Finite Group Databasebeta

← C23×#3416 #15671035

Subgroups — C23×#3416 #15671035

Order 2944.

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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
1C23×#578C211yesmaximal, characteristic, complemented
2C23×#576C211yesmaximal, characteristic, complemented
3C23×#510C213yesmaximal, complemented
4C23×#509C213yesmaximal, complemented
5C23×#508C213yesmaximal, complemented
6C2×C23×SD32C213yesmaximal, complemented
7C23×#504C211yesmaximal, characteristic, complemented
8C23×2−1+4—21no—
9C23×H(5,𝔽2)—21no—
10C23×#137—23no—
11C23×AGL(1,ℤ/8)—23no—
12C23×#135C2213yes—
13C23×#135C2213yescomplemented
14C2×Q16×C23C2213yescomplemented
15C2×D8×C23C2213yes—
16C23×#131C2211yescharacteristic
17C23×Q32C2213yescomplemented
18C23×SD32C2213yescomplemented
19C23×SD32C2216yescomplemented
20C23×SD32C2211yescharacteristic, complemented
21C23×D16C2213yescomplemented
22C23×M32C2213yescomplemented
23C2×C16×C23C2213yescomplemented
24#41×C23—23no—
25#41×C23—23no—
26#41×C23—41no—
27#41×C23—43no—
28#41×C23D411yescharacteristic
29C2×Q8×C23—23no—
30C2×Q8×C23—41no—
31C2×D4×C23—23no—
32C2×D4×C23—43no—
33Q16×C23—23no—
34Q16×C23C2313yes—
35Q16×C23C2311yescharacteristic
36SD16×C23—23no—
37SD16×C23—23no—
38D8×C23—23no—
39D8×C23C2311yescharacteristic
40D8×C23C2313yes—
41M16×C23D413yes—
42C2×C8×C23C2313yes—
43C16×C23C2311yescharacteristic
44C16×C23C2313yes—
45C23×C23—43no—
46Q8×C23—21no—
47Q8×C23—23no—
48Q8×C23—43no—
49Q8×C23D811yescharacteristic
50D4×C23—23no—
51D4×C23—21no—
52D4×C23—43no—
53D4×C23—43no—
54D4×C23—43no—
55D4×C23D813yes—
56C2×C4×C23—43no—
57C2×C4×C23—43no—
58C2×C4×C23—43no—
59C2×C4×C23C2×D413yes—
60C8×C23C2×D413yes—
61C8×C23C2411yescharacteristic
62#3416C2311yesmaximal, characteristic, direct product factor, Sylow
63C22×C23—41no—
64C22×C23—43no—
65C22×C23—163no—
66C22×C23C2×D813yes—
67C4×C23—41no—
68C4×C23—43no—
69C4×C23C2×D813yes—
70C4×C23C22×D411yescharacteristic
71#578C2×C2311yescharacteristic, complemented
72#576C2×C2311yescharacteristic, complemented
73#510C2×C2313yescomplemented
74#509C2×C2313yescomplemented
75#508C2×C2313yescomplemented
76C2×SD32C2×C2313yescomplemented
77#504C2×C2311yescharacteristic, complemented
78C2×C23—23no—
79C2×C23—81no—
80C2×C23—83no—
81C2×C23C22×D811yescharacteristic
822−1+4—21no—
83H(5,𝔽2)—21no—
84#137—23no—
85AGL(1,ℤ/8)—23no—
86#135C22×C2313yes—
87#135C22×C2313yescomplemented
88C2×Q16C22×C2313yescomplemented
89C2×D8C22×C2313yes—
90#131C22×C2311yescharacteristic
91Q32C22×C2313yescomplemented
92SD32C22×C2313yescomplemented
93SD32C22×C2316yescomplemented
94SD32C22×C2311yescharacteristic, complemented
95D16C22×C2313yescomplemented
96M32C22×C2313yescomplemented
97C2×C16C22×C2313yescomplemented
98C23#341611yescharacteristic, direct product factor, Sylow
99#41—23no—
100#41—23no—
101#41—41no—
102#41—43no—
103#41D4×C2311yescharacteristic
104C2×Q8—23no—
105C2×Q8—41no—
106C2×D4—23no—
107C2×D4—43no—
108Q16—23no—
109Q16C23×C2313yes—
110Q16C23×C2311yescharacteristic
111SD16—23no—
112SD16—23no—
113D8—23no—
114D8C23×C2311yescharacteristic
115D8C23×C2313yes—
116M16D4×C2313yes—
117C2×C8C23×C2313yes—
118C16C23×C2311yescharacteristic
119C16C23×C2313yes—
120C23—43no—
121Q8—21no—
122Q8—23no—
123Q8—43no—
124Q8D8×C2311yescharacteristic
125D4—23no—
126D4—21no—
127D4—43no—
128D4—43no—
129D4—43no—
130D4D8×C2313yes—
131C2×C4—43no—
132C2×C4—43no—
133C2×C4—43no—
134C2×C4C2×D4×C2313yes—
135C8C2×D4×C2313yes—
136C8C24×C2311yescharacteristic
137C22—41no—
138C22—43no—
139C22—163no—
140C22C2×D8×C2313yes—
141C4—41no—
142C4—43no—
143C4C2×D8×C2313yes—
144C4C22×D4×C2311yescharacteristic
145C2—23no—
146C2—83no—
147C2—81no—
148C2C22×D8×C2311yescharacteristic
subgroupG/H
center, socleC2×C23C22×D8
commutator, Frattini, cosocleC8C24×C23
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ #41×C23 ◁ C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#578C2semidirect
2. C23×#576C2semidirect
3. C23×#510C2semidirect
4. C23×#509C2semidirect
5. C23×#508C2semidirect
6. C2×C23×SD32C2semidirect
7. C23×#504C2semidirect
13. C23×#135C22semidirect
14. C2×Q16×C23C22semidirect
17. C23×Q32C22semidirect
18. C23×SD32C22semidirect
19. C23×SD32C22semidirect
20. C23×SD32C22semidirect
21. C23×D16C22semidirect
22. C23×M32C22semidirect
23. C2×C16×C23C22semidirect
62. #3416C23direct (N × H)
71. #578C2×C23semidirect
72. #576C2×C23semidirect
73. #510C2×C23semidirect
74. #509C2×C23semidirect
75. #508C2×C23semidirect
76. C2×SD32C2×C23semidirect
77. #504C2×C23semidirect
87. #135C22×C23semidirect
88. C2×Q16C22×C23semidirect
91. Q32C22×C23semidirect
92. SD32C22×C23semidirect
93. SD32C22×C23semidirect
94. SD32C22×C23semidirect
95. D16C22×C23semidirect
96. M32C22×C23semidirect
97. C2×C16C22×C23semidirect
98. C23#3416direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5782
C23×#5762
C23×#5102
C23×#5092
C23×#5082
C2×C23×SD322
C23×#5042
#341623