Finite Group Databasebeta

← C23×#34967 #19182593

Subgroups — C23×#34967 #19182593

Order 5888.

2164 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2522C214yesmaximal, complemented
2C23×#2516C214yesmaximal, complemented
3C23×#2464C214yesmaximal, complemented
4C23×#2457C214yesmaximal, complemented
5C23×#2402C211yesmaximal, characteristic, complemented
6C23×#2397C211yesmaximal, characteristic, complemented
7C23×#2392C212yesmaximal, complemented
8C4×C23×#123C214yesmaximal, complemented
9C4×C23×#121C214yesmaximal, complemented
10C23×#2298C211yesmaximal, characteristic, complemented
11C42×D4×C23C212yesmaximal, complemented
12C2×C23×#123C2214yes—
13C2×C23×#121C2214yes—
14C2×C4×D4×C23C2212yes—
15C2×C4×D4×C23C2214yes—
16C2×C4×D4×C23C2214yes—
17C2×C4×D4×C23C2214yes—
18C2×C23×#117C2211yescharacteristic
19C22×C42×C23C2214yes—
20C23×#400C2214yes—
21C23×#399C2214yes—
22C23×#398C2214yes—
23C23×#398C2212yes—
24C23×#397C2212yes—
25C23×#396C2214yes—
26C23×#394C2212yes—
27C23×#394C2214yes—
28C23×#392C2214yes—
29C23×#392C2212yes—
30C23×#390C2214yes—
31C23×#389C2214yes—
32C23×#388C2214yes—
33C23×#388C2214yes—
34C23×#388C2214yes—
35C23×#388C2214yes—
36C23×#388C2214yes—
37C23×#387C2214yes—
38C23×#387C2214yes—
39C23×#387C2214yes—
40C23×#386C2214yes—
41C23×#386C2214yes—
42C23×#386C2214yes—
43C23×#385C2214yes—
44C23×#381C2214yes—
45C23×#380C2214yes—
46C4×C4⋊C4×C23C2212yes—
47C4×C4⋊C4×C23C2214yes—
48C4×C4⋊C4×C23C2214yes—
49C4×C4⋊C4×C23C2212yes—
50C4×C22⋊C4×C23C2214yes—
51C4×C22⋊C4×C23C2214yes—
52C4×C22⋊C4×C23C2214yes—
53C4×C22⋊C4×C23C2212yes—
54C4×C22⋊C4×C23C2214yes—
55C23×#376C2211yescharacteristic
56C43×C23C2211yescharacteristic
57C22×D4×C23C2312yes—
58C22×D4×C23C2314yes—
59C22×D4×C23C2314yes—
60C23×C4×C23C2314yes—
61C23×C4×C23C2311yescharacteristic
62C23×C4×C23C2314yes—
63C23×C4×C23C2314yes—
64C23×#123—216no—
65C23×#121—216no—
66C4×D4×C23—216no—
67C4×D4×C23—28no—
68C4×D4×C23—48no—
69C4×D4×C23—48no—
70C23×#117—24no—
71C2×C4⋊C4×C23C2314yes—
72C2×C4⋊C4×C23C2312yes—
73C2×C4⋊C4×C23C2314yes—
74C2×C4⋊C4×C23C2314yes—
75C2×C4⋊C4×C23C2314yes—
76C2×C4⋊C4×C23C2312yes—
77C2×C4⋊C4×C23C2314yes—
78C2×C4⋊C4×C23C2314yes—
79C2×C22⋊C4×C23C2314yes—
80C2×C22⋊C4×C23C2314yes—
81C2×C22⋊C4×C23C2314yes—
82C2×C22⋊C4×C23C2314yes—
83C2×C22⋊C4×C23C2314yes—
84C2×C22⋊C4×C23C2312yes—
85C2×C22⋊C4×C23C2314yes—
86C2×C22⋊C4×C23C2314yes—
87C2×C22⋊C4×C23C2314yes—
88C2×C22⋊C4×C23C2314yes—
89C2×C22⋊C4×C23C2314yes—
90C2×C42×C23—48no—
91C2×C42×C23C2314yes—
92C2×C42×C23C2312yes—
93C2×C42×C23C2312yes—
94C2×C42×C23C2311yescharacteristic
95C2×C42×C23C2314yes—
96C2×C42×C23C2314yes—
97C2×C42×C23C2314yes—
98C2×C42×C23C2311yescharacteristic
