Finite Group Databasebeta

← C23×#10398 #19158024

Subgroups — C23×#10398 #19158024

Order 5888.

694 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2626C211yesmaximal, characteristic
2C23×#1872C211yesmaximal, characteristic, complemented
3C23×#1853C211yesmaximal, characteristic, complemented
4C2×C23×#328C211yesmaximal, characteristic, complemented
5C2×C23×#323C211yesmaximal, characteristic
6C23×#1439C211yesmaximal, characteristic
7C23×#1294C211yesmaximal, characteristic, complemented
8C2×C23×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4×C23C2211yescharacteristic
10C22×C22⋊C4×C23—21no—
11C22×C22⋊C4×C23C2211yescharacteristic
12C2×C23×#106—21no—
13C2×C23×#103—21no—
14C2×C23×C22⋊C8C2211yescharacteristic
15C2×C23×C22⋊C8C2211yescharacteristic
16C23×#399—21no—
17C23×#396—21no—
18C23×#396—21no—
19C23×#393—21no—
20C23×#385—21no—
21C23×#380C2211yescharacteristic
22C23×#379—21no—
23C2×C23×#95—21no—
24C2×C23×#95C2211yescharacteristic
25C23×#340—21no—
26C23×#340—21no—
27C23×#328—22no—
28C23×#323—22no—
29C25×C23D411yescharacteristic
30C22×Q8×C23D411yescharacteristic
31C23×C4×C23—21no—
32C23×C4×C23—21no—
33C23×C4×C23D411yescharacteristic
34C23×C4×C23C2311yescharacteristic
35C22×C8×C23—21no—
36C22×C8×C23—21no—
37C23×C22⋊Q8—21no—
38C23×C22⋊Q8—21no—
39C23×C22⋊Q8—21no—
40C23×C22⋊Q8—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23—42no—
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C4⋊C4×C23D411yescharacteristic
49C2×C4⋊C4×C23D411yescharacteristic
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
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—42no—
63C2×C22⋊C4×C23—42no—
64C2×C22⋊C4×C23—42no—
65C23×#106—42no—
66C23×#103—42no—
67C23×C22⋊C8—21no—
68C23×C22⋊C8—21no—
69C23×C22⋊C8—21no—
70C23×C22⋊C8D412yes—
71C23×#95—21no—
72C23×#95—21no—
73C23×#95—21no—
74C23×#95—21no—
75C23×#95—21no—
76C23×#95—42no—
77C24×C23—21no—
78C24×C23—21no—
79C24×C23—21no—
80C24×C23—21no—
81C24×C23—21no—
82C24×C23—21no—
83C24×C23—41no—
84C24×C23—41no—
85C24×C23—41no—
86C24×C23—41no—
87C24×C23SD1612yes—
88C24×C23C2×D411yescharacteristic
89C2×Q8×C23—21no—
90C2×Q8×C23—21no—
91C2×Q8×C23—41no—
92C2×Q8×C23—41no—
93C22×C4×C23—22no—
94C22×C4×C23—22no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—21no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—41no—
105C22×C4×C23—41no—
106C22×C4×C23—41no—
107C22×C4×C23—41no—
108C22×C4×C23—41no—
109C22×C4×C23—41no—
110C22×C4×C23—41no—
111C22×C4×C23—41no—
112C22×C4×C23—41no—
113C22×C4×C23—41no—
114C22×C4×C23—41no—
115C22×C4×C23—41no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23C2×D411yescharacteristic
119C22×C4×C23C2×D411yescharacteristic
120C22×C4×C23C2×D411yescharacteristic
121C22×C4×C23#4111yescharacteristic
122C22×C4×C23#4111yescharacteristic
123C22×C4×C23#4111yescharacteristic
124C2×C8×C23—22no—
125C2×C8×C23—41no—
126C2×C8×C23—41no—
127C2×C8×C23—41no—
128C2×C8×C23—41no—
129C2×C8×C23—41no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—21no—
133C4⋊C4×C23—21no—
134C4⋊C4×C23—41no—
135C4⋊C4×C23—41no—
136C4⋊C4×C23—41no—
137C4⋊C4×C23—41no—
138C4⋊C4×C23—41no—
139C4⋊C4×C23—41no—
140C4⋊C4×C23—81no—
141C4⋊C4×C23—81no—
142C22⋊C4×C23—48no—
143C22⋊C4×C23—41no—
144C22⋊C4×C23—41no—
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—81no—
152C22⋊C4×C23—81no—
153C22⋊C4×C23—81no—
154C22⋊C4×C23—81no—
