Finite Group Databasebeta

← C23×#33983 #19181609

Subgroups — C23×#33983 #19181609

Order 5888.

462 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 202 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×#3417C211yesmaximal, characteristic, complemented
2C23×#3415C211yesmaximal, characteristic, complemented
3C23×#2264C213yesmaximal, complemented
4C23×#2263C213yesmaximal, complemented
5C23×#2262C213yesmaximal, complemented
6C2×C23×SD64C213yesmaximal, complemented
7C23×#2258C211yesmaximal, characteristic, complemented
8C23×#578—21no—
9C23×#576—21no—
10C23×#510—23no—
11C23×#509—23no—
12C23×#508C2213yes—
13C23×#508C2213yescomplemented
14C2×C23×Q32C2213yescomplemented
15C2×C23×D16C2213yes—
16C23×#504C2211yescharacteristic
17C23×Q64C2213yescomplemented
18C23×SD64C2213yescomplemented
19C23×SD64C2216yescomplemented
20C23×SD64C2211yescharacteristic, complemented
21C23×D32C2213yescomplemented
22C23×M64C2213yescomplemented
23C2×C23×C32C2213yescomplemented
24C23×2−1+4—41no—
25C23×H(5,𝔽2)—41no—
26C23×#137—43no—
27C23×AGL(1,ℤ/8)—43no—
28C23×#135—23no—
29C23×#135—23no—
30C2×Q16×C23—23no—
31C2×D8×C23—23no—
32C23×#131D411yescharacteristic
33C23×Q32—23no—
34C23×Q32C2313yes—
35C23×Q32C2311yescharacteristic
36C23×SD32—23no—
37C23×SD32—23no—
38C23×D16—23no—
39C23×D16C2311yescharacteristic
40C23×D16C2313yes—
41C23×M32D413yes—
42C2×C16×C23C2313yes—
43C23×C32C2311yescharacteristic
44C23×C32C2313yes—
45#41×C23—43no—
46#41×C23—43no—
47#41×C23—81no—
48#41×C23—83no—
49#41×C23D811yescharacteristic
50C2×Q8×C23—43no—
51C2×Q8×C23—81no—
52C2×D4×C23—43no—
53C2×D4×C23—83no—
54Q16×C23—21no—
55Q16×C23—23no—
56Q16×C23—43no—
57SD16×C23—43no—
58SD16×C23—43no—
59D8×C23—23no—
60D8×C23—21no—
61D8×C23—43no—
62M16×C23D813yes—
63C2×C8×C23C2×D413yes—
64C16×C23C2×D413yes—
65C16×C23C2411yescharacteristic
66#33983C2311yesmaximal, characteristic, direct product factor, Sylow
67C23×C23—83no—
68Q8×C23—41no—
69Q8×C23—43no—
70Q8×C23—83no—
71Q8×C23D1611yescharacteristic
72D4×C23—43no—
73D4×C23—41no—
74D4×C23—83no—
75D4×C23—83no—
76D4×C23—83no—
77D4×C23D1613yes—
78C2×C4×C23—83no—
79C2×C4×C23—83no—
80C2×C4×C23—83no—
81C2×C4×C23C2×D813yes—
82C8×C23C2×D813yes—
83C8×C23C22×D411yescharacteristic
84#3417C2×C2311yescharacteristic, complemented
85#3415C2×C2311yescharacteristic, complemented
86#2264C2×C2313yescomplemented
87#2263C2×C2313yescomplemented
88#2262C2×C2313yescomplemented
89C2×SD64C2×C2313yescomplemented
90#2258C2×C2311yescharacteristic, complemented
91C22×C23—81no—
92C22×C23—83no—
93C22×C23—323no—
94C22×C23C2×D1613yes—
95C4×C23—81no—
96C4×C23—83no—
97C4×C23C2×D1613yes—
98C4×C23C22×D811yescharacteristic
99#578—21no—
100#576—21no—
101#510—23no—
102#509—23no—
103#508C22×C2313yes—
104#508C22×C2313yescomplemented
105C2×Q32C22×C2313yescomplemented
106C2×D16C22×C2313yes—
107#504C22×C2311yescharacteristic
108Q64C22×C2313yescomplemented
