Finite Group Databasebeta

← C23×#32871 #19180497

Subgroups — C23×#32871 #19180497

Order 5888.

492 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2103C211yesmaximal, characteristic, complemented
2C23×#2103C211yesmaximal, characteristic, complemented
3C23×#2098C211yesmaximal, characteristic, complemented
4C23×#2082C211yesmaximal, characteristic, complemented
5C23×#2081C211yesmaximal, characteristic, complemented
6C23×#2070C211yesmaximal, characteristic, complemented
7C23×#2069C211yesmaximal, characteristic, complemented
8C23×#2049C211yesmaximal, characteristic, complemented
9C23×#2048C211yesmaximal, characteristic
10C23×#2046C211yesmaximal, characteristic, complemented
11C23×#2046C211yesmaximal, characteristic, complemented
12C23×#2019C211yesmaximal, characteristic
13C23×#1903C211yesmaximal, characteristic
14C23×#1902C211yesmaximal, characteristic, complemented
15C23×#1858C211yesmaximal, characteristic, complemented
16C2×C23×#137C2211yescharacteristic
17C2×C23×AGL(1,ℤ/8)C2211yescharacteristic
18C22×SD16×C23C2211yescharacteristic
19C23×#568C2211yescharacteristic
20C23×#430C2211yescharacteristic
21C23×#430C2211yescharacteristic
22C23×#430C2211yescharacteristic
23C23×#430C2211yescharacteristic
24C23×#430C2211yescharacteristic
25C23×#424C2211yescharacteristic
26C23×#424C2211yescharacteristic
27C23×#421C2211yescharacteristic
28C23×#421C2211yescharacteristic
29C23×#419C2211yescharacteristic
30C23×#419C2211yescharacteristic
31C23×#418C2211yescharacteristic
32C23×#418C2211yescharacteristic
33C23×#413C2211yescharacteristic
34C2×C23×#101C2211yescharacteristic
35C2×C23×#101C2211yescharacteristic
36C2×C23×#101C2211yescharacteristic
37C2×C23×#100C2211yescharacteristic
38C2×C23×#100C2211yescharacteristic
39C2×C23×#100C2211yescharacteristic
40C23×#408C2211yescharacteristic
41C23×#344C2211yescharacteristic
42C23×#344C2211yescharacteristic
43C23×#344C2211yescharacteristic
44C23×#344C2211yescharacteristic
45C23×#341C2211yescharacteristic
46C23×#341C2211yescharacteristic
47C23×#341C2211yescharacteristic
48C23×#338C2211yescharacteristic
49C23×#338C2211yescharacteristic
50C23×H(3,ℤ/4)C2211yescharacteristic
51C2×#41×C23C2311yescharacteristic
52C22×Q8×C23C2311yescharacteristic
53C22×D4×C23C2311yescharacteristic
54C23×#137—41no—
55C23×#137—41no—
56C23×AGL(1,ℤ/8)—41no—
57C23×AGL(1,ℤ/8)—41no—
58C2×Q16×C23—21no—
59C2×SD16×C23—21no—
60C2×SD16×C23—21no—
61C2×SD16×C23—41no—
62C2×SD16×C23—41no—
63C2×SD16×C23—41no—
64C2×D8×C23—21no—
65C23×#131—41no—
66C23×#131—41no—
67C2×M16×C23—41no—
68C2×M16×C23C2311yescharacteristic
69C2×M16×C23C2311yescharacteristic
70C2×M16×C23C2311yescharacteristic
71C2×M16×C23C2311yescharacteristic
72C2×M16×C23C2311yescharacteristic
73C2×M16×C23C2311yescharacteristic
74C2×M16×C23C2311yescharacteristic
75C2×M16×C23C2311yescharacteristic
76C2×M16×C23C2311yescharacteristic
77C22×C8×C23C2311yescharacteristic
78C23×#117C2311yescharacteristic
79C23×#117C2311yescharacteristic
80C23×#108—41no—
