Finite Group Databasebeta

← C23×#9014 #19156640

Subgroups — C23×#9014 #19156640

Order 5888.

756 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C23×#395C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic
3C23×#1787C212yesmaximal
4C2×C23×#324C211yesmaximal, characteristic
5C23×#1294C212yesmaximal
6C23×Q8×C23C2211yescharacteristic
7C22×C4⋊C4×C23—21no—
8C22×C4⋊C4×C23C2211yescharacteristic
9C2×C23×#103—24no—
10C2×C23×C22⋊C8C2212yes—
11C23×#395—22no—
12C23×#395C414yescomplemented
13C23×#387—24no—
14C23×#381—21no—
15C23×#380C2212yes—
16C4×C22⋊C4×C23—21no—
17C2×C23×#95—21no—
18C2×C23×#95C2211yescharacteristic
19C23×#324—22no—
20C22×Q8×C23—21no—
21C22×Q8×C23—22no—
22C22×Q8×C23—22no—
23C22×Q8×C23—44no—
24C22×Q8×C23C2×C412yes—
25C23×C4×C23—21no—
26C23×C4×C23—21no—
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C23×C4×C23C2311yescharacteristic
30C22×C8×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—24no—
34C2×C4⋊C4×C23—44no—
35C2×C4⋊C4×C23C2×C412yes—
36C2×C4⋊C4×C23D412yes—
37C2×C4⋊C4×C23D414yes—
38C2×C22⋊C4×C23—22no—
39C2×C42×C23—22no—
40C23×#103—48no—
41C23×C22⋊C8—22no—
42C23×C22⋊C8—22no—
43C23×#95—22no—
44C23×#95—24no—
45C23×#95—24no—
46C23×#95—21no—
47C23×#95—22no—
48C23×#95C2×C412yes—
49C24×C23C2×D411yescharacteristic
50C2×Q8×C23—28no—
51C2×Q8×C23—24no—
52C2×Q8×C23—44no—
53C2×Q8×C23—44no—
54C2×Q8×C23—44no—
55C2×Q8×C23—48no—
56C2×Q8×C23—42no—
57C22×C4×C23—24no—
58C22×C4×C23—24no—
59C22×C4×C23—22no—
60C22×C4×C23—22no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—22no—
65C22×C4×C23—22no—
66C22×C4×C23—21no—
67C22×C4×C23—24no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23—41no—
72C22×C4×C23—41no—
73C22×C4×C23—42no—
74C22×C4×C23—42no—
75C22×C4×C23C22⋊C412yes—
76C22×C4×C23C22⋊C412yes—
77C22×C4×C23C22×C411yescharacteristic
78C22×C4×C23C2×D412yes—
79C22×C4×C23C2×D411yescharacteristic
80C22×C4×C23#4112yes—
81C2×C8×C23—44no—
82C2×C8×C23—42no—
83C4⋊C4×C23—24no—
84C4⋊C4×C23—24no—
85C4⋊C4×C23—44no—
86C4⋊C4×C23—48no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—42no—
89C42×C23—82no—
90#9014C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—22no—
92C23×C23—22no—
93C23×C23C2×C22⋊C411yescharacteristic
94C23×C23#12011yescharacteristic
95C23×C23#12311yescharacteristic
96C23×C23#13512yes—
97C23×C23#13712yes—
98Q8×C23—28no—
99Q8×C23—48no—
100Q8×C23—44no—
101C2×C4×C23—24no—
102C2×C4×C23—24no—
103C2×C4×C23—24no—
104C2×C4×C23—22no—
105C2×C4×C23—22no—
106C2×C4×C23—22no—
107C2×C4×C23—22no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—42no—
111C2×C4×C23—42no—
112C2×C4×C23—48no—
113C2×C4×C23—42no—
114C2×C4×C23—44no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—44no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—44no—
123C2×C4×C23—44no—
124C2×C4×C23—44no—
125C2×C4×C23—81no—
126C2×C4×C23C4×D412yes—
127C2×C4×C23#12112yes—
128C8×C23—82no—
129C2×#395C2×C2311yescharacteristic
130#2313C2×C2311yescharacteristic
131#1787C2×C2312yes—
132C2×#324C2×C2311yescharacteristic
133#1294C2×C2312yes—
