Finite Group Databasebeta

← C23×#16567 #19164193

Subgroups — C23×#16567 #19164193

Order 5888.

1194 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#400C211yesmaximal, characteristic
2C2×C23×#389C211yesmaximal, characteristic
3C2×C23×#387C211yesmaximal, characteristic
4C2×C23×#382C211yesmaximal, characteristic
5C23×#1445C211yesmaximal, characteristic
6C23×#1444C211yesmaximal, characteristic
7C23×#1442C212yesmaximal
8C23×#1441C211yesmaximal, characteristic
9C23×#1441C212yesmaximal
10C23×#1441C212yesmaximal
11C23×#1441C212yesmaximal
12C22×C4⋊C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2212yes—
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C4⋊C4×C23C2212yes—
18C22×C4⋊C4×C23C2212yes—
19C22×C42×C23C2211yescharacteristic
20C23×#400—24no—
21C23×#389—24no—
22C23×#387—24no—
23C23×#382—24no—
24C2×C23×#95C2212yes—
25C2×C23×#95C2211yescharacteristic
26C2×C23×#95C2211yescharacteristic
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2211yescharacteristic
32C2×C23×#95C2211yescharacteristic
33C2×C23×#95C2211yescharacteristic
34C2×C23×#95C2212yes—
35C2×C23×#95C2212yes—
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2211yescharacteristic
39C23×C4×C23C2311yescharacteristic
40C23×C4×C23C2312yes—
41C23×C4×C23C2312yes—
42C23×C4×C23C2311yescharacteristic
43C23×C4×C23C2312yes—
44C23×C4×C23C2311yescharacteristic
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2311yescharacteristic
47C23×C4×C23C2312yes—
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2311yescharacteristic
50C2×C4⋊C4×C23—22no—
51C2×C4⋊C4×C23—22no—
52C2×C4⋊C4×C23—24no—
53C2×C4⋊C4×C23—22no—
54C2×C4⋊C4×C23—24no—
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—24no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—22no—
59C2×C4⋊C4×C23—22no—
60C2×C4⋊C4×C23—24no—
61C2×C4⋊C4×C23—24no—
62C2×C4⋊C4×C23—22no—
63C2×C4⋊C4×C23—22no—
64C2×C4⋊C4×C23—42no—
65C2×C4⋊C4×C23—42no—
66C2×C4⋊C4×C23—42no—
67C2×C4⋊C4×C23—42no—
68C2×C4⋊C4×C23—42no—
69C2×C4⋊C4×C23Q814yes—
70C2×C42×C23—22no—
71C2×C42×C23—22no—
72C2×C42×C23D414yes—
73C23×#95—24no—
74C23×#95—22no—
75C23×#95—22no—
76C23×#95—22no—
77C23×#95—22no—
78C23×#95—22no—
79C23×#95—24no—
80C23×#95—24no—
81C23×#95—24no—
82C23×#95—24no—
83C23×#95—42no—
84C23×#95—42no—
85C23×#95—42no—
86C23×#95—42no—
87C23×#95D414yes—
88C24×C23C2411yescharacteristic
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—22no—
92C22×C4×C23—21no—
93C22×C4×C23—22no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—22no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—21no—
102C22×C4×C23—21no—
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23—21no—
107C22×C4×C23—22no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—22no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—21no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—21no—
121C22×C4×C23—22no—
122C22×C4×C23—21no—
123C22×C4×C23—22no—
124C22×C4×C23—21no—
125C22×C4×C23—22no—
126C22×C4×C23—21no—
127C22×C4×C23—22no—
128C22×C4×C23—21no—
129C22×C4×C23—22no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—22no—
133C22×C4×C23—21no—
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—22no—
141C22×C4×C23—22no—
142C22×C4×C23—21no—
143C22×C4×C23—22no—
144C22×C4×C23—22no—
145C22×C4×C23C2×D412yes—
146C22×C4×C23C2×D414yes—
147C22×C4×C23C2×D412yes—
148C22×C4×C23C2×D412yes—
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2×Q812yes—
