Finite Group Databasebeta

← C23×#16565 #19164191

Subgroups — C23×#16565 #19164191

Order 5888.

1258 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#395C211yesmaximal, characteristic
2C2×C23×#391C211yesmaximal, characteristic
3C2×C23×#387C211yesmaximal, characteristic
4C2×C23×#382C211yesmaximal, characteristic
5C23×#1446C211yesmaximal, characteristic
6C23×#1445C212yesmaximal
7C23×#1443C212yesmaximal
8C23×#1442C211yesmaximal, characteristic
9C23×#1441C212yesmaximal
10C23×#1441C212yesmaximal
11C23×#1441C211yesmaximal, characteristic
12C23×Q8×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2212yes—
15C22×C4⋊C4×C23C2212yes—
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C4⋊C4×C23C2211yescharacteristic
18C22×C42×C23C2211yescharacteristic
19C23×#395—24no—
20C23×#391—24no—
21C23×#387—24no—
22C23×#382—24no—
23C2×C23×#95C2211yescharacteristic
24C2×C23×#95C2212yes—
25C2×C23×#95C2212yes—
26C2×C23×#95C2212yes—
27C2×C23×#95C2211yescharacteristic
28C2×C23×#95C2211yescharacteristic
29C2×C23×#95C2211yescharacteristic
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2211yescharacteristic
33C2×C23×#95C2212yes—
34C2×C23×#95C2212yes—
35C2×C23×#95C2212yes—
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2211yescharacteristic
39C22×Q8×C23—22no—
40C22×Q8×C23—22no—
41C22×Q8×C23—24no—
42C22×Q8×C23—22no—
43C22×Q8×C23—42no—
44C23×C4×C23C2311yescharacteristic
45C23×C4×C23C2312yes—
46C23×C4×C23C2312yes—
47C23×C4×C23C2311yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2312yes—
50C23×C4×C23C2311yescharacteristic
51C23×C4×C23C2311yescharacteristic
52C23×C4×C23C2312yes—
53C23×C4×C23C2311yescharacteristic
54C23×C4×C23C2311yescharacteristic
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—24no—
58C2×C4⋊C4×C23—22no—
59C2×C4⋊C4×C23—22no—
60C2×C4⋊C4×C23—24no—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23—24no—
63C2×C4⋊C4×C23—24no—
64C2×C4⋊C4×C23—22no—
65C2×C4⋊C4×C23—42no—
66C2×C4⋊C4×C23—42no—
67C2×C4⋊C4×C23—42no—
68C2×C4⋊C4×C23Q814yescomplemented
69C2×C42×C23—22no—
70C2×C42×C23—22no—
71C2×C42×C23D414yes—
72C23×#95—24no—
73C23×#95—22no—
74C23×#95—22no—
75C23×#95—22no—
76C23×#95—22no—
77C23×#95—22no—
78C23×#95—24no—
79C23×#95—24no—
80C23×#95—24no—
81C23×#95—24no—
82C23×#95—42no—
83C23×#95—42no—
84C23×#95—42no—
85C23×#95—42no—
86C23×#95—42no—
87C23×#95D414yes—
88C24×C23C2411yescharacteristic
89C2×Q8×C23—28no—
90C2×Q8×C23—48no—
91C2×Q8×C23—48no—
92C2×Q8×C23—44no—
93C2×Q8×C23—44no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—22no—
97C22×C4×C23—21no—
98C22×C4×C23—22no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—21no—
102C22×C4×C23—22no—
103C22×C4×C23—21no—
104C22×C4×C23—21no—
105C22×C4×C23—22no—
106C22×C4×C23—21no—
107C22×C4×C23—21no—
108C22×C4×C23—21no—
109C22×C4×C23—22no—
110C22×C4×C23—22no—
111C22×C4×C23—21no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—21no—
120C22×C4×C23—22no—
121C22×C4×C23—22no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—21no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—22no—
129C22×C4×C23—22no—
130C22×C4×C23—21no—
131C22×C4×C23—21no—
132C22×C4×C23—22no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—22no—
138C22×C4×C23—22no—
139C22×C4×C23—22no—
140C22×C4×C23—21no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23—22no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—21no—
149C22×C4×C23—22no—
150C22×C4×C23C2×D412yes—
151C22×C4×C23C2×D414yes—
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2×D412yes—
154C22×C4×C23C2×D412yes—
155C22×C4×C23C2×Q812yes—
156C22×C4×C23C2×Q812yes—
