Finite Group Databasebeta

← C23×#9481 #19157107

Subgroups — C23×#9481 #19157107

Order 5888.

614 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2482C211yesmaximal, characteristic
2C23×#2312C211yesmaximal, characteristic
3C23×#1826C212yesmaximal, complemented
4C23×#1462C211yesmaximal, characteristic
5C23×#1284C212yesmaximal, complemented
6C2×C23×#117C2211yescharacteristic
7C22×C22⋊C4×C23C2211yescharacteristic
8C23×#407C2212yes—
9C23×#399C412yescomplemented
10C23×#397C412yescomplemented
11C23×#388—22no—
12C23×#388—24no—
13C23×#385—21no—
14C23×#381C2211yescharacteristic
15C23×#380—21no—
16C23×#379—21no—
17C23×#379—21no—
18C4×C22⋊C4×C23C2212yes—
19C23×#339—24no—
20C23×#323—22no—
21C25×C23D411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23C2311yescharacteristic
24C2×M16×C23—22no—
25C23×#117—22no—
26C23×#117—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23C2×C411yescharacteristic
30C2×C4⋊C4×C23C2×C411yescharacteristic
31C2×C22⋊C4×C23—22no—
32C2×C22⋊C4×C23—22no—
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—21no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23C2×C412yes—
41C2×C22⋊C4×C23D412yes—
42C2×C42×C23—21no—
43C2×C42×C23D414yes—
44C23×C22⋊C8—22no—
45C23×C22⋊C8—22no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—21no—
50C23×#95C2×C412yes—
51C24×C23—21no—
52C24×C23—21no—
53C24×C23—42no—
54C24×C23—42no—
55C24×C23—81no—
56C24×C23C22⋊C412yes—
57C24×C23C2×D411yescharacteristic
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—22no—
63C22×C4×C23—24no—
64C22×C4×C23—42no—
65C22×C4×C23—81no—
66C22×C4×C23C22⋊C412yes—
67C22×C4×C23C22×C411yescharacteristic
68C22×C4×C23C2×D412yes—
69C22×C4×C23C2×D411yescharacteristic
70C22×C4×C23#4112yes—
71M16×C23—44no—
72C2×C8×C23—42no—
73C4⋊C4×C23—22no—
74C4⋊C4×C23—22no—
75C4⋊C4×C23—82no—
76C4⋊C4×C23—82no—
77C22⋊C4×C23—24no—
78C22⋊C4×C23—24no—
79C22⋊C4×C23—44no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—44no—
82C22⋊C4×C23—41no—
83C22⋊C4×C23—41no—
84C22⋊C4×C23—82no—
85C22⋊C4×C23—82no—
86C22⋊C4×C23—82no—
87C42×C23—44no—
88C42×C23—42no—
89#9481C2311yesmaximal, characteristic, direct product factor, Sylow
90C23×C23—22no—
91C23×C23—22no—
92C23×C23—22no—
93C23×C23—22no—
94C23×C23—22no—
95C23×C23—41no—
96C23×C23—42no—
97C23×C23—42no—
98C23×C23—42no—
99C23×C23—42no—
100C23×C23—42no—
101C23×C23—42no—
102C23×C23—81no—
103C23×C23—81no—
104C23×C23—88no—
105C23×C23C2×C22⋊C411yescharacteristic
106C23×C23#12011yescharacteristic
107C23×C23#12311yescharacteristic
108C2×C4×C23—24no—
109C2×C4×C23—22no—
110C2×C4×C23—41no—
111C2×C4×C23—41no—
112C2×C4×C23—42no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—42no—
116C2×C4×C23—44no—
117C2×C4×C23—42no—
118C2×C4×C23—44no—
119C2×C4×C23—42no—
120C2×C4×C23—41no—
121C2×C4×C23—41no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—44no—
125C2×C4×C23—81no—
126C2×C4×C23C4×D412yes—
127C2×C4×C23#12112yes—
128C8×C23—82no—
129#2482C2×C2311yescharacteristic
130#2312C2×C2311yescharacteristic
131#1826C2×C2312yescomplemented
132#1462C2×C2311yescharacteristic
133#1284C2×C2312yescomplemented
134C22×C23—21no—
135C22×C23—21no—
136C22×C23—21no—
137C22×C23—21no—
138C22×C23—21no—
139C22×C23—21no—
140C22×C23—21no—
141C22×C23—44no—
142C22×C23—44no—
143C22×C23—42no—
144C22×C23—42no—
145C22×C23—42no—
146C22×C23—82no—
147C22×C23—84no—
148C22×C23—84no—
149C22×C23#38611yescharacteristic
150C22×C23#42112yes—
151C22×C23#45812yes—
152C4×C23—22no—
153C4×C23—22no—
154C4×C23—42no—
155C4×C23—84no—
156C4×C23—82no—
157C4×C23—81no—
158C4×C23—81no—
159C2×#117C22×C2311yescharacteristic
160C22×C22⋊C4C22×C2311yescharacteristic
161#407C22×C2312yes—
162#399C4×C2312yescomplemented
163#397C4×C2312yescomplemented
164#388—22no—
165#388—24no—
166#385—21no—
167#381C22×C2311yescharacteristic
168#380—21no—
169#379—21no—
170#379—21no—
171C4×C22⋊C4C22×C2312yes—
172#339—24no—
173#323—22no—
174C2×C23—22no—
175C2×C23—22no—
176C2×C23—41no—
177C2×C23—82no—
178C2×C23#189011yescharacteristic
179C2×C23#189711yescharacteristic
180C2×C23#191111yescharacteristic
181C25D4×C2311yescharacteristic
182C23×C4D4×C2311yescharacteristic
183C23×C4C23×C2311yescharacteristic
184C2×M16—22no—
185#117—22no—
186#117—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2×C4×C2311yescharacteristic
190C2×C4⋊C4C2×C4×C2311yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C2×C4×C2312yes—
201C2×C22⋊C4D4×C2312yes—
202C2×C42—21no—
203C2×C42D4×C2314yes—
204C22⋊C8—22no—
205C22⋊C8—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95C2×C4×C2312yes—
211C23#948111yescharacteristic, direct product factor, Sylow
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24C22⋊C4×C2312yes—
218C24C2×D4×C2311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4C22⋊C4×C2312yes—
228C22×C4C22×C4×C2311yescharacteristic
229C22×C4C2×D4×C2312yes—
230C22×C4C2×D4×C2311yescharacteristic
231C22×C4#41×C2312yes—
232M16—44no—
233C2×C8—42no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—44no—
249C42—42no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×C22⋊C4×C2311yescharacteristic
266C23C23×#12011yescharacteristic
267C23C23×#12311yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—81no—
286C2×C4C4×D4×C2312yes—
287C2×C4C23×#12112yes—
288C8—82no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22C23×#38611yescharacteristic
305C22C23×#42112yes—
306C22C23×#45812yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—82no—
312C4—81no—
313C4—81no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2C23×#189011yescharacteristic
319C2C23×#189711yescharacteristic
320C2C23×#191111yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1826C2semidirect
5. C23×#1284C2semidirect
9. C23×#399C4semidirect
10. C23×#397C4semidirect
89. #9481C23direct (N × H)
131. #1826C2×C23semidirect
133. #1284C2×C23semidirect
162. #399C4×C23semidirect
163. #397C4×C23semidirect
211. C23#9481direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24822
C23×#23122
C23×#18262
C23×#14622
C23×#12842
#948123