Finite Group Databasebeta

← C23×#7326 #19154952

Subgroups — C23×#7326 #19154952

Order 5888.

374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1463C212yesmaximal
2C23×#1445C211yesmaximal, characteristic
3C22×C4⋊C4×C23C2211yescharacteristic
4C23×#407—22no—
5C2×C23×#95—22no—
6C2×C23×#95C411yescharacteristic
7C2×C23×#95C411yescharacteristic
8C23×#324—22no—
9C23×#324C414yescomplemented
10C23×C4×C23—21no—
11C23×C4×C23—21no—
12C23×C4×C23C2×C411yescharacteristic
13C23×C4×C23D412yes—
14C2×M16×C23—42no—
15C2×C4⋊C4×C23—41no—
16C2×C4⋊C4×C23—41no—
17C2×C4⋊C4×C23C2×C412yes—
18C2×C4⋊C4×C23D411yescharacteristic
19C2×C4⋊C4×C23Q811yescharacteristic
20C23×C22⋊C8—24no—
21C23×C22⋊C8—42no—
22C23×#95—41no—
23C23×#95—41no—
24C23×#95—44no—
25C24×C23C22⋊C411yescharacteristic
26C22×C4×C23—22no—
27C22×C4×C23—21no—
28C22×C4×C23—21no—
29C22×C4×C23—22no—
30C22×C4×C23—22no—
31C22×C4×C23—22no—
32C22×C4×C23—42no—
33C22×C4×C23—42no—
34C22×C4×C23—42no—
35C22×C4×C23—42no—
36C22×C4×C23—42no—
37C22×C4×C23—41no—
38C22×C4×C23—41no—
39C22×C4×C23—42no—
40C22×C4×C23—42no—
41C22×C4×C23—41no—
42C22×C4×C23—41no—
43C22×C4×C23C4211yescharacteristic
44C22×C4×C23C22⋊C412yes—
45C22×C4×C23C4⋊C411yescharacteristic
46C22×C4×C23C4⋊C412yes—
47M16×C23—82no—
48M16×C23—82no—
49C2×C8×C23—44no—
50C4⋊C4×C23—42no—
51C4⋊C4×C23—41no—
52C4⋊C4×C23—41no—
53#7326C2311yesmaximal, characteristic, direct product factor, Sylow
54C23×C23—22no—
55C23×C23—21no—
56C23×C23—21no—
57C23×C23—21no—
58C23×C23—21no—
59C23×C23#9511yescharacteristic
60C23×C23#9911yescharacteristic
61C23×C23#9911yescharacteristic
62C2×C4×C23—22no—
63C2×C4×C23—21no—
64C2×C4×C23—21no—
65C2×C4×C23—22no—
66C2×C4×C23—22no—
67C2×C4×C23—22no—
68C2×C4×C23—42no—
69C2×C4×C23—42no—
70C2×C4×C23—42no—
71C2×C4×C23—42no—
72C2×C4×C23—42no—
73C2×C4×C23—42no—
74C2×C4×C23—44no—
75C2×C4×C23—44no—
76C2×C4×C23—42no—
77C2×C4×C23—42no—
78C2×C4×C23—82no—
79C2×C4×C23—82no—
80C2×C4×C23—82no—
81C2×C4×C23—81no—
82C2×C4×C23—81no—
83C2×C4×C23—81no—
84C2×C4×C23—81no—
85C8×C23—84no—
86#1463C2×C2312yes—
87#1445C2×C2311yescharacteristic
88C22×C23—22no—
89C22×C23—22no—
90C22×C23—22no—
91C22×C23—22no—
92C22×C23—21no—
93C22×C23—21no—
94C22×C23—21no—
95C22×C23—21no—
96C22×C23—21no—
97C22×C23—21no—
98C22×C23#34211yescharacteristic
99C22×C23#34412yes—
100C22×C23#35311yescharacteristic
101C22×C23#35411yescharacteristic
102C22×C23#35511yescharacteristic
103C22×C23#35611yescharacteristic
104C4×C23—41no—
105C4×C23—41no—
106C4×C23—42no—
107C4×C23—42no—
108C4×C23—82no—
109C4×C23—82no—
110C22×C4⋊C4C22×C2311yescharacteristic
111#407—22no—
112C2×#95—22no—
113C2×#95C4×C2311yescharacteristic
114C2×#95C4×C2311yescharacteristic
115#324—22no—
116#324C4×C2314yescomplemented
117C2×C23—21no—
118C2×C23—21no—
119C2×C23—22no—
120C2×C23—21no—
121C2×C23—21no—
122C2×C23#130611yescharacteristic
123C2×C23#139311yescharacteristic
124C2×C23#139411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C2×C4×C2311yescharacteristic
128C23×C4D4×C2312yes—
129C2×M16—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4C2×C4×C2312yes—
133C2×C4⋊C4D4×C2311yescharacteristic
134C2×C4⋊C4Q8×C2311yescharacteristic
135C22⋊C8—24no—
136C22⋊C8—42no—
137#95—41no—
138#95—41no—
139#95—44no—
140C23#732611yescharacteristic, direct product factor, Sylow
141C24C22⋊C4×C2311yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4C42×C2311yescharacteristic
160C22×C4C22⋊C4×C2312yes—
161C22×C4C4⋊C4×C2311yescharacteristic
162C22×C4C4⋊C4×C2312yes—
163M16—82no—
164M16—82no—
165C2×C8—44no—
166C4⋊C4—42no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23C23×#9511yescharacteristic
175C23C23×#9911yescharacteristic
176C23C23×#9911yescharacteristic
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C8—84no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22C23×#34211yescharacteristic
212C22C23×#34412yes—
213C22C23×#35311yescharacteristic
214C22C23×#35411yescharacteristic
215C22C23×#35511yescharacteristic
216C22C23×#35611yescharacteristic
217C4—42no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—82no—
223C2—21no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—21no—
228C2C23×#130611yescharacteristic
229C2C23×#139311yescharacteristic
230C2C23×#139411yescharacteristic
subgroupG/H
center, socleC22×C23#342
commutatorC22×C4C42×C23
Frattini, cosocleC22×C4⋊C4C22×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C4⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
9. C23×#324C4semidirect
53. #7326C23direct (N × H)
116. #324C4×C23semidirect
140. C23#7326direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#14632
C23×#14452
#732623