Finite Group Databasebeta

← C23×#11993 #19159619

Subgroups — C23×#11993 #19159619

Order 5888.

360 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3152C211yesmaximal, characteristic
2C23×#2165C212yesmaximal, complemented
3C23×#1359C212yesmaximal, complemented
4C23×#1358C212yesmaximal, complemented
5C2×C23×#137—22no—
6C23×#532C411yescharacteristic
7C2×C23×C4⋊Q8C411yescharacteristic
8C23×#497—22no—
9C23×#494C411yescharacteristic
10C23×#494C2211yescharacteristic
11C23×#492C411yescharacteristic
12C23×#492C2211yescharacteristic
13C4×M16×C23C2211yescharacteristic
14C23×#364C2214yes—
15C23×#349—22no—
16C23×#347—22no—
17C2×#41×C23—21no—
18C22×Q8×C23—21no—
19C23×#137—48no—
20C2×Q16×C23—22no—
21C2×Q16×C23—22no—
22C2×SD16×C23—22no—
23C2×SD16×C23—22no—
24C2×M16×C23D412yes—
25C23×C4⋊Q8—21no—
26C23×C4⋊Q8—41no—
27C23×C4⋊Q8C2×C411yescharacteristic
28C23×C4⋊Q8C2×C411yescharacteristic
29C23×C4⋊Q8C2×C411yescharacteristic
30C23×#127C2×C411yescharacteristic
31C23×#124—21no—
32C23×#121—41no—
33C4×D4×C23—41no—
34C2×C4⋊C4×C23—41no—
35C2×C42×C23C2×C411yescharacteristic
36C23×M32—24no—
37C23×#97D412yes—
38C4×C8×C23C2×C411yescharacteristic
39C4×C8×C23C2311yescharacteristic
40#41×C23—44no—
41C2×Q8×C23—21no—
42C2×Q8×C23—21no—
43C2×Q8×C23—21no—
44C2×Q8×C23—44no—
45C2×Q8×C23—41no—
46C2×D4×C23—21no—
47C2×D4×C23—41no—
48C2×D4×C23—41no—
49C22×C4×C23—21no—
50C22×C4×C23—41no—
51C22×C4×C23—41no—
52C22×C4×C23C22⋊C411yescharacteristic
53Q16×C23—44no—
54Q16×C23—44no—
55SD16×C23—44no—
56SD16×C23—44no—
57M16×C23—24no—
58C2×C8×C23C22⋊C412yes—
59C2×C8×C23C2×D412yes—
60C4⋊C4×C23—41no—
61C4⋊C4×C23—41no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—42no—
64C22⋊C4×C23—42no—
65C42×C23—21no—
66C42×C23C22⋊C411yescharacteristic
67C42×C23C22×C411yescharacteristic
68C16×C23—44no—
69#11993C2311yesmaximal, characteristic, direct product factor, Sylow
70C23×C23—41no—
71C23×C23#10011yescharacteristic
72Q8×C23—42no—
73Q8×C23—42no—
74Q8×C23—42no—
75Q8×C23—82no—
76D4×C23—42no—
77D4×C23—82no—
78D4×C23—82no—
79C2×C4×C23—21no—
80C2×C4×C23—22no—
81C2×C4×C23—21no—
82C2×C4×C23—42no—
83C2×C4×C23—41no—
84C2×C4×C23—41no—
85C2×C4×C23—41no—
86C2×C4×C23—82no—
87C2×C4×C23—82no—
88C2×C4×C23#10011yescharacteristic
89C2×C4×C23C2×C22⋊C411yescharacteristic
90C8×C23—22no—
91C8×C23AGL(1,ℤ/8)14yes—
92#3152C2×C2311yescharacteristic
93#2165C2×C2312yescomplemented
94#1359C2×C2312yescomplemented
95#1358C2×C2312yescomplemented
96C22×C23—21no—
97C22×C23—41no—
98C22×C23—82no—
99C22×C23—81no—
100C22×C23C2×#10011yescharacteristic
101C4×C23—22no—
102C4×C23—21no—
103C4×C23—41no—
104C4×C23—81no—
105C4×C23—81no—
106C4×C23—81no—
107C4×C23#41812yes—
108C2×#137—22no—
109#532C4×C2311yescharacteristic
110C2×C4⋊Q8C4×C2311yescharacteristic
111#497—22no—
112#494C4×C2311yescharacteristic
113#494C22×C2311yescharacteristic
114#492C4×C2311yescharacteristic
115#492C22×C2311yescharacteristic
116C4×M16C22×C2311yescharacteristic
117#364C22×C2314yes—
118#349—22no—
119#347—22no—
120C2×C23—21no—
121C2×C23—41no—
122C2×C23—161no—
123C2×C23#155511yescharacteristic
124C2×#41—21no—
125C22×Q8—21no—
126#137—48no—
127C2×Q16—22no—
128C2×Q16—22no—
129C2×SD16—22no—
130C2×SD16—22no—
131C2×M16D4×C2312yes—
132C4⋊Q8—21no—
133C4⋊Q8—41no—
134C4⋊Q8C2×C4×C2311yescharacteristic
135C4⋊Q8C2×C4×C2311yescharacteristic
136C4⋊Q8C2×C4×C2311yescharacteristic
137#127C2×C4×C2311yescharacteristic
138#124—21no—
139#121—41no—
140C4×D4—41no—
141C2×C4⋊C4—41no—
142C2×C42C2×C4×C2311yescharacteristic
143M32—24no—
144#97D4×C2312yes—
145C4×C8C2×C4×C2311yescharacteristic
146C4×C8C23×C2311yescharacteristic
147C23#1199311yescharacteristic, direct product factor, Sylow
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—44no—
153C2×Q8—41no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4C22⋊C4×C2311yescharacteristic
161Q16—44no—
162Q16—44no—
163SD16—44no—
164SD16—44no—
165M16—24no—
166C2×C8C22⋊C4×C2312yes—
167C2×C8C2×D4×C2312yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C42—21no—
174C42C22⋊C4×C2311yescharacteristic
175C42C22×C4×C2311yescharacteristic
176C16—44no—
177C23—41no—
178C23C23×#10011yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—42no—
182Q8—82no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4C23×#10011yescharacteristic
196C2×C4C2×C22⋊C4×C2311yescharacteristic
197C8—22no—
198C8C23×AGL(1,ℤ/8)14yes—
199C22—21no—
200C22—41no—
201C22—82no—
202C22—81no—
203C22C2×C23×#10011yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4C23×#41812yes—
211C2—21no—
212C2—41no—
213C2—161no—
214C2C23×#155511yescharacteristic
subgroupG/H
center, socleC2×C23#1555
commutatorC42C22×C4×C23
Frattini, cosocleC4×C8C23×C23
prime radicalC23×#532C4
LCSG ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C4×M16×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2165C2semidirect
3. C23×#1359C2semidirect
4. C23×#1358C2semidirect
69. #11993C23direct (N × H)
93. #2165C2×C23semidirect
94. #1359C2×C23semidirect
95. #1358C2×C23semidirect
147. C23#11993direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31522
C23×#21652
C23×#13592
C23×#13582
#1199323