—
106C22×C3—412no—
107C22×C3H(5,𝔽2)16yes—
108C22×C3C2511yescharacteristic
109C2×D3—212no—
110C2×D3—416no—
111C2×D3—48no—
112C3×C4—26no—
113Dic3—212no—
114C23—26no—
115C23—212no—
116C23—26no—
117C23—34no—
118C23—612no—
119C23—1212no—
120C23—1216no—
121C23C23×D312yes—
122C23C23×D316yes—
123C23C23×D311yescharacteristic
124D4—412no—
125D4—412no—
126D4—1224no—
127C2×C4—26no—
128C2×C4—312no—
129C2×C4—612no—
130C2×C4—1212no—
131C2×C4C23×D316yes—
132C2×C3—42no—
133C2×C3—46no—
134C2×C3#56014yescomplemented
135C2×C3C2×H(5,𝔽2)13yes—
136D3—44no—
137C22—26no—
138C22—212no—
139C22—26no—
140C22—44no—
141C22—412no—
142C22—612no—
143C22—1216no—
144C22—128no—
145C22#100416yes—
146C22C24×D311yescharacteristic
147C4—26no—
148C4—612no—
149C3C2×#56011yescharacteristic, complemented, Sylow
150C2—42no—
151C2—46no—
152C2—124no—
153C2#572014yesdirect product factor
154C2C2×#100413yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#127C2
socleC23×C3C24
hypercenterC2×#127D3
hypocenterC3C2×#560
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×C3×#127C2semidirect
2. C2×#939C2semidirect
3. C2×#937C2semidirect
4. C2×#935C2semidirect
5. #5720C2direct (N × H)
6. C2×#876C2semidirect
10. C22×C3×D4C22semidirect
12. C23×Dic3C22semidirect
13. C3×#127C22semidirect
14. C2×C3×C42C22semidirect
16. C2×#217C22semidirect
17. #939C22semidirect
18. #937C22semidirect
19. #935C22semidirect
21. C2×Dic3⋊C4C22semidirect
22. #876C22semidirect
24. C2×#127D3semidirect
37. C2×C3×D4C23semidirect
43. C22×Dic3C23semidirect
44. C3×C42C23semidirect
46. #217C23semidirect
48. Dic3⋊C4C23semidirect
52. C22×D4C2×D3semidirect
54. #127C2×D3semidirect
63. C2×C42C2×D3semidirect
92. C2×D4C22×D3semidirect
101. C42C22×D3semidirect
134. C2×C3#560semidirect
149. C3C2×#560semidirect
153. C2#5720direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1272
C2×#9392
C2×#9372
C2×#9352
#57202
C2×#8762
C2×#5603