155#10398C2311yesmaximal, characteristic, direct product factor, Sylow
156C23×C23—21no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—21no—
160C23×C23—22no—
161C23×C23—22no—
162C23×C23—22no—
163C23×C23—42no—
164C23×C23—42no—
165C23×C23—42no—
166C23×C23—42no—
167C23×C23—42no—
168C23×C23—42no—
169C23×C23—41no—
170C23×C23—41no—
171C23×C23—41no—
172C23×C23—41no—
173C23×C23—41no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—81no—
178C23×C23—81no—
179C23×C23—81no—
180C23×C23—81no—
181C23×C23—81no—
182C23×C23—81no—
183C23×C23#12011yescharacteristic
184C23×C23#12111yescharacteristic
185C23×C23#12311yescharacteristic
186C23×C23C2×SD1611yescharacteristic
187C23×C23#13511yescharacteristic
188C23×C23#13711yescharacteristic
189C23×C23#13711yescharacteristic
190Q8×C23—42no—
191Q8×C23—81no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—41no—
197C2×C4×C23—41no—
198C2×C4×C23—41no—
199C2×C4×C23—41no—
200C2×C4×C23—42no—
201C2×C4×C23—42no—
202C2×C4×C23—42no—
203C2×C4×C23—42no—
204C2×C4×C23—42no—
205C2×C4×C23—42no—
206C2×C4×C23—41no—
207C2×C4×C23—41no—
208C2×C4×C23—41no—
209C2×C4×C23—41no—
210C2×C4×C23—41no—
211C2×C4×C23—41no—
212C2×C4×C23—41no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—41no—
216C2×C4×C23—41no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—42no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—81no—
223C2×C4×C23—81no—
224C2×C4×C23—81no—
225C2×C4×C23—81no—
226C2×C4×C23—81no—
227C2×C4×C23—81no—
228C2×C4×C23—81no—
229C2×C4×C23—81no—
230C2×C4×C23—81no—
231C2×C4×C23#12111yescharacteristic
232C2×C4×C23#12111yescharacteristic
233C2×C4×C23#12311yescharacteristic
234C2×C4×C23#12411yescharacteristic
235C8×C23—42no—
236C8×C23—81no—
237#2626C2×C2311yescharacteristic
238#1872C2×C2311yescharacteristic, complemented
239#1853C2×C2311yescharacteristic, complemented
240C2×#328C2×C2311yescharacteristic, complemented
241C2×#323C2×C2311yescharacteristic
242#1439C2×C2311yescharacteristic
243#1294C2×C2311yescharacteristic, complemented
244C22×C23—21no—
245C22×C23—21no—
246C22×C23—21no—
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—21no—
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—44no—
257C22×C23—42no—
258C22×C23—42no—
259C22×C23—42no—
260C22×C23—82no—
261C22×C23—82no—
262C22×C23—82no—
263C22×C23—81no—
264C22×C23—81no—
265C22×C23—81no—
266C22×C23—81no—
267C22×C23#39411yescharacteristic
268C22×C23#44811yescharacteristic
269C22×C23#45211yescharacteristic
270C22×C23#45311yescharacteristic
271C22×C23#45511yescharacteristic
272C22×C23B(4,𝔽2)11yescharacteristic
273C22×C23#45811yescharacteristic
274C22×C23#46511yescharacteristic
275C22×C23#47011yescharacteristic
276C22×C23#48311yescharacteristic
277C22×C23#48511yescharacteristic
278C4×C23—21no—
279C4×C23—21no—
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—82no—
283C4×C23—82no—
284C4×C23—81no—
285C4×C23—81no—
286C4×C23—81no—
287C4×C23—81no—
288C4×C23—81no—
289C2×C22⋊Q8C22×C2311yescharacteristic
290C22×C4⋊C4C22×C2311yescharacteristic
291C22×C22⋊C4—21no—
292C22×C22⋊C4C22×C2311yescharacteristic
293C2×#106—21no—
294C2×#103—21no—
295C2×C22⋊C8C22×C2311yescharacteristic
296C2×C22⋊C8C22×C2311yescharacteristic
297#399—21no—
298#396—21no—
299#396—21no—
300#393—21no—
301#385—21no—
302#380C22×C2311yescharacteristic
303#379—21no—
304C2×#95—21no—
305C2×#95C22×C2311yescharacteristic
306#340—21no—
307#340—21no—
308#328—22no—
