Finite Group Databasebeta

← C23×#10213 #19157839

Subgroups — C23×#10213 #19157839

Order 5888.

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

normal subgroups only

This group has 498 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×#3253C211yesmaximal, characteristic
2C23×#3216C212yesmaximal, complemented
3C23×#1995C211yesmaximal, characteristic, complemented
4C23×#1505C212yesmaximal, complemented
5C23×#1307C211yesmaximal, characteristic, complemented
6C23×#533C2211yescharacteristic
7C23×#532C2212yes—
8C23×#497—22no—
9C23×#489—22no—
10C23×#476—22no—
11C23×#469—22no—
12C23×#462—22no—
13C23×#460—22no—
14C23×#459—22no—
15C23×#452—24no—
16C23×#432C2212yes—
17C23×#423C2211yescharacteristic
18C23×#376C2211yescharacteristic
19C23×#339—22no—
20C23×#327—24no—
21C2×#41×C23—22no—
22C2×Q16×C23—24no—
23C2×SD16×C23—24no—
24C2×D8×C23—22no—
25C2×M16×C23D412yes—
26C23×C4⋊Q8D412yes—
27C23×C4⋊Q8D412yes—
28C23×#127D412yes—
29C23×#125—21no—
30C23×#124—22no—
31C23×C22⋊Q8—22no—
32C23×#121—22no—
33C23×#121—22no—
34C4×Q8×C23—22no—
35C4×D4×C23—22no—
36C4×D4×C23—22no—
37C23×#117—21no—
38C2×C42×C23—21no—
39C2×C42×C23C2311yescharacteristic
40C23×#106—22no—
41C23×C4⋊C8—22no—
42C23×C4⋊C8—22no—
43C23×#103—24no—
44C23×#102—24no—
45C23×#102—24no—
46C23×C22⋊C8D414yes—
47C23×#97—22no—
48C23×#95—22no—
49#41×C23—84no—
50C2×Q8×C23—22no—
51C2×Q8×C23—22no—
52C2×D4×C23—22no—
53C2×D4×C23—22no—
54C2×D4×C23—22no—
55C2×D4×C23—42no—
56C22×C4×C23—41no—
57C22×C4×C23—42no—
58C22×C4×C23C2×D411yescharacteristic
59C22×C4×C23C2×D412yes—
60Q16×C23—84no—
61SD16×C23—84no—
62D8×C23—82no—
63M16×C23—42no—
64C2×C8×C23—22no—
65C2×C8×C23—24no—
66C4⋊C4×C23—22no—
67C4⋊C4×C23—22no—
68C4⋊C4×C23—22no—
69C4⋊C4×C23—22no—
70C4⋊C4×C23—42no—
71C22⋊C4×C23—41no—
72C22⋊C4×C23—42no—
73C22⋊C4×C23—42no—
74C42×C23—22no—
75C42×C23—41no—
76C42×C23—41no—
77C42×C23C2×D412yes—
78C42×C23C2×D411yescharacteristic
79C42×C23#4111yescharacteristic
80#10213C2311yesmaximal, characteristic, direct product factor, Sylow
81C23×C23—42no—
82C23×C23#12711yescharacteristic
83Q8×C23—42no—
84Q8×C23—42no—
85Q8×C23—44no—
86D4×C23—44no—
87D4×C23—42no—
88D4×C23—42no—
89D4×C23—82no—
90D4×C23—84no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—84no—
98C2×C4×C23—82no—
99C2×C4×C23#12012yes—
100C2×C4×C23#12011yescharacteristic
101C2×C4×C23#12111yescharacteristic
102C2×C4×C23#12112yes—
103C8×C23—44no—
104C8×C23—42no—
105#3253C2×C2311yescharacteristic
106#3216C2×C2312yescomplemented
107#1995C2×C2311yescharacteristic, complemented
108#1505C2×C2312yescomplemented
109#1307C2×C2311yescharacteristic, complemented
110C22×C23—21no—
111C22×C23—22no—
112C22×C23—82no—
113C22×C23—82no—
114C22×C23—82no—
115C22×C23#39211yescharacteristic
116C4×C23—21no—
117C4×C23—22no—
118C4×C23—22no—
119C4×C23—82no—
120C4×C23—82no—
121C4×C23—82no—
122C4×C23#45511yescharacteristic
123C4×C23#45611yescharacteristic
124#533C22×C2311yescharacteristic
125#532C22×C2312yes—
126#497—22no—
127#489—22no—
128#476—22no—
129#469—22no—
130#462—22no—
131#460—22no—
132#459—22no—
133#452—24no—
134#432C22×C2312yes—
135#423C22×C2311yescharacteristic
136#376C22×C2311yescharacteristic
137#339—22no—
138#327—24no—
139C2×C23—41no—
140C2×C23—162no—
141C2×C23#200711yescharacteristic
142C2×C23#200812yes—
143C2×#41—22no—
144C2×Q16—24no—
145C2×SD16—24no—
146C2×D8—22no—
147C2×M16D4×C2312yes—
148C4⋊Q8D4×C2312yes—
149C4⋊Q8D4×C2312yes—
150#127D4×C2312yes—
151#125—21no—
152#124—22no—
153C22⋊Q8—22no—
154#121—22no—
155#121—22no—
156C4×Q8—22no—
157C4×D4—22no—
158C4×D4—22no—
159#117—21no—
160C2×C42—21no—
161C2×C42C23×C2311yescharacteristic
162#106—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165#103—24no—
166#102—24no—
167#102—24no—
168C22⋊C8D4×C2314yes—
169#97—22no—
170#95—22no—
171C23#1021311yescharacteristic, direct product factor, Sylow
172#41—84no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4C2×D4×C2311yescharacteristic
182C22×C4C2×D4×C2312yes—
183Q16—84no—
184SD16—84no—
185D8—82no—
186M16—42no—
187C2×C8—24no—
188C2×C8—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C42—22no—
198C42—41no—
199C42—41no—
200C42C2×D4×C2311yescharacteristic
201C42C2×D4×C2312yes—
202C42#41×C2311yescharacteristic
203C23—42no—
204C23C23×#12711yescharacteristic
205Q8—42no—
206Q8—42no—
207Q8—44no—
208D4—44no—
209D4—42no—
210D4—42no—
211D4—82no—
212D4—84no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4C23×#12012yes—
222C2×C4C23×#12011yescharacteristic
223C2×C4C23×#12111yescharacteristic
224C2×C4C23×#12112yes—
225C8—44no—
226C8—42no—
227C22—21no—
228C22—22no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22C23×#39211yescharacteristic
233C4—21no—
234C4—22no—
235C4—22no—
236C4—82no—
237C4—82no—
238C4—82no—
239C4C23×#45511yescharacteristic
240C4C23×#45611yescharacteristic
241C2—41no—
242C2—162no—
243C2C23×#200711yescharacteristic
244C2C23×#200812yes—
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3253C2
LCSG ▷ C2×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×#3216C2semidirect
3. C23×#1995C2semidirect
4. C23×#1505C2semidirect
5. C23×#1307C2semidirect
80. #10213C23direct (N × H)
106. #3216C2×C23semidirect
107. #1995C2×C23semidirect
108. #1505C2×C23semidirect
109. #1307C2×C23semidirect
171. C23#10213direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32532
C23×#32162
C23×#19952
C23×#15052
C23×#13072
#1021323