Finite Group Databasebeta

← C23×#13344 #19160970

Subgroups — C23×#13344 #19160970

Order 5888.

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

normal subgroups only

This group has 892 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×#458C211yesmaximal, characteristic, complemented
2C2×C23×B(4,𝔽2)C211yesmaximal, characteristic
3C23×#2697C211yesmaximal, characteristic, complemented
4C2×C23×#352C211yesmaximal, characteristic, complemented
5C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6C23×#1792C211yesmaximal, characteristic, complemented
7C23×#1393C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#123—21no—
10C2×C23×#123C2211yescharacteristic
11C2×C23×#120—21no—
12C2×C23×#120C2211yescharacteristic
13C23×#458—22no—
14C23×#458—22no—
15C23×B(4,𝔽2)—22no—
16C23×B(4,𝔽2)—22no—
17C2×C23×#104—21no—
18C23×#417—21no—
19C2×C23×#100C2211yescharacteristic
20C2×C23×#99—21no—
21C2×C23×#99—21no—
22C2×C23×#99C2211yescharacteristic
23C2×C23×#99C2211yescharacteristic
24C23×#397—21no—
25C23×#396—21no—
26C23×#394—21no—
27C23×#390C2211yescharacteristic
28C23×#388—21no—
29C23×#386—21no—
30C23×#379—21no—
31C23×#352—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×D4×C23—21no—
42C22×D4×C23—21no—
43C22×D4×C23—21no—
44C22×D4×C23—41no—
45C22×D4×C23—41no—
46C22×D4×C23C2311yescharacteristic
47C23×C4×C23D411yescharacteristic, complemented
48C2×M16×C23—21no—
49C23×#123—24no—
50C23×#123—21no—
51C23×#123—21no—
52C23×#123—21no—
53C23×#123—21no—
54C23×#123—42no—
55C23×#120—24no—
56C23×#120—21no—
57C23×#120—21no—
58C23×#120—21no—
59C23×#120—21no—
60C23×#120—42no—
61C2×C4⋊C4×C23—21no—
62C2×C4⋊C4×C23—21no—
63C2×C4⋊C4×C23—41no—
64C2×C4⋊C4×C23D411yescharacteristic, complemented
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—42no—
81C23×#103—41no—
82C23×#102—41no—
83C23×#100—21no—
84C23×#100—21no—
85C23×#99—24no—
86C23×#99—24no—
87C23×#99—21no—
88C23×#99—21no—
89C23×#99D412yes—
90C23×#99D412yescomplemented
91C23×#95—21no—
92C23×#95—21no—
93C23×#95—21no—
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×D4×C23—21no—
115C2×D4×C23—22no—
116C2×D4×C23—22no—
117C2×D4×C23—22no—
118C2×D4×C23—22no—
119C2×D4×C23—41no—
120C2×D4×C23—41no—
121C2×D4×C23—42no—
122C2×D4×C23—42no—
123C2×D4×C23—41no—
124C2×D4×C23—41no—
125C2×D4×C23—41no—
126C2×D4×C23—41no—
127C2×D4×C23—42no—
128C2×D4×C23—42no—
129C2×D4×C23—41no—
130C2×D4×C23—41no—
131C2×D4×C23—41no—
132C2×D4×C23—44no—
133C2×D4×C23—42no—
134C2×D4×C23—41no—
135C2×D4×C23—41no—
136C2×D4×C23—81no—
137C2×D4×C23—82no—
138C2×D4×C23—82no—
139C2×D4×C23C2×D411yescharacteristic
140C2×D4×C23C2×D411yescharacteristic
141C2×D4×C23C2×D411yescharacteristic
142C2×D4×C23#4111yescharacteristic
143C22×C4×C23—21no—
144C22×C4×C23—21no—
145C22×C4×C23—21no—
146C22×C4×C23—21no—
147C22×C4×C23—21no—
148C22×C4×C23—21no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—21no—
152C22×C4×C23—41no—
153C22×C4×C23—41no—
154C22×C4×C23—41no—
155C22×C4×C23—41no—
156C22×C4×C23—41no—
157C22×C4×C23C2×D411yescharacteristic
158M16×C23—41no—
159M16×C23—41no—
160C2×C8×C23—41no—
161C4⋊C4×C23—41no—
162C4⋊C4×C23—44no—
163C4⋊C4×C23—41no—
164C4⋊C4×C23—41no—
165C4⋊C4×C23—81no—
166C22⋊C4×C23—21no—
167C22⋊C4×C23—21no—
168C22⋊C4×C23—22no—
169C22⋊C4×C23—22no—
170C22⋊C4×C23—21no—
171C22⋊C4×C23—21no—
172C22⋊C4×C23—44no—
173C22⋊C4×C23—42no—
174C22⋊C4×C23—41no—
175C22⋊C4×C23—41no—
176C22⋊C4×C23—44no—
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—81no—
186C42×C23—41no—
187#13344C2311yesmaximal, characteristic, direct product factor, Sylow
188C23×C23—21no—
189C23×C23—21no—
190C23×C23—21no—
191C23×C23—21no—
192C23×C23—22no—
193C23×C23—22no—
194C23×C23—22no—
195C23×C23—41no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—42no—
