Finite Group Databasebeta

← C23×#13348 #19160974

Subgroups — C23×#13348 #19160974

Order 5888.

900 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C23×B(4,𝔽2)C211yesmaximal, characteristic
2C2×C23×#455C211yesmaximal, characteristic, complemented
3C23×#2684C211yesmaximal, characteristic, complemented
4C2×C23×#354C211yesmaximal, characteristic, complemented
5C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6C23×#1790C211yesmaximal, characteristic, complemented
7C23×#1390C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#137—21no—
10C2×C23×#124C2211yescharacteristic
11C2×C23×#120—21no—
12C2×C23×#120C2211yescharacteristic
13C23×B(4,𝔽2)—22no—
14C23×B(4,𝔽2)—22no—
15C23×#455—22no—
16C23×#455—22no—
17C2×C23×#104—21no—
18C2×C23×#100C2211yescharacteristic
19C2×C23×#99—21no—
20C2×C23×#99—21no—
21C2×C23×#99C2211yescharacteristic
22C2×C23×#99C2211yescharacteristic
23C23×#397—21no—
24C23×#394—21no—
25C23×#393—21no—
26C23×#388—21no—
27C23×#386—21no—
28C23×#386—21no—
29C23×#386C2211yescharacteristic
30C23×#379—21no—
31C23×#354—22no—
32C23×AΣL(1,𝔽16)—22no—
33C25×C23D411yescharacteristic
34C23×H(5,𝔽2)—21no—
35C23×H(5,𝔽2)—21no—
36C23×H(5,𝔽2)—21no—
37C23×H(5,𝔽2)—42no—
38C23×H(5,𝔽2)D412yes—
39C2×#41×C23—21no—
40C2×#41×C23—41no—
41C22×Q8×C23D411yescharacteristic, complemented
42C22×D4×C23—21no—
43C22×D4×C23—21no—
44C22×D4×C23—21no—
45C22×D4×C23—41no—
46C22×D4×C23—41no—
47C22×D4×C23C2311yescharacteristic
48C23×C4×C23D411yescharacteristic, complemented
49C23×#137—24no—
50C23×#137—42no—
51C2×Q16×C23—41no—
52C2×SD16×C23—41no—
53C2×M16×C23—21no—
54C23×#124—21no—
55C23×#124—21no—
56C23×#124—21no—
57C23×#124—21no—
58C23×#120—24no—
59C23×#120—21no—
60C23×#120—21no—
61C23×#120—21no—
62C23×#120—21no—
63C23×#120—42no—
64C2×C4⋊C4×C23—21no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23—21no—
68C2×C22⋊C4×C23—21no—
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—21no—
71C2×C22⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—21no—
74C2×C22⋊C4×C23—21no—
75C2×C22⋊C4×C23—41no—
76C2×C22⋊C4×C23—41no—
77C2×C22⋊C4×C23D411yescharacteristic
78C2×C22⋊C4×C23D411yescharacteristic, complemented
79C2×C42×C23D411yescharacteristic, complemented
80C23×#104—24no—
81C23×#100—21no—
82C23×#100D412yescomplemented
83C23×#99—24no—
84C23×#99—21no—
85C23×#99—21no—
86C23×#99—21no—
87C23×#99—42no—
88C23×#99D412yes—
89C23×#95—21no—
90C23×#95—21no—
91C23×#95—21no—
92C23×#95—21no—
93C23×#95—41no—
94C24×C23—21no—
95C24×C23—21no—
96C24×C23—21no—
97C24×C23—21no—
98C24×C23—21no—
99C24×C23—41no—
100C24×C23—41no—
101C24×C23—42no—
102C24×C23—81no—
103C24×C23C2×D411yescharacteristic
104C24×C23C2×D411yescharacteristic
105#41×C23—22no—
106#41×C23—22no—
107#41×C23—42no—
108#41×C23—41no—
109#41×C23—42no—
110#41×C23—42no—
111#41×C23—42no—
112#41×C23—41no—
113C2×Q8×C23—21no—
114C2×Q8×C23—21no—
115C2×Q8×C23—21no—
116C2×Q8×C23—41no—
117C2×Q8×C23—41no—
118C2×D4×C23—21no—
119C2×D4×C23—22no—
120C2×D4×C23—22no—
121C2×D4×C23—22no—
122C2×D4×C23—22no—
123C2×D4×C23—41no—
124C2×D4×C23—41no—
125C2×D4×C23—42no—
