Finite Group Databasebeta

← C23×#13422 #19161048

Subgroups — C23×#13422 #19161048

Order 5888.

734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C23×#2669C211yesmaximal, characteristic
2C2×C23×#353C211yesmaximal, characteristic, complemented
3C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C23×#1889C211yesmaximal, characteristic, complemented
5C23×#1887C211yesmaximal, characteristic, complemented
6C23×#1867C211yesmaximal, characteristic, complemented
7C23×#1390C211yesmaximal, characteristic, complemented
8C2×C23×#127C2211yescharacteristic
9C2×C23×#121—21no—
10C2×C23×#120—21no—
11C2×C23×#120C2211yescharacteristic
12C23×#518C2211yescharacteristic
13C23×#418—21no—
14C2×C23×#100C2211yescharacteristic
15C2×C23×#99C2211yescharacteristic
16C2×C23×#99C2211yescharacteristic
17C23×#394—21no—
18C23×#392—21no—
19C23×#392—21no—
20C23×#388—21no—
21C23×#386C2211yescharacteristic
22C23×#385—21no—
23C23×#379—21no—
24C23×#353—22no—
25C23×AΣL(1,𝔽16)—22no—
26C23×#342—21no—
27C23×#342—21no—
28C23×#339—21no—
29C25×C23D411yescharacteristic
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C22×D4×C23—21no—
35C22×D4×C23D411yescharacteristic
36C22×D4×C23C2311yescharacteristic
37C23×C4×C23D411yescharacteristic
38C2×M16×C23—21no—
39C23×#127—21no—
40C23×#127—21no—
41C23×#127—21no—
42C23×#127—21no—
43C23×#121—44no—
44C23×#120—21no—
45C23×#120—21no—
46C23×#120—21no—
47C23×#120—21no—
48C23×#120—44no—
49C4×D4×C23—41no—
50C4×D4×C23—41no—
51C23×#117—21no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—41no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—41no—
65C2×C22⋊C4×C23—41no—
66C2×C22⋊C4×C23D411yescharacteristic
67C2×C22⋊C4×C23D411yescharacteristic
68C2×C42×C23D411yescharacteristic
69C23×#102—41no—
70C23×#102—41no—
71C23×#100—21no—
72C23×#100—21no—
73C23×#99—21no—
74C23×#99—21no—
75C23×#99—21no—
76C23×#99D412yescomplemented
77C23×#95—21no—
78C23×#95—21no—
79C23×#95—41no—
80C24×C23—21no—
81C24×C23—21no—
82C24×C23—21no—
83C24×C23—21no—
84C24×C23—41no—
85C24×C23—42no—
86C24×C23—41no—
87C24×C23—81no—
88C24×C23C2×D411yescharacteristic
89C24×C23C2×D411yescharacteristic
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—41no—
93C2×D4×C23—41no—
94C2×D4×C23—41no—
95C2×D4×C23—41no—
96C2×D4×C23—42no—
97C2×D4×C23—42no—
98C2×D4×C23—42no—
99C2×D4×C23—42no—
100C2×D4×C23—42no—
101C2×D4×C23—42no—
102C2×D4×C23—41no—
103C2×D4×C23—41no—
104C2×D4×C23—41no—
105C2×D4×C23—41no—
106C2×D4×C23—41no—
107C2×D4×C23—41no—
108C2×D4×C23—81no—
109C2×D4×C23—81no—
110C2×D4×C23—81no—
111C2×D4×C23—81no—
112C2×D4×C23—81no—
113C2×D4×C23—81no—
114C2×D4×C23C2×D411yescharacteristic
115C2×D4×C23#4111yescharacteristic
116C2×D4×C23#4111yescharacteristic
117C2×D4×C23#4111yescharacteristic
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—21no—
126C22×C4×C23—42no—
127C22×C4×C23—41no—
128C22×C4×C23—41no—
129C22×C4×C23—41no—
130C22×C4×C23C2×D411yescharacteristic
131M16×C23—41no—
132M16×C23—41no—
133C2×C8×C23—41no—
134C4⋊C4×C23—21no—
135C4⋊C4×C23—21no—
136C4⋊C4×C23—42no—
137C4⋊C4×C23—82no—
138C22⋊C4×C23—24no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—21no—
141C22⋊C4×C23—21no—
142C22⋊C4×C23—42no—
143C22⋊C4×C23—42no—
144C22⋊C4×C23—42no—
145C22⋊C4×C23—41no—
146C22⋊C4×C23—41no—
147C22⋊C4×C23—41no—
148C22⋊C4×C23—41no—
149C22⋊C4×C23—41no—
150C22⋊C4×C23—41no—
151C22⋊C4×C23—41no—
152C22⋊C4×C23—41no—
153C22⋊C4×C23—41no—
154C22⋊C4×C23—81no—