134C22×C23—24no—
135C22×C23—24no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23#38611yescharacteristic
141C22×C23#41611yescharacteristic
142C22×C23#41911yescharacteristic
143C22×C23#45212yes—
144C22×C23#45612yes—
145C22×C23B(4,𝔽2)12yes—
146C22×C23#48512yes—
147C4×C23—22no—
148C4×C23—22no—
149C4×C23—44no—
150C4×C23—42no—
151C4×C23—44no—
152C4×C23—42no—
153C4×C23—84no—
154C23×Q8C22×C2311yescharacteristic
155C22×C4⋊C4—21no—
156C22×C4⋊C4C22×C2311yescharacteristic
157C2×#103—24no—
158C2×C22⋊C8C22×C2312yes—
159#395—22no—
160#395C4×C2314yescomplemented
161#387—24no—
162#381—21no—
163#380C22×C2312yes—
164C4×C22⋊C4—21no—
165C2×#95—21no—
166C2×#95C22×C2311yescharacteristic
167#324—22no—
168C2×C23—21no—
169C2×C23—21no—
170C2×C23—22no—
171C2×C23#160314yes—
172C2×C23#187311yescharacteristic
173C2×C23#188611yescharacteristic
174C2×C23#188911yescharacteristic
175C22×Q8—21no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—44no—
179C22×Q8C2×C4×C2312yes—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4D4×C2311yescharacteristic
183C23×C4D4×C2311yescharacteristic
184C23×C4C23×C2311yescharacteristic
185C22×C8—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C2×C4×C2312yes—
191C2×C4⋊C4D4×C2312yes—
192C2×C4⋊C4D4×C2314yes—
193C2×C22⋊C4—22no—
194C2×C42—22no—
195#103—48no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—21no—
202#95—22no—
203#95C2×C4×C2312yes—
204C23#901411yescharacteristic, direct product factor, Sylow
205C24C2×D4×C2311yescharacteristic
206C2×Q8—28no—
207C2×Q8—24no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—48no—
212C2×Q8—42no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊C4×C2312yes—
232C22×C4C22⋊C4×C2312yes—
233C22×C4C22×C4×C2311yescharacteristic
234C22×C4C2×D4×C2312yes—
235C22×C4C2×D4×C2311yescharacteristic
236C22×C4#41×C2312yes—
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—48no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C42—82no—
246C23—22no—
247C23—22no—
248C23C2×C22⋊C4×C2311yescharacteristic
249C23C23×#12011yescharacteristic
250C23C23×#12311yescharacteristic
251C23C23×#13512yes—
252C23C23×#13712yes—
253Q8—28no—
254Q8—48no—
255Q8—44no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4C4×D4×C2312yes—
282C2×C4C23×#12112yes—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22C23×#38611yescharacteristic
291C22C23×#41611yescharacteristic
292C22C23×#41911yescharacteristic
293C22C23×#45212yes—
294C22C23×#45612yes—
295C22C23×B(4,𝔽2)12yes—
296C22C23×#48512yes—
297C4—22no—
298C4—22no—
299C4—44no—
300C4—44no—
301C4—42no—
302C4—42no—
303C4—84no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2C23×#160314yes—
308C2C23×#187311yescharacteristic
309C2C23×#188611yescharacteristic
310C2C23×#188911yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
12. C23×#395C4semidirect
90. #9014C23direct (N × H)
160. #395C4×C23semidirect
204. C23#9014direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3952
C23×#23132
C23×#17872
C2×C23×#3242
C23×#12942
#901423