Finite Group Databasebeta

← C23×#9983 #19157609

Subgroups — C23×#9983 #19157609

Order 5888.

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

normal subgroups only

This group has 602 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×#3094C212yesmaximal, complemented
2C23×#2815C211yesmaximal, characteristic
3C23×#1820C211yesmaximal, characteristic, complemented
4C23×#1510C212yesmaximal, complemented
5C23×#1306C211yesmaximal, characteristic, complemented
6C2×C23×#121C2212yes—
7C2×C23×#117C2211yescharacteristic
8C23×#482—24no—
9C23×#481—22no—
10C23×#480—22no—
11C23×#428—22no—
12C23×#418—22no—
13C23×#407C2212yes—
14C23×#397—22no—
15C23×#392—22no—
16C23×#392—22no—
17C23×#381C2211yescharacteristic
18C23×#381C2211yescharacteristic
19C23×#342—22no—
20C23×#327—24no—
21C22×D4×C23—22no—
22C22×D4×C23—22no—
23C22×D4×C23D412yes—
24C23×C4×C23C2311yescharacteristic
25C2×M16×C23—22no—
26C23×#121—22no—
27C23×#121—22no—
28C23×#121—22no—
29C23×#121—22no—
30C23×#117—22no—
31C23×#117—22no—
32C2×C4⋊C4×C23D412yes—
33C2×C4⋊C4×C23D412yescomplemented
34C2×C22⋊C4×C23—21no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—24no—
38C2×C22⋊C4×C23D412yes—
39C2×C42×C23—21no—
40C23×#107—24no—
41C23×#106—24no—
42C23×#102—24no—
43C23×#102—24no—
44C23×C22⋊C8—22no—
45C23×C22⋊C8D414yescomplemented
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—22no—
49C24×C23—22no—
50C24×C23C2×D411yescharacteristic
51C2×D4×C23—22no—
52C2×D4×C23—22no—
53C2×D4×C23—44no—
54C2×D4×C23—42no—
55C2×D4×C23—42no—
56C2×D4×C23—84no—
57C2×D4×C23—84no—
58C2×D4×C23—82no—
59C22×C4×C23—22no—
60C22×C4×C23—22no—
61C22×C4×C23—22no—
62C22×C4×C23—42no—
63C22×C4×C23—41no—
64C22×C4×C23C2×D412yes—
65C22×C4×C23C2×D411yescharacteristic
66C22×C4×C23C2×D412yes—
67C22×C4×C23#4111yescharacteristic
68M16×C23—44no—
69C2×C8×C23—24no—
70C4⋊C4×C23—22no—
71C4⋊C4×C23—22no—
72C4⋊C4×C23—22no—
73C4⋊C4×C23—22no—
74C22⋊C4×C23—41no—
75C22⋊C4×C23—41no—
76C22⋊C4×C23—42no—
77C22⋊C4×C23—42no—
78C22⋊C4×C23—42no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—84no—
81C42×C23—42no—
82#9983C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—22no—
84C23×C23—42no—
85C23×C23—42no—
86C23×C23—42no—
87C23×C23—44no—
88C23×C23—42no—
89C23×C23—44no—
90C23×C23#12011yescharacteristic
91C23×C23#12111yescharacteristic
92C23×C23#12711yescharacteristic
93D4×C23—44no—
94D4×C23—82no—
95D4×C23—84no—
96D4×C23—84no—
97D4×C23—88no—
98C2×C4×C23—24no—
99C2×C4×C23—41no—
100C2×C4×C23—42no—
101C2×C4×C23—41no—
102C2×C4×C23—44no—
103C2×C4×C23—44no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—82no—
108C2×C4×C23—81no—
109C2×C4×C23—82no—
110C2×C4×C23#12012yes—
111C2×C4×C23#12112yes—
112C8×C23—44no—
113#3094C2×C2312yescomplemented
114#2815C2×C2311yescharacteristic
115#1820C2×C2311yescharacteristic, complemented
116#1510C2×C2312yescomplemented
117#1306C2×C2311yescharacteristic, complemented
118C22×C23—22no—
119C22×C23—22no—
120C22×C23—22no—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—82no—
124C22×C23—84no—
125C22×C23—84no—
126C22×C23—82no—
127C22×C23—82no—
128C22×C23—82no—
129C22×C23#39211yescharacteristic
130C22×C23#45812yes—
131C4×C23—22no—
132C4×C23—22no—
133C4×C23—82no—
134C4×C23—82no—
135C4×C23—82no—
136C4×C23—161no—
137C2×#121C22×C2312yes—
138C2×#117C22×C2311yescharacteristic
139#482—24no—
140#481—22no—
141#480—22no—
142#428—22no—
143#418—22no—
144#407C22×C2312yes—
145#397—22no—
146#392—22no—
147#392—22no—
148#381C22×C2311yescharacteristic
149#381C22×C2311yescharacteristic
150#342—22no—
151#327—24no—
152C2×C23—22no—
153C2×C23—41no—
154C2×C23—41no—
155C2×C23—162no—
156C2×C23#200812yes—
157C2×C23#202211yescharacteristic
158C22×D4—22no—
159C22×D4—22no—
160C22×D4D4×C2312yes—
161C23×C4C23×C2311yescharacteristic
162C2×M16—22no—
163#121—22no—
164#121—22no—
165#121—22no—
166#121—22no—
167#117—22no—
168#117—22no—
169C2×C4⋊C4D4×C2312yes—
170C2×C4⋊C4D4×C2312yescomplemented
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4D4×C2312yes—
176C2×C42—21no—
177#107—24no—
178#106—24no—
179#102—24no—
180#102—24no—
181C22⋊C8—22no—
182C22⋊C8D4×C2314yescomplemented
183#95—22no—
184#95—22no—
185#95—22no—
186C23#998311yescharacteristic, direct product factor, Sylow
187C24—22no—
188C24C2×D4×C2311yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—84no—
195C2×D4—84no—
196C2×D4—82no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4C2×D4×C2312yes—
203C22×C4C2×D4×C2311yescharacteristic
204C22×C4C2×D4×C2312yes—
205C22×C4#41×C2311yescharacteristic
206M16—44no—
207C2×C8—24no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—84no—
219C42—42no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—44no—
225C23—42no—
226C23—44no—
227C23C23×#12011yescharacteristic
228C23C23×#12111yescharacteristic
229C23C23×#12711yescharacteristic
230D4—44no—
231D4—82no—
232D4—84no—
233D4—84no—
234D4—88no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4C23×#12012yes—
248C2×C4C23×#12112yes—
249C8—44no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—84no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22C23×#39211yescharacteristic
262C22C23×#45812yes—
263C4—22no—
264C4—22no—
265C4—82no—
266C4—82no—
267C4—82no—
268C4—161no—
269C2—22no—
270C2—41no—
271C2—41no—
272C2—162no—
273C2C23×#200812yes—
274C2C23×#202211yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2815C2
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
1. C23×#3094C2semidirect
3. C23×#1820C2semidirect
4. C23×#1510C2semidirect
5. C23×#1306C2semidirect
33. C2×C4⋊C4×C23D4semidirect
45. C23×C22⋊C8D4semidirect
82. #9983C23direct (N × H)
113. #3094C2×C23semidirect
115. #1820C2×C23semidirect
116. #1510C2×C23semidirect
117. #1306C2×C23semidirect
170. C2×C4⋊C4D4×C23semidirect
182. C22⋊C8D4×C23semidirect
186. C23#9983direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30942
C23×#28152
C23×#18202
C23×#15102
C23×#13062
#998323