151C22×C4×C23C2×Q812yes—
152C22×C4×C23C2×Q812yes—
153C22×C4×C23#4114yes—
154C22×C4×C23#4112yes—
155C22×C4×C23#4112yes—
156C4⋊C4×C23—28no—
157C4⋊C4×C23—28no—
158C4⋊C4×C23—48no—
159C4⋊C4×C23—48no—
160C4⋊C4×C23—48no—
161C4⋊C4×C23—44no—
162C4⋊C4×C23—44no—
163C42×C23—28no—
164#16567C2311yesmaximal, characteristic, direct product factor, Sylow
165C23×C23C22×D411yescharacteristic
166C23×C23C22×D411yescharacteristic
167C23×C23C22×Q811yescharacteristic
168C23×C23C2×#4111yescharacteristic
169C23×C23C2×#4111yescharacteristic
170C23×C23C2×#4112yes—
171C23×C23H(5,𝔽2)12yes—
172C23×C23H(5,𝔽2)11yescharacteristic
173C23×C232−1+412yes—
174C23×C232−1+412yes—
175C23×C232−1+411yescharacteristic
176C2×C4×C23—22no—
177C2×C4×C23—22no—
178C2×C4×C23—22no—
179C2×C4×C23—22no—
180C2×C4×C23—22no—
181C2×C4×C23—24no—
182C2×C4×C23—22no—
183C2×C4×C23—24no—
184C2×C4×C23—22no—
185C2×C4×C23—22no—
186C2×C4×C23—22no—
187C2×C4×C23—22no—
188C2×C4×C23—22no—
189C2×C4×C23—24no—
190C2×C4×C23—22no—
191C2×C4×C23—22no—
192C2×C4×C23—24no—
193C2×C4×C23—24no—
194C2×C4×C23—24no—
195C2×C4×C23—24no—
196C2×C4×C23—24no—
197C2×C4×C23—24no—
198C2×C4×C23—22no—
199C2×C4×C23—22no—
200C2×C4×C23—41no—
201C2×C4×C23—42no—
202C2×C4×C23—41no—
203C2×C4×C23—41no—
204C2×C4×C23—42no—
205C2×C4×C23—42no—
206C2×C4×C23—41no—
207C2×C4×C23—42no—
208C2×C4×C23—42no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—42no—
212C2×C4×C23—42no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—42no—
216C2×C4×C23—42no—
217C2×C4×C23—42no—
218C2×C4×C23—41no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—42no—
222C2×C4×C23—41no—
223C2×C4×C23—42no—
224C2×C4×C23—41no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—41no—
228C2×C4×C23—42no—
229C2×C4×C23—42no—
230C2×C4×C23—42no—
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#12114yes—
242C2×C4×C23C22⋊Q814yes—
243C2×#400C2×C2311yescharacteristic
244C2×#389C2×C2311yescharacteristic
245C2×#387C2×C2311yescharacteristic
246C2×#382C2×C2311yescharacteristic
247#1445C2×C2311yescharacteristic
248#1444C2×C2311yescharacteristic
249#1442C2×C2312yes—
250#1441C2×C2311yescharacteristic
251#1441C2×C2312yes—
252#1441C2×C2312yes—
253#1441C2×C2312yes—
254C22×C23C2×#12111yescharacteristic
255C22×C23C2×C22⋊Q811yescharacteristic
256C22×C23#52511yescharacteristic
257C22×C23#53511yescharacteristic
258C22×C23#53611yescharacteristic
259C22×C23#53611yescharacteristic
260C22×C23#53711yescharacteristic
261C22×C23#53712yes—
262C22×C23#53812yes—
263C22×C23#53912yes—
264C22×C23#54212yes—
265C22×C23#54212yes—
266C22×C23#54412yes—
267C22×C23#54611yescharacteristic
268C22×C23#54712yes—
269C22×C23D4×Q811yescharacteristic
270C22×C23#55212yes—
271C22×C23#55211yescharacteristic
272C22×C23#55411yescharacteristic
273C22×C23#55612yes—
274C22×C23#55712yes—
275C22×C23#56312yes—
276C22×C23#56412yes—
277C4×C23—24no—
278C4×C23—24no—
279C4×C23—42no—
280C4×C23—42no—
281C4×C23—44no—
282C4×C23—42no—
283C4×C23—44no—
284C4×C23—82no—
285C4×C23—81no—
286C4×C23—81no—
287C4×C23—82no—
288C22×C4⋊C4C22×C2311yescharacteristic
289C22×C4⋊C4C22×C2311yescharacteristic
290C22×C4⋊C4C22×C2311yescharacteristic
291C22×C4⋊C4C22×C2312yes—
292C22×C4⋊C4C22×C2311yescharacteristic
293C22×C4⋊C4C22×C2312yes—
294C22×C4⋊C4C22×C2312yes—
295C22×C42C22×C2311yescharacteristic
296#400—24no—
297#389—24no—
298#387—24no—
299#382—24no—
300C2×#95C22×C2312yes—
301C2×#95C22×C2312yes—
