Finite Group Databasebeta

← C23×#9407 #19157033

Subgroups — C23×#9407 #19157033

Order 5888.

726 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2561C211yesmaximal, characteristic
2C23×#2273C211yesmaximal, characteristic
3C23×#1508C212yesmaximal, complemented
4C23×#1462C211yesmaximal, characteristic
5C23×#1297C212yesmaximal, complemented
6C2×C23×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C4×C23—21no—
8C22×C22⋊C4×C23C2211yescharacteristic
9C23×#407C2212yes—
10C23×#397C412yescomplemented
11C23×#393C412yescomplemented
12C23×#388—22no—
13C23×#381—21no—
14C23×#379—21no—
15C4×C22⋊C4×C23—24no—
16C4×C22⋊C4×C23—21no—
17C4×C22⋊C4×C23C2211yescharacteristic
18C4×C22⋊C4×C23C2212yes—
19C23×#328—24no—
20C23×#323—22no—
21C25×C23D411yescharacteristic
22C22×Q8×C23C2×C411yescharacteristic, complemented
23C23×C4×C23—22no—
24C23×C4×C23D411yescharacteristic
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—22no—
27C23×C22⋊Q8—22no—
28C23×C22⋊Q8—22no—
29C2×C4⋊C4×C23—21no—
30C2×C4⋊C4×C23C2×C411yescharacteristic, complemented
31C2×C22⋊C4×C23—22no—
32C2×C22⋊C4×C23—28no—
33C2×C22⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23C2×C412yes—
41C2×C22⋊C4×C23D412yes—
42C2×C42×C23—22no—
43C2×C42×C23—22no—
44C2×C42×C23—21no—
45C23×C22⋊C8—22no—
46C23×C22⋊C8—22no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—22no—
50C23×#95C2×C412yescomplemented
51C23×#95D414yes—
52C24×C23—21no—
53C24×C23—24no—
54C24×C23—21no—
55C24×C23—42no—
56C24×C23—81no—
57C24×C23C22⋊C412yes—
58C24×C23C2×D411yescharacteristic
59C2×Q8×C23—22no—
60C2×Q8×C23—81no—
61C22×C4×C23—24no—
62C22×C4×C23—21no—
63C22×C4×C23—28no—
64C22×C4×C23—21no—
65C22×C4×C23—22no—
66C22×C4×C23—21no—
67C22×C4×C23—21no—
68C22×C4×C23—42no—
69C22×C4×C23—81no—
70C22×C4×C23—82no—
71C22×C4×C23—82no—
72C22×C4×C23C22⋊C412yes—
73C22×C4×C23C22×C411yescharacteristic
74C22×C4×C23C2×D412yes—
75C22×C4×C23C2×D411yescharacteristic
76C22×C4×C23#4112yes—
77M16×C23—82no—
78C2×C8×C23—42no—
79C4⋊C4×C23—22no—
80C4⋊C4×C23—42no—
81C22⋊C4×C23—28no—
82C22⋊C4×C23—28no—
83C22⋊C4×C23—24no—
84C22⋊C4×C23—24no—
85C22⋊C4×C23—42no—
86C22⋊C4×C23—44no—
87C22⋊C4×C23—41no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—82no—
90C42×C23—82no—
91C42×C23—81no—
92C42×C23—82no—
93#9407C2311yesmaximal, characteristic, direct product factor, Sylow
94C23×C23—22no—
95C23×C23—22no—
96C23×C23—24no—
97C23×C23—22no—
98C23×C23—22no—
99C23×C23—22no—
100C23×C23—22no—
101C23×C23—41no—
102C23×C23—42no—
103C23×C23—42no—
104C23×C23—42no—
105C23×C23—42no—
106C23×C23—81no—
107C23×C23—81no—
108C23×C23—88no—
109C23×C23#10414yes—
110C23×C23C2×C22⋊C411yescharacteristic
111C23×C23#12011yescharacteristic
112C23×C23#12311yescharacteristic
113Q8×C23—82no—
114C2×C4×C23—28no—
115C2×C4×C23—22no—
116C2×C4×C23—41no—
117C2×C4×C23—42no—
118C2×C4×C23—42no—
119C2×C4×C23—41no—
120C2×C4×C23—44no—
121C2×C4×C23—42no—
122C2×C4×C23—44no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—41no—
128C2×C4×C23—41no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—81no—
132C2×C4×C23—88no—
133C2×C4×C23C4×D412yes—
134C2×C4×C23#12112yes—
135C8×C23—82no—
136#2561C2×C2311yescharacteristic
137#2273C2×C2311yescharacteristic
138#1508C2×C2312yescomplemented
139#1462C2×C2311yescharacteristic
140#1297C2×C2312yescomplemented
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—21no—