200C23×C23—41no—
201C23×C23—41no—
202C23×C23—41no—
203C23×C23—41no—
204C23×C23—41no—
205C23×C23—42no—
206C23×C23—42no—
207C23×C23—41no—
208C23×C23—41no—
209C23×C23—41no—
210C23×C23—41no—
211C23×C23—81no—
212C23×C23—82no—
213C23×C23—81no—
214C23×C23—82no—
215C23×C23—84no—
216C23×C23—81no—
217C23×C23—81no—
218C23×C23—81no—
219C23×C23—81no—
220C23×C23—81no—
221C23×C23#12011yescharacteristic
222C23×C23#12011yescharacteristic
223C23×C23#12011yescharacteristic
224C23×C23#12111yescharacteristic
225C23×C23#12111yescharacteristic
226Q8×C23—22no—
227Q8×C23—41no—
228D4×C23—22no—
229D4×C23—41no—
230D4×C23—41no—
231D4×C23—41no—
232D4×C23—41no—
233D4×C23—41no—
234D4×C23—42no—
235D4×C23—42no—
236D4×C23—42no—
237D4×C23—42no—
238D4×C23—42no—
239D4×C23—42no—
240D4×C23—82no—
241D4×C23—82no—
242D4×C23—88no—
243C2×C4×C23—21no—
244C2×C4×C23—21no—
245C2×C4×C23—22no—
246C2×C4×C23—22no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—41no—
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—41no—
265C2×C4×C23—41no—
266C2×C4×C23—41no—
267C2×C4×C23—41no—
268C2×C4×C23—41no—
269C2×C4×C23—41no—
270C2×C4×C23—81no—
271C2×C4×C23—81no—
272C2×C4×C23—81no—
273C2×C4×C23—81no—
274C2×C4×C23—81no—
275C2×C4×C23#12111yescharacteristic
276C2×C4×C23#12711yescharacteristic
277C8×C23—81no—
278C2×#458C2×C2311yescharacteristic, complemented
279C2×B(4,𝔽2)C2×C2311yescharacteristic
280#2697C2×C2311yescharacteristic, complemented
281C2×#352C2×C2311yescharacteristic, complemented
282C2×AΣL(1,𝔽16)C2×C2311yescharacteristic, complemented
283#1792C2×C2311yescharacteristic, complemented
284#1393C2×C2311yescharacteristic, complemented
285C22×C23—21no—
286C22×C23—21no—
287C22×C23—21no—
288C22×C23—21no—
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—41no—
292C22×C23—41no—
293C22×C23—41no—
294C22×C23—41no—
295C22×C23—42no—
296C22×C23—42no—
297C22×C23—41no—
298C22×C23—41no—
299C22×C23—41no—
300C22×C23—41no—
301C22×C23—44no—
302C22×C23—42no—
303C22×C23—42no—
304C22×C23—81no—
305C22×C23—81no—
306C22×C23—81no—
307C22×C23—81no—
308C22×C23—82no—
309C22×C23—82no—
310C22×C23—82no—
311C22×C23—82no—
312C22×C23—82no—
313C22×C23—81no—
314C22×C23—81no—
315C22×C23—81no—
316C22×C23—81no—
317C22×C23—161no—
318C22×C23—161no—
319C22×C23#39211yescharacteristic
320C22×C23B(4,𝔽2)11yescharacteristic
321C22×C23B(4,𝔽2)11yescharacteristic
322C4×C23—21no—
323C4×C23—21no—
324C4×C23—42no—
325C4×C23—81no—
326C4×C23—81no—
327C4×C23—81no—
328C4×C23—82no—
329C4×C23—82no—
330C4×C23—81no—
331C4×C23—81no—
332C2×H(5,𝔽2)C22×C2311yescharacteristic
333C2×#123—21no—
334C2×#123C22×C2311yescharacteristic
335C2×#120—21no—
336C2×#120C22×C2311yescharacteristic
337#458—22no—
338#458—22no—
339B(4,𝔽2)—22no—
340B(4,𝔽2)—22no—
341C2×#104—21no—
342#417—21no—
343C2×#100C22×C2311yescharacteristic
344C2×#99—21no—
345C2×#99—21no—
346C2×#99C22×C2311yescharacteristic
347C2×#99C22×C2311yescharacteristic
348#397—21no—
349#396—21no—
350#394—21no—
351#390C22×C2311yescharacteristic
352#388—21no—
353#386—21no—
354#379—21no—
355#352—22no—
356AΣL(1,𝔽16)—22no—
357C2×C23—21no—
358C2×C23—21no—
359C2×C23—41no—
360C2×C23—41no—
361C2×C23—41no—
362C2×C23—41no—
363C2×C23—82no—
364C2×C23—82no—
365C2×C23#202111yescharacteristic
366C2×C23#219711yescharacteristic
367C2×C23#220011yescharacteristic
368C25D4×C2311yescharacteristic
369H(5,𝔽2)—21no—
370H(5,𝔽2)—21no—
371H(5,𝔽2)—21no—
372H(5,𝔽2)—42no—
373H(5,𝔽2)D4×C2312yes—
374C2×#41—21no—
375C2×#41—41no—
376C22×D4—21no—
377C22×D4—21no—
378C22×D4—21no—
379C22×D4—41no—
380C22×D4—41no—
381C22×D4C23×C2311yescharacteristic
382C23×C4D4×C2311yescharacteristic, complemented
383C2×M16—21no—
384#123—24no—
385#123—21no—
386#123—21no—
387#123—21no—
388#123—21no—
389#123—42no—
390#120—24no—
391#120—21no—
392#120—21no—
