Finite Group Databasebeta

← C23×#10387 #19158013

Subgroups — C23×#10387 #19158013

Order 5888.

414 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2607C211yesmaximal, characteristic
2C23×#1826C212yesmaximal
3C23×#1820C211yesmaximal, characteristic
4C23×#1306C211yesmaximal, characteristic
5C23×#1284C212yesmaximal
6C2×C23×#117C2211yescharacteristic
7C23×#407C2212yes—
8C23×#399—21no—
9C23×#393—21no—
10C23×#391—22no—
11C23×#387—22no—
12C23×#381C2211yescharacteristic
13C23×#381C2211yescharacteristic
14C4×C22⋊C4×C23C2212yes—
15C23×#342—21no—
16C23×#342—21no—
17C23×#339—22no—
18C23×#339—22no—
19C22×Q8×C23—21no—
20C23×C4×C23C2311yescharacteristic
21C2×M16×C23—22no—
22C23×#117—22no—
23C23×#117—22no—
24C2×C4⋊C4×C23—22no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23Q812yescomplemented
27C2×C22⋊C4×C23—21no—
28C2×C22⋊C4×C23—22no—
29C2×C22⋊C4×C23—21no—
30C2×C22⋊C4×C23D411yescharacteristic
31C2×C22⋊C4×C23D411yescharacteristic
32C2×C42×C23—21no—
33C2×C42×C23D412yes—
34C2×C42×C23D412yes—
35C23×C22⋊C8—22no—
36C23×C22⋊C8—22no—
37C23×#95—21no—
38C23×#95—22no—
39C23×#95—21no—
40C23×#95—22no—
41C23×#95—22no—
42C23×#95—22no—
43C24×C23C2×D411yescharacteristic
44C2×Q8×C23—44no—
45C2×Q8×C23—81no—
46C22×C4×C23—22no—
47C22×C4×C23—22no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—22no—
51C22×C4×C23—41no—
52C22×C4×C23—41no—
53C22×C4×C23—41no—
54C22×C4×C23C2×D412yes—
55C22×C4×C23C2×Q811yescharacteristic
56C22×C4×C23#4112yes—
57C22×C4×C23#4111yescharacteristic
58M16×C23—42no—
59M16×C23—42no—
60C2×C8×C23—42no—
61C4⋊C4×C23—24no—
62C4⋊C4×C23—44no—
63C4⋊C4×C23—81no—
64C22⋊C4×C23—44no—
65C22⋊C4×C23—41no—
66C22⋊C4×C23—41no—
67C22⋊C4×C23—41no—
68C22⋊C4×C23—41no—
69C22⋊C4×C23—41no—
70C22⋊C4×C23—41no—
71C42×C23—42no—
72C42×C23—42no—
73C42×C23—42no—
74#10387C2311yesmaximal, characteristic, direct product factor, Sylow
75C23×C23—21no—
76C23×C23—21no—
77C23×C23—42no—
78C23×C23#12011yescharacteristic
79C23×C23C22⋊Q811yescharacteristic
80C23×C23#12411yescharacteristic
81Q8×C23—84no—
82C2×C4×C23—22no—
83C2×C4×C23—22no—
84C2×C4×C23—24no—
85C2×C4×C23—41no—
86C2×C4×C23—41no—
87C2×C4×C23—42no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—42no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—41no—
97C2×C4×C23—41no—
98C2×C4×C23—82no—
99C2×C4×C23—81no—
100C2×C4×C23—82no—
101C2×C4×C23C22⋊Q812yes—
102C2×C4×C23#12412yes—
103C8×C23—82no—
104#2607C2×C2311yescharacteristic
105#1826C2×C2312yes—
106#1820C2×C2311yescharacteristic
107#1306C2×C2311yescharacteristic
108#1284C2×C2312yes—
109C22×C23—22no—
110C22×C23—21no—
111C22×C23—21no—
112C22×C23—22no—
113C22×C23—22no—
114C22×C23—81no—
115C22×C23—81no—
116C22×C23#39311yescharacteristic
117C22×C23#45811yescharacteristic
118C22×C23#45811yescharacteristic
119C4×C23—22no—
120C4×C23—22no—
121C4×C23—44no—
122C4×C23—82no—
123C4×C23—81no—
124C4×C23—81no—
125C4×C23—82no—
126C4×C23—161no—
127C2×#117C22×C2311yescharacteristic
128#407C22×C2312yes—
129#399—21no—
130#393—21no—
131#391—22no—
132#387—22no—
133#381C22×C2311yescharacteristic
134#381C22×C2311yescharacteristic
135C4×C22⋊C4C22×C2312yes—
136#342—21no—
137#342—21no—
138#339—22no—
139#339—22no—
140C2×C23—21no—
141C2×C23—21no—
142C2×C23—41no—
143C2×C23—41no—
144C2×C23#203011yescharacteristic
145C2×C23#203112yes—
146C22×Q8—21no—
147C23×C4C23×C2311yescharacteristic
148C2×M16—22no—
149#117—22no—
150#117—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4Q8×C2312yescomplemented
154C2×C22⋊C4—21no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4D4×C2311yescharacteristic
158C2×C22⋊C4D4×C2311yescharacteristic
159C2×C42—21no—
160C2×C42D4×C2312yes—
161C2×C42D4×C2312yes—
162C22⋊C8—22no—
163C22⋊C8—22no—
164#95—21no—
165#95—22no—
166#95—21no—
167#95—22no—
168#95—22no—
169#95—22no—
170C23#1038711yescharacteristic, direct product factor, Sylow
171C24C2×D4×C2311yescharacteristic
172C2×Q8—44no—
173C2×Q8—81no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4C2×D4×C2312yes—
183C22×C4C2×Q8×C2311yescharacteristic
184C22×C4#41×C2312yes—
185C22×C4#41×C2311yescharacteristic
186M16—42no—
187M16—42no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—44no—
191C4⋊C4—81no—
192C22⋊C4—44no—
193C22⋊C4—41no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—41no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C42—42no—
200C42—42no—
201C42—42no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23C23×#12011yescharacteristic
206C23C23×C22⋊Q811yescharacteristic
207C23C23×#12411yescharacteristic
208Q8—84no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4C23×C22⋊Q812yes—
229C2×C4C23×#12412yes—
230C8—82no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22C23×#39311yescharacteristic
239C22C23×#45811yescharacteristic
240C22C23×#45811yescharacteristic
241C4—22no—
242C4—22no—
243C4—44no—
244C4—81no—
245C4—82no—
246C4—81no—
247C4—82no—
248C4—161no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2C23×#203011yescharacteristic
254C2C23×#203112yes—
subgroupG/H
center, socleC22×C23#393
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
26. C2×C4⋊C4×C23Q8semidirect
74. #10387C23direct (N × H)
153. C2×C4⋊C4Q8×C23semidirect
170. C23#10387direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26072
C23×#18262
C23×#18202
C23×#13062
C23×#12842
#1038723