Finite Group Databasebeta

← C23×#11726 #19159352

Subgroups — C23×#11726 #19159352

Order 5888.

606 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2397C211yesmaximal, characteristic
2C4×C23×#99C211yesmaximal, characteristic, complemented
3C23×#1459C211yesmaximal, characteristic, complemented
4C23×#1352C212yesmaximal, complemented
5C23×#1343C212yesmaximal, complemented
6C2×C4×D4×C23—21no—
7C2×C4×D4×C23—21no—
8C2×C4×D4×C23C411yescharacteristic, complemented
9C2×C4×D4×C23C2211yescharacteristic
10C2×C23×#99—21no—
11C2×C23×C22⋊C8—21no—
12C23×#399—21no—
13C23×#399C411yescharacteristic, complemented
14C23×#399C2211yescharacteristic
15C23×#392—21no—
16C23×#392C411yescharacteristic, complemented
17C23×#392C2211yescharacteristic
18C4×C22⋊C4×C23—21no—
19C23×#376C411yescharacteristic, complemented
20C23×#342C2212yes—
21C23×#324—21no—
22C23×#323C2212yes—
23C22×D4×C23—21no—
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C22×D4×C23D411yescharacteristic
27C23×C4×C23—21no—
28C23×C4×C23D411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C22×C8×C23—41no—
31C4×D4×C23—41no—
32C4×D4×C23—41no—
33C4×D4×C23—44no—
34C4×D4×C23—44no—
35C4×D4×C23—41no—
36C4×D4×C23—41no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23C2×C411yescharacteristic
48C2×C22⋊C4×C23C2311yescharacteristic
49C2×C42×C23—21no—
50C2×C42×C23—41no—
51C2×C42×C23C2×C411yescharacteristic
52C23×#99—24no—
53C23×C22⋊C8—22no—
54C23×C22⋊C8—41no—
55C23×#95—21no—
56C23×#95—41no—
57C23×#95C2×C411yescharacteristic
58C23×#95C2×C411yescharacteristic
59C24×C23—21no—
60C24×C23C22⋊C411yescharacteristic
61C24×C23C2×D411yescharacteristic
62C2×D4×C23—22no—
63C2×D4×C23—22no—
64C2×D4×C23—41no—
65C2×D4×C23—41no—
66C2×D4×C23—42no—
67C2×D4×C23—42no—
68C2×D4×C23—42no—
69C2×D4×C23—42no—
70C2×D4×C23—42no—
71C2×D4×C23—42no—
72C2×D4×C23—42no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—42no—
81C22×C4×C23—42no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23C22⋊C411yescharacteristic
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22×C411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C2×C8×C23—42no—
93C2×C8×C23—82no—
94C4⋊C4×C23—41no—
95C4⋊C4×C23—41no—
96C4⋊C4×C23—42no—
97C4⋊C4×C23—42no—
98C22⋊C4×C23—24no—
99C22⋊C4×C23—41no—
100C22⋊C4×C23—41no—
101C22⋊C4×C23—42no—
102C22⋊C4×C23—42no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—42no—
106C22⋊C4×C23—42no—
107C42×C23—24no—
108C42×C23—21no—
109C42×C23—21no—
110C42×C23—82no—
111#11726C2311yesmaximal, characteristic, direct product factor, Sylow
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—22no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23#10412yes—
128C23×C23C2×C22⋊C411yescharacteristic
129C23×C23#13512yes—
130D4×C23—44no—
131D4×C23—42no—
132D4×C23—48no—
133D4×C23—48no—
134D4×C23—42no—
135C2×C4×C23—22no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—44no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—81no—
158C2×C4×C23—81no—
159C2×C4×C23—82no—
160C2×C4×C23—82no—
161C2×C4×C23—82no—
162C2×C4×C23—82no—
163C2×C4×C23#9911yescharacteristic
164C2×C4×C23#9911yescharacteristic
165C8×C23—82no—
166#2397C2×C2311yescharacteristic
167C4×#99C2×C2311yescharacteristic, complemented
168#1459C2×C2311yescharacteristic, complemented
169#1352C2×C2312yescomplemented
170#1343C2×C2312yescomplemented
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—22no—
174C22×C23—22no—
175C22×C23—42no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—42no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23—42no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23—81no—
188C22×C23—81no—
189C22×C23—81no—
190C22×C23C2×#9911yescharacteristic
191C22×C23#41611yescharacteristic
192C22×C23C2×#10411yescharacteristic
