Finite Group Databasebeta

← C23×#8513 #19156139

Subgroups — C23×#8513 #19156139

Order 5888.

724 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×#3559C211yesmaximal, characteristic
2C23×#1989C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic, complemented
4C23×#1551C211yesmaximal, characteristic, complemented
5C23×#1528C211yesmaximal, characteristic, complemented
6C23×#1509C211yesmaximal, characteristic, complemented
7C23×#1483C211yesmaximal, characteristic, complemented
8C23×#563C412yescomplemented
9C23×#562C412yescomplemented
10C23×#554—21no—
11C23×#553—21no—
12C23×#553—21no—
13C23×#547—22no—
14C23×#547—21no—
15C23×#544—21no—
16C23×#540—21no—
17C23×#539—22no—
18C23×#539—21no—
19C23×#537—21no—
20C23×#532C2211yescharacteristic
21C23×#529C2211yescharacteristic
22C2×C23×#125C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C2×C23×C4⋊C8C2211yescharacteristic
25C2×C23×#104—22no—
26C4×M16×C23C2211yescharacteristic
27C4×C4⋊C4×C23C2211yescharacteristic
28C23×#340—21no—
29C23×#340—21no—
30C23×#333—21no—
31C23×#331—21no—
32C23×#330—22no—
33C23×#328—21no—
34C23×#327—21no—
35C23×Q8⋊C8—21no—
36C23×D4⋊C8—21no—
37C2×#41×C23—22no—
38C2×#41×C23D412yes—
39C2×M16×C23—21no—
40C22×C8×C23—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—22no—
43C23×C4⋊Q8D411yescharacteristic
44C23×#127D411yescharacteristic
45C23×#126—22no—
46C23×#126—22no—
47C23×#126—21no—
48C23×#125—21no—
49C23×#125—22no—
50C23×#125—21no—
51C23×#125—21no—
52C23×#125C2×C412yes—
53C23×#125C2×C411yescharacteristic
54C23×#125D412yes—
55C23×#124C2×C411yescharacteristic
56C23×#124C2×C412yes—
57C23×#123—22no—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—21no—
62C23×#123—22no—
63C23×#123—22no—
64C23×C22⋊Q8—22no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8—22no—
70C23×C22⋊Q8D411yescharacteristic
71C23×#121—22no—
72C23×#121—21no—
73C23×#121—21no—
74C23×#121—22no—
75C23×#121—22no—
76C23×#121—22no—
77C23×#121—21no—
78C23×#121—21no—
79C23×#121D411yescharacteristic
80C4×Q8×C23D411yescharacteristic
81C4×D4×C23—22no—
82C4×D4×C23—22no—
83C4×D4×C23—21no—
84C4×D4×C23—21no—
85C4×D4×C23—21no—
86C4×D4×C23—21no—
87C4×D4×C23—21no—
88C4×D4×C23D411yescharacteristic
89C23×#117—21no—
90C23×#117—21no—
91C2×C4⋊C4×C23—21no—
92C2×C4⋊C4×C23—22no—
93C2×C4⋊C4×C23—21no—
94C2×C4⋊C4×C23—21no—
95C2×C4⋊C4×C23—21no—
96C2×C4⋊C4×C23D411yescharacteristic
97C2×C4⋊C4×C23D411yescharacteristic
98C2×C42×C23—21no—
99C2×C42×C23C2311yescharacteristic
100C23×C4⋊C8—21no—
101C23×C4⋊C8—21no—
102C23×C4⋊C8—21no—
103C23×#104—44no—
104C23×C22⋊C8—21no—
105C23×#97—21no—
106C23×#97—21no—
107C4×C8×C23—21no—
108C23×#95—21no—
109#41×C23—42no—
110#41×C23—42no—
111#41×C23—82no—
112#41×C23—82no—
113C2×Q8×C23—22no—
114C2×Q8×C23C22⋊C411yescharacteristic
115C2×Q8×C23C22⋊C412yes—
116C2×D4×C23—22no—
117C2×D4×C23—21no—
118C2×D4×C23—22no—
119C2×D4×C23—22no—
120C2×D4×C23—21no—
121C2×D4×C23—21no—
122C2×D4×C23C22⋊C411yescharacteristic
123C2×D4×C23C22⋊C412yes—
124C22×C4×C23—21no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23—21no—
129C22×C4×C23—21no—
130C22×C4×C23—22no—
131C22×C4×C23—41no—
132C22×C4×C23C2×D411yescharacteristic
133C22×C4×C23C2×D411yescharacteristic
134C22×C4×C23C2×D411yescharacteristic
135M16×C23—22no—
136M16×C23—41no—
137C2×C8×C23—41no—
138C2×C8×C23—42no—
139C2×C8×C23—41no—
140C2×C8×C23—41no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—22no—
146C4⋊C4×C23—22no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—21no—
149C4⋊C4×C23—22no—
150C4⋊C4×C23—21no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—21no—
153C4⋊C4×C23—22no—
154C4⋊C4×C23—21no—
155C4⋊C4×C23—21no—
