Finite Group Databasebeta

← C23×#13219 #19160845

Subgroups — C23×#13219 #19160845

Order 5888.

624 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2534C211yesmaximal, characteristic
2C2×C23×#352C212yesmaximal, complemented
3C2×C23×AΣL(1,𝔽16)C212yesmaximal
4C23×#1799C211yesmaximal, characteristic, complemented
5C4×C23×#100C211yesmaximal, characteristic, complemented
6C2×C23×#123C2211yescharacteristic
7C2×C23×#120C2211yescharacteristic
8C2×C4×D4×C23C2211yescharacteristic
9C22×C22⋊C4×C23C411yescharacteristic, complemented
10C2×C23×#100C2212yes—
11C2×C23×#99C2212yes—
12C4×M16×C23—21no—
13C23×#396—21no—
14C23×#394—21no—
15C23×#388—21no—
16C23×#386—21no—
17C23×#381C411yescharacteristic, complemented
18C23×#380—21no—
19C23×#380C411yescharacteristic, complemented
20C23×#379C411yescharacteristic, complemented
21C23×#352—24no—
22C23×AΣL(1,𝔽16)—24no—
23C23×#343—21no—
24C23×#342—21no—
25C25×C23C2×C411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23C2311yescharacteristic
28C23×C4×C23C2×C411yescharacteristic
29C23×C4×C23C2×C411yescharacteristic
30C23×C4×C23D411yescharacteristic
31C2×M16×C23—22no—
32C23×#123—22no—
33C23×#123—22no—
34C23×#120—22no—
35C23×#120—22no—
36C4×D4×C23—41no—
37C4×D4×C23—41no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23—41no—
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×C23—21no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23—41no—
50C2×C22⋊C4×C23—41no—
51C2×C22⋊C4×C23C2×C411yescharacteristic
52C2×C22⋊C4×C23C2×C411yescharacteristic
53C2×C22⋊C4×C23C2×C411yescharacteristic
54C2×C22⋊C4×C23D411yescharacteristic
55C2×C42×C23D411yescharacteristic
56C23×#100—22no—
57C23×#100D414yes—
58C23×#99—22no—
59C23×#99—22no—
60C23×#97—41no—
61C4×C8×C23—41no—
62C23×#95—21no—
63C23×#95—21no—
64C23×#95—21no—
65C23×#95—41no—
66C24×C23—21no—
67C24×C23—22no—
68C24×C23—41no—
69C24×C23—42no—
70C24×C23—41no—
71C24×C23—81no—
72C24×C23C22×C411yescharacteristic
73C24×C23C2×D411yescharacteristic
74C2×D4×C23—41no—
75C2×D4×C23—41no—
76C2×D4×C23—42no—
77C2×D4×C23—41no—
78C2×D4×C23—41no—
79C2×D4×C23—81no—
80C2×D4×C23C2×D412yes—
81C2×D4×C23#4112yes—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23—22no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—42no—
93C22×C4×C23—81no—
94C22×C4×C23—81no—
95C22×C4×C23C22⋊C411yescharacteristic
96C22×C4×C23C22⋊C411yescharacteristic
97C22×C4×C23C22⋊C411yescharacteristic
98C22×C4×C23C22⋊C411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100M16×C23—24no—
101M16×C23—42no—
102C2×C8×C23—42no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23—81no—
106C4⋊C4×C23—82no—
107C22⋊C4×C23—22no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—42no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23—41no—
114C22⋊C4×C23—41no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—42no—
118C22⋊C4×C23—42no—
119C22⋊C4×C23—42no—
120C22⋊C4×C23—81no—
121C42×C23—21no—
122C42×C23—21no—
123#13219C2311yesmaximal, characteristic, direct product factor, Sylow
124C23×C23—21no—
125C23×C23—21no—
126C23×C23—21no—
127C23×C23—21no—
128C23×C23—22no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—41no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—41no—
136C23×C23—42no—
137C23×C23—42no—
138C23×C23—41no—
139C23×C23—41no—
140C23×C23—41no—
141C23×C23—82no—
142C23×C23—84no—
143C23×C23—84no—
144C23×C23—82no—
145C23×C23C2×C22⋊C411yescharacteristic
146C23×C23C4×D412yes—
147C23×C23#12411yescharacteristic
148C23×C23#12711yescharacteristic
149D4×C23—42no—
150D4×C23—42no—
151D4×C23—82no—
152D4×C23—82no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—41no—
159C2×C4×C23—42no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—81no—
173C2×C4×C23—81no—
174C2×C4×C23—81no—
175C2×C4×C23—82no—
176C2×C4×C23—82no—
177C2×C4×C23—82no—
178C2×C4×C23—82no—
179C2×C4×C23#9911yescharacteristic
180C2×C4×C23#9911yescharacteristic
181C2×C4×C23#12112yes—
182C8×C23—44no—
183#2534C2×C2311yescharacteristic
184C2×#352C2×C2312yescomplemented
185C2×AΣL(1,𝔽16)C2×C2312yes—
186#1799C2×C2311yescharacteristic, complemented
187C4×#100C2×C2311yescharacteristic, complemented
