Finite Group Databasebeta

← C4×C23×#555 #19181791

Subgroups — C4×C23×#555 #19181791

Order 5888.

2564 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#555C211yesmaximal, characteristic
2C23×#2372C214yesmaximal, complemented
3C23×#2369C211yesmaximal, characteristic, complemented
4C23×#2363C214yesmaximal, complemented
5C23×#2354C212yesmaximal, complemented
6C23×#2346C214yesmaximal, complemented
7C4×C23×C4⋊Q8C211yesmaximal, characteristic, complemented
8C4×C23×#124C214yesmaximal, complemented
9C4×C23×#121C214yesmaximal, complemented
10C42×D4×C23C212yesmaximal, complemented
11C4×C23×#117C214yesmaximal, complemented
12C23×#555C4116yesdirect product factor
13C2×C23×C4⋊Q8C2211yescharacteristic
14C2×C23×#124C2214yes—
15C2×C23×#121C2214yes—
16C2×C4×Q8×C23C2212yes—
17C2×C4×D4×C23C2212yes—
18C2×C4×D4×C23C2212yes—
19C2×C4×D4×C23C2214yes—
20C2×C4×D4×C23C2212yes—
21C2×C23×#117C2214yes—
22C2×C23×#117C2214yes—
23C22×C42×C23C2214yes—
24C23×#391C2214yes—
25C23×#391C2212yes—
26C23×#390C2214yes—
27C23×#390C2214yes—
28C23×#389C2214yes—
29C23×#389C2214yes—
30C23×#388C2218yes—
31C23×#388C2218yes—
32C23×#388C2218yes—
33C23×#386C2218yes—
34C23×#384C2211yescharacteristic
35C23×#382C2214yes—
36C23×#380C2214yes—
37C4×C4⋊C4×C23C2212yes—
38C4×C4⋊C4×C23C2214yes—
39C4×C4⋊C4×C23C2214yes—
40C4×C4⋊C4×C23C2214yes—
41C4×C4⋊C4×C23C2212yes—
42C4×C4⋊C4×C23C2212yes—
43C4×C22⋊C4×C23C2218yes—
44C4×C22⋊C4×C23C2218yes—
45C4×C22⋊C4×C23C2214yes—
46C4×C22⋊C4×C23C2214yes—
47C4×C22⋊C4×C23C2214yes—
48C23×#376C2218yes—
49C43×C23C2214yes—
50C43×C23C2211yescharacteristic
51C22×Q8×C23C2312yes—
52C22×D4×C23C2312yes—
53C22×D4×C23C2314yes—
54C23×C4×C23C2314yes—
55C23×C4×C23C2314yes—
56C23×C4×C23C2314yes—
57C23×C4⋊Q8C2×C418yescomplemented
58C23×#124C2×C4132yescomplemented
59C23×#121C2×C4132yescomplemented
60C4×Q8×C23—44no—
61C4×D4×C23—28no—
62C4×D4×C23—28no—
63C4×D4×C23—48no—
64C4×D4×C23C2×C4116yescomplemented
65C23×#117—216no—
66C23×#117C2×C4132yescomplemented
67C2×C4⋊C4×C23C2312yes—
68C2×C4⋊C4×C23C2314yes—
69C2×C4⋊C4×C23C2314yes—
70C2×C4⋊C4×C23C2314yes—
71C2×C4⋊C4×C23C2312yes—
72C2×C4⋊C4×C23C2312yes—
73C2×C4⋊C4×C23C2314yes—
74C2×C4⋊C4×C23C2312yes—
75C2×C22⋊C4×C23C2314yes—
76C2×C22⋊C4×C23C2318yes—
77C2×C22⋊C4×C23C2314yes—
78C2×C22⋊C4×C23C2318yes—
79C2×C22⋊C4×C23C2318yes—
80C2×C22⋊C4×C23C2314yes—
81C2×C42×C23—216no—
82C2×C42×C23C2318yes—
83C2×C42×C23C2312yes—
84C2×C42×C23C2314yes—
85C2×C42×C23C2314yes—
86C2×C42×C23C2314yes—
87C2×C42×C23C2311yescharacteristic
88C2×C42×C23C2314yes—
89C2×C42×C23C2311yescharacteristic
90C2×C42×C23C2314yes—
91C2×C42×C23C2314yes—
92C2×C42×C23C2311yescharacteristic
93C2×C42×C23C2312yes—
94C23×#95C2314yes—
95C23×#95C2318yes—
96C23×#95C2318yes—
97C23×#95C2318yes—
98C23×#95C2314yes—
99C23×#95C2314yes—
100C24×C23C2414yes—
101C2×Q8×C23—42no—