193C4×C23—21no—
194C4×C23—42no—
195C4×C23—42no—
196C4×C23—41no—
197C4×C23—41no—
198C4×C23—41no—
199C4×C23—41no—
200C4×C23—84no—
201C4×C23—81no—
202C4×C23#35312yes—
203C2×C4×D4—21no—
204C2×C4×D4—21no—
205C2×C4×D4C4×C2311yescharacteristic, complemented
206C2×C4×D4C22×C2311yescharacteristic
207C2×#99—21no—
208C2×C22⋊C8—21no—
209#399—21no—
210#399C4×C2311yescharacteristic, complemented
211#399C22×C2311yescharacteristic
212#392—21no—
213#392C4×C2311yescharacteristic, complemented
214#392C22×C2311yescharacteristic
215C4×C22⋊C4—21no—
216#376C4×C2311yescharacteristic, complemented
217#342C22×C2312yes—
218#324—21no—
219#323C22×C2312yes—
220C2×C23—21no—
221C2×C23—21no—
222C2×C23—42no—
223C2×C23—42no—
224C2×C23—82no—
225C2×C23#150111yescharacteristic
226C2×C23#212011yescharacteristic
227C2×C23C2×#35311yescharacteristic
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2×C4×C2311yescharacteristic
231C22×D4D4×C2311yescharacteristic
232C23×C4—21no—
233C23×C4D4×C2311yescharacteristic
234C23×C4D4×C2311yescharacteristic
235C22×C8—41no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—41no—
241C4×D4—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C2×C4×C2311yescharacteristic
245C2×C4⋊C4D4×C2311yescharacteristic
246C2×C22⋊C4—22no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4C2×C4×C2311yescharacteristic
253C2×C22⋊C4C23×C2311yescharacteristic
254C2×C42—21no—
255C2×C42—41no—
256C2×C42C2×C4×C2311yescharacteristic
257#99—24no—
258C22⋊C8—22no—
259C22⋊C8—41no—
260#95—21no—
261#95—41no—
262#95C2×C4×C2311yescharacteristic
263#95C2×C4×C2311yescharacteristic
264C23#1172611yescharacteristic, direct product factor, Sylow
265C24—21no—
266C24C22⋊C4×C2311yescharacteristic
267C24C2×D4×C2311yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C22⋊C4×C2311yescharacteristic
294C22×C4C22⋊C4×C2311yescharacteristic
295C22×C4C22⋊C4×C2311yescharacteristic
296C22×C4C22×C4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298C2×C8—42no—
299C2×C8—82no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—24no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C42—24no—
314C42—21no—
315C42—21no—
316C42—82no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C23×#10412yes—
333C23C2×C22⋊C4×C2311yescharacteristic
334C23C23×#13512yes—
335D4—44no—
336D4—42no—
337D4—48no—
338D4—48no—
339D4—42no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—44no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4C23×#9911yescharacteristic
369C2×C4C23×#9911yescharacteristic
370C8—82no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×C23×#9911yescharacteristic
391C22C23×#41611yescharacteristic
392C22C2×C23×#10411yescharacteristic
393C4—21no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—84no—
401C4—81no—
402C4C23×#35312yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—82no—
408C2C23×#150111yescharacteristic
409C2C23×#212011yescharacteristic
410C2C2×C23×#35311yescharacteristic
subgroupG/H
centerC2×C4×C23#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
socleC22×C23C2×#99
prime radicalC23×#2397C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C23×#99C2semidirect
3. C23×#1459C2semidirect
4. C23×#1352C2semidirect
5. C23×#1343C2semidirect
8. C2×C4×D4×C23C4semidirect
13. C23×#399C4semidirect
16. C23×#392C4semidirect
19. C23×#376C4semidirect
111. #11726C23direct (N × H)
167. C4×#99C2×C23semidirect
168. #1459C2×C23semidirect
169. #1352C2×C23semidirect
170. #1343C2×C23semidirect
205. C2×C4×D4C4×C23semidirect
210. #399C4×C23semidirect
213. #392C4×C23semidirect
216. #376C4×C23semidirect
264. C23#11726direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23972
C4×C23×#992
C23×#14592
C23×#13522
C23×#13432
#1172623