Finite Group Databasebeta

← C23×#9024 #19156650

Subgroups — C23×#9024 #19156650

Order 5888.

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

normal subgroups only

This group has 740 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×#399C211yesmaximal, characteristic
2C23×#2312C211yesmaximal, characteristic
3C23×#1824C212yesmaximal, complemented
4C2×C23×#323C211yesmaximal, characteristic
5C23×#1294C212yesmaximal, complemented
6C22×C4⋊C4×C23—21no—
7C22×C4⋊C4×C23C2211yescharacteristic
8C22×C22⋊C4×C23—21no—
9C22×C22⋊C4×C23C2211yescharacteristic
10C2×C23×C22⋊C8C2212yes—
11C23×#399—22no—
12C23×#399C414yescomplemented
13C23×#388—24no—
14C23×#380C2212yes—
15C23×#379—21no—
16C4×C22⋊C4×C23—21no—
17C2×C23×#95C2211yescharacteristic
18C23×#340—24no—
19C23×#323—22no—
20C25×C23D411yescharacteristic
21C23×C4×C23—21no—
22C23×C4×C23—21no—
23C23×C4×C23D411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C22×C8×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—24no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23—44no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23D414yes—
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—24no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—44no—
40C2×C22⋊C4×C23C2×C412yes—
41C2×C22⋊C4×C23D412yes—
42C2×C42×C23—22no—
43C23×C22⋊C8—22no—
44C23×C22⋊C8—22no—
45C23×#95—22no—
46C23×#95—21no—
47C23×#95—22no—
48C23×#95C2×C412yes—
49C24×C23—21no—
50C24×C23—22no—
51C24×C23—21no—
52C24×C23—22no—
53C24×C23—42no—
54C24×C23—42no—
55C24×C23C22⋊C412yes—
56C24×C23C2×D411yescharacteristic
57C22×C4×C23—24no—
58C22×C4×C23—22no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—24no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23—42no—
69C22×C4×C23—41no—
70C22×C4×C23—41no—
71C22×C4×C23C22⋊C412yes—
72C22×C4×C23C22×C411yescharacteristic
73C22×C4×C23C2×D412yes—
74C22×C4×C23C2×D411yescharacteristic
75C22×C4×C23#4112yes—
76C2×C8×C23—44no—
77C2×C8×C23—42no—
78C4⋊C4×C23—24no—
79C4⋊C4×C23—44no—
80C4⋊C4×C23—48no—
81C4⋊C4×C23—42no—
82C22⋊C4×C23—24no—
83C22⋊C4×C23—24no—
84C22⋊C4×C23—48no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—42no—
87C22⋊C4×C23—82no—
88C22⋊C4×C23—82no—
89C42×C23—82no—
90#9024C2311yesmaximal, characteristic, direct product factor, Sylow
91C23×C23—22no—
92C23×C23—22no—
93C23×C23—22no—
94C23×C23—22no—
95C23×C23—22no—
96C23×C23—42no—
97C23×C23—42no—
98C23×C23—48no—
99C23×C23—42no—
100C23×C23—42no—
101C23×C23—42no—
102C23×C23—42no—
103C23×C23—81no—
104C23×C23—81no—
105C23×C23—84no—
106C23×C23C2×C22⋊C411yescharacteristic
107C23×C23#12011yescharacteristic
108C23×C23#12311yescharacteristic
109C23×C23#13512yes—
110C23×C23#13712yes—
111C2×C4×C23—24no—
112C2×C4×C23—24no—
113C2×C4×C23—22no—
114C2×C4×C23—41no—
115C2×C4×C23—41no—
116C2×C4×C23—42no—
117C2×C4×C23—48no—
118C2×C4×C23—42no—
119C2×C4×C23—42no—
120C2×C4×C23—42no—
121C2×C4×C23—42no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—44no—
126C2×C4×C23—44no—
127C2×C4×C23—44no—
128C2×C4×C23—81no—
129C2×C4×C23—82no—
130C2×C4×C23—82no—
131C2×C4×C23C4×D412yes—
132C2×C4×C23#12112yes—
133C8×C23—82no—
134C2×#399C2×C2311yescharacteristic
135#2312C2×C2311yescharacteristic
136#1824C2×C2312yescomplemented
137C2×#323C2×C2311yescharacteristic
138#1294C2×C2312yescomplemented
139C22×C23—24no—
140C22×C23—24no—
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—44no—
146C22×C23—42no—
