Finite Group Databasebeta

← C23×#10431 #19158057

Subgroups — C23×#10431 #19158057

Order 5888.

614 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2537C211yesmaximal, characteristic
2C23×#1834C211yesmaximal, characteristic, complemented
3C23×#1802C211yesmaximal, characteristic, complemented
4C23×#1802C211yesmaximal, characteristic, complemented
5C2×C23×D4⋊C8C211yesmaximal, characteristic, complemented
6C2×C23×D4⋊C8C211yesmaximal, characteristic, complemented
7C23×#1296C211yesmaximal, characteristic, complemented
8C2×C23×#121—21no—
9C2×C23×#121—21no—
10C2×C4×D4×C23C2211yescharacteristic
11C2×C4×D4×C23C2211yescharacteristic
12C2×C23×C4⋊C8C2211yescharacteristic
13C2×C23×C4⋊C8C2211yescharacteristic
14C2×C23×#102—21no—
15C2×C23×#102—21no—
16C2×C4×C8×C23C2211yescharacteristic
17C23×#394—22no—
18C23×#384C2211yescharacteristic
19C23×#384C2211yescharacteristic
20C23×#380—21no—
21C23×#380—21no—
22C23×#340—21no—
23C23×#340—21no—
24C23×#340—22no—
25C23×D4⋊C8—22no—
26C23×D4⋊C8—22no—
27C22×D4×C23—22no—
28C22×D4×C23D411yescharacteristic
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C23×C4×C23—21no—
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C23×#121—42no—
36C23×#121—42no—
37C23×#121—42no—
38C23×#121—42no—
39C4×D4×C23—21no—
40C4×D4×C23—21no—
41C4×D4×C23—21no—
42C4×D4×C23—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23—21no—
46C4×D4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C4⋊C4×C23D411yescharacteristic
53C2×C4⋊C4×C23D411yescharacteristic
54C2×C4⋊C4×C23D412yes—
55C2×C22⋊C4×C23—22no—
56C2×C22⋊C4×C23—22no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C42×C23C2311yescharacteristic
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×#102—42no—
65C23×#102—42no—
66C4×C8×C23—21no—
67C4×C8×C23D412yescomplemented
68C23×#95—22no—
69C24×C23—21no—
70C24×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—21no—
75C2×D4×C23—21no—
76C2×D4×C23—21no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—44no—
82C2×D4×C23—44no—
83C2×D4×C23—82no—
84C2×D4×C23D812yes—
85C2×D4×C23SD1612yes—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—22no—
89C22×C4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—41no—
97C22×C4×C23—41no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23C2×D411yescharacteristic
105C2×C8×C23—22no—
106C2×C8×C23—41no—
107C2×C8×C23—41no—
108C2×C8×C23—41no—
109C2×C8×C23—41no—
110C2×C8×C23—41no—
111C2×C8×C23—41no—
112C2×C8×C23—41no—
113C2×C8×C23—41no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—42no—
119C4⋊C4×C23—42no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23—41no—
122C4⋊C4×C23—41no—
123C4⋊C4×C23—41no—
124C4⋊C4×C23—41no—
125C4⋊C4×C23—41no—
126C4⋊C4×C23—41no—
127C4⋊C4×C23—41no—
128C22⋊C4×C23—44no—
129C22⋊C4×C23—44no—
130C22⋊C4×C23—41no—
131C22⋊C4×C23—41no—
132C22⋊C4×C23—41no—
133C22⋊C4×C23—41no—
134C42×C23C2×D411yescharacteristic
135C42×C23#4111yescharacteristic
136C42×C23#4111yescharacteristic
137C42×C23#4111yescharacteristic
138#10431C2311yesmaximal, characteristic, direct product factor, Sylow
139C23×C23—22no—
140C23×C23—22no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23—41no—
144C23×C23—41no—
145C23×C23—41no—
146C23×C23—41no—
147C23×C23—41no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23#12011yescharacteristic
154D4×C23—22no—
155D4×C23—22no—
156D4×C23—22no—
157D4×C23—22no—
158D4×C23—41no—
159D4×C23—41no—
160D4×C23—41no—
161D4×C23—41no—
162D4×C23—88no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—42no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—42no—
172C2×C4×C23—41no—
173C2×C4×C23—42no—
174C2×C4×C23—81no—
175C2×C4×C23—81no—
176C2×C4×C23—81no—
177C2×C4×C23—81no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23—81no—
181C2×C4×C23—81no—
182C2×C4×C23—81no—
183C2×C4×C23—81no—
184C2×C4×C23—81no—
185C2×C4×C23#12111yescharacteristic
186C2×C4×C23#12111yescharacteristic
187C2×C4×C23#12111yescharacteristic
188C2×C4×C23#12311yescharacteristic
189C2×C4×C23#12311yescharacteristic
190C2×C4×C23#12411yescharacteristic
191C2×C4×C23C2×D811yescharacteristic
192C2×C4×C23C2×SD1611yescharacteristic
193C2×C4×C23#13511yescharacteristic
194C2×C4×C23#13511yescharacteristic
195C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C23#13711yescharacteristic
199C8×C23—42no—
200C8×C23—81no—
201C8×C23—81no—
202#2537C2×C2311yescharacteristic
203#1834C2×C2311yescharacteristic, complemented
204#1802C2×C2311yescharacteristic, complemented
205#1802C2×C2311yescharacteristic, complemented
206C2×D4⋊C8C2×C2311yescharacteristic, complemented
207C2×D4⋊C8C2×C2311yescharacteristic, complemented
208#1296C2×C2311yescharacteristic, complemented
209C22×C23—42no—
210C22×C23—42no—
211C22×C23—42no—
212C22×C23—42no—
213C22×C23—42no—
214C22×C23—42no—
215C22×C23—81no—
216C22×C23—81no—
217C22×C23—81no—
218C22×C23—81no—
219C22×C23—81no—
