Finite Group Databasebeta

← C23×#9919 #19157545

Subgroups — C23×#9919 #19157545

Order 5888.

716 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#480C211yesmaximal, characteristic
2C23×#2618C211yesmaximal, characteristic
3C2×C23×#398C211yesmaximal, characteristic, complemented
4C23×#1823C211yesmaximal, characteristic, complemented
5C2×C23×#327C211yesmaximal, characteristic
6C2×C23×#324C211yesmaximal, characteristic, complemented
7C23×#1294C211yesmaximal, characteristic, complemented
8C2×C23×#123—21no—
9C2×C23×#121C2211yescharacteristic
10C22×C4⋊C4×C23—21no—
11C22×C4⋊C4×C23C2211yescharacteristic
12C22×C4⋊C4×C23C2211yescharacteristic
13C23×#480—22no—
14C23×#480—22no—
15C2×C23×#107—21no—
16C2×C23×#102—21no—
17C2×C23×C22⋊C8C2211yescharacteristic
18C2×C23×C22⋊C8C2211yescharacteristic
19C23×#398—22no—
20C23×#398—21no—
21C23×#398—22no—
22C23×#396—22no—
23C23×#394—21no—
24C23×#386—21no—
25C23×#380C2211yescharacteristic
26C2×C23×#95—21no—
27C2×C23×#95C2211yescharacteristic
28C23×#340—22no—
29C23×#327—22no—
30C23×#324—22no—
31C22×D4×C23—21no—
32C22×D4×C23D411yescharacteristic
33C23×C4×C23—21no—
34C23×C4×C23—21no—
35C23×C4×C23D411yescharacteristic
36C23×C4×C23D411yescharacteristic
37C23×C4×C23C2311yescharacteristic
38C22×C8×C23—21no—
39C22×C8×C23—21no—
40C23×#123—42no—
41C23×#123—42no—
42C23×#121—21no—
43C23×#121—21no—
44C23×#121—21no—
45C23×#121—21no—
46C2×C4⋊C4×C23—21no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—22no—
51C2×C4⋊C4×C23—24no—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23—21no—
54C2×C4⋊C4×C23—21no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23—21no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23—21no—
59C2×C4⋊C4×C23—42no—
60C2×C4⋊C4×C23—42no—
61C2×C4⋊C4×C23D411yescharacteristic
62C2×C4⋊C4×C23D412yes—
63C2×C4⋊C4×C23D412yes—
64C2×C22⋊C4×C23—22no—
65C2×C22⋊C4×C23—21no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23—22no—
68C23×#107—24no—
69C23×#102—24no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8D412yes—
74C23×#95—24no—
75C23×#95—22no—
76C23×#95—21no—
77C23×#95—21no—
78C23×#95D412yes—
79C24×C23—21no—
80C24×C23C2×D411yescharacteristic
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23—42no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C22×C4×C23—24no—
91C22×C4×C23—21no—
92C22×C4×C23—22no—
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—22no—
99C22×C4×C23—21no—
100C22×C4×C23—22no—
101C22×C4×C23—21no—
102C22×C4×C23—21no—
103C22×C4×C23—21no—
104C22×C4×C23—42no—
105C22×C4×C23—42no—
106C22×C4×C23—41no—
107C22×C4×C23—41no—
108C22×C4×C23—42no—
109C22×C4×C23—41no—
110C22×C4×C23—41no—
111C22×C4×C23—41no—
112C22×C4×C23—41no—
113C22×C4×C23—41no—
114C22×C4×C23—41no—
115C22×C4×C23D812yes—
116C22×C4×C23C2×D411yescharacteristic
117C22×C4×C23C2×D411yescharacteristic
118C22×C4×C23C2×D411yescharacteristic
119C22×C4×C23C2×D411yescharacteristic
120C22×C4×C23C2×D411yescharacteristic
121C22×C4×C23#4111yescharacteristic
122C2×C8×C23—22no—
123C2×C8×C23—42no—
124C2×C8×C23—42no—
125C2×C8×C23—41no—
126C4⋊C4×C23—22no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—22no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—42no—
