Finite Group Databasebeta

← C23×#9401 #19157027

Subgroups — C23×#9401 #19157027

Order 5888.

534 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2975C211yesmaximal, characteristic
2C23×#2547C211yesmaximal, characteristic
3C23×#1800C211yesmaximal, characteristic
4C23×#1546C212yesmaximal, complemented
5C23×#1303C212yesmaximal, complemented
6C2×C23×C4⋊Q8C2211yescharacteristic
7C2×C23×#124—21no—
8C2×C4×D4×C23C2211yescharacteristic
9C23×#436—24no—
10C23×#431C2212yes—
11C23×#423C2212yes—
12C2×C23×#100—21no—
13C23×#407—22no—
14C23×#397C414yes—
15C23×#391—21no—
16C23×#380—22no—
17C4×C4⋊C4×C23C2211yescharacteristic
18C23×#343—21no—
19C23×#332—24no—
20C22×Q8×C23—21no—
21C22×D4×C23D411yescharacteristic
22C23×C4×C23D412yes—
23C2×M16×C23—24no—
24C2×M16×C23—22no—
25C23×C4⋊Q8—22no—
26C23×C4⋊Q8—22no—
27C23×#124—44no—
28C4×D4×C23—22no—
29C4×D4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23C2×C412yes—
33C2×C4⋊C4×C23D411yescharacteristic
34C2×C22⋊C4×C23—24no—
35C2×C22⋊C4×C23C2×C412yes—
36C2×C42×C23—21no—
37C2×C42×C23C2311yescharacteristic
38C23×C4⋊C8—22no—
39C23×C4⋊C8—22no—
40C23×C4⋊C8D414yes—
41C23×#100—42no—
42C23×C22⋊C8—24no—
43C23×C22⋊C8—22no—
44C4×C8×C23—22no—
45C23×#95—24no—
46C23×#95C2×C412yes—
47C24×C23C22⋊C412yes—
48C2×Q8×C23—42no—
49C2×Q8×C23—42no—
50C2×Q8×C23—41no—
51C2×Q8×C23—41no—
52C2×D4×C23—22no—
53C2×D4×C23—21no—
54C2×D4×C23—22no—
55C2×D4×C23—21no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—22no—
59C22×C4×C23—22no—
60C22×C4×C23—44no—
61C22×C4×C23C22⋊C412yes—
62C22×C4×C23C22×C411yescharacteristic
63C22×C4×C23C2×D411yescharacteristic
64C22×C4×C23C2×D411yescharacteristic
65M16×C23—42no—
66M16×C23—42no—
67M16×C23—84no—
68C2×C8×C23—24no—
69C2×C8×C23—42no—
70C4⋊C4×C23—22no—
71C4⋊C4×C23—24no—
72C4⋊C4×C23—42no—
73C4⋊C4×C23—41no—
74C4⋊C4×C23—41no—
75C22⋊C4×C23—24no—
76C22⋊C4×C23—48no—
77C42×C23—42no—
78C42×C23C2×D412yes—
79C42×C23#4112yes—
80#9401C2311yesmaximal, characteristic, direct product factor, Sylow
81C23×C23—22no—
82C23×C23—22no—
83C23×C23—22no—
84C23×C23—42no—
85C23×C23—42no—
86C23×C23C2×C22⋊C411yescharacteristic
87Q8×C23—82no—
88Q8×C23—82no—
89D4×C23—42no—
90D4×C23—42no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—41no—
94C2×C4×C23—44no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—82no—
103C2×C4×C23C4×D412yes—
104C2×C4×C23#12011yescharacteristic
105C2×C4×C23#12112yes—
106C2×C4×C23#12311yescharacteristic
107C2×C4×C23AGL(1,ℤ/8)12yes—
108C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C23#13711yescharacteristic
110C8×C23—44no—
111C8×C23—82no—
112#2975C2×C2311yescharacteristic
113#2547C2×C2311yescharacteristic
114#1800C2×C2311yescharacteristic
115#1546C2×C2312yescomplemented
116#1303C2×C2312yescomplemented
117C22×C23—21no—
118C22×C23—21no—
119C22×C23—44no—
120C22×C23—42no—
121C22×C23—42no—
122C22×C23—42no—
123C22×C23—82no—
124C22×C23#38611yescharacteristic
125C22×C23#41711yescharacteristic
126C22×C23#41811yescharacteristic
127C4×C23—21no—
128C4×C23—22no—
129C4×C23—42no—
130C4×C23—82no—
131C4×C23—82no—
132C4×C23#45512yes—
133C4×C23#46812yes—
134C4×C23#46912yes—
135C2×C4⋊Q8C22×C2311yescharacteristic
136C2×#124—21no—
137C2×C4×D4C22×C2311yescharacteristic
138#436—24no—
139#431C22×C2312yes—
140#423C22×C2312yes—
141C2×#100—21no—
142#407—22no—
143#397C4×C2314yes—
144#391—21no—
145#380—22no—
146C4×C4⋊C4C22×C2311yescharacteristic
147#343—21no—
148#332—24no—
149C2×C23—21no—
150C2×C23—21no—
151C2×C23—82no—
152C2×C23#188511yescharacteristic
153C2×C23#189511yescharacteristic
154C2×C23#191511yescharacteristic
155C22×Q8—21no—
156C22×D4D4×C2311yescharacteristic
157C23×C4D4×C2312yes—
158C2×M16—24no—
159C2×M16—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162#124—44no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C2×C4×C2312yes—
168C2×C4⋊C4D4×C2311yescharacteristic
169C2×C22⋊C4—24no—
170C2×C22⋊C4C2×C4×C2312yes—
171C2×C42—21no—
172C2×C42C23×C2311yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8D4×C2314yes—
176#100—42no—
177C22⋊C8—24no—
178C22⋊C8—22no—
179C4×C8—22no—
180#95—24no—
181#95C2×C4×C2312yes—
182C23#940111yescharacteristic, direct product factor, Sylow
183C24C22⋊C4×C2312yes—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—44no—
197C22×C4C22⋊C4×C2312yes—
198C22×C4C22×C4×C2311yescharacteristic
199C22×C4C2×D4×C2311yescharacteristic
200C22×C4C2×D4×C2311yescharacteristic
201M16—42no—
202M16—42no—
203M16—84no—
204C2×C8—24no—
205C2×C8—42no—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C22⋊C4—24no—
212C22⋊C4—48no—
213C42—42no—
214C42C2×D4×C2312yes—
215C42#41×C2312yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×C22⋊C4×C2311yescharacteristic
222Q8—82no—
223Q8—82no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4C4×D4×C2312yes—
239C2×C4C23×#12011yescharacteristic
240C2×C4C23×#12112yes—
241C2×C4C23×#12311yescharacteristic
242C2×C4C23×AGL(1,ℤ/8)12yes—
243C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
244C2×C4C23×#13711yescharacteristic
245C8—44no—
246C8—82no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22C23×#38611yescharacteristic
255C22C23×#41711yescharacteristic
256C22C23×#41811yescharacteristic
257C4—21no—
258C4—22no—
259C4—42no—
260C4—82no—
261C4—82no—
262C4C23×#45512yes—
263C4C23×#46812yes—
264C4C23×#46912yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2C23×#188511yescharacteristic
269C2C23×#189511yescharacteristic
270C2C23×#191511yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1546C2semidirect
5. C23×#1303C2semidirect
80. #9401C23direct (N × H)
115. #1546C2×C23semidirect
116. #1303C2×C23semidirect
182. C23#9401direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29752
C23×#25472
C23×#18002
C23×#15462
C23×#13032
#940123