Finite Group Databasebeta

← C23×#9230 #19156856

Subgroups — C23×#9230 #19156856

Order 5888.

570 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#460C211yesmaximal, characteristic
2C2×C23×#446C211yesmaximal, characteristic, complemented
3C23×#1957C211yesmaximal, characteristic
4C23×#1803C211yesmaximal, characteristic, complemented
5C2×C23×#331C211yesmaximal, characteristic
6C23×#1296C211yesmaximal, characteristic, complemented
7C23×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C23—21no—
9C2×C23×#127C2211yescharacteristic
10C2×C4×Q8×C23C2211yescharacteristic
11C23×#460—22no—
12C23×#460C414yescomplemented
13C23×#446—22no—
14C23×#446—22no—
15C2×C23×C4⋊C8—21no—
16C2×C23×C4⋊C8C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C2×C23×C4⋊C8C2211yescharacteristic
19C2×C23×#103—21no—
20C2×C23×#102—21no—
21C2×C23×#102—21no—
22C2×C23×#102—21no—
23C2×C23×#97—21no—
24C2×C4×C8×C23C2211yescharacteristic
25C23×#384C2211yescharacteristic
26C23×#340—21no—
27C23×#331—22no—
28C22×Q8×C23D411yescharacteristic
29C22×D4×C23—21no—
30C22×D4×C23—21no—
31C22×D4×C23—21no—
32C2×SD16×C23—24no—
33C2×SD16×C23—42no—
34C2×SD16×C23—42no—
35C22×C8×C23—21no—
36C22×C8×C23—21no—
37C22×C8×C23—21no—
38C22×C8×C23D411yescharacteristic
39C22×C8×C23D411yescharacteristic
40C23×#127—21no—
41C23×#127—21no—
42C23×#127—21no—
43C23×#127C2×C412yes—
44C4×Q8×C23—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23C2×C412yes—
47C4×Q8×C23D412yes—
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×C42×C23—21no—
54C2×C42×C23C2311yescharacteristic
55C23×C4⋊C8—24no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8C2×C412yes—
61C23×C4⋊C8D412yes—
62C23×#103—42no—
63C23×#102—24no—
64C23×#102—42no—
65C23×#102—42no—
66C23×#97—24no—
67C4×C8×C23—21no—
68C4×C8×C23—21no—
69C24×C23—41no—
70C2×Q8×C23—21no—
71C2×Q8×C23—21no—
72C2×Q8×C23—21no—
73C2×Q8×C23—21no—
74C2×Q8×C23—21no—
75C2×Q8×C23C22⋊C412yes—
76C2×D4×C23—22no—
77C2×D4×C23—22no—
78C2×D4×C23—22no—
79C2×D4×C23—41no—
80C2×D4×C23—41no—
81C2×D4×C23—41no—
82C2×D4×C23—41no—
83C2×D4×C23—41no—
84C2×D4×C23—41no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—41no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—81no—
91C2×D4×C23—81no—
92C22×C4×C23—21no—
93C22×C4×C23—21no—
94C22×C4×C23—41no—
95C22×C4×C23C2×D411yescharacteristic
96C22×C4×C23C2×D411yescharacteristic
97C22×C4×C23#4111yescharacteristic
98SD16×C23—48no—
99C2×C8×C23—21no—
100C2×C8×C23—21no—
101C2×C8×C23—22no—
102C2×C8×C23—21no—
103C2×C8×C23—22no—
104C2×C8×C23—21no—
105C2×C8×C23—21no—
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×C23SD1612yescomplemented
114C4⋊C4×C23—22no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—22no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23—41no—
120C4⋊C4×C23—41no—
121C4⋊C4×C23—81no—
122C4⋊C4×C23C22⋊C412yes—
123C42×C23—22no—
124C42×C23—22no—
125C42×C23C22×C411yescharacteristic
126C42×C23C2×D411yescharacteristic
127C42×C23C2×D411yescharacteristic
128C42×C23#4111yescharacteristic
129#9230C2311yesmaximal, characteristic, direct product factor, Sylow
130C23×C23—42no—
131C23×C23—81no—
132C23×C23—81no—
133C23×C23—81no—
134C23×C23—81no—
135C23×C23—81no—
136C23×C23—81no—
137C23×C23#12111yescharacteristic