220C22×C23—81no—
221C22×C23—81no—
222C22×C23—81no—
223C22×C23#39411yescharacteristic
224C22×C23#44711yescharacteristic
225C22×C23#44911yescharacteristic
226C22×C23#45011yescharacteristic
227C22×C23#45211yescharacteristic
228C22×C23#45411yescharacteristic
229C22×C23#45511yescharacteristic
230C4×C23—21no—
231C4×C23—21no—
232C4×C23—82no—
233C4×C23—81no—
234C4×C23—81no—
235C4×C23—161no—
236C4×C23#46511yescharacteristic
237C4×C23#46611yescharacteristic
238C4×C23#46811yescharacteristic
239C4×C23#46911yescharacteristic
240C4×C23#48011yescharacteristic
241C4×C23#48111yescharacteristic
242C4×C23#48211yescharacteristic
243C4×C23#48211yescharacteristic
244C2×#121—21no—
245C2×#121—21no—
246C2×C4×D4C22×C2311yescharacteristic
247C2×C4×D4C22×C2311yescharacteristic
248C2×C4⋊C8C22×C2311yescharacteristic
249C2×C4⋊C8C22×C2311yescharacteristic
250C2×#102—21no—
251C2×#102—21no—
252C2×C4×C8C22×C2311yescharacteristic
253#394—22no—
254#384C22×C2311yescharacteristic
255#384C22×C2311yescharacteristic
256#380—21no—
257#380—21no—
258#340—21no—
259#340—21no—
260#340—22no—
261D4⋊C8—22no—
262D4⋊C8—22no—
263C2×C23—82no—
264C2×C23—82no—
265C2×C23#162511yescharacteristic
266C2×C23#163511yescharacteristic
267C2×C23#163911yescharacteristic
268C2×C23#164611yescharacteristic
269C2×C23#203311yescharacteristic
270C2×C23#203411yescharacteristic
271C2×C23#204011yescharacteristic
272C22×D4—22no—
273C22×D4D4×C2311yescharacteristic
274C22×D4D4×C2311yescharacteristic
275C23×C4—21no—
276C23×C4—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280#121—42no—
281#121—42no—
282#121—42no—
283#121—42no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4D4×C2311yescharacteristic
298C2×C4⋊C4D4×C2311yescharacteristic
299C2×C4⋊C4D4×C2312yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C42C23×C2311yescharacteristic
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309#102—42no—
310#102—42no—
311C4×C8—21no—
312C4×C8D4×C2312yescomplemented
313#95—22no—
314C23#1043111yescharacteristic, direct product factor, Sylow
315C24—21no—
316C24—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—82no—
330C2×D4D8×C2312yes—
331C2×D4SD16×C2312yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×D4×C2311yescharacteristic
349C22×C4C2×D4×C2311yescharacteristic
350C22×C4C2×D4×C2311yescharacteristic
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42C2×D4×C2311yescharacteristic
381C42#41×C2311yescharacteristic
382C42#41×C2311yescharacteristic
383C42#41×C2311yescharacteristic
384C23—22no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C23×#12011yescharacteristic
399D4—22no—
400D4—22no—
401D4—22no—
402D4—22no—
403D4—41no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—88no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4C23×#12111yescharacteristic
431C2×C4C23×#12111yescharacteristic
432C2×C4C23×#12111yescharacteristic
433C2×C4C23×#12311yescharacteristic
434C2×C4C23×#12311yescharacteristic
435C2×C4C23×#12411yescharacteristic
436C2×C4C2×D8×C2311yescharacteristic
437C2×C4C2×SD16×C2311yescharacteristic
438C2×C4C23×#13511yescharacteristic
439C2×C4C23×#13511yescharacteristic
440C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
441C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C23×#13711yescharacteristic
444C8—42no—
445C8—81no—
446C8—81no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22C23×#39411yescharacteristic
462C22C23×#44711yescharacteristic
463C22C23×#44911yescharacteristic
464C22C23×#45011yescharacteristic
465C22C23×#45211yescharacteristic
466C22C23×#45411yescharacteristic
467C22C23×#45511yescharacteristic
468C4—21no—
469C4—21no—
470C4—82no—
471C4—81no—
472C4—81no—
473C4—161no—
474C4C23×#46511yescharacteristic
475C4C23×#46611yescharacteristic
476C4C23×#46811yescharacteristic
477C4C23×#46911yescharacteristic
478C4C23×#48011yescharacteristic
479C4C23×#48111yescharacteristic
480C4C23×#48211yescharacteristic
481C4C23×#48211yescharacteristic
482C2—82no—
483C2—82no—
484C2C23×#162511yescharacteristic
485C2C23×#163511yescharacteristic
486C2C23×#163911yescharacteristic
487C2C23×#164611yescharacteristic
488C2C23×#203311yescharacteristic
489C2C23×#203411yescharacteristic
490C2C23×#204011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#2537C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1834C2semidirect
3. C23×#1802C2semidirect
4. C23×#1802C2semidirect
5. C2×C23×D4⋊C8C2semidirect
6. C2×C23×D4⋊C8C2semidirect
7. C23×#1296C2semidirect
67. C4×C8×C23D4semidirect
138. #10431C23direct (N × H)
203. #1834C2×C23semidirect
204. #1802C2×C23semidirect
205. #1802C2×C23semidirect
206. C2×D4⋊C8C2×C23semidirect
207. C2×D4⋊C8C2×C23semidirect
208. #1296C2×C23semidirect
312. C4×C8D4×C23semidirect
314. C23#10431direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25372
C23×#18342
C23×#18022
C2×C23×D4⋊C82
C23×#12962
#1043123