Finite Group Databasebeta

← C23×#36695 #19184321

Subgroups — C23×#36695 #19184321

Order 5888.

1664 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2845C211yesmaximal, characteristic, complemented
2C23×#2837C214yesmaximal, complemented
3C23×#2831C214yesmaximal, complemented
4C23×#2829C214yesmaximal, complemented
5C23×#2820C212yesmaximal, complemented
6C23×#2819C211yesmaximal, characteristic
7C23×#2396C214yesmaximal, complemented
8C23×#2395C214yesmaximal, complemented
9C23×#2392C214yesmaximal, complemented
10C4×C23×#125C211yesmaximal, characteristic, complemented
11C4×C23×#124C212yesmaximal, complemented
12C2×C23×#125C2211yescharacteristic
13C2×C23×#124C2212yes—
14C2×C4×Q8×C23C2212yes—
15C2×C4×D4×C23C2212yes—
16C2×C4×D4×C23C2214yes—
17C2×C23×#117C2214yes—
18C23×#400C2214yes—
19C23×#400C2214yes—
20C23×#399C2214yes—
21C23×#399C2214yes—
22C23×#398C2214yes—
23C23×#398C2214yes—
24C23×#398C2214yes—
25C23×#398C2214yes—
26C23×#397C2214yes—
27C23×#397C2214yes—
28C23×#397C2214yes—
29C23×#397C2214yes—
30C23×#396C2214yes—
31C23×#396C2214yes—
32C23×#396C2214yes—
33C23×#396C2214yes—
34C23×#395C2214yes—
35C23×#394C2214yes—
36C23×#394C2214yes—
37C23×#394C2214yes—
38C23×#394C2214yes—
39C23×#394C2214yes—
40C23×#393C2214yes—
41C23×#392C2214yes—
42C23×#391C2212yes—
43C23×#390C2212yes—
44C23×#389C2212yes—
45C23×#388C2214yes—
46C23×#388C2214yes—
47C23×#387C2214yes—
48C23×#382C2212yes—
49C23×#382C2211yescharacteristic
50C4×C4⋊C4×C23C2214yes—
51C4×C4⋊C4×C23C2212yes—
52C4×C4⋊C4×C23C2214yes—
53C4×C22⋊C4×C23C2214yes—
54C4×C22⋊C4×C23C2214yes—
55C4×C22⋊C4×C23C2214yes—
56C43×C23C2211yescharacteristic
57C22×Q8×C23C2312yes—
58C22×D4×C23C2312yes—
59C22×D4×C23C2314yes—
60C23×C4×C23C2314yes—
61C23×#125—24no—
62C23×#124—28no—
63C4×Q8×C23—44no—
64C4×D4×C23—48no—
65C4×D4×C23—44no—
66C23×#117—48no—
67C2×C4⋊C4×C23C2314yes—
68C2×C4⋊C4×C23C2312yes—
69C2×C4⋊C4×C23C2312yes—
70C2×C4⋊C4×C23C2314yes—
71C2×C4⋊C4×C23C2314yes—
72C2×C4⋊C4×C23C2314yes—
73C2×C4⋊C4×C23C2312yes—
74C2×C4⋊C4×C23C2312yes—
75C2×C4⋊C4×C23C2314yes—
76C2×C4⋊C4×C23C2314yes—
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×C23C2314yes—
85C2×C22⋊C4×C23C2314yes—
86C2×C22⋊C4×C23C2314yes—
87C2×C22⋊C4×C23C2314yes—
88C2×C22⋊C4×C23C2314yes—
89C2×C22⋊C4×C23C2314yes—
90C2×C42×C23C2311yescharacteristic
91C2×C42×C23C2312yes—
92C2×C42×C23C2314yes—
93C2×C42×C23C2314yes—
94C2×C42×C23C2312yes—
95C2×C42×C23C2311yescharacteristic
96C2×C42×C23C2311yescharacteristic
97C23×#95C2314yes—
98C23×#95C2314yes—
99C23×#95C2314yes—
100C23×#95C2314yes—
101C23×#95C2314yes—