156C4⋊C4×C23—21no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—41no—
159C4⋊C4×C23C22⋊C412yes—
160C4⋊C4×C23C22⋊C412yes—
161C4⋊C4×C23C22⋊C411yescharacteristic
162C4⋊C4×C23C22⋊C411yescharacteristic
163C22⋊C4×C23—22no—
164C22⋊C4×C23—22no—
165C22⋊C4×C23—22no—
166C22⋊C4×C23—22no—
167C22⋊C4×C23—22no—
168C22⋊C4×C23—21no—
169C22⋊C4×C23—21no—
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—21no—
173C22⋊C4×C23—21no—
174C22⋊C4×C23—21no—
175C42×C23—22no—
176C42×C23—21no—
177C42×C23—42no—
178C42×C23C22×C411yescharacteristic
179C42×C23C2×D411yescharacteristic
180C42×C23C2×D411yescharacteristic
181C42×C23C2×D411yescharacteristic
182#8513C2311yesmaximal, characteristic, direct product factor, Sylow
183C23×C23—22no—
184C23×C23—21no—
185C23×C23#12011yescharacteristic
186Q8×C23—41no—
187Q8×C23—42no—
188Q8×C23—82no—
189D4×C23—41no—
190D4×C23—42no—
191D4×C23—42no—
192D4×C23—41no—
193D4×C23—42no—
194D4×C23—41no—
195D4×C23—41no—
196D4×C23—41no—
197D4×C23—81no—
198D4×C23—82no—
199D4×C23—82no—
200C2×C4×C23—21no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—21no—
206C2×C4×C23—22no—
207C2×C4×C23—22no—
208C2×C4×C23—21no—
209C2×C4×C23—22no—
210C2×C4×C23—41no—
211C2×C4×C23—41no—
212C2×C4×C23—41no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—42no—
216C2×C4×C23—82no—
217C2×C4×C23—81no—
218C2×C4×C23—82no—
219C2×C4×C23—82no—
220C2×C4×C23—82no—
221C2×C4×C23—81no—
222C2×C4×C23—81no—
223C2×C4×C23C2×C22⋊C411yescharacteristic
224C2×C4×C23C2×C22⋊C411yescharacteristic
225C2×C4×C23C2×C22⋊C411yescharacteristic
226C2×C4×C23#12011yescharacteristic
227C2×C4×C23#12011yescharacteristic
228C2×C4×C23#12011yescharacteristic
229C8×C23—42no—
230C8×C23—81no—
231C8×C23—81no—
232#3559C2×C2311yescharacteristic
233#1989C2×C2311yescharacteristic, complemented
234#1980C2×C2311yescharacteristic, complemented
235#1551C2×C2311yescharacteristic, complemented
236#1528C2×C2311yescharacteristic, complemented
237#1509C2×C2311yescharacteristic, complemented
238#1483C2×C2311yescharacteristic, complemented
239C22×C23—21no—
240C22×C23—21no—
241C22×C23—42no—
242C22×C23—41no—
243C22×C23—41no—
244C22×C23—42no—
245C22×C23—42no—
246C22×C23—41no—
247C22×C23#37911yescharacteristic
248C22×C23#45311yescharacteristic
249C22×C23#45511yescharacteristic
250C4×C23—21no—
251C4×C23—21no—
252C4×C23—21no—
253C4×C23—21no—
254C4×C23—21no—
255C4×C23—42no—
256C4×C23—42no—
257C4×C23—41no—
258C4×C23—41no—
259C4×C23—42no—
260C4×C23—41no—
261C4×C23—82no—
262C4×C23#45411yescharacteristic
263C4×C23#45611yescharacteristic
264#563C4×C2312yescomplemented
265#562C4×C2312yescomplemented
266#554—21no—
267#553—21no—
268#553—21no—
269#547—22no—
270#547—21no—
271#544—21no—
272#540—21no—
273#539—22no—
274#539—21no—
275#537—21no—
276#532C22×C2311yescharacteristic
277#529C22×C2311yescharacteristic
278C2×#125C22×C2311yescharacteristic
279#432C22×C2311yescharacteristic
280C2×C4⋊C8C22×C2311yescharacteristic
281C2×#104—22no—
282C4×M16C22×C2311yescharacteristic
283C4×C4⋊C4C22×C2311yescharacteristic
284#340—21no—
285#340—21no—
286#333—21no—
287#331—21no—
288#330—22no—
289#328—21no—
290#327—21no—
291Q8⋊C8—21no—
292D4⋊C8—21no—
293C2×C23—21no—
294C2×C23—21no—
295C2×C23—82no—
296C2×C23—81no—
297C2×C23#179211yescharacteristic
298C2×C23#179311yescharacteristic
299C2×C23#179411yescharacteristic
300C2×#41—22no—
301C2×#41D4×C2312yes—
302C2×M16—21no—
303C22×C8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—22no—
306C4⋊Q8D4×C2311yescharacteristic
307#127D4×C2311yescharacteristic
308#126—22no—
309#126—22no—
310#126—21no—
311#125—22no—
312#125—21no—
313#125—21no—
314#125—21no—
315#125C2×C4×C2312yes—
