Finite Group Databasebeta

← C23×#8517 #19156143

Subgroups — C23×#8517 #19156143

Order 5888.

1040 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#537C211yesmaximal, characteristic
2C23×#1785C211yesmaximal, characteristic, complemented
3C23×#1782C211yesmaximal, characteristic, complemented
4C23×#1511C212yesmaximal, complemented
5C23×#1510C212yesmaximal, complemented
6C22×#41×C23C411yescharacteristic, complemented
7C22×#41×C23C2211yescharacteristic
8C23×#537—24no—
9C23×#537—24no—
10C2×C23×C22⋊Q8—21no—
11C2×C23×C22⋊Q8C411yescharacteristic, complemented
12C2×C23×C22⋊Q8C2211yescharacteristic
13C2×C23×#121—21no—
14C2×C23×#121—21no—
15C2×C23×#121—21no—
16C2×C23×#121C411yescharacteristic, complemented
17C2×C23×#121C2211yescharacteristic
18C22×C4⋊C4×C23C411yescharacteristic, complemented
19C2×C23×#101—21no—
20C2×C23×#100—21no—
21C2×C23×#99—21no—
22C2×C23×#99—21no—
23C23×#407C2212yes—
24C23×#381C2212yes—
25C23×#328—24no—
26C23×#327—24no—
27C2×#41×C23—22no—
28C2×#41×C23—22no—
29C2×#41×C23—22no—
30C2×#41×C23—22no—
31C2×#41×C23—42no—
32C2×#41×C23—42no—
33C2×#41×C23—42no—
34C2×#41×C23—42no—
35C22×Q8×C23C2×C411yescharacteristic
36C22×Q8×C23D411yescharacteristic
37C22×D4×C23—21no—
38C22×D4×C23—21no—
39C22×D4×C23—21no—
40C22×D4×C23C2×C411yescharacteristic
41C22×D4×C23D411yescharacteristic
42C22×D4×C23D411yescharacteristic
43C22×D4×C23D411yescharacteristic
44C23×C4×C23—21no—
45C23×C4×C23—21no—
46C23×C4×C23C2×C411yescharacteristic
47C23×C4×C23C2×C411yescharacteristic
48C23×C4×C23C2311yescharacteristic
49C2×M16×C23—22no—
50C23×C22⋊Q8—28no—
51C23×C22⋊Q8—22no—
52C23×C22⋊Q8—22no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8D414yes—
55C23×#121—28no—
56C23×#121—28no—
57C23×#121—28no—
58C23×#121—22no—
59C23×#121—22no—
60C23×#121—22no—
61C23×#121D414yes—
62C2×C4⋊C4×C23—21no—
63C2×C4⋊C4×C23—21no—
64C2×C4⋊C4×C23—21no—
65C2×C4⋊C4×C23—22no—
66C2×C4⋊C4×C23—22no—
67C2×C4⋊C4×C23C2×C411yescharacteristic
68C2×C4⋊C4×C23C2×C411yescharacteristic
69C2×C22⋊C4×C23—21no—
70C2×C22⋊C4×C23—21no—
71C2×C22⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—22no—
74C23×#101—42no—
75C23×#100—42no—
76C23×#99—42no—
77C23×#99—42no—
78C23×C22⋊C8—22no—
79C23×C22⋊C8—22no—
80C23×#95—22no—
81C23×#95—22no—
82C24×C23—21no—
83C24×C23C22⋊C411yescharacteristic
84C24×C23C22⋊C411yescharacteristic
85C24×C23C2×D411yescharacteristic
86#41×C23—48no—
87#41×C23—48no—
88#41×C23—48no—
89#41×C23—48no—
90C2×Q8×C23—22no—
91C2×Q8×C23—21no—
92C2×Q8×C23—22no—
93C2×Q8×C23—41no—
94C2×Q8×C23—41no—
95C2×Q8×C23—41no—
96C2×Q8×C23C22⋊C412yes—
97C2×D4×C23—24no—
98C2×D4×C23—22no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—24no—
103C2×D4×C23—24no—
104C2×D4×C23—22no—
105C2×D4×C23—21no—
106C2×D4×C23—22no—
107C2×D4×C23—41no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—42no—
113C2×D4×C23—42no—
114C2×D4×C23—82no—
115C2×D4×C23—82no—
116C2×D4×C23C22⋊C412yes—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—24no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—24no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—41no—
129C22×C4×C23—41no—
130C22×C4×C23—41no—
131C22×C4×C23—41no—
132C22×C4×C23—42no—
133C22×C4×C23—42no—
134C22×C4×C23—42no—
135C22×C4×C23—82no—
136C22×C4×C23C22⋊C411yescharacteristic