102C23×#95C2314yes—
103C23×#95C2314yes—
104C23×#95C2314yes—
105C23×#95C2314yes—
106C23×#95C2314yes—
107C23×#95C2314yes—
108C24×C23C2414yes—
109C2×Q8×C23—24no—
110C2×Q8×C23—42no—
111C2×Q8×C23—42no—
112C2×D4×C23—24no—
113C2×D4×C23—42no—
114C2×D4×C23—44no—
115C2×D4×C23—42no—
116C2×D4×C23—44no—
117C2×D4×C23—44no—
118C22×C4×C23—28no—
119C22×C4×C23—28no—
120C22×C4×C23—28no—
121C22×C4×C23C2412yes—
122C22×C4×C23C2411yescharacteristic
123C22×C4×C23C2414yes—
124C22×C4×C23C2414yes—
125C22×C4×C23C2414yes—
126C22×C4×C23C2414yes—
127C22×C4×C23C2414yes—
128C22×C4×C23C2411yescharacteristic
129C22×C4×C23C2412yes—
130C22×C4×C23C2411yescharacteristic
131C4⋊C4×C23—24no—
132C4⋊C4×C23—28no—
133C4⋊C4×C23—42no—
134C4⋊C4×C23—44no—
135C4⋊C4×C23—44no—
136C4⋊C4×C23—44no—
137C4⋊C4×C23—44no—
138C4⋊C4×C23—44no—
139C4⋊C4×C23—42no—
140C4⋊C4×C23—44no—
141C4⋊C4×C23—42no—
142C4⋊C4×C23—44no—
143C22⋊C4×C23—28no—
144C22⋊C4×C23—28no—
145C22⋊C4×C23—44no—
146C22⋊C4×C23—44no—
147C22⋊C4×C23—44no—
148C22⋊C4×C23—44no—
149C22⋊C4×C23—44no—
150C22⋊C4×C23—44no—
151C22⋊C4×C23—44no—
152C22⋊C4×C23—44no—
153C22⋊C4×C23—44no—
154C42×C23—28no—
155C42×C23—28no—
156C42×C23—24no—
157C42×C23—22no—
158C42×C23—41no—
159C42×C23—42no—
160C42×C23#4114yes—
161#36695C2311yesmaximal, characteristic, direct product factor, Sylow
162C23×C23—24no—
163C23×C23—24no—
164C23×C23—24no—
165C23×C23—24no—
166C23×C23—24no—
167C23×C23—24no—
168C23×C23—24no—
169C23×C23C2511yescharacteristic
170Q8×C23—48no—
171D4×C23—48no—
172D4×C23—416no—
173C2×C4×C23—22no—
174C2×C4×C23—21no—
175C2×C4×C23—21no—
176C2×C4×C23—22no—
177C2×C4×C23—24no—
178C2×C4×C23—24no—
179C2×C4×C23—24no—
180C2×C4×C23—22no—
181C2×C4×C23—24no—
182C2×C4×C23—24no—
183C2×C4×C23—24no—
184C2×C4×C23—24no—
185C2×C4×C23—24no—
186C2×C4×C23—21no—
187C2×C4×C23—24no—
188C2×C4×C23—22no—
189C2×C4×C23—22no—
190C2×C4×C23—22no—
191C2×C4×C23—24no—
192C2×C4×C23—24no—
193C2×C4×C23—24no—
194C2×C4×C23—24no—
195C2×C4×C23—24no—
196C2×C4×C23—24no—
197C2×C4×C23—416no—
198C2×C4×C23C2×#4112yes—
199C2×C4×C23C2×#4114yes—
200C2×C4×C23H(5,𝔽2)14yes—
201C2×C4×C232−1+412yes—
202#2845C2×C2311yescharacteristic, complemented
203#2837C2×C2314yescomplemented
204#2831C2×C2314yescomplemented
205#2829C2×C2314yescomplemented
206#2820C2×C2312yescomplemented
207#2819C2×C2311yescharacteristic
208#2396C2×C2314yescomplemented
209#2395C2×C2314yescomplemented
210#2392C2×C2314yescomplemented
211C4×#125C2×C2311yescharacteristic, complemented
212C4×#124C2×C2312yescomplemented
213C22×C23—44no—
214C22×C23—44no—
215C22×C23—44no—
216C22×C23—44no—
217C22×C23—44no—