155C22⋊C4×C23—82no—
156C22⋊C4×C23—82no—
157C42×C23—21no—
158C42×C23—21no—
159C42×C23—41no—
160#13422C2311yesmaximal, characteristic, direct product factor, Sylow
161C23×C23—21no—
162C23×C23—21no—
163C23×C23—21no—
164C23×C23—21no—
165C23×C23—22no—
166C23×C23—41no—
167C23×C23—41no—
168C23×C23—41no—
169C23×C23—41no—
170C23×C23—41no—
171C23×C23—41no—
172C23×C23—41no—
173C23×C23—42no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—41no—
178C23×C23—42no—
179C23×C23—42no—
180C23×C23—41no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—81no—
187C23×C23—82no—
188C23×C23—81no—
189C23×C23—82no—
190C23×C23—84no—
191C23×C23—81no—
192C23×C23#12011yescharacteristic
193C23×C23#12111yescharacteristic
194C23×C23#12111yescharacteristic
195C23×C23#12311yescharacteristic
196C23×C23#12411yescharacteristic
197D4×C23—41no—
198D4×C23—41no—
199D4×C23—41no—
200D4×C23—41no—
201D4×C23—42no—
202D4×C23—81no—
203D4×C23—82no—
204D4×C23—82no—
205D4×C23—82no—
206D4×C23—82no—
207D4×C23—84no—
208D4×C23—82no—
209D4×C23—82no—
210D4×C23—84no—
211C2×C4×C23—21no—
212C2×C4×C23—21no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—41no—
216C2×C4×C23—41no—
217C2×C4×C23—41no—
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—41no—
225C2×C4×C23—41no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—81no—
230C2×C4×C23—81no—
231C2×C4×C23—81no—
232C2×C4×C23—81no—
233C2×C4×C23—81no—
234C2×C4×C23—81no—
235C2×C4×C23—81no—
236C2×C4×C23—81no—
237C2×C4×C23—81no—
238C2×C4×C23—81no—
239C2×C4×C23#12111yescharacteristic
240C2×C4×C23#12311yescharacteristic
241C8×C23—81no—
242#2669C2×C2311yescharacteristic
243C2×#353C2×C2311yescharacteristic, complemented
244C2×AΣL(1,𝔽16)C2×C2311yescharacteristic, complemented
245#1889C2×C2311yescharacteristic, complemented
246#1887C2×C2311yescharacteristic, complemented
247#1867C2×C2311yescharacteristic, complemented
248#1390C2×C2311yescharacteristic, complemented
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—41no—
256C22×C23—41no—
257C22×C23—41no—
258C22×C23—41no—
259C22×C23—42no—
260C22×C23—42no—
261C22×C23—41no—
262C22×C23—41no—
263C22×C23—41no—
264C22×C23—41no—
265C22×C23—44no—
266C22×C23—42no—
267C22×C23—81no—
268C22×C23—81no—
269C22×C23—81no—
270C22×C23—81no—
271C22×C23—81no—
272C22×C23—81no—
273C22×C23—81no—
274C22×C23—81no—
275C22×C23—82no—
276C22×C23—82no—
277C22×C23—82no—
278C22×C23—81no—
279C22×C23—81no—
280C22×C23—81no—
281C22×C23—81no—
282C22×C23—81no—
283C22×C23#39411yescharacteristic
284C22×C23B(4,𝔽2)11yescharacteristic
285C22×C23#45811yescharacteristic
286C4×C23—21no—
287C4×C23—21no—
288C4×C23—42no—
289C4×C23—42no—
290C4×C23—81no—
291C4×C23—81no—
292C4×C23—82no—
293C4×C23—81no—
294C4×C23—81no—
295C4×C23—161no—
296C2×#127C22×C2311yescharacteristic
297C2×#121—21no—
298C2×#120—21no—
299C2×#120C22×C2311yescharacteristic
300#518C22×C2311yescharacteristic
301#418—21no—
302C2×#100C22×C2311yescharacteristic
303C2×#99C22×C2311yescharacteristic
304C2×#99C22×C2311yescharacteristic
305#394—21no—
306#392—21no—
307#392—21no—
308#388—21no—
309#386C22×C2311yescharacteristic
310#385—21no—
311#379—21no—
312#353—22no—
313AΣL(1,𝔽16)—22no—
314#342—21no—
315#342—21no—
316#339—21no—
317C2×C23—21no—
318C2×C23—21no—
319C2×C23—41no—
320C2×C23—41no—
321C2×C23—41no—
322C2×C23—41no—
323C2×C23—82no—
324C2×C23—161no—
325C2×C23#204411yescharacteristic
326C2×C23#219611yescharacteristic
327C2×C23#220011yescharacteristic