138Q8×C23—22no—
139Q8×C23—22no—
140Q8×C23—41no—
141Q8×C23—41no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23—42no—
145D4×C23—42no—
146D4×C23—82no—
147D4×C23—82no—
148D4×C23—81no—
149D4×C23—81no—
150D4×C23—81no—
151D4×C23—81no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—22no—
155C2×C4×C23—22no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—81no—
161C2×C4×C23—81no—
162C2×C4×C23—81no—
163C2×C4×C23C2×C22⋊C411yescharacteristic
164C2×C4×C23C4×D411yescharacteristic
165C2×C4×C23C4×D411yescharacteristic
166C2×C4×C23#12011yescharacteristic
167C2×C4×C23#12111yescharacteristic
168C2×C4×C23#12311yescharacteristic
169C2×C4×C23C2×SD1611yescharacteristic
170C2×C4×C23#13511yescharacteristic
171C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C23#13711yescharacteristic
176C2×C4×C23#13711yescharacteristic
177C8×C23—22no—
178C8×C23—42no—
179C8×C23—41no—
180C8×C23—42no—
181C8×C23—81no—
182C2×#460C2×C2311yescharacteristic
183C2×#446C2×C2311yescharacteristic, complemented
184#1957C2×C2311yescharacteristic
185#1803C2×C2311yescharacteristic, complemented
186C2×#331C2×C2311yescharacteristic
187#1296C2×C2311yescharacteristic, complemented
188#1277C2×C2311yescharacteristic, complemented
189C22×C23—82no—
190C22×C23—82no—
191C22×C23—82no—
192C22×C23—161no—
193C22×C23—161no—
194C22×C23—161no—
195C22×C23—161no—
196C22×C23#38611yescharacteristic
197C22×C23#46011yescharacteristic
198C22×C23#46411yescharacteristic
199C22×C23#46811yescharacteristic
200C22×C23#46911yescharacteristic
201C22×C23#47111yescharacteristic
202C22×C23#47211yescharacteristic
203C4×C23—21no—
204C4×C23—21no—
205C4×C23—42no—
206C4×C23—42no—
207C4×C23—161no—
208C4×C23#41711yescharacteristic
209C4×C23#41811yescharacteristic
210C4×C23C4×SD1611yescharacteristic
211C4×C23#44011yescharacteristic
212C4×C23#44811yescharacteristic
213C4×C23#44911yescharacteristic
214C4×C23#46811yescharacteristic
215C4×C23#46911yescharacteristic
216C22×SD16—21no—
217C2×#127C22×C2311yescharacteristic
218C2×C4×Q8C22×C2311yescharacteristic
219#460—22no—
220#460C4×C2314yescomplemented
221#446—22no—
222#446—22no—
223C2×C4⋊C8—21no—
224C2×C4⋊C8C22×C2311yescharacteristic
225C2×C4⋊C8C22×C2311yescharacteristic
226C2×C4⋊C8C22×C2311yescharacteristic
227C2×#103—21no—
228C2×#102—21no—
229C2×#102—21no—
230C2×#102—21no—
231C2×#97—21no—
232C2×C4×C8C22×C2311yescharacteristic
233#384C22×C2311yescharacteristic
234#340—21no—
235#331—22no—
236C2×C23—162no—
237C2×C23#168611yescharacteristic
238C2×C23#168811yescharacteristic
239C2×C23#169811yescharacteristic
240C2×C23#170211yescharacteristic
241C2×C23#188011yescharacteristic
242C2×C23#189511yescharacteristic
243C2×C23#190711yescharacteristic
244C22×Q8D4×C2311yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—21no—
248C2×SD16—24no—
249C2×SD16—42no—
250C2×SD16—42no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8D4×C2311yescharacteristic
255C22×C8D4×C2311yescharacteristic
256#127—21no—
257#127—21no—
258#127—21no—
259#127C2×C4×C2312yes—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8C2×C4×C2312yes—
263C4×Q8D4×C2312yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4D4×C2311yescharacteristic
269C2×C42—21no—
270C2×C42C23×C2311yescharacteristic