218C22×C23—44no—
219C22×C23—44no—
220C22×C23C22×#4111yescharacteristic
221C22×C23C2×H(5,𝔽2)12yes—
222C22×C23C2×2−1+411yescharacteristic
223C22×C23#58512yes—
224C22×C23#58511yescharacteristic
225C4×C23—24no—
226C4×C23—22no—
227C4×C23—44no—
228C4×C23—44no—
229C4×C23—44no—
230C4×C23—42no—
231C4×C23—44no—
232C4×C23—44no—
233C4×C23—41no—
234C4×C23#56214yes—
235C2×#125C22×C2311yescharacteristic
236C2×#124C22×C2312yes—
237C2×C4×Q8C22×C2312yes—
238C2×C4×D4C22×C2314yes—
239C2×C4×D4C22×C2312yes—
240C2×#117C22×C2314yes—
241#400C22×C2314yes—
242#400C22×C2314yes—
243#399C22×C2314yes—
244#399C22×C2314yes—
245#398C22×C2314yes—
246#398C22×C2314yes—
247#398C22×C2314yes—
248#398C22×C2314yes—
249#397C22×C2314yes—
250#397C22×C2314yes—
251#397C22×C2314yes—
252#397C22×C2314yes—
253#396C22×C2314yes—
254#396C22×C2314yes—
255#396C22×C2314yes—
256#396C22×C2314yes—
257#395C22×C2314yes—
258#394C22×C2314yes—
259#394C22×C2314yes—
260#394C22×C2314yes—
261#394C22×C2314yes—
262#394C22×C2314yes—
263#393C22×C2314yes—
264#392C22×C2314yes—
265#391C22×C2312yes—
266#390C22×C2312yes—
267#389C22×C2312yes—
268#388C22×C2314yes—
269#388C22×C2314yes—
270#387C22×C2314yes—
271#382C22×C2311yescharacteristic
272#382C22×C2312yes—
273C4×C4⋊C4C22×C2314yes—
274C4×C4⋊C4C22×C2312yes—
275C4×C4⋊C4C22×C2314yes—
276C4×C22⋊C4C22×C2314yes—
277C4×C22⋊C4C22×C2314yes—
278C4×C22⋊C4C22×C2314yes—
279C43C22×C2311yescharacteristic
280C2×C23—84no—
281C2×C23#346012yes—
282C2×C23#346111yescharacteristic
283C2×C23C2×#56211yescharacteristic
284C2×C23#352912yes—
285C2×C23#353111yescharacteristic
286C22×Q8C23×C2312yes—
287C22×D4C23×C2312yes—
288C22×D4C23×C2314yes—
289C23×C4C23×C2314yes—
290#125—24no—
291#124—28no—
292C4×Q8—44no—
293C4×D4—48no—
294C4×D4—44no—
295#117—48no—
296C2×C4⋊C4C23×C2314yes—
297C2×C4⋊C4C23×C2312yes—
298C2×C4⋊C4C23×C2314yes—
299C2×C4⋊C4C23×C2314yes—
300C2×C4⋊C4C23×C2312yes—
301C2×C4⋊C4C23×C2314yes—
302C2×C4⋊C4C23×C2312yes—
303C2×C4⋊C4C23×C2314yes—
304C2×C4⋊C4C23×C2312yes—
305C2×C4⋊C4C23×C2314yes—
306C2×C4⋊C4C23×C2314yes—
307C2×C4⋊C4C23×C2314yes—
308C2×C22⋊C4C23×C2314yes—
309C2×C22⋊C4C23×C2314yes—
310C2×C22⋊C4C23×C2314yes—
311C2×C22⋊C4C23×C2314yes—
312C2×C22⋊C4C23×C2314yes—
313C2×C22⋊C4C23×C2314yes—
314C2×C22⋊C4C23×C2314yes—
315C2×C22⋊C4C23×C2314yes—
316C2×C22⋊C4C23×C2314yes—
317C2×C22⋊C4C23×C2314yes—
318C2×C22⋊C4C23×C2314yes—
319C2×C42C23×C2314yes—
320C2×C42C23×C2314yes—
321C2×C42C23×C2312yes—
322C2×C42C23×C2311yescharacteristic
323C2×C42C23×C2312yes—
324C2×C42C23×C2311yescharacteristic
325C2×C42C23×C2311yescharacteristic
326#95C23×C2314yes—