302C2×#95C22×C2312yes—
303C2×#95C22×C2311yescharacteristic
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×C2312yes—
313C2×#95C22×C2312yes—
314C2×#95C22×C2311yescharacteristic
315C2×C23#241311yescharacteristic
316C2×C23#241711yescharacteristic
317C2×C23#258111yescharacteristic
318C2×C23#262011yescharacteristic
319C2×C23#262811yescharacteristic
320C2×C23#263611yescharacteristic
321C2×C23#271512yes—
322C2×C23#272612yes—
323C2×C23#276712yes—
324C2×C23#277412yes—
325C2×C23#282211yescharacteristic
326C23×C4C23×C2311yescharacteristic
327C23×C4C23×C2312yes—
328C23×C4C23×C2311yescharacteristic
329C23×C4C23×C2312yes—
330C23×C4C23×C2312yes—
331C23×C4C23×C2311yescharacteristic
332C23×C4C23×C2311yescharacteristic
333C23×C4C23×C2311yescharacteristic
334C23×C4C23×C2312yes—
335C23×C4C23×C2311yescharacteristic
336C23×C4C23×C2311yescharacteristic
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—42no—
354C2×C4⋊C4—42no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4Q8×C2314yes—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42D4×C2314yes—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95D4×C2314yes—
375C23#1656711yescharacteristic, direct product factor, Sylow
376C24C24×C2311yescharacteristic
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4C2×D4×C2314yes—
434C22×C4C2×D4×C2312yes—
435C22×C4C2×D4×C2312yes—
436C22×C4C2×D4×C2312yes—
437C22×C4C2×D4×C2312yes—
438C22×C4C2×Q8×C2312yes—
439C22×C4C2×Q8×C2312yes—
440C22×C4C2×Q8×C2312yes—
441C22×C4#41×C2314yes—
442C22×C4#41×C2312yes—
443C22×C4#41×C2312yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C42—28no—
452C23C22×D4×C2311yescharacteristic
453C23C22×D4×C2311yescharacteristic
454C23C22×Q8×C2311yescharacteristic
455C23C2×#41×C2312yes—
456C23C2×#41×C2311yescharacteristic
457C23C2×#41×C2311yescharacteristic
458C23C23×H(5,𝔽2)11yescharacteristic
459C23C23×H(5,𝔽2)12yes—
460C23C23×2−1+412yes—
461C23C23×2−1+411yescharacteristic
462C23C23×2−1+412yes—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4C23×#12114yes—
529C2×C4C23×C22⋊Q814yes—
530C22C2×C23×#12111yescharacteristic
531C22C2×C23×C22⋊Q811yescharacteristic
532C22C23×#52511yescharacteristic
533C22C23×#53511yescharacteristic
534C22C23×#53611yescharacteristic
535C22C23×#53611yescharacteristic
536C22C23×#53711yescharacteristic
537C22C23×#53712yes—
538C22C23×#53812yes—
539C22C23×#53912yes—
540C22C23×#54212yes—
541C22C23×#54212yes—
542C22C23×#54412yes—
543C22C23×#54611yescharacteristic
544C22C23×#54712yes—
545C22D4×Q8×C2311yescharacteristic
546C22C23×#55212yes—
547C22C23×#55211yescharacteristic
548C22C23×#55411yescharacteristic
549C22C23×#55612yes—
550C22C23×#55712yes—
551C22C23×#56312yes—
552C22C23×#56412yes—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—81no—
561C4—82no—
562C4—82no—
563C4—81no—
564C2C23×#241311yescharacteristic
565C2C23×#241711yescharacteristic
566C2C23×#258111yescharacteristic
567C2C23×#262011yescharacteristic
568C2C23×#262811yescharacteristic
569C2C23×#263611yescharacteristic
570C2C23×#271512yes—
571C2C23×#272612yes—
572C2C23×#276712yes—
573C2C23×#277412yes—
574C2C23×#282211yescharacteristic
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
164. #16567C23direct (N × H)
375. C23#16567direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4002
C2×C23×#3892
C2×C23×#3872
C2×C23×#3822
C23×#14452
C23×#14442
C23×#14422
C23×#14412
#1656723