157C22×C4×C23C2×Q812yes—
158C22×C4×C23#4114yes—
159C22×C4×C23#4112yes—
160C22×C4×C23#4112yes—
161C4⋊C4×C23—28no—
162C4⋊C4×C23—48no—
163C4⋊C4×C23—48no—
164C4⋊C4×C23—44no—
165C4⋊C4×C23—44no—
166C42×C23—28no—
167#16565C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23C22×D411yescharacteristic
169C23×C23C22×D411yescharacteristic
170C23×C23C22×Q811yescharacteristic
171C23×C23C2×#4111yescharacteristic
172C23×C23C2×#4111yescharacteristic
173C23×C23C2×#4112yes—
174C23×C23H(5,𝔽2)12yes—
175C23×C23H(5,𝔽2)11yescharacteristic
176C23×C232−1+412yes—
177C23×C232−1+412yes—
178C23×C232−1+411yescharacteristic
179Q8×C23—416no—
180C2×C4×C23—22no—
181C2×C4×C23—22no—
182C2×C4×C23—22no—
183C2×C4×C23—22no—
184C2×C4×C23—22no—
185C2×C4×C23—24no—
186C2×C4×C23—22no—
187C2×C4×C23—24no—
188C2×C4×C23—22no—
189C2×C4×C23—22no—
190C2×C4×C23—22no—
191C2×C4×C23—22no—
192C2×C4×C23—22no—
193C2×C4×C23—24no—
194C2×C4×C23—22no—
195C2×C4×C23—22no—
196C2×C4×C23—24no—
197C2×C4×C23—24no—
198C2×C4×C23—24no—
199C2×C4×C23—24no—
200C2×C4×C23—24no—
201C2×C4×C23—22no—
202C2×C4×C23—24no—
203C2×C4×C23—22no—
204C2×C4×C23—41no—
205C2×C4×C23—42no—
206C2×C4×C23—41no—
207C2×C4×C23—42no—
208C2×C4×C23—42no—
209C2×C4×C23—41no—
210C2×C4×C23—41no—
211C2×C4×C23—42no—
212C2×C4×C23—42no—
213C2×C4×C23—42no—
214C2×C4×C23—42no—
215C2×C4×C23—41no—
216C2×C4×C23—42no—
217C2×C4×C23—42no—
218C2×C4×C23—41no—
219C2×C4×C23—42no—
220C2×C4×C23—42no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—42no—
225C2×C4×C23—41no—
226C2×C4×C23—42no—
227C2×C4×C23—41no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—41no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—42no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23#12114yes—
246C2×C4×C23C22⋊Q814yes—
247C2×#395C2×C2311yescharacteristic
248C2×#391C2×C2311yescharacteristic
249C2×#387C2×C2311yescharacteristic
250C2×#382C2×C2311yescharacteristic
251#1446C2×C2311yescharacteristic
252#1445C2×C2312yes—
253#1443C2×C2312yes—
254#1442C2×C2311yescharacteristic
255#1441C2×C2311yescharacteristic
256#1441C2×C2312yes—
257#1441C2×C2312yes—
258C22×C23C2×#12111yescharacteristic
259C22×C23C2×C22⋊Q811yescharacteristic
260C22×C23#52511yescharacteristic
261C22×C23#53511yescharacteristic
262C22×C23#53611yescharacteristic
263C22×C23#53611yescharacteristic
264C22×C23#53612yes—
265C22×C23#53711yescharacteristic
266C22×C23#54112yes—
267C22×C23#54112yes—
268C22×C23#54212yes—
269C22×C23#54212yes—
270C22×C23#54412yes—
271C22×C23#54612yes—
272C22×C23#54611yescharacteristic
273C22×C23D4×Q811yescharacteristic
274C22×C23#55112yes—
275C22×C23#55211yescharacteristic
276C22×C23#55411yescharacteristic
277C22×C23#55612yes—
278C22×C23#55712yes—
279C22×C23#56212yes—
280C22×C23#56312yes—
281C4×C23—24no—
282C4×C23—24no—
283C4×C23—42no—
284C4×C23—42no—
285C4×C23—44no—
286C4×C23—42no—
287C4×C23—44no—
288C4×C23—82no—
289C4×C23—81no—
290C4×C23—81no—
291C4×C23—82no—
292C23×Q8C22×C2311yescharacteristic
293C22×C4⋊C4C22×C2311yescharacteristic
294C22×C4⋊C4C22×C2312yes—
295C22×C4⋊C4C22×C2312yes—
296C22×C4⋊C4C22×C2311yescharacteristic
297C22×C4⋊C4C22×C2311yescharacteristic
298C22×C42C22×C2311yescharacteristic
299#395—24no—
300#391—24no—
301#387—24no—
302#382—24no—
303C2×#95C22×C2312yes—
304C2×#95C22×C2311yescharacteristic
305C2×#95C22×C2312yes—
306C2×#95C22×C2312yes—
307C2×#95C22×C2311yescharacteristic
308C2×#95C22×C2311yescharacteristic
309C2×#95C22×C2311yescharacteristic
310C2×#95C22×C2312yes—
311C2×#95C22×C2312yes—
312C2×#95C22×C2311yescharacteristic
313C2×#95C22×C2312yes—
314C2×#95C22×C2312yes—
