Finite Group Databasebeta

← C23×#11471 #19159097

Subgroups — C23×#11471 #19159097

Order 5888.

756 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C23×#120C211yesmaximal, characteristic
2C23×#1460C211yesmaximal, characteristic, complemented
3C23×#1459C211yesmaximal, characteristic, complemented
4C23×#1326C212yesmaximal, complemented
5C23×#1316C212yesmaximal, complemented
6C24×C4×C23C411yescharacteristic
7C2×C23×#120C411yescharacteristic
8C2×C4×D4×C23—21no—
9C2×C4×D4×C23C2211yescharacteristic
10C23×#431—21no—
11C2×C23×C22⋊C8—21no—
12C23×#386—21no—
13C23×#386C411yescharacteristic
14C23×#385—21no—
15C23×#385C2211yescharacteristic
16C23×#379C2211yescharacteristic
17C4×C22⋊C4×C23—21no—
18C4×C22⋊C4×C23C411yescharacteristic
19C23×#324—21no—
20C23×#323—21no—
21C23×#323C2212yes—
22C23×#323C2212yes—
23C25×C23C2×C411yescharacteristic
24C22×D4×C23—21no—
25C22×D4×C23C2×C411yescharacteristic
26C23×C4×C23—21no—
27C23×C4×C23—21no—
28C23×C4×C23—44no—
29C23×C4×C23—41no—
30C23×C4×C23—41no—
31C23×C4×C23C2×C411yescharacteristic
32C23×C4×C23C2×C411yescharacteristic
33C23×C4×C23D411yescharacteristic
34C22×C8×C23—41no—
35C23×#120—22no—
36C23×#120C814yescomplemented
37C4×D4×C23—41no—
38C4×D4×C23—41no—
39C4×D4×C23—44no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23D411yescharacteristic
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23C2×C411yescharacteristic
47C2×C22⋊C4×C23C2×C411yescharacteristic
48C2×C22⋊C4×C23D411yescharacteristic
49C2×C22⋊C4×C23C2311yescharacteristic
50C2×C42×C23—21no—
51C2×C42×C23D411yescharacteristic
52C23×C4⋊C8—41no—
53C23×C22⋊C8—22no—
54C23×C22⋊C8—22no—
55C23×C22⋊C8—41no—
56C4×C8×C23—41no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—21no—
60C24×C23—21no—
61C24×C23—21no—
62C24×C23—42no—
63C24×C23—41no—
64C24×C23—42no—
65C24×C23—41no—
66C24×C23C22⋊C411yescharacteristic
67C24×C23C2×C812yescomplemented
68C24×C23C22×C411yescharacteristic
69C2×D4×C23—24no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—42no—
73C2×D4×C23—42no—
74C2×D4×C23C2×C812yes—
75C2×D4×C23M1612yes—
76C22×C4×C23—24no—
77C22×C4×C23—21no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23—42no—
87C22×C4×C23—42no—
88C22×C4×C23—42no—
89C22×C4×C23—42no—
90C22×C4×C23—42no—
91C22×C4×C23—44no—
92C22×C4×C23—44no—
93C22×C4×C23—44no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—41no—
97C22×C4×C23—44no—
98C22×C4×C23C22⋊C411yescharacteristic
99C22×C4×C23C22⋊C411yescharacteristic
100C22×C4×C23C22⋊C411yescharacteristic
101C22×C4×C23C2×D411yescharacteristic
102C22×C4×C23C2×D411yescharacteristic
103C2×C8×C23—42no—
104C2×C8×C23—42no—
105C2×C8×C23—82no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—42no—
108C22⋊C4×C23—21no—
109C22⋊C4×C23—24no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23—42no—
114C22⋊C4×C23—42no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23C2×C812yescomplemented
117C42×C23—21no—
118C42×C23—21no—
119C42×C23—42no—
120#11471C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—22no—
122C23×C23—22no—
123C23×C23—22no—
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23—41no—
129C23×C23—44no—
130C23×C23—42no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—44no—
135C23×C23—42no—
136C23×C23—41no—
137C23×C23—41no—
138C23×C23—42no—
139C23×C23—42no—
140C23×C23—42no—
141C23×C23—48no—
142C23×C23—42no—
143C23×C23—41no—
144C23×C23—41no—
145C23×C23C22⋊C812yes—
146C23×C23C2×C22⋊C411yescharacteristic
147C23×C23C22×C811yescharacteristic
148C23×C23C2×M1611yescharacteristic
149D4×C23—42no—
150D4×C23—42no—
151D4×C23—48no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—22no—
155C2×C4×C23—22no—
156C2×C4×C23—22no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—44no—
161C2×C4×C23—42no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—48no—
166C2×C4×C23—42no—
167C2×C4×C23—41no—
168C2×C4×C23—42no—
169C2×C4×C23—44no—
170C2×C4×C23—42no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—42no—
175C2×C4×C23—41no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—81no—
179C2×C4×C23C22⋊C812yes—
180C2×C4×C23#9911yescharacteristic
181C2×C4×C23#9911yescharacteristic
182C8×C23—82no—
183C8×C23—82no—
184C4×#120C2×C2311yescharacteristic
185#1460C2×C2311yescharacteristic, complemented
186#1459C2×C2311yescharacteristic, complemented
187#1326C2×C2312yescomplemented
188#1316C2×C2312yescomplemented
189C22×C23—21no—
190C22×C23—21no—
191C22×C23—24no—
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—22no—
195C22×C23—42no—
196C22×C23—41no—
197C22×C23—41no—
198C22×C23—42no—
199C22×C23—42no—
200C22×C23—42no—