81C23×#108—41no—
82C23×#108—41no—
83C23×#108—41no—
84C23×#108—41no—
85C23×C4⋊C8—41no—
86C23×C4⋊C8—41no—
87C23×#104—41no—
88C23×#104—41no—
89C23×#103—41no—
90C23×#103—41no—
91C23×#102—41no—
92C23×#102—41no—
93C23×#101—21no—
94C23×#101—41no—
95C23×#101—41no—
96C23×#101—41no—
97C23×#100—21no—
98C23×#100—41no—
99C23×#100—41no—
100C23×#100—41no—
101C23×C22⋊C8—41no—
102C24×C23—21no—
103#41×C23—21no—
104#41×C23—21no—
105#41×C23—21no—
106#41×C23—21no—
107C2×Q8×C23—21no—
108C2×Q8×C23—21no—
109C2×Q8×C23—21no—
110C2×Q8×C23—21no—
111C2×Q8×C23—21no—
112C2×Q8×C23—21no—
113C2×Q8×C23#4111yescharacteristic
114C2×D4×C23—21no—
115C2×D4×C23—21no—
116C2×D4×C23—21no—
117C2×D4×C23—21no—
118C2×D4×C23—21no—
119C2×D4×C23—21no—
120C2×D4×C23—21no—
121C2×D4×C23#4111yescharacteristic
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23C2411yescharacteristic
125Q16×C23—42no—
126SD16×C23—42no—
127SD16×C23—42no—
128SD16×C23—44no—
129D8×C23—42no—
130M16×C23—21no—
131M16×C23—21no—
132M16×C23—21no—
133M16×C23—21no—
134M16×C23—21no—
135M16×C23—21no—
136M16×C23—21no—
137M16×C23—21no—
138M16×C23—21no—
139M16×C23—21no—
140M16×C23—21no—
141M16×C23—21no—
142M16×C23—21no—
143M16×C23—21no—
144M16×C23—21no—
145M16×C23—21no—
146M16×C23—21no—
147M16×C23—21no—
148M16×C23—42no—
149C2×C8×C23—21no—
150C2×C8×C23—21no—
151C2×C8×C23—21no—
152C2×C8×C23—21no—
153C2×C8×C23—21no—
154C2×C8×C23—21no—
155C2×C8×C23—21no—
156C2×C8×C23—21no—
157C2×C8×C23—21no—
158C2×C8×C23—21no—
159C2×C8×C23—21no—
160C2×C8×C23—21no—
161C2×C8×C23—42no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—21no—
164C22⋊C4×C23—21no—
165C22⋊C4×C23—21no—
166C42×C23—21no—
167C42×C23—21no—
168#32871C2311yesmaximal, characteristic, direct product factor, Sylow
169C23×C23—21no—
170C23×C23—41no—
171C23×C23—41no—
172C23×C23—41no—
173C23×C23—161no—
174C23×C232−1+411yescharacteristic
175Q8×C23—21no—
176Q8×C23—21no—
177Q8×C23—41no—
178Q8×C23—41no—
179Q8×C23—41no—
180Q8×C23—41no—
181D4×C23—21no—
182D4×C23—21no—
183D4×C23—41no—
184D4×C23—41no—
185D4×C23—41no—
186D4×C23—41no—
187D4×C23—41no—
188D4×C23—41no—
189C2×C4×C23—21no—
190C2×C4×C23—41no—
191C2×C4×C23—41no—
192C2×C4×C23—41no—
193C2×C4×C23—41no—
194C2×C4×C23—41no—
195C2×C4×C23—41no—
196C2×C4×C23—41no—
197C2×C4×C23C2×#4111yescharacteristic
198C2×C4×C23H(5,𝔽2)11yescharacteristic
199C2×C4×C23H(5,𝔽2)11yescharacteristic
200C2×C4×C23H(5,𝔽2)11yescharacteristic
201C2×C4×C23H(5,𝔽2)11yescharacteristic
202C2×C4×C23H(5,𝔽2)11yescharacteristic
203C8×C23—41no—
204C8×C23—41no—
205C8×C23—41no—
206C8×C23—41no—
207C8×C23—41no—
208C8×C23—41no—
209C8×C23—41no—
210C8×C23—41no—
211C8×C23—41no—
212C8×C23—41no—
213#2103C2×C2311yescharacteristic, complemented
214#2103C2×C2311yescharacteristic, complemented
215#2098C2×C2311yescharacteristic, complemented
216#2082C2×C2311yescharacteristic, complemented
217#2081C2×C2311yescharacteristic, complemented
218#2070C2×C2311yescharacteristic, complemented
219#2069C2×C2311yescharacteristic, complemented