99C23×#95C2314yes—
100C23×#95C2314yes—
101C23×#95C2314yes—
102C23×#95C2312yes—
103C23×#95C2312yes—
104C23×#95C2314yes—
105C23×#95C2314yes—
106C23×#95C2314yes—
107C23×#95C2314yes—
108C23×#95C2314yes—
109C23×#95C2314yes—
110C24×C23C2411yescharacteristic
111C24×C23C2414yes—
112C2×D4×C23—28no—
113C2×D4×C23—28no—
114C2×D4×C23—28no—
115C2×D4×C23—28no—
116C2×D4×C23—24no—
117C2×D4×C23—28no—
118C2×D4×C23—44no—
119C2×D4×C23—44no—
120C22×C4×C23—28no—
121C22×C4×C23—22no—
122C22×C4×C23—28no—
123C22×C4×C23—28no—
124C22×C4×C23—28no—
125C22×C4×C23—28no—
126C22×C4×C23—28no—
127C22×C4×C23—28no—
128C22×C4×C23—24no—
129C22×C4×C23—44no—
130C22×C4×C23—44no—
131C22×C4×C23C2414yes—
132C22×C4×C23C2414yes—
133C22×C4×C23C2412yes—
134C22×C4×C23C2414yes—
135C22×C4×C23C2411yescharacteristic
136C22×C4×C23C2414yes—
137C22×C4×C23C2412yes—
138C22×C4×C23C2414yes—
139C22×C4×C23C2411yescharacteristic
140C4⋊C4×C23—24no—
141C4⋊C4×C23—28no—
142C4⋊C4×C23—24no—
143C4⋊C4×C23—28no—
144C4⋊C4×C23—28no—
145C4⋊C4×C23—28no—
146C4⋊C4×C23—44no—
147C4⋊C4×C23—44no—
148C22⋊C4×C23—28no—
149C22⋊C4×C23—24no—
150C22⋊C4×C23—28no—
151C22⋊C4×C23—28no—
152C22⋊C4×C23—28no—
153C22⋊C4×C23—28no—
154C22⋊C4×C23—44no—
155C22⋊C4×C23—44no—
156C22⋊C4×C23—44no—
157C22⋊C4×C23—44no—
158C22⋊C4×C23#41116yes—
159C42×C23—22no—
160C42×C23—28no—
161C42×C23—24no—
162C42×C23—28no—
163C42×C23—24no—
164C42×C23—44no—
165C42×C23—44no—
166C42×C23#4114yes—
167#34967C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—24no—
169C23×C23—24no—
170C23×C23—24no—
171C23×C23—24no—
172C23×C23—24no—
173C23×C23—24no—
174C23×C23—21no—
175C23×C23—24no—
176C23×C23—22no—
177C23×C23C2×#4118yes—
178C23×C23C2511yescharacteristic
179D4×C23—216no—
180D4×C23—416no—
181D4×C23—416no—
182C2×C4×C23—24no—
183C2×C4×C23—24no—
184C2×C4×C23—24no—
185C2×C4×C23—24no—
186C2×C4×C23—24no—
187C2×C4×C23—22no—
188C2×C4×C23—24no—
189C2×C4×C23—24no—
190C2×C4×C23—24no—
191C2×C4×C23—24no—
192C2×C4×C23—24no—
193C2×C4×C23—21no—
194C2×C4×C23—24no—
195C2×C4×C23—24no—
196C2×C4×C23—24no—
197C2×C4×C23—24no—
198C2×C4×C23—24no—
199C2×C4×C23—22no—
200C2×C4×C23—416no—
201C2×C4×C23—416no—
202C2×C4×C23—416no—
203C2×C4×C23—44no—
204C2×C4×C23C2×#4118yes—
205C2×C4×C23C2×#4114yes—
206C2×C4×C23C2×#4118yes—
207C2×C4×C23C2×#4112yes—
208C2×C4×C23H(5,𝔽2)14yes—
209#2522C2×C2314yescomplemented
210#2516C2×C2314yescomplemented
211#2464C2×C2314yescomplemented
212#2457C2×C2314yescomplemented
213#2402C2×C2311yescharacteristic, complemented
214#2397C2×C2311yescharacteristic, complemented
215#2392C2×C2312yescomplemented
216C4×#123C2×C2314yescomplemented
217C4×#121C2×C2314yescomplemented
218#2298C2×C2311yescharacteristic, complemented
