Finite Group Databasebeta

← C23×#19035 #19166661

Subgroups — C23×#19035 #19166661

Order 5888.

824 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic
3C23×#3165C211yesmaximal, characteristic, complemented
4C2×C23×#491C211yesmaximal, characteristic, complemented
5C23×#1664C214yesmaximal, complemented
6C23×#1663C214yesmaximal, complemented
7C23×#1531C211yesmaximal, characteristic, complemented
8C23×#1528C211yesmaximal, characteristic, complemented
9C23×#1524C211yesmaximal, characteristic
10C23×#564C2212yes—
11C23×#563C2212yes—
12C23×#557—21no—
13C23×#554—21no—
14C23×#552—21no—
15C23×#552—22no—
16C23×#552—21no—
17C23×#544—21no—
18C23×#544—21no—
19C23×#541—21no—
20C23×#541—22no—
21C23×#539—21no—
22C23×#533C2211yescharacteristic
23C2×C23×C4⋊Q8C2211yescharacteristic
24C23×#529C2211yescharacteristic
25C2×C23×#125C2211yescharacteristic
26C23×#501C2214yes—
27C23×#501C2212yes—
28C23×#500—21no—
29C23×#498—21no—
30C23×#491—22no—
31C23×#491C2214yes—
32C23×#489C2212yes—
33C23×#484—21no—
34C23×#481—21no—
35C23×#475—21no—
36C23×#474—21no—
37C23×#432C2211yescharacteristic
38C23×#428—22no—
39C2×C23×#103—21no—
40C2×C23×#103—21no—
41C4×M16×C23C2211yescharacteristic
42C2×C23×#97C2211yescharacteristic
43C23×#330C2212yes—
44C23×#330C2214yes—
45C23×#330C2214yes—
46C23×#329C2212yes—
47C22×Q8×C23—21no—
48C2×M16×C23—21no—
49C22×C8×C23—21no—
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8—22no—
52C23×C4⋊Q8—22no—
53C23×C4⋊Q8—21no—
54C23×C4⋊Q8—21no—
55C23×C4⋊Q8—21no—
56C23×C4⋊Q8D411yescharacteristic
57C23×C4⋊Q8D411yescharacteristic
58C23×C4⋊Q8C2312yes—
59C23×#126—22no—
60C23×#126—22no—
61C23×#126—22no—
62C23×#126—22no—
63C23×#126—21no—
64C23×#125—22no—
65C23×#125—21no—
66C23×#125—22no—
67C23×#125—22no—
68C23×#125—22no—
69C23×#125—22no—
70C23×#125—22no—
71C23×#125—22no—
72C23×#125—22no—
73C23×#125—21no—
74C23×#125—21no—
75C23×#125D412yes—
76C23×#125C2312yes—
77C23×#125C2312yes—
78C23×#125C2311yescharacteristic
79C23×#124C2311yescharacteristic
80C23×#123—22no—
81C23×#123—22no—
82C23×#123—21no—
83C23×C22⋊Q8—21no—
84C23×C22⋊Q8—21no—
85C23×C22⋊Q8—22no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—21no—
88C23×C22⋊Q8—21no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8D411yescharacteristic
92C23×#121D411yescharacteristic
93C4×Q8×C23—21no—
94C4×Q8×C23—22no—
95C4×Q8×C23—21no—
96C4×Q8×C23—21no—
97C4×Q8×C23—21no—
98C4×Q8×C23D411yescharacteristic
99C4×D4×C23—22no—
100C4×D4×C23—21no—
101C4×D4×C23D411yescharacteristic
102C23×#117—22no—
103C23×#117—21no—
104C23×#117—22no—
105C23×#117—21no—
106C23×#117D412yes—
107C2×C4⋊C4×C23—21no—
108C2×C4⋊C4×C23—21no—
109C2×C4⋊C4×C23—21no—
110C2×C4⋊C4×C23—21no—
111C2×C4⋊C4×C23—21no—
112C2×C4⋊C4×C23D411yescharacteristic
113C2×C4⋊C4×C23D411yescharacteristic
114C2×C42×C23C2311yescharacteristic
115C23×#107—24no—
116C23×#107—22no—
117C23×#106—24no—
118C23×#106—22no—
119C23×C4⋊C8—21no—
120C23×#103—22no—
121C23×#103—24no—
122C23×#103—24no—
123C23×#102—22no—
124C23×C22⋊C8—21no—
125C23×#97—21no—
126C23×#97C2312yes—
127C23×#97C2312yes—
128C23×#97C2312yes—
129C4×C8×C23—21no—
130C2×Q8×C23—22no—
131C2×Q8×C23—21no—
132C2×Q8×C23—21no—
133C2×Q8×C23—22no—
134C2×Q8×C23—44no—
135C2×Q8×C23—41no—
136C2×Q8×C23C2×D411yescharacteristic
137C2×D4×C23—21no—
138C2×D4×C23C2×D411yescharacteristic
139C22×C4×C23—21no—
140C22×C4×C23—22no—
141C22×C4×C23—21no—
142C22×C4×C23—21no—
143C22×C4×C23—41no—
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D411yescharacteristic
146C22×C4×C23C2×D411yescharacteristic
