Finite Group Databasebeta

← C23×#9390 #19157016

Subgroups — C23×#9390 #19157016

Order 5888.

446 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic
2C23×#2611C211yesmaximal, characteristic
3C23×#1528C212yesmaximal, complemented
4C23×#1464C211yesmaximal, characteristic
5C23×#1300C212yesmaximal, complemented
6C2×C23×#125C2211yescharacteristic
7C2×C23×#117C2211yescharacteristic
8C23×#433—24no—
9C23×#432C2212yes—
10C23×#423—21no—
11C23×#407—21no—
12C4×M16×C23C2212yes—
13C23×#400C411yescharacteristic
14C23×#399C411yescharacteristic
15C23×#398C411yescharacteristic
16C23×#397C411yescharacteristic
17C23×#389—21no—
18C23×#388—21no—
19C23×#387—21no—
20C23×#385—21no—
21C23×#376C2211yescharacteristic
22C23×#330—24no—
23C23×#324—21no—
24C23×#323—21no—
25C23×C4×C23D411yescharacteristic
26C2×M16×C23—22no—
27C23×#125—22no—
28C23×#125—22no—
29C23×#117—22no—
30C23×#117—22no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23D411yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23C2×C411yescharacteristic
42C2×C22⋊C4×C23D411yescharacteristic
43C2×C42×C23—21no—
44C2×C42×C23D411yescharacteristic
45C2×C42×C23C2311yescharacteristic
46C23×C4⋊C8—24no—
47C23×C4⋊C8—22no—
48C23×C22⋊C8—24no—
49C23×C22⋊C8—22no—
50C23×#97—22no—
51C23×#97D414yes—
52C4×C8×C23—22no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95C2×C411yescharacteristic
59C23×#95C2×C411yescharacteristic
60C24×C23C22⋊C411yescharacteristic
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—21no—
64C22×C4×C23—21no—
65C22×C4×C23—21no—
66C22×C4×C23—21no—
67C22×C4×C23—41no—
68C22×C4×C23—41no—
69C22×C4×C23C22⋊C411yescharacteristic
70C22×C4×C23C22⋊C411yescharacteristic
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22×C411yescharacteristic
73C22×C4×C23C2×D411yescharacteristic
74C22×C4×C23C2×D411yescharacteristic
75M16×C23—42no—
76M16×C23—42no—
77C2×C8×C23—24no—
78C2×C8×C23—42no—
79C4⋊C4×C23—22no—
80C4⋊C4×C23—22no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—22no—
83C4⋊C4×C23—42no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—41no—
86C4⋊C4×C23—42no—
87C4⋊C4×C23—41no—
88C22⋊C4×C23—22no—
89C22⋊C4×C23—22no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C42×C23—21no—
93C42×C23—21no—
94C42×C23—42no—
95C42×C23C2×D412yes—
96C42×C23#4112yes—
97#9390C2311yesmaximal, characteristic, direct product factor, Sylow
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—41no—
102C23×C23—41no—
103C23×C23C2×C22⋊C411yescharacteristic
104C2×C4×C23—21no—
105C2×C4×C23—21no—
106C2×C4×C23—22no—
107C2×C4×C23—21no—
108C2×C4×C23—21no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—21no—
112C2×C4×C23—41no—
113C2×C4×C23—41no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—41no—
117C2×C4×C23—41no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—41no—
123C2×C4×C23—41no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—41no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23C4×D412yes—
130C2×C4×C23#12011yescharacteristic
131C2×C4×C23#12112yes—
132C2×C4×C23#12311yescharacteristic
133C8×C23—44no—
134C8×C23—82no—
135#2921C2×C2311yescharacteristic
136#2611C2×C2311yescharacteristic
137#1528C2×C2312yescomplemented
138#1464C2×C2311yescharacteristic
139#1300C2×C2312yescomplemented
140C22×C23—21no—
141C22×C23—21no—
142C22×C23—42no—
143C22×C23—41no—
144C22×C23—41no—
145C22×C23—41no—
146C22×C23—81no—
147C22×C23#38611yescharacteristic
148C22×C23#42111yescharacteristic
149C22×C23#42111yescharacteristic
150C4×C23—22no—
151C4×C23—21no—
152C4×C23—21no—
153C4×C23—22no—
154C4×C23—42no—
155C4×C23—41no—
156C4×C23—41no—
157C4×C23—41no—
158C4×C23—81no—
159C4×C23—81no—
160C4×C23—81no—
161C2×#125C22×C2311yescharacteristic
162C2×#117C22×C2311yescharacteristic
163#433—24no—
164#432C22×C2312yes—
165#423—21no—
166#407—21no—
167C4×M16C22×C2312yes—
168#400C4×C2311yescharacteristic
169#399C4×C2311yescharacteristic
170#398C4×C2311yescharacteristic
171#397C4×C2311yescharacteristic
172#389—21no—
173#388—21no—
174#387—21no—
175#385—21no—
176#376C22×C2311yescharacteristic
177#330—24no—
178#324—21no—
179#323—21no—
180C2×C23—21no—
181C2×C23—21no—
182C2×C23—81no—
183C2×C23#188411yescharacteristic
184C2×C23#189711yescharacteristic
185C2×C23#191511yescharacteristic
186C23×C4D4×C2311yescharacteristic
187C2×M16—22no—
188#125—22no—
189#125—22no—
190#117—22no—
191#117—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×C4×C2311yescharacteristic
197C2×C4⋊C4C2×C4×C2311yescharacteristic
198C2×C4⋊C4C2×C4×C2311yescharacteristic
199C2×C4⋊C4D4×C2311yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4C2×C4×C2311yescharacteristic
203C2×C22⋊C4D4×C2311yescharacteristic
204C2×C42—21no—
205C2×C42D4×C2311yescharacteristic
206C2×C42C23×C2311yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—22no—
209C22⋊C8—24no—
210C22⋊C8—22no—
211#97—22no—
212#97D4×C2314yes—
213C4×C8—22no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95C2×C4×C2311yescharacteristic
220#95C2×C4×C2311yescharacteristic
221C23#939011yescharacteristic, direct product factor, Sylow
222C24C22⋊C4×C2311yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊C4×C2311yescharacteristic
232C22×C4C22⋊C4×C2311yescharacteristic
233C22×C4C22⋊C4×C2311yescharacteristic
234C22×C4C22×C4×C2311yescharacteristic
235C22×C4C2×D4×C2311yescharacteristic
236C22×C4C2×D4×C2311yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—42no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42C2×D4×C2312yes—
258C42#41×C2312yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊C4×C2311yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4C4×D4×C2312yes—
291C2×C4C23×#12011yescharacteristic
292C2×C4C23×#12112yes—
293C2×C4C23×#12311yescharacteristic
294C8—44no—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C23×#38611yescharacteristic
304C22C23×#42111yescharacteristic
305C22C23×#42111yescharacteristic
306C4—21no—
307C4—21no—
308C4—22no—
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2C23×#188411yescharacteristic
321C2C23×#189711yescharacteristic
322C2C23×#191511yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC24×C23C22⋊C4
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
3. C23×#1528C2semidirect
5. C23×#1300C2semidirect
97. #9390C23direct (N × H)
137. #1528C2×C23semidirect
139. #1300C2×C23semidirect
221. C23#9390direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29212
C23×#26112
C23×#15282
C23×#14642
C23×#13002
#939023