Finite Group Databasebeta

← C23×#9429 #19157055

Subgroups — C23×#9429 #19157055

Order 5888.

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2381C211yesmaximal, characteristic, complemented
3C23×#1558C211yesmaximal, characteristic, complemented
4C23×#1514C212yesmaximal
5C23×#1288C212yesmaximal, complemented
6C2×C23×#137—22no—
7C23×#532C2211yescharacteristic
8C2×C23×C4⋊Q8C2211yescharacteristic
9C23×#497—22no—
10C23×#494C412yescomplemented
11C23×#492C412yescomplemented
12C23×#432C2212yes—
13C4×M16×C23C2211yescharacteristic
14C23×#391—22no—
15C23×#387—24no—
16C23×#383C2212yes—
17C23×#333—22no—
18C23×#327—24no—
19C2×#41×C23—21no—
20C22×Q8×C23D412yes—
21C23×#137—44no—
22C23×#137—44no—
23C2×Q16×C23—24no—
24C2×SD16×C23—24no—
25C2×M16×C23—21no—
26C23×C4⋊Q8—22no—
27C23×C4⋊Q8C2×C411yescharacteristic
28C23×C4⋊Q8C2×C412yes—
29C23×C4⋊Q8D412yes—
30C23×#127C2×C411yescharacteristic
31C23×#124—21no—
32C23×#121—22no—
33C4×D4×C23—22no—
34C2×C4⋊C4×C23—22no—
35C2×C4⋊C4×C23—22no—
36C2×C42×C23—22no—
37C2×C42×C23C2311yescharacteristic
38C23×C4⋊C8—22no—
39C23×C22⋊C8—22no—
40C23×#97—21no—
41C23×#97—22no—
42C4×C8×C23C2×C412yes—
43C23×#95—24no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95D414yes—
47#41×C23—42no—
48#41×C23—42no—
49C2×Q8×C23—22no—
50C2×Q8×C23—21no—
51C2×Q8×C23—42no—
52C2×Q8×C23—42no—
53C2×Q8×C23C22⋊C414yes—
54C2×D4×C23—21no—
55C2×D4×C23—22no—
56C2×D4×C23—41no—
57C22×C4×C23—22no—
58C22×C4×C23—24no—
59C22×C4×C23—41no—
60C22×C4×C23C2×D411yescharacteristic
61C22×C4×C23C2×D412yes—
62Q16×C23—48no—
63SD16×C23—48no—
64M16×C23—24no—
65C2×C8×C23—22no—
66C2×C8×C23—42no—
67C4⋊C4×C23—24no—
68C4⋊C4×C23—24no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—82no—
71C22⋊C4×C23—42no—
72C42×C23—82no—
73C42×C23C22×C411yescharacteristic
74C42×C23C2×D411yescharacteristic
75C42×C23#4112yes—
76#9429C2311yesmaximal, characteristic, direct product factor, Sylow
77C23×C23—41no—
78C23×C23#12011yescharacteristic
79Q8×C23—44no—
80Q8×C23—42no—
81Q8×C23—82no—
82D4×C23—42no—
83D4×C23—82no—
84D4×C23—82no—
85C2×C4×C23—22no—
86C2×C4×C23—24no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—48no—
91C2×C4×C23—41no—
92C2×C4×C23—44no—
93C2×C4×C23—81no—
94C2×C4×C23—81no—
95C2×C4×C23C2×C22⋊C411yescharacteristic
96C2×C4×C23C4×D412yes—
97C2×C4×C23#12112yes—
98C2×C4×C23#12311yescharacteristic
99C8×C23—24no—
100C8×C23—82no—
101#3152C2×C2311yescharacteristic
102#2381C2×C2311yescharacteristic, complemented
103#1558C2×C2311yescharacteristic, complemented
104#1514C2×C2312yes—
105#1288C2×C2312yescomplemented
106C22×C23—22no—
107C22×C23—81no—
108C22×C23—82no—
109C22×C23#38611yescharacteristic
110C22×C23#45312yes—
111C4×C23—21no—
112C4×C23—21no—
113C4×C23—22no—
114C4×C23—22no—
115C4×C23—44no—
116C4×C23—84no—
117C4×C23—81no—
118C2×#137—22no—
119#532C22×C2311yescharacteristic
120C2×C4⋊Q8C22×C2311yescharacteristic
121#497—22no—
122#494C4×C2312yescomplemented
123#492C4×C2312yescomplemented
124#432C22×C2312yes—
125C4×M16C22×C2311yescharacteristic
126#391—22no—
127#387—24no—
128#383C22×C2312yes—
129#333—22no—
130#327—24no—
131C2×C23—22no—
132C2×C23—161no—
133C2×C23#188711yescharacteristic
134C2×C23#190512yes—
135C2×#41—21no—
136C22×Q8D4×C2312yes—
137#137—44no—
138#137—44no—
139C2×Q16—24no—
140C2×SD16—24no—
141C2×M16—21no—
142C4⋊Q8—22no—
143C4⋊Q8C2×C4×C2311yescharacteristic
144C4⋊Q8C2×C4×C2312yes—
145C4⋊Q8D4×C2312yes—
146#127C2×C4×C2311yescharacteristic
147#124—21no—
148#121—22no—
149C4×D4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C42—22no—
153C2×C42C23×C2311yescharacteristic
154C4⋊C8—22no—
155C22⋊C8—22no—
156#97—21no—
157#97—22no—
158C4×C8C2×C4×C2312yes—
159#95—24no—
160#95—22no—
161#95—22no—
162#95D4×C2314yes—
163C23#942911yescharacteristic, direct product factor, Sylow
164#41—42no—
165#41—42no—
166C2×Q8—22no—
167C2×Q8—21no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8C22⋊C4×C2314yes—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—41no—
177C22×C4C2×D4×C2311yescharacteristic
178C22×C4C2×D4×C2312yes—
179Q16—48no—
180SD16—48no—
181M16—24no—
182C2×C8—22no—
183C2×C8—42no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—82no—
188C22⋊C4—42no—
189C42—82no—
190C42C22×C4×C2311yescharacteristic
191C42C2×D4×C2311yescharacteristic
192C42#41×C2312yes—
193C23—41no—
194C23C23×#12011yescharacteristic
195Q8—44no—
196Q8—42no—
197Q8—82no—
198D4—42no—
199D4—82no—
200D4—82no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—41no—
208C2×C4—44no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4C2×C22⋊C4×C2311yescharacteristic
212C2×C4C4×D4×C2312yes—
213C2×C4C23×#12112yes—
214C2×C4C23×#12311yescharacteristic
215C8—24no—
216C8—82no—
217C22—22no—
218C22—81no—
219C22—82no—
220C22C23×#38611yescharacteristic
221C22C23×#45312yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—84no—
228C4—81no—
229C2—22no—
230C2—161no—
231C2C23×#188711yescharacteristic
232C2C23×#190512yes—
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2381C2semidirect
3. C23×#1558C2semidirect
5. C23×#1288C2semidirect
10. C23×#494C4semidirect
11. C23×#492C4semidirect
76. #9429C23direct (N × H)
102. #2381C2×C23semidirect
103. #1558C2×C23semidirect
105. #1288C2×C23semidirect
122. #494C4×C23semidirect
123. #492C4×C23semidirect
163. C23#9429direct (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×#23812
C23×#15582
C23×#15142
C23×#12882
#942923