147M16×C23—24no—
148C2×C8×C23—22no—
149C2×C8×C23—22no—
150C2×C8×C23—22no—
151C2×C8×C23—42no—
152C4⋊C4×C23—22no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—22no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—21no—
165C4⋊C4×C23—21no—
166C4⋊C4×C23—21no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—21no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—41no—
177C4⋊C4×C23—41no—
178C4⋊C4×C23C2×D412yes—
179C4⋊C4×C23C2×D412yes—
180C4⋊C4×C23C2×D412yes—
181C4⋊C4×C23C2×D411yescharacteristic
182C4⋊C4×C23C2×D412yes—
183C4⋊C4×C23C2×D411yescharacteristic
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—21no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—21no—
190C22⋊C4×C23—21no—
191C22⋊C4×C23—21no—
192C22⋊C4×C23—21no—
193C42×C23—22no—
194C42×C23—22no—
195C42×C23—22no—
196C42×C23—22no—
197C42×C23—21no—
198C42×C23C2×D411yescharacteristic
199C42×C23C2×D411yescharacteristic
200C42×C23C2×D411yescharacteristic
201C42×C23C2411yescharacteristic
202#19035C2311yesmaximal, characteristic, direct product factor, Sylow
203C23×C23—21no—
204C23×C23#12011yescharacteristic
205Q8×C23—41no—
206Q8×C23—44no—
207Q8×C23—42no—
208Q8×C23—81no—
209Q8×C23—82no—
210Q8×C23—82no—
211D4×C23—41no—
212D4×C23—42no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—21no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—41no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—81no—
227C2×C4×C23—82no—
228C2×C4×C23—82no—
229C2×C4×C23#12011yescharacteristic
230C2×C4×C23#12011yescharacteristic
231C2×C4×C23#12011yescharacteristic
232C2×C4×C23C22×D411yescharacteristic
233C2×C4×C23C22×D411yescharacteristic
234C2×C4×C23C22×D411yescharacteristic
235C8×C23—24no—
236C8×C23—42no—
237C8×C23—42no—
238#3563C2×C2311yescharacteristic
239#3239C2×C2311yescharacteristic
240#3165C2×C2311yescharacteristic, complemented
241C2×#491C2×C2311yescharacteristic, complemented
242#1664C2×C2314yescomplemented
243#1663C2×C2314yescomplemented
244#1531C2×C2311yescharacteristic, complemented
245#1528C2×C2311yescharacteristic, complemented
246#1524C2×C2311yescharacteristic
247C22×C23—21no—
248C22×C23—21no—
249C22×C23—41no—
250C22×C23—41no—
251C22×C23—41no—
252C22×C23#45612yes—
253C22×C23C2×#12011yescharacteristic
254C4×C23—21no—
255C4×C23—21no—
256C4×C23—21no—
257C4×C23—21no—
258C4×C23—21no—
259C4×C23—41no—
260C4×C23—42no—
261C4×C23—42no—
262C4×C23—41no—
263C4×C23—42no—
264C4×C23—42no—
265C4×C23—41no—
266C4×C23—82no—
267C4×C23#45412yes—
268#564C22×C2312yes—
269#563C22×C2312yes—
270#557—21no—
271#554—21no—
272#552—21no—
273#552—22no—
274#552—21no—
275#544—21no—
276#544—21no—
277#541—21no—
278#541—22no—
279#539—21no—
280#533C22×C2311yescharacteristic
281C2×C4⋊Q8C22×C2311yescharacteristic
282#529C22×C2311yescharacteristic
283C2×#125C22×C2311yescharacteristic
284#501C22×C2314yes—
285#501C22×C2312yes—
286#500—21no—
287#498—21no—
288#491—22no—
289#491C22×C2314yes—
290#489C22×C2312yes—
291#484—21no—
292#481—21no—
293#475—21no—
294#474—21no—
295#432C22×C2311yescharacteristic
296#428—22no—
297C2×#103—21no—
298C2×#103—21no—
299C4×M16C22×C2311yescharacteristic
300C2×#97C22×C2311yescharacteristic
301#330C22×C2312yes—
302#330C22×C2314yes—
303#330C22×C2314yes—
304#329C22×C2312yes—
305C2×C23—22no—
306C2×C23—81no—
307C2×C23C2×#45411yescharacteristic
308C2×C23C2×#45611yescharacteristic
309C2×C23#302211yescharacteristic
310C22×Q8—21no—
311C2×M16—21no—
312C22×C8—21no—
313C4⋊Q8—22no—
314C4⋊Q8—21no—
315C4⋊Q8—22no—
316C4⋊Q8—21no—