126C2×D4×C23—42no—
127C2×D4×C23—41no—
128C2×D4×C23—41no—
129C2×D4×C23—41no—
130C2×D4×C23—41no—
131C2×D4×C23—42no—
132C2×D4×C23—42no—
133C2×D4×C23—41no—
134C2×D4×C23—41no—
135C2×D4×C23—41no—
136C2×D4×C23—44no—
137C2×D4×C23—42no—
138C2×D4×C23—41no—
139C2×D4×C23—41no—
140C2×D4×C23—81no—
141C2×D4×C23—82no—
142C2×D4×C23—82no—
143C2×D4×C23C2×D411yescharacteristic
144C2×D4×C23C2×D411yescharacteristic
145C2×D4×C23C2×D411yescharacteristic
146C2×D4×C23#4111yescharacteristic
147C22×C4×C23—21no—
148C22×C4×C23—21no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—21no—
153C22×C4×C23—21no—
154C22×C4×C23—41no—
155C22×C4×C23—41no—
156C22×C4×C23—41no—
157C22×C4×C23—41no—
158C22×C4×C23—81no—
159C22×C4×C23C2×D411yescharacteristic
160Q16×C23—44no—
161SD16×C23—44no—
162M16×C23—22no—
163M16×C23—41no—
164C2×C8×C23—41no—
165C4⋊C4×C23—81no—
166C22⋊C4×C23—21no—
167C22⋊C4×C23—22no—
168C22⋊C4×C23—21no—
169C22⋊C4×C23—44no—
170C22⋊C4×C23—42no—
171C22⋊C4×C23—41no—
172C22⋊C4×C23—41no—
173C22⋊C4×C23—41no—
174C22⋊C4×C23—41no—
175C22⋊C4×C23—41no—
176C22⋊C4×C23—41no—
177C22⋊C4×C23—41no—
178C22⋊C4×C23—41no—
179C22⋊C4×C23—41no—
180C22⋊C4×C23—41no—
181C22⋊C4×C23—81no—
182C22⋊C4×C23—81no—
183C22⋊C4×C23—81no—
184C22⋊C4×C23—81no—
185C22⋊C4×C23—82no—
186C42×C23—21no—
187C42×C23—21no—
188#13348C2311yesmaximal, characteristic, direct product factor, Sylow
189C23×C23—21no—
190C23×C23—21no—
191C23×C23—21no—
192C23×C23—21no—
193C23×C23—22no—
194C23×C23—22no—
195C23×C23—22no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—41no—
200C23×C23—42no—
201C23×C23—41no—
202C23×C23—41no—
203C23×C23—41no—
204C23×C23—41no—
205C23×C23—41no—
206C23×C23—42no—
207C23×C23—42no—
208C23×C23—41no—
209C23×C23—41no—
210C23×C23—41no—
211C23×C23—41no—
212C23×C23—81no—
213C23×C23—82no—
214C23×C23—81no—
215C23×C23—82no—
216C23×C23—84no—
217C23×C23—81no—
218C23×C23—81no—
219C23×C23—81no—
220C23×C23—81no—
221C23×C23—81no—
222C23×C23#12011yescharacteristic
223C23×C23#12011yescharacteristic
224C23×C23#12111yescharacteristic
225C23×C23#12111yescharacteristic
226C23×C23#12711yescharacteristic
227Q8×C23—22no—
228Q8×C23—41no—
229Q8×C23—42no—
230Q8×C23—81no—
231D4×C23—22no—
232D4×C23—41no—
233D4×C23—41no—
234D4×C23—41no—
235D4×C23—41no—
236D4×C23—41no—
237D4×C23—42no—
238D4×C23—42no—
239D4×C23—42no—
240D4×C23—42no—
241D4×C23—42no—
242D4×C23—42no—
243D4×C23—82no—
244D4×C23—82no—
245D4×C23—88no—
246C2×C4×C23—21no—
247C2×C4×C23—21no—
248C2×C4×C23—22no—
249C2×C4×C23—22no—
250C2×C4×C23—42no—
251C2×C4×C23—42no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—41no—
255C2×C4×C23—42no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—41no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—42no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—41no—
270C2×C4×C23—41no—
271C2×C4×C23—41no—
272C2×C4×C23—81no—
273C2×C4×C23—81no—
274C2×C4×C23—81no—
275C2×C4×C23—81no—
276C2×C4×C23—81no—
277C2×C4×C23—81no—
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12111yescharacteristic
280C8×C23—42no—
281C2×B(4,𝔽2)C2×C2311yescharacteristic