393#120—21no—
394#120—21no—
395#120—42no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4D4×C2311yescharacteristic, complemented
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊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—41no—
411C2×C22⋊C4—41no—
412C2×C22⋊C4D4×C2311yescharacteristic
413C2×C22⋊C4D4×C2311yescharacteristic, complemented
414C2×C42D4×C2311yescharacteristic, complemented
415#104—42no—
416#103—41no—
417#102—41no—
418#100—21no—
419#100—21no—
420#99—24no—
421#99—24no—
422#99—21no—
423#99—21no—
424#99D4×C2312yes—
425#99D4×C2312yescomplemented
426#95—21no—
427#95—21no—
428#95—21no—
429C23#1334411yescharacteristic, direct product factor, Sylow
430C24—21no—
431C24—21no—
432C24—21no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—41no—
437C24—42no—
438C24—81no—
439C24C2×D4×C2311yescharacteristic
440C24C2×D4×C2311yescharacteristic
441#41—22no—
442#41—22no—
443#41—42no—
444#41—41no—
445#41—42no—
446#41—42no—
447#41—42no—
448#41—41no—
449C2×Q8—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—41no—
471C2×D4—41no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4C2×D4×C2311yescharacteristic
476C2×D4C2×D4×C2311yescharacteristic
477C2×D4C2×D4×C2311yescharacteristic
478C2×D4#41×C2311yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×D4×C2311yescharacteristic
494M16—41no—
495M16—41no—
496C2×C8—41no—
497C4⋊C4—41no—
498C4⋊C4—44no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—81no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—44no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C42—41no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—81no—
547C23—82no—
548C23—81no—
549C23—82no—
550C23—84no—
551C23—81no—
552C23—81no—
553C23—81no—
554C23—81no—
555C23—81no—
556C23C23×#12011yescharacteristic
557C23C23×#12011yescharacteristic
558C23C23×#12011yescharacteristic
559C23C23×#12111yescharacteristic
560C23C23×#12111yescharacteristic
561Q8—22no—
562Q8—41no—
563D4—22no—
564D4—41no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—82no—
576D4—82no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
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—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4C23×#12111yescharacteristic
611C2×C4C23×#12711yescharacteristic
612C8—81no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—42no—
624C22—42no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—161no—
646C22—161no—
647C22C23×#39211yescharacteristic
648C22C23×B(4,𝔽2)11yescharacteristic
649C22C23×B(4,𝔽2)11yescharacteristic
650C4—21no—
651C4—21no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—82no—
658C4—81no—
659C4—81no—
660C2—21no—
661C2—21no—
662C2—41no—
663C2—41no—
664C2—41no—
665C2—41no—
666C2—82no—
667C2—82no—
668C2C23×#202111yescharacteristic
669C2C23×#219711yescharacteristic
670C2C23×#220011yescharacteristic
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
1. C2×C23×#458C2semidirect
3. C23×#2697C2semidirect
4. C2×C23×#352C2semidirect
5. C2×C23×AΣL(1,𝔽16)C2semidirect
6. C23×#1792C2semidirect
7. C23×#1393C2semidirect
47. C23×C4×C23D4semidirect
64. C2×C4⋊C4×C23D4semidirect
78. C2×C22⋊C4×C23D4semidirect
79. C2×C42×C23D4semidirect
90. C23×#99D4semidirect
187. #13344C23direct (N × H)
278. C2×#458C2×C23semidirect
280. #2697C2×C23semidirect
281. C2×#352C2×C23semidirect
282. C2×AΣL(1,𝔽16)C2×C23semidirect
283. #1792C2×C23semidirect
284. #1393C2×C23semidirect
382. C23×C4D4×C23semidirect
399. C2×C4⋊C4D4×C23semidirect
413. C2×C22⋊C4D4×C23semidirect
414. C2×C42D4×C23semidirect
425. #99D4×C23semidirect
429. C23#13344direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4582
C2×C23×B(4,𝔽2)2
C23×#26972
C2×C23×#3522
C2×C23×AΣL(1,𝔽16)2
C23×#17922
C23×#13932
#1334423