131C4⋊C4×C23—48no—
132C4⋊C4×C23—44no—
133C4⋊C4×C23—41no—
134C4⋊C4×C23—41no—
135C4⋊C4×C23—41no—
136C4⋊C4×C23—41no—
137C4⋊C4×C23—41no—
138C4⋊C4×C23—41no—
139C22⋊C4×C23—44no—
140C22⋊C4×C23—41no—
141C22⋊C4×C23—41no—
142C22⋊C4×C23—41no—
143C22⋊C4×C23—41no—
144C22⋊C4×C23—82no—
145#9919C2311yesmaximal, characteristic, direct product factor, Sylow
146C23×C23—22no—
147C23×C23—22no—
148C23×C23—41no—
149C23×C23—41no—
150C23×C23—41no—
151C23×C23—42no—
152C23×C23—42no—
153C23×C23#12011yescharacteristic
154C23×C23#12011yescharacteristic
155C23×C23#12111yescharacteristic
156C23×C23C2×D811yescharacteristic
157C23×C23#13511yescharacteristic
158C23×C23AGL(1,ℤ/8)11yescharacteristic
159C23×C23AGL(1,ℤ/8)11yescharacteristic
160D4×C23—42no—
161D4×C23—81no—
162D4×C23—81no—
163D4×C23—81no—
164D4×C23—81no—
165D4×C23—81no—
166C2×C4×C23—22no—
167C2×C4×C23—22no—
168C2×C4×C23—24no—
169C2×C4×C23—22no—
170C2×C4×C23—22no—
171C2×C4×C23—22no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—44no—
177C2×C4×C23—44no—
178C2×C4×C23—42no—
179C2×C4×C23—41no—
180C2×C4×C23—42no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—42no—
185C2×C4×C23—41no—
186C2×C4×C23—42no—
187C2×C4×C23—41no—
188C2×C4×C23—42no—
189C2×C4×C23—42no—
190C2×C4×C23—41no—
191C2×C4×C23—42no—
192C2×C4×C23—41no—
193C2×C4×C23—42no—
194C2×C4×C23—42no—
195C2×C4×C23—42no—
196C2×C4×C23—81no—
197C2×C4×C23—81no—
198C2×C4×C23—81no—
199C2×C4×C23—81no—
200C2×C4×C23—81no—
201C2×C4×C23#12011yescharacteristic
202C2×C4×C23#12111yescharacteristic
203C2×C4×C23#12111yescharacteristic
204C2×C4×C23#12711yescharacteristic
205C8×C23—42no—
206C8×C23—81no—
207C2×#480C2×C2311yescharacteristic
208#2618C2×C2311yescharacteristic
209C2×#398C2×C2311yescharacteristic, complemented
210#1823C2×C2311yescharacteristic, complemented
211C2×#327C2×C2311yescharacteristic
212C2×#324C2×C2311yescharacteristic, complemented
213#1294C2×C2311yescharacteristic, complemented
214C22×C23—22no—
215C22×C23—22no—
216C22×C23—22no—
217C22×C23—22no—
218C22×C23—21no—
219C22×C23—21no—
220C22×C23—21no—
221C22×C23—21no—
222C22×C23—82no—
223C22×C23—82no—
224C22×C23—81no—
225C22×C23—81no—
226C22×C23—81no—
227C22×C23#39211yescharacteristic
228C22×C23#44711yescharacteristic
229C22×C23#44711yescharacteristic
230C22×C23#44911yescharacteristic
231C22×C23#44911yescharacteristic
232C22×C23#45612yes—
233C22×C23#45812yes—
234C22×C23#46611yescharacteristic
235C22×C23#46911yescharacteristic
236C4×C23—21no—
237C4×C23—21no—
238C4×C23—21no—
239C4×C23—21no—
240C4×C23—44no—
241C4×C23—42no—
242C4×C23—82no—
243C4×C23—82no—
244C4×C23—81no—
245C4×C23—81no—
246C2×#123—21no—
247C2×#121C22×C2311yescharacteristic
248C22×C4⋊C4—21no—
249C22×C4⋊C4C22×C2311yescharacteristic
250C22×C4⋊C4C22×C2311yescharacteristic
251#480—22no—
252#480—22no—
253C2×#107—21no—
254C2×#102—21no—
255C2×C22⋊C8C22×C2311yescharacteristic
256C2×C22⋊C8C22×C2311yescharacteristic
257#398—21no—
258#398—22no—
259#398—22no—
260#396—22no—
261#394—21no—
262#386—21no—
263#380C22×C2311yescharacteristic
264C2×#95—21no—
265C2×#95C22×C2311yescharacteristic
266#340—22no—
267#327—22no—
268#324—22no—
269C2×C23—21no—