315C2×#95C22×C2312yes—
316C2×#95C22×C2312yes—
317C2×#95C22×C2312yes—
318C2×#95C22×C2311yescharacteristic
319C2×C23#241111yescharacteristic
320C2×C23#241711yescharacteristic
321C2×C23#258111yescharacteristic
322C2×C23#262111yescharacteristic
323C2×C23#262811yescharacteristic
324C2×C23#263711yescharacteristic
325C2×C23#270012yes—
326C2×C23#271312yes—
327C2×C23#275512yes—
328C2×C23#276212yes—
329C2×C23#282011yescharacteristic
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8—24no—
333C22×Q8—22no—
334C22×Q8—42no—
335C23×C4C23×C2311yescharacteristic
336C23×C4C23×C2312yes—
337C23×C4C23×C2312yes—
338C23×C4C23×C2311yescharacteristic
339C23×C4C23×C2312yes—
340C23×C4C23×C2311yescharacteristic
341C23×C4C23×C2311yescharacteristic
342C23×C4C23×C2311yescharacteristic
343C23×C4C23×C2312yes—
344C23×C4C23×C2311yescharacteristic
345C23×C4C23×C2311yescharacteristic
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4—42no—
359C2×C4⋊C4Q8×C2314yescomplemented
360C2×C42—22no—
361C2×C42—22no—
362C2×C42D4×C2314yes—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—42no—
378#95D4×C2314yes—
379C23#1656511yescharacteristic, direct product factor, Sylow
380C24C24×C2311yescharacteristic
381C2×Q8—28no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4C2×D4×C2312yes—
443C22×C4C2×D4×C2314yes—
444C22×C4C2×D4×C2312yes—
445C22×C4C2×D4×C2312yes—
446C22×C4C2×D4×C2312yes—
447C22×C4C2×Q8×C2312yes—
448C22×C4C2×Q8×C2312yes—
449C22×C4C2×Q8×C2312yes—
450C22×C4#41×C2314yes—
451C22×C4#41×C2312yes—
452C22×C4#41×C2312yes—
453C4⋊C4—28no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C42—28no—
459C23C22×D4×C2311yescharacteristic
460C23C22×D4×C2311yescharacteristic
461C23C22×Q8×C2311yescharacteristic
462C23C2×#41×C2311yescharacteristic
463C23C2×#41×C2312yes—
464C23C2×#41×C2311yescharacteristic
465C23C23×H(5,𝔽2)12yes—
466C23C23×H(5,𝔽2)11yescharacteristic
467C23C23×2−1+411yescharacteristic
468C23C23×2−1+412yes—
469C23C23×2−1+412yes—
470Q8—416no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4C23×#12114yes—
537C2×C4C23×C22⋊Q814yes—
538C22C2×C23×#12111yescharacteristic
539C22C2×C23×C22⋊Q811yescharacteristic
540C22C23×#52511yescharacteristic
541C22C23×#53511yescharacteristic
542C22C23×#53612yes—
543C22C23×#53611yescharacteristic
544C22C23×#53611yescharacteristic
545C22C23×#53711yescharacteristic
546C22C23×#54112yes—
547C22C23×#54112yes—
548C22C23×#54212yes—
549C22C23×#54212yes—
550C22C23×#54412yes—
551C22C23×#54612yes—
552C22C23×#54611yescharacteristic
553C22D4×Q8×C2311yescharacteristic
554C22C23×#55112yes—
555C22C23×#55211yescharacteristic
556C22C23×#55411yescharacteristic
557C22C23×#55612yes—
558C22C23×#55712yes—
559C22C23×#56212yes—
560C22C23×#56312yes—
561C4—24no—
562C4—24no—
563C4—44no—
564C4—42no—
565C4—44no—
566C4—42no—
567C4—42no—
568C4—81no—
569C4—82no—
570C4—82no—
571C4—81no—
572C2C23×#241111yescharacteristic
573C2C23×#241711yescharacteristic
574C2C23×#258111yescharacteristic
575C2C23×#262111yescharacteristic
576C2C23×#262811yescharacteristic
577C2C23×#263711yescharacteristic
578C2C23×#270012yes—
579C2C23×#271312yes—
580C2C23×#275512yes—
581C2C23×#276212yes—
582C2C23×#282011yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C23C24
commutator, Frattini, cosocleC24C24×C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
68. C2×C4⋊C4×C23Q8semidirect
167. #16565C23direct (N × H)
359. C2×C4⋊C4Q8×C23semidirect
379. C23#16565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3952
C2×C23×#3912
C2×C23×#3872
C2×C23×#3822
C23×#14462
C23×#14452
C23×#14432
C23×#14422
C23×#14412
#1656523