309#323—22no—
310C2×C23—21no—
311C2×C23—21no—
312C2×C23—21no—
313C2×C23—21no—
314C2×C23—82no—
315C2×C23#159611yescharacteristic
316C2×C23#160611yescharacteristic
317C2×C23#160811yescharacteristic
318C2×C23#161811yescharacteristic
319C2×C23#203511yescharacteristic
320C2×C23#203711yescharacteristic
321C2×C23#204411yescharacteristic
322C25D4×C2311yescharacteristic
323C22×Q8D4×C2311yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4D4×C2311yescharacteristic
327C23×C4C23×C2311yescharacteristic
328C22×C8—21no—
329C22×C8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—42no—
340C2×C4⋊C4D4×C2311yescharacteristic
341C2×C4⋊C4D4×C2311yescharacteristic
342C2×C4⋊C4D4×C2311yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358#106—42no—
359#103—42no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—21no—
363C22⋊C8D4×C2312yes—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—42no—
370C23#1039811yescharacteristic, direct product factor, Sylow
371C24—21no—
372C24—21no—
373C24—21no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24SD16×C2312yes—
382C24C2×D4×C2311yescharacteristic
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—41no—
386C2×Q8—41no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4C2×D4×C2311yescharacteristic
413C22×C4C2×D4×C2311yescharacteristic
414C22×C4C2×D4×C2311yescharacteristic
415C22×C4#41×C2311yescharacteristic
416C22×C4#41×C2311yescharacteristic
417C22×C4#41×C2311yescharacteristic
418C2×C8—22no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C22⋊C4—48no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23C23×#12011yescharacteristic
477C23C23×#12111yescharacteristic
478C23C23×#12311yescharacteristic
479C23C2×SD16×C2311yescharacteristic
480C23C23×#13511yescharacteristic
481C23C23×#13711yescharacteristic
482C23C23×#13711yescharacteristic
483Q8—42no—
484Q8—81no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4C23×#12111yescharacteristic
525C2×C4C23×#12111yescharacteristic
526C2×C4C23×#12311yescharacteristic
527C2×C4C23×#12411yescharacteristic
528C8—42no—
529C8—81no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C23×#39411yescharacteristic
554C22C23×#44811yescharacteristic
555C22C23×#45211yescharacteristic
556C22C23×#45311yescharacteristic
557C22C23×#45511yescharacteristic
558C22C23×B(4,𝔽2)11yescharacteristic
559C22C23×#45811yescharacteristic
560C22C23×#46511yescharacteristic
561C22C23×#47011yescharacteristic
562C22C23×#48311yescharacteristic
563C22C23×#48511yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—82no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—82no—
573C4—81no—
574C4—81no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—82no—
580C2C23×#159611yescharacteristic
581C2C23×#160611yescharacteristic
582C2C23×#160811yescharacteristic
583C2C23×#161811yescharacteristic
584C2C23×#203511yescharacteristic
585C2C23×#203711yescharacteristic
586C2C23×#204411yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1872C2semidirect
3. C23×#1853C2semidirect
4. C2×C23×#328C2semidirect
7. C23×#1294C2semidirect
155. #10398C23direct (N × H)
238. #1872C2×C23semidirect
239. #1853C2×C23semidirect
240. C2×#328C2×C23semidirect
243. #1294C2×C23semidirect
370. C23#10398direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26262
C23×#18722
C23×#18532
C2×C23×#3282
C2×C23×#3232
C23×#14392
C23×#12942
#1039823