270C2×C23—21no—
271C2×C23—21no—
272C2×C23—21no—
273C2×C23—161no—
274C2×C23#160912yes—
275C2×C23#161012yes—
276C2×C23#199911yescharacteristic
277C2×C23#200511yescharacteristic
278C2×C23#202211yescharacteristic
279C22×D4—21no—
280C22×D4D4×C2311yescharacteristic
281C23×C4—21no—
282C23×C4—21no—
283C23×C4D4×C2311yescharacteristic
284C23×C4D4×C2311yescharacteristic
285C23×C4C23×C2311yescharacteristic
286C22×C8—21no—
287C22×C8—21no—
288#123—42no—
289#123—42no—
290#121—21no—
291#121—21no—
292#121—21no—
293#121—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4D4×C2312yes—
310C2×C4⋊C4D4×C2311yescharacteristic
311C2×C4⋊C4D4×C2312yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316#107—24no—
317#102—24no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8D4×C2312yes—
322#95—24no—
323#95—22no—
324#95—21no—
325#95—21no—
326#95D4×C2312yes—
327C23#991911yescharacteristic, direct product factor, Sylow
328C24—21no—
329C24C2×D4×C2311yescharacteristic
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4D8×C2312yes—
365C22×C4C2×D4×C2311yescharacteristic
366C22×C4C2×D4×C2311yescharacteristic
367C22×C4C2×D4×C2311yescharacteristic
368C22×C4C2×D4×C2311yescharacteristic
369C22×C4C2×D4×C2311yescharacteristic
370C22×C4#41×C2311yescharacteristic
371C2×C8—22no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—41no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C22⋊C4—44no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—82no—
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23C23×#12011yescharacteristic
402C23C23×#12011yescharacteristic
403C23C23×#12111yescharacteristic
404C23C2×D8×C2311yescharacteristic
405C23C23×#13511yescharacteristic
406C23C23×AGL(1,ℤ/8)11yescharacteristic
407C23C23×AGL(1,ℤ/8)11yescharacteristic
408D4—42no—
409D4—81no—
410D4—81no—
411D4—81no—
412D4—81no—
413D4—81no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4C23×#12011yescharacteristic
450C2×C4C23×#12111yescharacteristic
451C2×C4C23×#12111yescharacteristic
452C2×C4C23×#12711yescharacteristic
453C8—42no—
454C8—81no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—82no—
467C22—82no—
468C22C23×#39211yescharacteristic
469C22C23×#44711yescharacteristic
470C22C23×#44711yescharacteristic
471C22C23×#44911yescharacteristic
472C22C23×#44911yescharacteristic
473C22C23×#45612yes—
474C22C23×#45812yes—
475C22C23×#46611yescharacteristic
476C22C23×#46911yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—42no—
483C4—82no—
484C4—81no—
485C4—81no—
486C4—82no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—161no—
492C2C23×#160912yes—
493C2C23×#161012yes—
494C2C23×#199911yescharacteristic
495C2C23×#200511yescharacteristic
496C2C23×#202211yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#398C2semidirect
4. C23×#1823C2semidirect
6. C2×C23×#324C2semidirect
7. C23×#1294C2semidirect
145. #9919C23direct (N × H)
209. C2×#398C2×C23semidirect
210. #1823C2×C23semidirect
212. C2×#324C2×C23semidirect
213. #1294C2×C23semidirect
327. C23#9919direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4802
C23×#26182
C2×C23×#3982
C23×#18232
C2×C23×#3272
C2×C23×#3242
C23×#12942
#991923