282C2×#455C2×C2311yescharacteristic, complemented
283#2684C2×C2311yescharacteristic, complemented
284C2×#354C2×C2311yescharacteristic, complemented
285C2×AΣL(1,𝔽16)C2×C2311yescharacteristic, complemented
286#1790C2×C2311yescharacteristic, complemented
287#1390C2×C2311yescharacteristic, complemented
288C22×C23—21no—
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—21no—
294C22×C23—41no—
295C22×C23—41no—
296C22×C23—41no—
297C22×C23—41no—
298C22×C23—42no—
299C22×C23—42no—
300C22×C23—41no—
301C22×C23—41no—
302C22×C23—41no—
303C22×C23—41no—
304C22×C23—44no—
305C22×C23—42no—
306C22×C23—42no—
307C22×C23—81no—
308C22×C23—81no—
309C22×C23—81no—
310C22×C23—81no—
311C22×C23—82no—
312C22×C23—82no—
313C22×C23—82no—
314C22×C23—82no—
315C22×C23—82no—
316C22×C23—81no—
317C22×C23—81no—
318C22×C23—81no—
319C22×C23—81no—
320C22×C23—161no—
321C22×C23—161no—
322C22×C23#39211yescharacteristic
323C22×C23B(4,𝔽2)11yescharacteristic
324C22×C23B(4,𝔽2)11yescharacteristic
325C4×C23—21no—
326C4×C23—21no—
327C4×C23—42no—
328C4×C23—81no—
329C4×C23—81no—
330C4×C23—81no—
331C4×C23—82no—
332C4×C23—81no—
333C4×C23—81no—
334C4×C23—161no—
335C2×H(5,𝔽2)C22×C2311yescharacteristic
336C2×#137—21no—
337C2×#124C22×C2311yescharacteristic
338C2×#120—21no—
339C2×#120C22×C2311yescharacteristic
340B(4,𝔽2)—22no—
341B(4,𝔽2)—22no—
342#455—22no—
343#455—22no—
344C2×#104—21no—
345C2×#100C22×C2311yescharacteristic
346C2×#99—21no—
347C2×#99—21no—
348C2×#99C22×C2311yescharacteristic
349C2×#99C22×C2311yescharacteristic
350#397—21no—
351#394—21no—
352#393—21no—
353#388—21no—
354#386—21no—
355#386—21no—
356#386C22×C2311yescharacteristic
357#379—21no—
358#354—22no—
359AΣL(1,𝔽16)—22no—
360C2×C23—21no—
361C2×C23—21no—
362C2×C23—41no—
363C2×C23—41no—
364C2×C23—41no—
365C2×C23—41no—
366C2×C23—82no—
367C2×C23—82no—
368C2×C23#202111yescharacteristic
369C2×C23#219711yescharacteristic
370C2×C23#219911yescharacteristic
371C25D4×C2311yescharacteristic
372H(5,𝔽2)—21no—
373H(5,𝔽2)—21no—
374H(5,𝔽2)—21no—
375H(5,𝔽2)—42no—
376H(5,𝔽2)D4×C2312yes—
377C2×#41—21no—
378C2×#41—41no—
379C22×Q8D4×C2311yescharacteristic, complemented
380C22×D4—21no—
381C22×D4—21no—
382C22×D4—21no—
383C22×D4—41no—
384C22×D4—41no—
385C22×D4C23×C2311yescharacteristic
386C23×C4D4×C2311yescharacteristic, complemented
387#137—24no—
388#137—42no—
389C2×Q16—41no—
390C2×SD16—41no—
391C2×M16—21no—
392#124—21no—
393#124—21no—
394#124—21no—
395#124—21no—
396#120—24no—
397#120—21no—
398#120—21no—
399#120—21no—
400#120—21no—
401#120—42no—
402C2×C4⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—41no—
414C2×C22⋊C4—41no—
415C2×C22⋊C4D4×C2311yescharacteristic
416C2×C22⋊C4D4×C2311yescharacteristic, complemented
417C2×C42D4×C2311yescharacteristic, complemented
418#104—24no—
419#100—21no—
420#100D4×C2312yescomplemented
421#99—24no—
422#99—21no—
423#99—21no—
424#99—21no—
425#99—42no—
426#99D4×C2312yes—
427#95—21no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—41no—
432C23#1334811yescharacteristic, direct product factor, Sylow
433C24—21no—
434C24—21no—
435C24—21no—