220#2049C2×C2311yescharacteristic, complemented
221#2048C2×C2311yescharacteristic
222#2046C2×C2311yescharacteristic, complemented
223#2046C2×C2311yescharacteristic, complemented
224#2019C2×C2311yescharacteristic
225#1903C2×C2311yescharacteristic
226#1902C2×C2311yescharacteristic, complemented
227#1858C2×C2311yescharacteristic, complemented
228C22×C23—41no—
229C22×C23—41no—
230C22×C23—81no—
231C22×C23—81no—
232C22×C23—161no—
233C22×C23—161no—
234C22×C23—161no—
235C22×C23#53911yescharacteristic
236C22×C23#56211yescharacteristic
237C22×C23#56211yescharacteristic
238C4×C23—41no—
239C4×C23—81no—
240C4×C23—81no—
241C4×C23—81no—
242C4×C23#53811yescharacteristic
243C4×C23#53811yescharacteristic
244C4×C23#56011yescharacteristic
245C4×C23#56011yescharacteristic
246C2×#137C22×C2311yescharacteristic
247C2×AGL(1,ℤ/8)C22×C2311yescharacteristic
248C22×SD16C22×C2311yescharacteristic
249#568C22×C2311yescharacteristic
250#430C22×C2311yescharacteristic
251#430C22×C2311yescharacteristic
252#430C22×C2311yescharacteristic
253#430C22×C2311yescharacteristic
254#430C22×C2311yescharacteristic
255#424C22×C2311yescharacteristic
256#424C22×C2311yescharacteristic
257#421C22×C2311yescharacteristic
258#421C22×C2311yescharacteristic
259#419C22×C2311yescharacteristic
260#419C22×C2311yescharacteristic
261#418C22×C2311yescharacteristic
262#418C22×C2311yescharacteristic
263#413C22×C2311yescharacteristic
264C2×#101C22×C2311yescharacteristic
265C2×#101C22×C2311yescharacteristic
266C2×#101C22×C2311yescharacteristic
267C2×#100C22×C2311yescharacteristic
268C2×#100C22×C2311yescharacteristic
269C2×#100C22×C2311yescharacteristic
270#408C22×C2311yescharacteristic
271#344C22×C2311yescharacteristic
272#344C22×C2311yescharacteristic
273#344C22×C2311yescharacteristic
274#344C22×C2311yescharacteristic
275#341C22×C2311yescharacteristic
276#341C22×C2311yescharacteristic
277#341C22×C2311yescharacteristic
278#338C22×C2311yescharacteristic
279#338C22×C2311yescharacteristic
280H(3,ℤ/4)C22×C2311yescharacteristic
281C2×C23—21no—
282C2×C23—21no—
283C2×C23—21no—
284C2×C23—81no—
285C2×C23—161no—
286C2×C23#282611yescharacteristic
287C2×#41C23×C2311yescharacteristic
288C22×Q8C23×C2311yescharacteristic
289C22×D4C23×C2311yescharacteristic
290#137—41no—
291#137—41no—
292AGL(1,ℤ/8)—41no—
293AGL(1,ℤ/8)—41no—
294C2×Q16—21no—
295C2×SD16—21no—
296C2×SD16—21no—
297C2×SD16—41no—
298C2×SD16—41no—
299C2×SD16—41no—
300C2×D8—21no—
301#131—41no—
302#131—41no—
303C2×M16—41no—
304C2×M16C23×C2311yescharacteristic
305C2×M16C23×C2311yescharacteristic
306C2×M16C23×C2311yescharacteristic
307C2×M16C23×C2311yescharacteristic
308C2×M16C23×C2311yescharacteristic
309C2×M16C23×C2311yescharacteristic
310C2×M16C23×C2311yescharacteristic
311C2×M16C23×C2311yescharacteristic
312C2×M16C23×C2311yescharacteristic
313C22×C8C23×C2311yescharacteristic
314#117C23×C2311yescharacteristic
315#117C23×C2311yescharacteristic
316#108—41no—
317#108—41no—
318#108—41no—
319#108—41no—
320#108—41no—
321C4⋊C8—41no—