137C22×C4×C23C22⋊C411yescharacteristic
138C22×C4×C23C22×C411yescharacteristic
139C22×C4×C23C2×D411yescharacteristic
140C22×C4×C23C2×D412yes—
141C22×C4×C23C2×D412yes—
142M16×C23—42no—
143M16×C23—42no—
144C2×C8×C23—42no—
145C4⋊C4×C23—24no—
146C4⋊C4×C23—24no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—24no—
150C4⋊C4×C23C22⋊C412yes—
151C4⋊C4×C23C22⋊C412yes—
152C22⋊C4×C23—24no—
153C22⋊C4×C23—24no—
154C22⋊C4×C23—24no—
155C22⋊C4×C23—24no—
156C22⋊C4×C23—42no—
157C22⋊C4×C23—42no—
158#8517C2311yesmaximal, characteristic, direct product factor, Sylow
159C23×C23—21no—
160C23×C23—21no—
161C23×C23—21no—
162C23×C23—21no—
163C23×C23—21no—
164C23×C23—22no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—41no—
168C23×C23—41no—
169C23×C23—41no—
170C23×C23—41no—
171C23×C23—42no—
172C23×C23—42no—
173C23×C23—41no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23#9912yes—
178C23×C23C2×C22⋊C411yescharacteristic
179C23×C23#12012yes—
180Q8×C23—44no—
181Q8×C23—42no—
182Q8×C23—42no—
183D4×C23—44no—
184D4×C23—44no—
185D4×C23—42no—
186D4×C23—42no—
187D4×C23—44no—
188D4×C23—48no—
189D4×C23—84no—
190D4×C23—84no—
191C2×C4×C23—24no—
192C2×C4×C23—24no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—21no—
197C2×C4×C23—21no—
198C2×C4×C23—21no—
199C2×C4×C23—22no—
200C2×C4×C23—22no—
201C2×C4×C23—22no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—42no—
206C2×C4×C23—42no—
207C2×C4×C23—48no—
208C2×C4×C23—42no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—42no—
212C2×C4×C23—41no—
213C2×C4×C23—41no—
214C2×C4×C23—84no—
215C2×C4×C23—82no—
216C2×C4×C23—82no—
217C2×C4×C23—82no—
218C2×C4×C23#9912yes—
219C2×C4×C23C2×C22⋊C411yescharacteristic
220C2×C4×C23C2×C22⋊C411yescharacteristic
221C2×C4×C23#12012yes—
222C8×C23—82no—
223C2×#537C2×C2311yescharacteristic
224#1785C2×C2311yescharacteristic, complemented
225#1782C2×C2311yescharacteristic, complemented
226#1511C2×C2312yescomplemented
227#1510C2×C2312yescomplemented
228C22×C23—21no—
229C22×C23—21no—
230C22×C23—22no—
231C22×C23—22no—
232C22×C23—22no—
233C22×C23—22no—
234C22×C23—42no—
235C22×C23—42no—
236C22×C23—41no—
237C22×C23—41no—
238C22×C23—41no—
239C22×C23—42no—
240C22×C23—41no—
241C22×C23—42no—
242C22×C23—41no—
243C22×C23—42no—
244C22×C23—81no—
245C22×C23—81no—
246C22×C23—81no—
247C22×C23—81no—
248C22×C23—82no—
249C22×C23—82no—
250C22×C23#37911yescharacteristic
251C22×C23C2×#9911yescharacteristic
252C22×C23C2×#9911yescharacteristic
253C4×C23—22no—
254C4×C23—22no—
255C4×C23—21no—
256C4×C23—21no—
257C4×C23—42no—
258C4×C23—42no—
259C4×C23—41no—
260C4×C23—42no—
261C4×C23—162no—
262C22×#41C4×C2311yescharacteristic, complemented
263C22×#41C22×C2311yescharacteristic
264#537—24no—
265#537—24no—
266C2×C22⋊Q8—21no—
267C2×C22⋊Q8C4×C2311yescharacteristic, complemented
268C2×C22⋊Q8C22×C2311yescharacteristic
269C2×#121—21no—
270C2×#121—21no—
271C2×#121—21no—
272C2×#121C4×C2311yescharacteristic, complemented
273C2×#121C22×C2311yescharacteristic
274C22×C4⋊C4C4×C2311yescharacteristic, complemented
275C2×#101—21no—
276C2×#100—21no—
277C2×#99—21no—
278C2×#99—21no—
279#407C22×C2312yes—
280#381C22×C2312yes—
281#328—24no—
282#327—24no—
283C2×C23—21no—
284C2×C23—21no—
285C2×C23—42no—
286C2×C23—42no—
287C2×C23—81no—
288C2×C23—82no—