271C4⋊C8—24no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8C2×C4×C2312yes—
277C4⋊C8D4×C2312yes—
278#103—42no—
279#102—24no—
280#102—42no—
281#102—42no—
282#97—24no—
283C4×C8—21no—
284C4×C8—21no—
285C23#923011yescharacteristic, direct product factor, Sylow
286C24—41no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C22⋊C4×C2312yes—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4—81no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4C2×D4×C2311yescharacteristic
313C22×C4C2×D4×C2311yescharacteristic
314C22×C4#41×C2311yescharacteristic
315SD16—48no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8SD16×C2312yescomplemented
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—81no—
339C4⋊C4C22⋊C4×C2312yes—
340C42—22no—
341C42—22no—
342C42C22×C4×C2311yescharacteristic
343C42C2×D4×C2311yescharacteristic
344C42C2×D4×C2311yescharacteristic
345C42#41×C2311yescharacteristic
346C23—42no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23C23×#12111yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—82no—
363D4—82no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4C2×C22⋊C4×C2311yescharacteristic
380C2×C4C4×D4×C2311yescharacteristic
381C2×C4C4×D4×C2311yescharacteristic
382C2×C4C23×#12011yescharacteristic
383C2×C4C23×#12111yescharacteristic
384C2×C4C23×#12311yescharacteristic
385C2×C4C2×SD16×C2311yescharacteristic
386C2×C4C23×#13511yescharacteristic
387C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
388C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
391C2×C4C23×#13711yescharacteristic
392C2×C4C23×#13711yescharacteristic
393C8—22no—
394C8—42no—
395C8—41no—
396C8—42no—
397C8—81no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22C23×#38611yescharacteristic
406C22C23×#46011yescharacteristic
407C22C23×#46411yescharacteristic
408C22C23×#46811yescharacteristic
409C22C23×#46911yescharacteristic
410C22C23×#47111yescharacteristic
411C22C23×#47211yescharacteristic
412C4—21no—
413C4—21no—
414C4—42no—
415C4—42no—
416C4—161no—
417C4C23×#41711yescharacteristic
418C4C23×#41811yescharacteristic
419C4C4×SD16×C2311yescharacteristic
420C4C23×#44011yescharacteristic
421C4C23×#44811yescharacteristic
422C4C23×#44911yescharacteristic
423C4C23×#46811yescharacteristic
424C4C23×#46911yescharacteristic
425C2—162no—
426C2C23×#168611yescharacteristic
427C2C23×#168811yescharacteristic
428C2C23×#169811yescharacteristic
429C2C23×#170211yescharacteristic
430C2C23×#188011yescharacteristic
431C2C23×#189511yescharacteristic
432C2C23×#190711yescharacteristic
subgroupG/H
center, socleC23×C23#121
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×C23×#127C22
LCSG ▷ 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. C2×C23×#446C2semidirect
4. C23×#1803C2semidirect
6. C23×#1296C2semidirect
7. C23×#1277C2semidirect
12. C23×#460C4semidirect
113. C2×C8×C23SD16semidirect
129. #9230C23direct (N × H)
183. C2×#446C2×C23semidirect
185. #1803C2×C23semidirect
187. #1296C2×C23semidirect
188. #1277C2×C23semidirect
220. #460C4×C23semidirect
285. C23#9230direct (N × H)
330. C2×C8SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#4602
C2×C23×#4462
C23×#19572
C23×#18032
C2×C23×#3312
C23×#12962
C23×#12772
#923023