327#95C23×C2314yes—
328#95C23×C2314yes—
329#95C23×C2314yes—
330#95C23×C2314yes—
331#95C23×C2314yes—
332#95C23×C2314yes—
333#95C23×C2314yes—
334#95C23×C2314yes—
335#95C23×C2314yes—
336#95C23×C2314yes—
337C23#3669511yescharacteristic, direct product factor, Sylow
338C24C24×C2314yes—
339C2×Q8—24no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—42no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4C24×C2311yescharacteristic
352C22×C4C24×C2314yes—
353C22×C4C24×C2314yes—
354C22×C4C24×C2314yes—
355C22×C4C24×C2312yes—
356C22×C4C24×C2314yes—
357C22×C4C24×C2314yes—
358C22×C4C24×C2311yescharacteristic
359C22×C4C24×C2312yes—
360C22×C4C24×C2311yescharacteristic
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C22⋊C4—28no—
374C22⋊C4—28no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C42—28no—
385C42—28no—
386C42—24no—
387C42—22no—
388C42—41no—
389C42—42no—
390C42#41×C2314yes—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23C25×C2311yescharacteristic
399Q8—48no—
400D4—48no—
401D4—416no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—416no—
427C2×C4C2×#41×C2312yes—
428C2×C4C2×#41×C2314yes—
429C2×C4C23×H(5,𝔽2)14yes—
430C2×C4C23×2−1+412yes—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22C22×#41×C2311yescharacteristic
439C22C2×C23×H(5,𝔽2)12yes—
440C22C2×C23×2−1+411yescharacteristic
441C22C23×#58512yes—
442C22C23×#58511yescharacteristic
443C4—24no—
444C4—22no—
445C4—44no—
446C4—44no—
447C4—44no—
448C4—44no—
449C4—42no—
450C4—44no—
451C4—41no—
452C4C23×#56214yes—
453C2—84no—
454C2C23×#346012yes—
455C2C23×#346111yescharacteristic
456C2C2×C23×#56211yescharacteristic
457C2C23×#352912yes—
458C2C23×#353111yescharacteristic
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC23×#2819C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2845C2semidirect
2. C23×#2837C2semidirect
3. C23×#2831C2semidirect
4. C23×#2829C2semidirect
5. C23×#2820C2semidirect
7. C23×#2396C2semidirect
8. C23×#2395C2semidirect
9. C23×#2392C2semidirect
10. C4×C23×#125C2semidirect
11. C4×C23×#124C2semidirect
161. #36695C23direct (N × H)
202. #2845C2×C23semidirect
203. #2837C2×C23semidirect
204. #2831C2×C23semidirect
205. #2829C2×C23semidirect
206. #2820C2×C23semidirect
208. #2396C2×C23semidirect
209. #2395C2×C23semidirect
210. #2392C2×C23semidirect
211. C4×#125C2×C23semidirect
212. C4×#124C2×C23semidirect
337. C23#36695direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28452
C23×#28372
C23×#28312
C23×#28292
C23×#28202
C23×#28192
C23×#23962
C23×#23952
C23×#23922
C4×C23×#1252
C4×C23×#1242
#3669523