436C24—21no—
437C24—21no—
438C24—41no—
439C24—41no—
440C24—42no—
441C24—81no—
442C24C2×D4×C2311yescharacteristic
443C24C2×D4×C2311yescharacteristic
444#41—22no—
445#41—22no—
446#41—42no—
447#41—41no—
448#41—42no—
449#41—42no—
450#41—42no—
451#41—41no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—41no—
456C2×Q8—41no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—41no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—41no—
475C2×D4—44no—
476C2×D4—42no—
477C2×D4—41no—
478C2×D4—41no—
479C2×D4—81no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4C2×D4×C2311yescharacteristic
483C2×D4C2×D4×C2311yescharacteristic
484C2×D4C2×D4×C2311yescharacteristic
485C2×D4#41×C2311yescharacteristic
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—81no—
498C22×C4C2×D4×C2311yescharacteristic
499Q16—44no—
500SD16—44no—
501M16—22no—
502M16—41no—
503C2×C8—41no—
504C4⋊C4—81no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—82no—
525C42—21no—
526C42—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—81no—
551C23—82no—
552C23—81no—
553C23—82no—
554C23—84no—
555C23—81no—
556C23—81no—
557C23—81no—
558C23—81no—
559C23—81no—
560C23C23×#12011yescharacteristic
561C23C23×#12011yescharacteristic
562C23C23×#12111yescharacteristic
563C23C23×#12111yescharacteristic
564C23C23×#12711yescharacteristic
565Q8—22no—
566Q8—41no—
567Q8—42no—
568Q8—81no—
569D4—22no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—42no—
581D4—82no—
582D4—82no—
583D4—88no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4C23×#12011yescharacteristic
617C2×C4C23×#12111yescharacteristic
618C8—42no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—42no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—161no—
652C22—161no—
653C22C23×#39211yescharacteristic
654C22C23×B(4,𝔽2)11yescharacteristic
655C22C23×B(4,𝔽2)11yescharacteristic
656C4—21no—
657C4—21no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4—161no—
666C2—21no—
667C2—21no—
668C2—41no—
669C2—41no—
670C2—41no—
671C2—41no—
672C2—82no—
673C2—82no—
674C2C23×#202111yescharacteristic
675C2C23×#219711yescharacteristic
676C2C23×#219911yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×B(4,𝔽2)C2
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#455C2semidirect
3. C23×#2684C2semidirect
4. C2×C23×#354C2semidirect
5. C2×C23×AΣL(1,𝔽16)C2semidirect
6. C23×#1790C2semidirect
7. C23×#1390C2semidirect
41. C22×Q8×C23D4semidirect
48. C23×C4×C23D4semidirect
78. C2×C22⋊C4×C23D4semidirect
79. C2×C42×C23D4semidirect
82. C23×#100D4semidirect
188. #13348C23direct (N × H)
282. C2×#455C2×C23semidirect
283. #2684C2×C23semidirect
284. C2×#354C2×C23semidirect
285. C2×AΣL(1,𝔽16)C2×C23semidirect
286. #1790C2×C23semidirect
287. #1390C2×C23semidirect
379. C22×Q8D4×C23semidirect
386. C23×C4D4×C23semidirect
416. C2×C22⋊C4D4×C23semidirect
417. C2×C42D4×C23semidirect
420. #100D4×C23semidirect
432. C23#13348direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×B(4,𝔽2)2
C2×C23×#4552
C23×#26842
C2×C23×#3542
C2×C23×AΣL(1,𝔽16)2
C23×#17902
C23×#13902
#1334823