Finite Group Databasebeta

← C23×#13465 #19161091

Subgroups — C23×#13465 #19161091

Order 5888.

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

normal subgroups only

This group has 384 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×#2809C211yesmaximal, characteristic
2C23×#1857C212yesmaximal
3C23×#1857C211yesmaximal, characteristic
4C23×#1394C212yesmaximal
5C23×#1394C211yesmaximal, characteristic
6C23×#519C2211yescharacteristic
7C23×#428—22no—
8C23×#428—21no—
9C2×C23×#101C2212yes—
10C2×C23×#101C2211yescharacteristic
11C23×#400—21no—
12C23×#398—22no—
13C23×#391C2212yes—
14C23×#391C2211yescharacteristic
15C23×#389—21no—
16C23×#387—22no—
17C23×#339—22no—
18C23×#339—21no—
19C22×Q8×C23C2311yescharacteristic
20C2×M16×C23—22no—
21C2×M16×C23—21no—
22C4×Q8×C23—41no—
23C4×Q8×C23—41no—
24C23×#117—22no—
25C23×#117—21no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23D412yes—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C42×C23D412yes—
33C2×C42×C23D411yescharacteristic
34C23×#107—42no—
35C23×#107—41no—
36C23×#106—42no—
37C23×#106—41no—
38C23×#101—22no—
39C23×#101—21no—
40C23×#101Q814yes—
41C23×#101Q812yes—
42C23×#95—21no—
43C23×#95—22no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95—22no—
47C2×Q8×C23—41no—
48C2×Q8×C23—41no—
49C2×Q8×C23C2×Q812yes—
50C2×Q8×C23C2×Q811yescharacteristic
51C2×Q8×C23#4111yescharacteristic
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—21no—
56C22×C4×C23C2×D412yes—
57C22×C4×C23C2×D411yescharacteristic
58M16×C23—24no—
59M16×C23—22no—
60M16×C23—41no—
61M16×C23—42no—
62C2×C8×C23—42no—
63C2×C8×C23—41no—
64C4⋊C4×C23—22no—
65C4⋊C4×C23—21no—
66C4⋊C4×C23—21no—
67C4⋊C4×C23—22no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—41no—
70C4⋊C4×C23—44no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—81no—
73C4⋊C4×C23—81no—
74C22⋊C4×C23—41no—
75C42×C23—41no—
76C42×C23—42no—
77C42×C23—42no—
78C42×C23—41no—
79#13465C2311yesmaximal, characteristic, direct product factor, Sylow
80C23×C23#12011yescharacteristic
81Q8×C23—42no—
82Q8×C23—41no—
83Q8×C23—41no—
84C2×C4×C23—22no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—41no—
90C2×C4×C23—41no—
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—41no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23C22⋊Q812yes—
102C2×C4×C23C22⋊Q811yescharacteristic
103C2×C4×C23C4⋊Q812yes—
104C2×C4×C23C4⋊Q811yescharacteristic
105C8×C23—44no—
106C8×C23—42no—
107#2809C2×C2311yescharacteristic
108#1857C2×C2312yes—
109#1857C2×C2311yescharacteristic
110#1394C2×C2311yescharacteristic
111#1394C2×C2312yes—
112C22×C23—21no—
113C22×C23—21no—
114C22×C23#39511yescharacteristic
115C22×C23B(4,𝔽2)11yescharacteristic
116C22×C23#45811yescharacteristic
117C4×C23—22no—
118C4×C23—21no—
119C4×C23—22no—
120C4×C23—21no—
121C4×C23—42no—
122C4×C23—82no—
123C4×C23—81no—
124C4×C23—81no—
125C4×C23—81no—
126C4×C23—82no—
127#519C22×C2311yescharacteristic
128#428—22no—
129#428—21no—
130C2×#101C22×C2312yes—
131C2×#101C22×C2311yescharacteristic
132#400—21no—
133#398—22no—
134#391C22×C2311yescharacteristic
135#391C22×C2312yes—
136#389—21no—
137#387—22no—
138#339—22no—
139#339—21no—
140C2×C23—21no—
141C2×C23—21no—
142C2×C23#206211yescharacteristic
143C2×C23#220311yescharacteristic
144C2×C23#220411yescharacteristic
145C22×Q8C23×C2311yescharacteristic
146C2×M16—22no—
147C2×M16—21no—
148C4×Q8—41no—
149C4×Q8—41no—
150#117—22no—
151#117—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4D4×C2312yes—
157C2×C4⋊C4D4×C2311yescharacteristic
158C2×C42D4×C2312yes—
159C2×C42D4×C2311yescharacteristic
160#107—42no—
161#107—41no—
162#106—42no—
163#106—41no—
164#101—22no—
165#101—21no—
166#101Q8×C2314yes—
167#101Q8×C2312yes—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173C23#1346511yescharacteristic, direct product factor, Sylow
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C2×Q8×C2312yes—
177C2×Q8C2×Q8×C2311yescharacteristic
178C2×Q8#41×C2311yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×D4×C2312yes—
184C22×C4C2×D4×C2311yescharacteristic
185M16—24no—
186M16—22no—
187M16—41no—
188M16—42no—
189C2×C8—42no—
190C2×C8—41no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—41no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C22⋊C4—41no—
202C42—41no—
203C42—42no—
204C42—42no—
205C42—41no—
206C23C23×#12011yescharacteristic
207Q8—42no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4C23×C22⋊Q812yes—
228C2×C4C23×C22⋊Q811yescharacteristic
229C2×C4C23×C4⋊Q812yes—
230C2×C4C23×C4⋊Q811yescharacteristic
231C8—44no—
232C8—42no—
233C22—21no—
234C22—21no—
235C22C23×#39511yescharacteristic
236C22C23×B(4,𝔽2)11yescharacteristic
237C22C23×#45811yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4—81no—
246C4—82no—
247C4—81no—
248C2—21no—
249C2—21no—
250C2C23×#206211yescharacteristic
251C2C23×#220311yescharacteristic
252C2C23×#220411yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C23
prime radicalC23×C23#120
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
79. #13465C23direct (N × H)
173. C23#13465direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28092
C23×#18572
C23×#13942
#1346523