Finite Group Databasebeta

← C23×#7408 #19155034

Subgroups — C23×#7408 #19155034

Order 5888.

298 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 46 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×#2006C211yesmaximal, characteristic
2C23×#1393C211yesmaximal, characteristic, complemented
3C23×#1270C211yesmaximal, characteristic, complemented
4C2×C23×AGL(1,ℤ/8)—21no—
5C23×#537C411yescharacteristic, complemented
6C2×C23×#104—21no—
7C2×C23×#99—21no—
8C23×#390C2211yescharacteristic
9C23×#343C411yescharacteristic, complemented
10C23×#336—21no—
11C2×#41×C23—21no—
12C22×D4×C23D411yescharacteristic
13C23×AGL(1,ℤ/8)—42no—
14C23×AGL(1,ℤ/8)—41no—
15C23×AGL(1,ℤ/8)—41no—
16C2×SD16×C23—41no—
17C2×D8×C23—41no—
18C2×M16×C23—21no—
19C22×C8×C23—41no—
20C23×C22⋊Q8—41no—
21C23×#121—41no—
22C23×#121—41no—
23C2×C4⋊C4×C23C2×C411yescharacteristic
24C2×C22⋊C4×C23—41no—
25C2×C22⋊C4×C23—41no—
26C2×C42×C23D411yescharacteristic
27C23×#104—42no—
28C23×#99—22no—
29C23×#99—41no—
30C24×C23—21no—
31#41×C23—41no—
32#41×C23—41no—
33#41×C23—41no—
34#41×C23—41no—
35C2×Q8×C23—21no—
36C2×D4×C23—21no—
37C2×D4×C23—21no—
38C2×D4×C23—41no—
39C2×D4×C23—41no—
40C2×D4×C23—41no—
41C2×D4×C23—41no—
42C2×D4×C23D811yescharacteristic
43C2×D4×C23SD1611yescharacteristic
44C22×C4×C23—21no—
45C22×C4×C23—21no—
46C22×C4×C23—41no—
47C22×C4×C23—81no—
48C22×C4×C23C22⋊C411yescharacteristic
49SD16×C23—42no—
50SD16×C23—42no—
51D8×C23—42no—
52D8×C23—42no—
53M16×C23—22no—
54M16×C23—41no—
55C2×C8×C23—41no—
56C2×C8×C23—42no—
57C2×C8×C23—81no—
58C2×C8×C23—81no—
59C4⋊C4×C23—41no—
60C4⋊C4×C23—41no—
61C22⋊C4×C23—42no—
62C22⋊C4×C23—42no—
63C22⋊C4×C23—81no—
64C22⋊C4×C23—81no—
65C22⋊C4×C23—81no—
66C42×C23—41no—
67#7408C2311yesmaximal, characteristic, direct product factor, Sylow
68C23×C23—21no—
69C23×C23—21no—
70C23×C23—41no—
71C23×C23—41no—
72C23×C23—41no—
73C23×C23—81no—
74C23×C23—81no—
75C23×C23#9911yescharacteristic
76Q8×C23—41no—
77Q8×C23—41no—
78D4×C23—41no—
79D4×C23—41no—
80D4×C23—41no—
81D4×C23—41no—
82D4×C23—81no—
83D4×C23—81no—
84D4×C23—81no—
85D4×C23—161no—
86C2×C4×C23—21no—
87C2×C4×C23—21no—
88C2×C4×C23—21no—
89C2×C4×C23—21no—
90C2×C4×C23—41no—
91C2×C4×C23—41no—
92C2×C4×C23—41no—
93C2×C4×C23—41no—
94C2×C4×C23—81no—
95C2×C4×C23—81no—
96C2×C4×C23—81no—
97C2×C4×C23—82no—
98C2×C4×C23—161no—
99C2×C4×C23—161no—
100C2×C4×C23#10211yescharacteristic
101C2×C4×C23#10411yescharacteristic
102C8×C23—42no—
103C8×C23—82no—
104#2006C2×C2311yescharacteristic
105#1393C2×C2311yescharacteristic, complemented
106#1270C2×C2311yescharacteristic, complemented
107C22×C23—21no—
108C22×C23—21no—
109C22×C23—41no—
110C22×C23—41no—
111C22×C23—81no—
112C22×C23—81no—
113C22×C23—81no—
114C22×C23—81no—
115C22×C23—81no—
116C22×C23—81no—
117C22×C23—81no—
118C22×C23—161no—
119C22×C23#32711yescharacteristic
120C22×C23AΣL(1,𝔽16)11yescharacteristic
121C22×C23#35311yescharacteristic