289C2×C23#178111yescharacteristic
290C2×C23#179511yescharacteristic
291C2×C23#179611yescharacteristic
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—42no—
297C2×#41—42no—
298C2×#41—42no—
299C2×#41—42no—
300C22×Q8C2×C4×C2311yescharacteristic
301C22×Q8D4×C2311yescharacteristic
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4C2×C4×C2311yescharacteristic
306C22×D4D4×C2311yescharacteristic
307C22×D4D4×C2311yescharacteristic
308C22×D4D4×C2311yescharacteristic
309C23×C4—21no—
310C23×C4—21no—
311C23×C4C2×C4×C2311yescharacteristic
312C23×C4C2×C4×C2311yescharacteristic
313C23×C4C23×C2311yescharacteristic
314C2×M16—22no—
315C22⋊Q8—28no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8D4×C2314yes—
320#121—28no—
321#121—28no—
322#121—28no—
323#121—22no—
324#121—22no—
325#121—22no—
326#121D4×C2314yes—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2×C4×C2311yescharacteristic
333C2×C4⋊C4C2×C4×C2311yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339#101—42no—
340#100—42no—
341#99—42no—
342#99—42no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345#95—22no—
346#95—22no—
347C23#851711yescharacteristic, direct product factor, Sylow
348C24—21no—
349C24C22⋊C4×C2311yescharacteristic
350C24C22⋊C4×C2311yescharacteristic
351C24C2×D4×C2311yescharacteristic
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8C22⋊C4×C2312yes—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4C22⋊C4×C2312yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4C22⋊C4×C2311yescharacteristic
403C22×C4C22⋊C4×C2311yescharacteristic
404C22×C4C22×C4×C2311yescharacteristic
405C22×C4C2×D4×C2311yescharacteristic
406C22×C4C2×D4×C2312yes—
407C22×C4C2×D4×C2312yes—
408M16—42no—
409M16—42no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—24no—
416C4⋊C4C22⋊C4×C2312yes—
417C4⋊C4C22⋊C4×C2312yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C23×#9912yes—
443C23C2×C22⋊C4×C2311yescharacteristic
444C23C23×#12012yes—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—44no—
453D4—48no—
454D4—84no—
455D4—84no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4C23×#9912yes—
484C2×C4C2×C22⋊C4×C2311yescharacteristic
485C2×C4C2×C22⋊C4×C2311yescharacteristic
486C2×C4C23×#12012yes—
487C8—82no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22C23×#37911yescharacteristic
511C22C2×C23×#9911yescharacteristic
512C22C2×C23×#9911yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—162no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—42no—
526C2—82no—
527C2—81no—
528C2C23×#178111yescharacteristic
529C2C23×#179511yescharacteristic
530C2C23×#179611yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1785C2semidirect
3. C23×#1782C2semidirect
4. C23×#1511C2semidirect
5. C23×#1510C2semidirect
6. C22×#41×C23C4semidirect
11. C2×C23×C22⋊Q8C4semidirect
16. C2×C23×#121C4semidirect
18. C22×C4⋊C4×C23C4semidirect
158. #8517C23direct (N × H)
224. #1785C2×C23semidirect
225. #1782C2×C23semidirect
226. #1511C2×C23semidirect
227. #1510C2×C23semidirect
262. C22×#41C4×C23semidirect
267. C2×C22⋊Q8C4×C23semidirect
272. C2×#121C4×C23semidirect
274. C22×C4⋊C4C4×C23semidirect
347. C23#8517direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5372
C23×#17852
C23×#17822
C23×#15112
C23×#15102
#851723