322C4⋊C8—41no—
323#104—41no—
324#104—41no—
325#103—41no—
326#103—41no—
327#102—41no—
328#102—41no—
329#101—21no—
330#101—41no—
331#101—41no—
332#101—41no—
333#100—21no—
334#100—41no—
335#100—41no—
336#100—41no—
337C22⋊C8—41no—
338C23#3287111yescharacteristic, direct product factor, Sylow
339C24—21no—
340#41—21no—
341#41—21no—
342#41—21no—
343#41—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8#41×C2311yescharacteristic
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4#41×C2311yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4C24×C2311yescharacteristic
362Q16—42no—
363SD16—42no—
364SD16—42no—
365SD16—44no—
366D8—42no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—21no—
380M16—21no—
381M16—21no—
382M16—21no—
383M16—21no—
384M16—21no—
385M16—42no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—42no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C42—21no—
404C42—21no—
405C23—21no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—161no—
410C23C23×2−1+411yescharacteristic
411Q8—21no—
412Q8—21no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4C2×#41×C2311yescharacteristic
434C2×C4C23×H(5,𝔽2)11yescharacteristic
435C2×C4C23×H(5,𝔽2)11yescharacteristic
436C2×C4C23×H(5,𝔽2)11yescharacteristic
437C2×C4C23×H(5,𝔽2)11yescharacteristic
438C2×C4C23×H(5,𝔽2)11yescharacteristic
439C8—41no—
440C8—41no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—41no—
445C8—41no—
446C8—41no—
447C8—41no—
448C8—41no—
449C22—41no—
450C22—41no—
451C22—81no—
452C22—81no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22C23×#53911yescharacteristic
457C22C23×#56211yescharacteristic
458C22C23×#56211yescharacteristic
459C4—41no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4C23×#53811yescharacteristic
464C4C23×#53811yescharacteristic
465C4C23×#56011yescharacteristic
466C4C23×#56011yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—81no—
471C2—161no—
472C2C23×#282611yescharacteristic
subgroupG/H
center, socleC2×C23#2826
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC2×C23×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2103C2semidirect
2. C23×#2103C2semidirect
3. C23×#2098C2semidirect
4. C23×#2082C2semidirect
5. C23×#2081C2semidirect
6. C23×#2070C2semidirect
7. C23×#2069C2semidirect
8. C23×#2049C2semidirect
10. C23×#2046C2semidirect
11. C23×#2046C2semidirect
14. C23×#1902C2semidirect
15. C23×#1858C2semidirect
168. #32871C23direct (N × H)
213. #2103C2×C23semidirect
214. #2103C2×C23semidirect
215. #2098C2×C23semidirect
216. #2082C2×C23semidirect
217. #2081C2×C23semidirect
218. #2070C2×C23semidirect
219. #2069C2×C23semidirect
220. #2049C2×C23semidirect
222. #2046C2×C23semidirect
223. #2046C2×C23semidirect
226. #1902C2×C23semidirect
227. #1858C2×C23semidirect
338. C23#32871direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#21032
C23×#20982
C23×#20822
C23×#20812
C23×#20702
C23×#20692
C23×#20492
C23×#20482
C23×#20462
C23×#20192
C23×#19032
C23×#19022
C23×#18582
#3287123