188C22×C23—21no—
189C22×C23—21no—
190C22×C23—21no—
191C22×C23—22no—
192C22×C23—21no—
193C22×C23—41no—
194C22×C23—41no—
195C22×C23—42no—
196C22×C23—42no—
197C22×C23—42no—
198C22×C23—42no—
199C22×C23—41no—
200C22×C23—41no—
201C22×C23—41no—
202C22×C23—41no—
203C22×C23—42no—
204C22×C23—81no—
205C22×C23—81no—
206C22×C23—81no—
207C22×C23—84no—
208C22×C23—84no—
209C22×C23—81no—
210C22×C23—81no—
211C22×C23#39011yescharacteristic
212C22×C23C2×#9911yescharacteristic
213C22×C23#41011yescharacteristic
214C4×C23—21no—
215C4×C23—21no—
216C4×C23—22no—
217C4×C23—41no—
218C4×C23—41no—
219C4×C23—41no—
220C4×C23—81no—
221C4×C23—81no—
222C4×C23—81no—
223C4×C23—162no—
224C2×#123C22×C2311yescharacteristic
225C2×#120C22×C2311yescharacteristic
226C2×C4×D4C22×C2311yescharacteristic
227C22×C22⋊C4C4×C2311yescharacteristic, complemented
228C2×#100C22×C2312yes—
229C2×#99C22×C2312yes—
230C4×M16—21no—
231#396—21no—
232#394—21no—
233#388—21no—
234#386—21no—
235#381C4×C2311yescharacteristic, complemented
236#380—21no—
237#380C4×C2311yescharacteristic, complemented
238#379C4×C2311yescharacteristic, complemented
239#352—24no—
240AΣL(1,𝔽16)—24no—
241#343—21no—
242#342—21no—
243C2×C23—21no—
244C2×C23—21no—
245C2×C23—41no—
246C2×C23—42no—
247C2×C23—41no—
248C2×C23—82no—
249C2×C23#196411yescharacteristic
250C2×C23#212011yescharacteristic
251C2×C23#212111yescharacteristic
252C25C2×C4×C2311yescharacteristic
253C22×D4—21no—
254C22×D4C23×C2311yescharacteristic
255C23×C4C2×C4×C2311yescharacteristic
256C23×C4C2×C4×C2311yescharacteristic
257C23×C4D4×C2311yescharacteristic
258C2×M16—22no—
259#123—22no—
260#123—22no—
261#120—22no—
262#120—22no—
263C4×D4—41no—
264C4×D4—41no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4D4×C2311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4C2×C4×C2311yescharacteristic
279C2×C22⋊C4C2×C4×C2311yescharacteristic
280C2×C22⋊C4C2×C4×C2311yescharacteristic
281C2×C22⋊C4D4×C2311yescharacteristic
282C2×C42D4×C2311yescharacteristic
283#100—22no—
284#100D4×C2314yes—
285#99—22no—
286#99—22no—
287#97—41no—
288C4×C8—41no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—41no—
293C23#1321911yescharacteristic, direct product factor, Sylow
294C24—21no—
295C24—22no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—81no—
300C24C22×C4×C2311yescharacteristic
301C24C2×D4×C2311yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4C2×D4×C2312yes—
309C2×D4#41×C2312yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4C22⋊C4×C2311yescharacteristic
324C22×C4C22⋊C4×C2311yescharacteristic
325C22×C4C22⋊C4×C2311yescharacteristic
326C22×C4C22⋊C4×C2311yescharacteristic
327C22×C4C2×D4×C2311yescharacteristic
328M16—24no—
329M16—42no—
330C2×C8—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—82no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—81no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23C2×C22⋊C4×C2311yescharacteristic
373C23C4×D4×C2312yes—
374C23C23×#12411yescharacteristic
375C23C23×#12711yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4C23×#9911yescharacteristic
407C2×C4C23×#9911yescharacteristic
408C2×C4C23×#12112yes—
409C8—44no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22C23×#39011yescharacteristic
434C22C2×C23×#9911yescharacteristic
435C22C23×#41011yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—162no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2C23×#196411yescharacteristic
453C2C23×#212011yescharacteristic
454C2C23×#212111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×#120C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#352C2semidirect
4. C23×#1799C2semidirect
5. C4×C23×#100C2semidirect
9. C22×C22⋊C4×C23C4semidirect
17. C23×#381C4semidirect
19. C23×#380C4semidirect
20. C23×#379C4semidirect
123. #13219C23direct (N × H)
184. C2×#352C2×C23semidirect
186. #1799C2×C23semidirect
187. C4×#100C2×C23semidirect
227. C22×C22⋊C4C4×C23semidirect
235. #381C4×C23semidirect
237. #380C4×C23semidirect
238. #379C4×C23semidirect
293. C23#13219direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25342
C2×C23×#3522
C2×C23×AΣL(1,𝔽16)2
C23×#17992
C4×C23×#1002
#1321923