219C42×D4C2×C2312yescomplemented
220C22×C23—28no—
221C22×C23—28no—
222C22×C23—28no—
223C22×C23—44no—
224C22×C23—44no—
225C22×C23—44no—
226C22×C23—44no—
227C22×C23—42no—
228C22×C23—41no—
229C22×C23—44no—
230C22×C23C22×#4111yescharacteristic
231C22×C23C22×#4114yes—
232C22×C23C2×H(5,𝔽2)12yes—
233C4×C23—24no—
234C4×C23—28no—
235C4×C23—28no—
236C4×C23—42no—
237C4×C23—44no—
238C4×C23—41no—
239C4×C23—44no—
240C4×C23—44no—
241C4×C23#54014yes—
242C2×#123C22×C2314yes—
243C2×#121C22×C2314yes—
244C2×C4×D4C22×C2312yes—
245C2×C4×D4C22×C2314yes—
246C2×C4×D4C22×C2314yes—
247C2×C4×D4C22×C2314yes—
248C2×#117C22×C2311yescharacteristic
249C22×C42C22×C2314yes—
250#400C22×C2314yes—
251#399C22×C2314yes—
252#398C22×C2312yes—
253#398C22×C2314yes—
254#397C22×C2312yes—
255#396C22×C2314yes—
256#394C22×C2312yes—
257#394C22×C2314yes—
258#392C22×C2312yes—
259#392C22×C2314yes—
260#390C22×C2314yes—
261#389C22×C2314yes—
262#388C22×C2314yes—
263#388C22×C2314yes—
264#388C22×C2314yes—
265#388C22×C2314yes—
266#388C22×C2314yes—
267#387C22×C2314yes—
268#387C22×C2314yes—
269#387C22×C2314yes—
270#386C22×C2314yes—
271#386C22×C2314yes—
272#386C22×C2314yes—
273#385C22×C2314yes—
274#381C22×C2314yes—
275#380C22×C2314yes—
276C4×C4⋊C4C22×C2312yes—
277C4×C4⋊C4C22×C2314yes—
278C4×C4⋊C4C22×C2314yes—
279C4×C4⋊C4C22×C2312yes—
280C4×C22⋊C4C22×C2314yes—
281C4×C22⋊C4C22×C2314yes—
282C4×C22⋊C4C22×C2314yes—
283C4×C22⋊C4C22×C2314yes—
284C4×C22⋊C4C22×C2312yes—
285#376C22×C2311yescharacteristic
286C43C22×C2311yescharacteristic
287C2×C23—48no—
288C2×C23—81no—
289C2×C23#344412yes—
290C2×C23C2×#54011yescharacteristic
291C2×C23#348114yes—
292C22×D4C23×C2312yes—
293C22×D4C23×C2314yes—
294C22×D4C23×C2314yes—
295C23×C4C23×C2311yescharacteristic
296C23×C4C23×C2314yes—
297C23×C4C23×C2314yes—
298C23×C4C23×C2314yes—
299#123—216no—
300#121—216no—
301C4×D4—216no—
302C4×D4—28no—
303C4×D4—48no—
304C4×D4—48no—
305#117—24no—
306C2×C4⋊C4C23×C2312yes—
307C2×C4⋊C4C23×C2314yes—
308C2×C4⋊C4C23×C2312yes—
309C2×C4⋊C4C23×C2314yes—
310C2×C4⋊C4C23×C2314yes—
311C2×C4⋊C4C23×C2314yes—
312C2×C4⋊C4C23×C2314yes—
313C2×C4⋊C4C23×C2314yes—
314C2×C22⋊C4C23×C2312yes—
315C2×C22⋊C4C23×C2314yes—
316C2×C22⋊C4C23×C2314yes—
317C2×C22⋊C4C23×C2314yes—
318C2×C22⋊C4C23×C2314yes—
319C2×C22⋊C4C23×C2314yes—
320C2×C22⋊C4C23×C2314yes—
321C2×C22⋊C4C23×C2314yes—
322C2×C22⋊C4C23×C2314yes—
323C2×C22⋊C4C23×C2314yes—
324C2×C22⋊C4C23×C2314yes—
325C2×C42—48no—
326C2×C42C23×C2311yescharacteristic
327C2×C42C23×C2312yes—
328C2×C42C23×C2314yes—
329C2×C42C23×C2314yes—
330C2×C42C23×C2312yes—
331C2×C42C23×C2314yes—
332C2×C42C23×C2314yes—
333C2×C42C23×C2311yescharacteristic