316#125C2×C4×C2311yescharacteristic
317#125D4×C2312yes—
318#124C2×C4×C2311yescharacteristic
319#124C2×C4×C2312yes—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—21no—
325#123—22no—
326#123—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8D4×C2311yescharacteristic
334#121—22no—
335#121—22no—
336#121—21no—
337#121—21no—
338#121—22no—
339#121—22no—
340#121—21no—
341#121—21no—
342#121D4×C2311yescharacteristic
343C4×Q8D4×C2311yescharacteristic
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4D4×C2311yescharacteristic
352#117—21no—
353#117—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4D4×C2311yescharacteristic
360C2×C4⋊C4D4×C2311yescharacteristic
361C2×C42—21no—
362C2×C42C23×C2311yescharacteristic
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366#104—44no—
367C22⋊C8—21no—
368#97—21no—
369#97—21no—
370C4×C8—21no—
371#95—21no—
372C23#851311yescharacteristic, direct product factor, Sylow
373#41—42no—
374#41—42no—
375#41—82no—
376#41—82no—
377C2×Q8—22no—
378C2×Q8C22⋊C4×C2311yescharacteristic
379C2×Q8C22⋊C4×C2312yes—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C22⋊C4×C2311yescharacteristic
387C2×D4C22⋊C4×C2312yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4C2×D4×C2311yescharacteristic
397C22×C4C2×D4×C2311yescharacteristic
398C22×C4C2×D4×C2311yescharacteristic
399M16—22no—
400M16—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—41no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—41no—
423C4⋊C4C22⋊C4×C2312yes—
424C4⋊C4C22⋊C4×C2312yes—
425C4⋊C4C22⋊C4×C2311yescharacteristic
426C4⋊C4C22⋊C4×C2311yescharacteristic
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C42—22no—
440C42—21no—
441C42—42no—
442C42C22×C4×C2311yescharacteristic
443C42C2×D4×C2311yescharacteristic
444C42C2×D4×C2311yescharacteristic
445C42C2×D4×C2311yescharacteristic
446C23—22no—
447C23—21no—
448C23C23×#12011yescharacteristic
449Q8—42no—
450Q8—41no—
451Q8—82no—
452D4—42no—
453D4—41no—
454D4—42no—
455D4—41no—
456D4—42no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—81no—
461D4—82no—
462D4—82no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C2×C22⋊C4×C2311yescharacteristic
487C2×C4C2×C22⋊C4×C2311yescharacteristic
488C2×C4C2×C22⋊C4×C2311yescharacteristic
489C2×C4C23×#12011yescharacteristic
490C2×C4C23×#12011yescharacteristic
491C2×C4C23×#12011yescharacteristic
492C8—42no—
493C8—81no—
494C8—81no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—41no—
499C22—41no—
500C22—42no—
501C22—42no—
502C22—41no—
503C22C23×#37911yescharacteristic
504C22C23×#45311yescharacteristic
505C22C23×#45511yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—42no—
512C4—42no—
513C4—41no—
514C4—41no—
515C4—42no—
516C4—41no—
517C4—82no—
518C4C23×#45411yescharacteristic
519C4C23×#45611yescharacteristic
520C2—21no—
521C2—21no—
522C2—82no—
523C2—81no—
524C2C23×#179211yescharacteristic
525C2C23×#179311yescharacteristic
526C2C23×#179411yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3559C2
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
2. C23×#1989C2semidirect
3. C23×#1980C2semidirect
4. C23×#1551C2semidirect
5. C23×#1528C2semidirect
6. C23×#1509C2semidirect
7. C23×#1483C2semidirect
8. C23×#563C4semidirect
9. C23×#562C4semidirect
182. #8513C23direct (N × H)
233. #1989C2×C23semidirect
234. #1980C2×C23semidirect
235. #1551C2×C23semidirect
236. #1528C2×C23semidirect
237. #1509C2×C23semidirect
238. #1483C2×C23semidirect
264. #563C4×C23semidirect
265. #562C4×C23semidirect
372. C23#8513direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35592
C23×#19892
C23×#19802
C23×#15512
C23×#15282
C23×#15092
C23×#14832
#851323