328C25D4×C2311yescharacteristic
329C22×D4—21no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4D4×C2311yescharacteristic
335C22×D4C23×C2311yescharacteristic
336C23×C4D4×C2311yescharacteristic
337C2×M16—21no—
338#127—21no—
339#127—21no—
340#127—21no—
341#127—21no—
342#121—44no—
343#120—21no—
344#120—21no—
345#120—21no—
346#120—21no—
347#120—44no—
348C4×D4—41no—
349C4×D4—41no—
350#117—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—41no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—41no—
364C2×C22⋊C4—41no—
365C2×C22⋊C4D4×C2311yescharacteristic
366C2×C22⋊C4D4×C2311yescharacteristic
367C2×C42D4×C2311yescharacteristic
368#102—41no—
369#102—41no—
370#100—21no—
371#100—21no—
372#99—21no—
373#99—21no—
374#99—21no—
375#99D4×C2312yescomplemented
376#95—21no—
377#95—21no—
378#95—41no—
379C23#1342211yescharacteristic, direct product factor, Sylow
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—41no—
385C24—42no—
386C24—41no—
387C24—81no—
388C24C2×D4×C2311yescharacteristic
389C24C2×D4×C2311yescharacteristic
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4C2×D4×C2311yescharacteristic
415C2×D4#41×C2311yescharacteristic
416C2×D4#41×C2311yescharacteristic
417C2×D4#41×C2311yescharacteristic
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4C2×D4×C2311yescharacteristic
431M16—41no—
432M16—41no—
433C2×C8—41no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—42no—
437C4⋊C4—82no—
438C22⋊C4—24no—
439C22⋊C4—22no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—81no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C42—21no—
458C42—21no—
459C42—41no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—81no—
486C23—82no—
487C23—81no—
488C23—82no—
489C23—84no—
490C23—81no—
491C23C23×#12011yescharacteristic
492C23C23×#12111yescharacteristic
493C23C23×#12111yescharacteristic
494C23C23×#12311yescharacteristic
495C23C23×#12411yescharacteristic
496D4—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—42no—
501D4—81no—
502D4—82no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—84no—
507D4—82no—
508D4—82no—
509D4—84no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4C23×#12111yescharacteristic
539C2×C4C23×#12311yescharacteristic
540C8—81no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—44no—
558C22—42no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22C23×#39411yescharacteristic
576C22C23×B(4,𝔽2)11yescharacteristic
577C22C23×#45811yescharacteristic
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—161no—
588C2—21no—
589C2—21no—
590C2—41no—
591C2—41no—
592C2—41no—
593C2—41no—
594C2—82no—
595C2—161no—
596C2C23×#204411yescharacteristic
597C2C23×#219611yescharacteristic
598C2C23×#220011yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C23
prime radicalC23×#2669C2
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×#353C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1889C2semidirect
5. C23×#1887C2semidirect
6. C23×#1867C2semidirect
7. C23×#1390C2semidirect
76. C23×#99D4semidirect
160. #13422C23direct (N × H)
243. C2×#353C2×C23semidirect
244. C2×AΣL(1,𝔽16)C2×C23semidirect
245. #1889C2×C23semidirect
246. #1887C2×C23semidirect
247. #1867C2×C23semidirect
248. #1390C2×C23semidirect
375. #99D4×C23semidirect
379. C23#13422direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26692
C2×C23×#3532
C2×C23×AΣL(1,𝔽16)2
C23×#18892
C23×#18872
C23×#18672
C23×#13902
#1342223