334#95C23×C2314yes—
335#95C23×C2314yes—
336#95C23×C2314yes—
337#95C23×C2312yes—
338#95C23×C2312yes—
339#95C23×C2314yes—
340#95C23×C2314yes—
341#95C23×C2314yes—
342#95C23×C2314yes—
343#95C23×C2314yes—
344#95C23×C2314yes—
345C23#3496711yescharacteristic, direct product factor, Sylow
346C24C24×C2311yescharacteristic
347C24C24×C2314yes—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—44no—
355C2×D4—44no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C24×C2314yes—
368C22×C4C24×C2312yes—
369C22×C4C24×C2314yes—
370C22×C4C24×C2314yes—
371C22×C4C24×C2311yescharacteristic
372C22×C4C24×C2314yes—
373C22×C4C24×C2312yes—
374C22×C4C24×C2314yes—
375C22×C4C24×C2311yescharacteristic
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—28no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4#41×C23116yes—
395C42—22no—
396C42—28no—
397C42—24no—
398C42—28no—
399C42—24no—
400C42—44no—
401C42—44no—
402C42#41×C2314yes—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—21no—
410C23—24no—
411C23—22no—
412C23C2×#41×C2318yes—
413C23C25×C2311yescharacteristic
414D4—216no—
415D4—416no—
416D4—416no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—416no—
436C2×C4—416no—
437C2×C4—416no—
438C2×C4—44no—
439C2×C4C2×#41×C2318yes—
440C2×C4C2×#41×C2314yes—
441C2×C4C2×#41×C2318yes—
442C2×C4C2×#41×C2312yes—
443C2×C4C23×H(5,𝔽2)14yes—
444C22—28no—
445C22—28no—
446C22—28no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—42no—
452C22—41no—
453C22—44no—
454C22C22×#41×C2311yescharacteristic
455C22C22×#41×C2314yes—
456C22C2×C23×H(5,𝔽2)12yes—
457C4—24no—
458C4—28no—
459C4—28no—
460C4—42no—
461C4—44no—
462C4—41no—
463C4—44no—
464C4—44no—
465C4C23×#54014yes—
466C2—48no—
467C2—81no—
468C2C23×#344412yes—
469C2C2×C23×#54011yescharacteristic
470C2C23×#348114yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2522C2semidirect
2. C23×#2516C2semidirect
3. C23×#2464C2semidirect
4. C23×#2457C2semidirect
5. C23×#2402C2semidirect
6. C23×#2397C2semidirect
7. C23×#2392C2semidirect
8. C4×C23×#123C2semidirect
9. C4×C23×#121C2semidirect
10. C23×#2298C2semidirect
11. C42×D4×C23C2semidirect
167. #34967C23direct (N × H)
209. #2522C2×C23semidirect
210. #2516C2×C23semidirect
211. #2464C2×C23semidirect
212. #2457C2×C23semidirect
213. #2402C2×C23semidirect
214. #2397C2×C23semidirect
215. #2392C2×C23semidirect
216. C4×#123C2×C23semidirect
217. C4×#121C2×C23semidirect
218. #2298C2×C23semidirect
219. C42×D4C2×C23semidirect
345. C23#34967direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25222
C23×#25162
C23×#24642
C23×#24572
C23×#24022
C23×#23972
C23×#23922
C4×C23×#1232
C4×C23×#1212
C23×#22982
C42×D4×C232
#3496723