144C22×C23—21no—
145C22×C23—21no—
146C22×C23—24no—
147C22×C23—21no—
148C22×C23—21no—
149C22×C23—44no—
150C22×C23—44no—
151C22×C23—44no—
152C22×C23—44no—
153C22×C23—42no—
154C22×C23—42no—
155C22×C23—82no—
156C22×C23—84no—
157C22×C23#38611yescharacteristic
158C22×C23C2×#10412yes—
159C22×C23#45512yes—
160C4×C23—22no—
161C4×C23—22no—
162C4×C23—48no—
163C4×C23—42no—
164C4×C23—82no—
165C4×C23—81no—
166C4×C23—81no—
167C2×C22⋊Q8C22×C2311yescharacteristic
168C22×C22⋊C4—21no—
169C22×C22⋊C4C22×C2311yescharacteristic
170#407C22×C2312yes—
171#397C4×C2312yescomplemented
172#393C4×C2312yescomplemented
173#388—22no—
174#381—21no—
175#379—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—24no—
178C4×C22⋊C4C22×C2311yescharacteristic
179C4×C22⋊C4C22×C2312yes—
180#328—24no—
181#323—22no—
182C2×C23—22no—
183C2×C23—22no—
184C2×C23—41no—
185C2×C23—44no—
186C2×C23#188511yescharacteristic
187C2×C23#189611yescharacteristic
188C2×C23#191311yescharacteristic
189C25D4×C2311yescharacteristic
190C22×Q8C2×C4×C2311yescharacteristic, complemented
191C23×C4—22no—
192C23×C4D4×C2311yescharacteristic
193C23×C4C23×C2311yescharacteristic
194C2×M16—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×C4×C2311yescharacteristic, complemented
199C2×C22⋊C4—28no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C2×C4×C2312yes—
209C2×C22⋊C4D4×C2312yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95C2×C4×C2312yescomplemented
219#95D4×C2314yes—
220C23#940711yescharacteristic, direct product factor, Sylow
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24C22⋊C4×C2312yes—
227C24C2×D4×C2311yescharacteristic
228C2×Q8—22no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C22⋊C4×C2312yes—
242C22×C4C22×C4×C2311yescharacteristic
243C22×C4C2×D4×C2312yes—
244C22×C4C2×D4×C2311yescharacteristic
245C22×C4#41×C2312yes—
246M16—82no—
247C2×C8—42no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C22⋊C4—28no—
251C22⋊C4—28no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—44no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—82no—
259C42—82no—
260C42—81no—
261C42—82no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23C23×#10414yes—
278C23C2×C22⋊C4×C2311yescharacteristic
279C23C23×#12011yescharacteristic
280C23C23×#12311yescharacteristic
281Q8—82no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—88no—
301C2×C4C4×D4×C2312yes—
302C2×C4C23×#12112yes—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22C23×#38611yescharacteristic
321C22C2×C23×#10412yes—
322C22C23×#45512yes—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—82no—
328C4—81no—
329C4—81no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2C23×#188511yescharacteristic
335C2C23×#189611yescharacteristic
336C2C23×#191311yescharacteristic
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×#1508C2semidirect
5. C23×#1297C2semidirect
10. C23×#397C4semidirect
11. C23×#393C4semidirect
22. C22×Q8×C23C2×C4semidirect
30. C2×C4⋊C4×C23C2×C4semidirect
50. C23×#95C2×C4semidirect
93. #9407C23direct (N × H)
138. #1508C2×C23semidirect
140. #1297C2×C23semidirect
171. #397C4×C23semidirect
172. #393C4×C23semidirect
190. C22×Q8C2×C4×C23semidirect
198. C2×C4⋊C4C2×C4×C23semidirect
218. #95C2×C4×C23semidirect
220. C23#9407direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25612
C23×#22732
C23×#15082
C23×#14622
C23×#12972
#940723