122C4×C23—21no—
123C4×C23—21no—
124C4×C23—41no—
125C4×C23—41no—
126C4×C23—81no—
127C4×C23—81no—
128C4×C23—162no—
129C2×AGL(1,ℤ/8)—21no—
130#537C4×C2311yescharacteristic, complemented
131C2×#104—21no—
132C2×#99—21no—
133#390C22×C2311yescharacteristic
134#343C4×C2311yescharacteristic, complemented
135#336—21no—
136C2×C23—21no—
137C2×C23—21no—
138C2×C23—81no—
139C2×C23—81no—
140C2×C23—161no—
141C2×C23#134311yescharacteristic
142C2×C23#140211yescharacteristic
143C2×C23#140411yescharacteristic
144C2×#41—21no—
145C22×D4D4×C2311yescharacteristic
146AGL(1,ℤ/8)—42no—
147AGL(1,ℤ/8)—41no—
148AGL(1,ℤ/8)—41no—
149C2×SD16—41no—
150C2×D8—41no—
151C2×M16—21no—
152C22×C8—41no—
153C22⋊Q8—41no—
154#121—41no—
155#121—41no—
156C2×C4⋊C4C2×C4×C2311yescharacteristic
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C42D4×C2311yescharacteristic
160#104—42no—
161#99—22no—
162#99—41no—
163C23#740811yescharacteristic, direct product factor, Sylow
164C24—21no—
165#41—41no—
166#41—41no—
167#41—41no—
168#41—41no—
169C2×Q8—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4D8×C2311yescharacteristic
177C2×D4SD16×C2311yescharacteristic
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4C22⋊C4×C2311yescharacteristic
183SD16—42no—
184SD16—42no—
185D8—42no—
186D8—42no—
187M16—22no—
188M16—41no—
189C2×C8—41no—
190C2×C8—42no—
191C2×C8—81no—
192C2×C8—81no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—81no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—41no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—81no—
207C23—81no—
208C23C23×#9911yescharacteristic
209Q8—41no—
210Q8—41no—
211D4—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—81no—
218D4—161no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—82no—
231C2×C4—161no—
232C2×C4—161no—
233C2×C4C23×#10211yescharacteristic
234C2×C4C23×#10411yescharacteristic
235C8—42no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—161no—
249C22C23×#32711yescharacteristic
250C22C23×AΣL(1,𝔽16)11yescharacteristic
251C22C23×#35311yescharacteristic
252C4—21no—
253C4—21no—
254C4—41no—
255C4—41no—
256C4—81no—
257C4—81no—
258C4—162no—
259C2—21no—
260C2—21no—
261C2—81no—
262C2—81no—
263C2—161no—
264C2C23×#134311yescharacteristic
265C2C23×#140211yescharacteristic
266C2C23×#140411yescharacteristic
subgroupG/H
center, socleC22×C23AΣL(1,𝔽16)
commutatorC2×C4⋊C4C2×C4×C23
Frattini, cosocle#390C22×C23
prime radicalC23×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×C4×C23 ◁ C23×#390 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1393C2semidirect
3. C23×#1270C2semidirect
5. C23×#537C4semidirect
9. C23×#343C4semidirect
67. #7408C23direct (N × H)
105. #1393C2×C23semidirect
106. #1270C2×C23semidirect
130. #537C4×C23semidirect
134. #343C4×C23semidirect
163. C23#7408direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#20062
C23×#13932
C23×#12702
#740823