201C22×C23—42no—
202C22×C23—44no—
203C22×C23—44no—
204C22×C23—44no—
205C22×C23—44no—
206C22×C23—42no—
207C22×C23—41no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—42no—
211C22×C23—41no—
212C22×C23—42no—
213C22×C23—81no—
214C22×C23C2×C22⋊C811yescharacteristic
215C22×C23C2×#9911yescharacteristic
216C22×C23#41311yescharacteristic
217C4×C23—21no—
218C4×C23—22no—
219C4×C23—42no—
220C4×C23—41no—
221C4×C23—42no—
222C4×C23—41no—
223C4×C23—41no—
224C4×C23—81no—
225C4×C23AΣL(1,𝔽16)12yes—
226C24×C4C4×C2311yescharacteristic
227C2×#120C4×C2311yescharacteristic
228C2×C4×D4—21no—
229C2×C4×D4C22×C2311yescharacteristic
230#431—21no—
231C2×C22⋊C8—21no—
232#386—21no—
233#386C4×C2311yescharacteristic
234#385—21no—
235#385C22×C2311yescharacteristic
236#379C22×C2311yescharacteristic
237C4×C22⋊C4—21no—
238C4×C22⋊C4C4×C2311yescharacteristic
239#324—21no—
240#323—21no—
241#323C22×C2312yes—
242#323C22×C2312yes—
243C2×C23—21no—
244C2×C23—21no—
245C2×C23—44no—
246C2×C23—41no—
247C2×C23—41no—
248C2×C23—42no—
249C2×C23#145911yescharacteristic
250C2×C23C2×AΣL(1,𝔽16)11yescharacteristic
251C2×C23#212011yescharacteristic
252C25C2×C4×C2311yescharacteristic
253C22×D4—21no—
254C22×D4C2×C4×C2311yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—44no—
258C23×C4—41no—
259C23×C4—41no—
260C23×C4C2×C4×C2311yescharacteristic
261C23×C4C2×C4×C2311yescharacteristic
262C23×C4D4×C2311yescharacteristic
263C22×C8—41no—
264#120—22no—
265#120C8×C2314yescomplemented
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4D4×C2311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4C2×C4×C2311yescharacteristic
276C2×C22⋊C4C2×C4×C2311yescharacteristic
277C2×C22⋊C4D4×C2311yescharacteristic
278C2×C22⋊C4C23×C2311yescharacteristic
279C2×C42—21no—
280C2×C42D4×C2311yescharacteristic
281C4⋊C8—41no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C4×C8—41no—
286#95—21no—
287#95—21no—
288#95—21no—
289C23#1147111yescharacteristic, direct product factor, Sylow
290C24—21no—
291C24—21no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24C22⋊C4×C2311yescharacteristic
297C24C2×C8×C2312yescomplemented
298C24C22×C4×C2311yescharacteristic
299C2×D4—24no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C2×C8×C2312yes—
305C2×D4M16×C2312yes—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4C22⋊C4×C2311yescharacteristic
329C22×C4C22⋊C4×C2311yescharacteristic
330C22×C4C22⋊C4×C2311yescharacteristic
331C22×C4C2×D4×C2311yescharacteristic
332C22×C4C2×D4×C2311yescharacteristic
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—82no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C22⋊C4—21no—
339C22⋊C4—24no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4C2×C8×C2312yescomplemented
347C42—21no—
348C42—21no—
349C42—42no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—44no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—44no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—48no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23C23×C22⋊C812yes—
375C23C2×C22⋊C4×C2311yescharacteristic
376C23C22×C8×C2311yescharacteristic
377C23C2×M16×C2311yescharacteristic
378D4—42no—
379D4—42no—
380D4—48no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4C23×C22⋊C812yes—
409C2×C4C23×#9911yescharacteristic
410C2×C4C23×#9911yescharacteristic
411C8—82no—
412C8—82no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22C2×C23×C22⋊C811yescharacteristic
439C22C2×C23×#9911yescharacteristic
440C22C23×#41311yescharacteristic
441C4—22no—
442C4—21no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—41no—
447C4—41no—
448C4—81no—
449C4C23×AΣL(1,𝔽16)12yes—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—42no—
456C2C23×#145911yescharacteristic
457C2C2×C23×AΣL(1,𝔽16)11yescharacteristic
458C2C23×#212011yescharacteristic
subgroupG/H
centerC2×C4×C23#99
commutatorC23C22×C8×C23
Frattini, cosocleC2×C22⋊C4C23×C23
socleC22×C23C2×#99
prime radicalC2×C23×#120C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1460C2semidirect
3. C23×#1459C2semidirect
4. C23×#1326C2semidirect
5. C23×#1316C2semidirect
36. C23×#120C8semidirect
67. C24×C23C2×C8semidirect
116. C22⋊C4×C23C2×C8semidirect
120. #11471C23direct (N × H)
185. #1460C2×C23semidirect
186. #1459C2×C23semidirect
187. #1326C2×C23semidirect
188. #1316C2×C23semidirect
265. #120C8×C23semidirect
289. C23#11471direct (N × H)
297. C24C2×C8×C23semidirect
346. C22⋊C4C2×C8×C23semidirect

Maximal subgroups

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

subgroupindex
C4×C23×#1202
C23×#14602
C23×#14592
C23×#13262
C23×#13162
#1147123