317C4⋊Q8—21no—
318C4⋊Q8—21no—
319C4⋊Q8D4×C2311yescharacteristic
320C4⋊Q8D4×C2311yescharacteristic
321C4⋊Q8C23×C2312yes—
322#126—22no—
323#126—22no—
324#126—22no—
325#126—22no—
326#126—21no—
327#125—22no—
328#125—22no—
329#125—21no—
330#125—22no—
331#125—22no—
332#125—22no—
333#125—22no—
334#125—22no—
335#125—22no—
336#125—21no—
337#125—21no—
338#125D4×C2312yes—
339#125C23×C2312yes—
340#125C23×C2311yescharacteristic
341#125C23×C2312yes—
342#124C23×C2311yescharacteristic
343#123—22no—
344#123—22no—
345#123—21no—
346C22⋊Q8—22no—
347C22⋊Q8—21no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8D4×C2311yescharacteristic
355#121D4×C2311yescharacteristic
356C4×Q8—21no—
357C4×Q8—22no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8D4×C2311yescharacteristic
362C4×D4—22no—
363C4×D4—21no—
364C4×D4D4×C2311yescharacteristic
365#117—22no—
366#117—21no—
367#117—22no—
368#117—21no—
369#117D4×C2312yes—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4D4×C2311yescharacteristic
376C2×C4⋊C4D4×C2311yescharacteristic
377C2×C42C23×C2311yescharacteristic
378#107—24no—
379#107—22no—
380#106—24no—
381#106—22no—
382C4⋊C8—21no—
383#103—22no—
384#103—24no—
385#103—24no—
386#102—22no—
387C22⋊C8—21no—
388#97—21no—
389#97C23×C2312yes—
390#97C23×C2312yes—
391#97C23×C2312yes—
392C4×C8—21no—
393C23#1903511yescharacteristic, direct product factor, Sylow
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—44no—
399C2×Q8—41no—
400C2×Q8C2×D4×C2311yescharacteristic
401C2×D4—21no—
402C2×D4C2×D4×C2311yescharacteristic
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—41no—
408C22×C4C2×D4×C2311yescharacteristic
409C22×C4C2×D4×C2311yescharacteristic
410C22×C4C2×D4×C2311yescharacteristic
411M16—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×D4×C2312yes—
443C4⋊C4C2×D4×C2312yes—
444C4⋊C4C2×D4×C2312yes—
445C4⋊C4C2×D4×C2311yescharacteristic
446C4⋊C4C2×D4×C2312yes—
447C4⋊C4C2×D4×C2311yescharacteristic
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—21no—
452C22⋊C4—22no—
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42C2×D4×C2311yescharacteristic
463C42C2×D4×C2311yescharacteristic
464C42C2×D4×C2311yescharacteristic
465C42C24×C2311yescharacteristic
466C23—21no—
467C23C23×#12011yescharacteristic
468Q8—41no—
469Q8—44no—
470Q8—42no—
471Q8—81no—
472Q8—82no—
473Q8—82no—
474D4—41no—
475D4—42no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4C23×#12011yescharacteristic
493C2×C4C23×#12011yescharacteristic
494C2×C4C23×#12011yescharacteristic
495C2×C4C22×D4×C2311yescharacteristic
496C2×C4C22×D4×C2311yescharacteristic
497C2×C4C22×D4×C2311yescharacteristic
498C8—24no—
499C8—42no—
500C8—42no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22C23×#45612yes—
507C22C2×C23×#12011yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—82no—
521C4C23×#45412yes—
522C2—22no—
523C2—81no—
524C2C2×C23×#45411yescharacteristic
525C2C2×C23×#45611yescharacteristic
526C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#3165C2semidirect
4. C2×C23×#491C2semidirect
5. C23×#1664C2semidirect
6. C23×#1663C2semidirect
7. C23×#1531C2semidirect
8. C23×#1528C2semidirect
202. #19035C23direct (N × H)
240. #3165C2×C23semidirect
241. C2×#491C2×C23semidirect
242. #1664C2×C23semidirect
243. #1663C2×C23semidirect
244. #1531C2×C23semidirect
245. #1528C2×C23semidirect
393. C23#19035direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#32392
C23×#31652
C2×C23×#4912
C23×#16642
C23×#16632
C23×#15312
C23×#15282
C23×#15242
#1903523