109SD64C22×C2313yescomplemented
110SD64C22×C2316yescomplemented
111SD64C22×C2311yescharacteristic, complemented
112D32C22×C2313yescomplemented
113M64C22×C2313yescomplemented
114C2×C32C22×C2313yescomplemented
115C2×C23—23no—
116C2×C23—161no—
117C2×C23—163no—
118C2×C23C22×D1611yescharacteristic
1192−1+4—41no—
120H(5,𝔽2)—41no—
121#137—43no—
122AGL(1,ℤ/8)—43no—
123#135—23no—
124#135—23no—
125C2×Q16—23no—
126C2×D8—23no—
127#131D4×C2311yescharacteristic
128Q32—23no—
129Q32C23×C2313yes—
130Q32C23×C2311yescharacteristic
131SD32—23no—
132SD32—23no—
133D16—23no—
134D16C23×C2311yescharacteristic
135D16C23×C2313yes—
136M32D4×C2313yes—
137C2×C16C23×C2313yes—
138C32C23×C2311yescharacteristic
139C32C23×C2313yes—
140C23#3398311yescharacteristic, direct product factor, Sylow
141#41—43no—
142#41—43no—
143#41—81no—
144#41—83no—
145#41D8×C2311yescharacteristic
146C2×Q8—43no—
147C2×Q8—81no—
148C2×D4—43no—
149C2×D4—83no—
150Q16—21no—
151Q16—23no—
152Q16—43no—
153SD16—43no—
154SD16—43no—
155D8—23no—
156D8—21no—
157D8—43no—
158M16D8×C2313yes—
159C2×C8C2×D4×C2313yes—
160C16C2×D4×C2313yes—
161C16C24×C2311yescharacteristic
162C23—83no—
163Q8—41no—
164Q8—43no—
165Q8—83no—
166Q8C23×D1611yescharacteristic
167D4—43no—
168D4—41no—
169D4—83no—
170D4—83no—
171D4—83no—
172D4C23×D1613yes—
173C2×C4—83no—
174C2×C4—83no—
175C2×C4—83no—
176C2×C4C2×D8×C2313yes—
177C8C2×D8×C2313yes—
178C8C22×D4×C2311yescharacteristic
179C22—81no—
180C22—83no—
181C22—323no—
182C22C2×C23×D1613yes—
183C4—81no—
184C4—83no—
185C4C2×C23×D1613yes—
186C4C22×D8×C2311yescharacteristic
187C2—23no—
188C2—163no—
189C2—161no—
190C2C22×C23×D1611yescharacteristic
subgroupG/H
center, socleC2×C23C22×D16
commutator, Frattini, cosocleC16C24×C23
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ #41×C23 ◁ C23×#131 ◁ C23×#504 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3417C2semidirect
2. C23×#3415C2semidirect
3. C23×#2264C2semidirect
4. C23×#2263C2semidirect
5. C23×#2262C2semidirect
6. C2×C23×SD64C2semidirect
7. C23×#2258C2semidirect
13. C23×#508C22semidirect
14. C2×C23×Q32C22semidirect
17. C23×Q64C22semidirect
18. C23×SD64C22semidirect
19. C23×SD64C22semidirect
20. C23×SD64C22semidirect
21. C23×D32C22semidirect
22. C23×M64C22semidirect
23. C2×C23×C32C22semidirect
66. #33983C23direct (N × H)
84. #3417C2×C23semidirect
85. #3415C2×C23semidirect
86. #2264C2×C23semidirect
87. #2263C2×C23semidirect
88. #2262C2×C23semidirect
89. C2×SD64C2×C23semidirect
90. #2258C2×C23semidirect
104. #508C22×C23semidirect
105. C2×Q32C22×C23semidirect
108. Q64C22×C23semidirect
109. SD64C22×C23semidirect
110. SD64C22×C23semidirect
111. SD64C22×C23semidirect
112. D32C22×C23semidirect
113. M64C22×C23semidirect
114. C2×C32C22×C23semidirect
140. C23#33983direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34172
C23×#34152
C23×#22642
C23×#22632
C23×#22622
C2×C23×SD642
C23×#22582
#3398323