147C22×C23—42no—
148C22×C23—42no—
149C22×C23—82no—
150C22×C23—84no—
151C22×C23—84no—
152C22×C23#38611yescharacteristic
153C22×C23#41611yescharacteristic
154C22×C23#41911yescharacteristic
155C22×C23#45212yes—
156C22×C23#45312yes—
157C22×C23#45812yes—
158C22×C23#48512yes—
159C4×C23—22no—
160C4×C23—22no—
161C4×C23—44no—
162C4×C23—42no—
163C4×C23—84no—
164C4×C23—82no—
165C22×C4⋊C4—21no—
166C22×C4⋊C4C22×C2311yescharacteristic
167C22×C22⋊C4—21no—
168C22×C22⋊C4C22×C2311yescharacteristic
169C2×C22⋊C8C22×C2312yes—
170#399—22no—
171#399C4×C2314yescomplemented
172#388—24no—
173#380C22×C2312yes—
174#379—21no—
175C4×C22⋊C4—21no—
176C2×#95C22×C2311yescharacteristic
177#340—24no—
178#323—22no—
179C2×C23—21no—
180C2×C23—21no—
181C2×C23—22no—
182C2×C23—82no—
183C2×C23#161814yes—
184C2×C23#187311yescharacteristic
185C2×C23#188711yescharacteristic
186C2×C23#189011yescharacteristic
187C25D4×C2311yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4D4×C2311yescharacteristic
191C23×C4C23×C2311yescharacteristic
192C22×C8—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C2×C4×C2312yes—
199C2×C4⋊C4D4×C2314yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4C2×C4×C2312yes—
208C2×C22⋊C4D4×C2312yes—
209C2×C42—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95C2×C4×C2312yes—
216C23#902411yescharacteristic, direct product factor, Sylow
217C24—21no—
218C24—22no—
219C24—21no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24C22⋊C4×C2312yes—
224C24C2×D4×C2311yescharacteristic
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4C22⋊C4×C2312yes—
240C22×C4C22×C4×C2311yescharacteristic
241C22×C4C2×D4×C2312yes—
242C22×C4C2×D4×C2311yescharacteristic
243C22×C4#41×C2312yes—
244C2×C8—44no—
245C2×C8—42no—
246C4⋊C4—24no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—42no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—48no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—82no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23—48no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—84no—
273C23C2×C22⋊C4×C2311yescharacteristic
274C23C23×#12011yescharacteristic
275C23C23×#12311yescharacteristic
276C23C23×#13512yes—
277C23C23×#13712yes—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—48no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4C4×D4×C2312yes—
299C2×C4C23×#12112yes—
300C8—82no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—24no—
307C22—44no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—82no—
312C22—84no—
313C22—84no—
314C22C23×#38611yescharacteristic
315C22C23×#41611yescharacteristic
316C22C23×#41911yescharacteristic
317C22C23×#45212yes—
318C22C23×#45312yes—
319C22C23×#45812yes—
320C22C23×#48512yes—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—82no—
326C4—84no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—82no—
331C2C23×#161814yes—
332C2C23×#187311yescharacteristic
333C2C23×#188711yescharacteristic
334C2C23×#189011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23D4
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
3. C23×#1824C2semidirect
5. C23×#1294C2semidirect
12. C23×#399C4semidirect
90. #9024C23direct (N × H)
136. #1824C2×C23semidirect
138. #1294C2×C23semidirect
171. #399C4×C23semidirect
216. C23#9024direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3992
C23×#23122
C23×#18242
C2×C23×#3232
C23×#12942
#902423