Finite Group Databasebeta

← C23×#6551 #14267686

Subgroups — C23×#6551 #14267686

Order 1792.

59806 conjugacy classes of subgroups; only the 32139 normal subgroups are shown.

normal subgroups only

This group has 59806 conjugacy classes of subgroups — too many to draw. Only its 32139 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
1C24×C7×Q8C211yesmaximal, characteristic, complemented
2C22×#6551C21112yesmaximal, direct product factor
3C23×Q8×D7C214yesmaximal, complemented
4C23×#1161C214yesmaximal, complemented
5C23×#1157C216yesmaximal, complemented
6C23×C7×Q8C2214yes—
7C23×C7×Q8C22156yescomplemented
8C24×C4×C7C2213yes—
9C23×C22⋊D7C2214yes—
10C2×#6551C2211792yesdirect product factor
11C22×#1161C221224yescomplemented
12C22×Q8×D7C221224yescomplemented
13C22×#1157C221336yescomplemented
14C23×D28C2216yes—
15C23×C4×D7C22112yes—
16C23×Dic14C2216yes—
17C23×2−1+4—71nomaximal, Sylow
18C25×C7C2311yescharacteristic
19C24×D7C2314yes—
20C22×C7×Q8C231112yes—
21C22×C7×Q8C231448yescomplemented
22C23×C4×C7C2316yes—
23C23×C4×C7C23184yes—
24C22×C22⋊D7C231112yes—
25C23×Dic7C2314yes—
26#6551C2314096yesdirect product factor
27C2×#1161C2311792yescomplemented
28C2×Q8×D7C2311792yescomplemented
29C2×#1157C2312688yescomplemented
30C22×D28C231168yes—
31C22×C4×D7C231336yes—
32C22×Dic14C231168yes—
33C22×2−1+4—7112no—
34C23×#41—74no—
35C23×#41—76no—
36C24×Q8—74no—
37C24×Q8D711yescharacteristic, complemented
38C24×C7—28no—
39C24×C7C24114yes—
40C24×C7C2411yescharacteristic
41C23×D7—232no—
42C23×D7C24156yes—
43C2×C7×Q8C241448yes—
44C2×C7×Q8C241512yescomplemented
45C22×C4×C7C24184yes—
46C22×C4×C7C241336yes—
47C2×C22⋊D7C241448yes—
48C22×Dic7C24156yes—
49#1161C2412048yescomplemented
50Q8×D7C2412048yescomplemented
51#1157C2413072yescomplemented
52C2×D28C241672yes—
53C2×C4×D7C2411344yes—
54C2×Dic14C241672yes—
55C2×2−1+4—71792no—
56C22×#41—7224no—
57C22×#41—7336no—
58C23×Q8—7224no—
59C23×Q8—76no—
60C23×Q8C2×D714yes—
61C23×Q8C2×D7156yescomplemented
62C23×D4—74no—
63C23×D4—76no—
64C24×C4—712no—
65C24×C4C2×D713yes—
66C23×C7—256no—
67C23×C72−1+418yescomplemented
68C23×C7C25128yes—
69C23×C7C2517yes—
70C22×D7—2224no—
71C22×D7C251112yes—
72C7×Q8C251256yes—
73C2×C4×C7C251168yes—
74C2×C4×C7C251192yes—
75C22⋊D7C251256yes—
76C2×Dic7C251112yes—
77D28C251384yes—
78C4×D7C251768yes—
79Dic14C251384yes—
80C25—74no—
81C25C22×D711yescharacteristic
822−1+4—74096no—
83C2×#41—71792no—
84C2×#41—72688no—
85C22×Q8—71792no—
86C22×Q8—7168no—
87C22×Q8C22×D71112yes—
88C22×Q8C22×D71448yescomplemented
89C22×D4—7112no—
90C22×D4—7168no—
91C23×C4—74no—
92C23×C4—7336no—
93C23×C4C22×D7184yes—
94C23×C4C22×D716yes—
95C22×C7—256no—
96C22×C7C2×2−1+4128yescomplemented
97C22×C7C2618yes—
98C22×C7C2617yes—
99C2×D7—2224no—
100C2×D7C26132yes—
101C4×C7C26148yes—
102Dic7C26132yes—
103C24—28no—
104C24—756no—
105C24—1432no—
106C24C23×D7114yes—
107C24C23×D711yescharacteristic
108#41—72048no—
109#41—73072no—
110C2×Q8—72048no—
111C2×Q8—7672no—
112C2×Q8C23×D71448yes—
113C2×Q8C23×D71512yescomplemented
114C2×D4—7448no—
115C2×D4—7672no—
116C22×C4—756no—
117C22×C4—71344no—
118C22×C4C23×D7184yes—
119C22×C4C23×D71336yes—
120C2×C7—28no—
121C2×C7C22×2−1+4114yescomplemented
122C2×C7C2711yescharacteristic
123D7—232no—
124C23—256no—
125C23—7112no—
126C23—14224no—
127C23#655118yesdirect product factor
128C23C24×D7128yes—
129C23C24×D717yes—
130Q8—7384no—
131Q8C24×D71256yes—
132D4—7256no—
133D4—7384no—
134C2×C4—7112no—
135C2×C4—7768no—
136C2×C4C24×D71168yes—
137C2×C4C24×D71192yes—
138C7C23×2−1+411yescharacteristic, complemented, Sylow
139C22—256no—
140C22—732no—
141C22—14224no—
142C22C2×#6551128yesdirect product factor
143C22C25×D718yes—
144C22C25×D717yes—
145C4—732no—
146C4C25×D7148yes—
147C2—28no—
148C2—1432no—
149C2C22×#6551114yesdirect product factor
150C2C26×D711yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC2×C7C27
FrattiniC2C26×D7
FittingC24×C7×Q8C2
socleC24×C7C24
hypercenterC24×Q8D7
hypocenterC7C23×2−1+4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C24×C7×Q8C2semidirect
2. C22×#6551C2direct (N × H)
3. C23×Q8×D7C2semidirect
4. C23×#1161C2semidirect
5. C23×#1157C2semidirect
7. C23×C7×Q8C22semidirect
10. C2×#6551C22direct (N × H)
11. C22×#1161C22semidirect
12. C22×Q8×D7C22semidirect
13. C22×#1157C22semidirect
21. C22×C7×Q8C23semidirect
26. #6551C23direct (N × H)
27. C2×#1161C23semidirect
28. C2×Q8×D7C23semidirect
29. C2×#1157C23semidirect
37. C24×Q8D7semidirect
44. C2×C7×Q8C24semidirect
49. #1161C24semidirect
50. Q8×D7C24semidirect
51. #1157C24semidirect
61. C23×Q8C2×D7semidirect
67. C23×C72−1+4semidirect
88. C22×Q8C22×D7semidirect
96. C22×C7C2×2−1+4semidirect
113. C2×Q8C23×D7semidirect
121. C2×C7C22×2−1+4semidirect
127. C23#6551direct (N × H)
138. C7C23×2−1+4semidirect
142. C22C2×#6551direct (N × H)
149. C2C22×#6551direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C7×Q82
C22×#65512
C23×Q8×D72
C23×#11612
C23×#11572
C23×2−1+47