102C2×Q8×C23—42no—
103C2×Q8×C23C22×C418yes—
104C2×D4×C23—24no—
105C2×D4×C23—24no—
106C2×D4×C23—44no—
107C2×D4×C23—44no—
108C2×D4×C23C22×C418yes—
109C2×D4×C23C22×C4116yes—
110C22×C4×C23—28no—
111C22×C4×C23—28no—
112C22×C4×C23—28no—
113C22×C4×C23—28no—
114C22×C4×C23—28no—
115C22×C4×C23—28no—
116C22×C4×C23—28no—
117C22×C4×C23—44no—
118C22×C4×C23C22×C4116yes—
119C22×C4×C23C2411yescharacteristic
120C22×C4×C23C2414yes—
121C22×C4×C23C2414yes—
122C22×C4×C23C2414yes—
123C22×C4×C23C2411yescharacteristic
124C22×C4×C23C2414yes—
125C22×C4×C23C2414yes—
126C22×C4×C23C2412yes—
127C22×C4×C23C2412yes—
128C22×C4×C23C2411yescharacteristic
129C4⋊C4×C23—28no—
130C4⋊C4×C23—24no—
131C4⋊C4×C23—44no—
132C4⋊C4×C23—44no—
133C4⋊C4×C23—42no—
134C4⋊C4×C23C22×C418yes—
135C4⋊C4×C23C22×C4116yes—
136C4⋊C4×C23#4118yes—
137C22⋊C4×C23—216no—
138C22⋊C4×C23—28no—
139C22⋊C4×C23—28no—
140C22⋊C4×C23—48no—
141C22⋊C4×C23C22×C4132yes—
142C22⋊C4×C23C22×C4116yes—
143C42×C23—28no—
144C42×C23—28no—
145C42×C23—28no—
146C42×C23—216no—
147C42×C23—24no—
148C42×C23—28no—
149C42×C23—22no—
150C42×C23—44no—
151C42×C23—41no—
152C42×C23C22×C4116yes—
153C42×C23C22×C414yes—
154C42×C23#4118yes—
155C4×#555C2311yesmaximal, characteristic, direct product factor, Sylow
156C23×C23—24no—
157C23×C23—24no—
158C23×C23—24no—
159C23×C23—24no—
160C23×C23—24no—
161C23×C23—24no—
162C23×C23C23×C418yes—
163C23×C23C2511yescharacteristic
164Q8×C23—48no—
165D4×C23—216no—
166D4×C23—416no—
167C2×C4×C23—24no—
168C2×C4×C23—24no—
169C2×C4×C23—24no—
170C2×C4×C23—22no—
171C2×C4×C23—24no—
172C2×C4×C23—21no—
173C2×C4×C23—24no—
174C2×C4×C23—22no—
175C2×C4×C23—24no—
176C2×C4×C23—21no—
177C2×C4×C23—232no—
178C2×C4×C23—24no—
179C2×C4×C23—22no—
180C2×C4×C23—28no—
181C2×C4×C23—24no—
182C2×C4×C23—24no—
183C2×C4×C23—21no—
184C2×C4×C23—24no—
185C2×C4×C23—416no—
186C2×C4×C23—416no—
187C2×C4×C23C23×C418yes—
188C2×C4×C23C23×C412yes—
189C2×C4×C23C23×C418yes—
190C2×C4×C23C23×C414yes—
191C2×C4×C23C2×#4114yes—
192C2×C4×C23C2×#4118yes—
193C2×C4×C23C2×#4114yes—
194C2×C4×C23C2×#4114yes—
195C2×C4×C23C2×#4114yes—
196C2×C4×C23H(5,𝔽2)12yes—
197C2×#555C2×C2311yescharacteristic
198#2372C2×C2314yescomplemented
199#2369C2×C2311yescharacteristic, complemented
200#2363C2×C2314yescomplemented
201#2354C2×C2312yescomplemented
202#2346C2×C2314yescomplemented
203C4×C4⋊Q8C2×C2311yescharacteristic, complemented
204C4×#124C2×C2314yescomplemented
205C4×#121C2×C2314yescomplemented
206C42×D4C2×C2312yescomplemented
207C4×#117C2×C2314yescomplemented
208C22×C23—28no—
209C22×C23—28no—
210C22×C23—28no—
211C22×C23—44no—
212C22×C23—44no—
213C22×C23—44no—
214C22×C23—44no—
215C22×C23C24×C411yescharacteristic
216C22×C23C22×#4112yes—
217C22×C23C22×#4112yes—
218C22×C23C2×H(5,𝔽2)11yescharacteristic
219C22×C23#58511yescharacteristic
220C4×C23—28no—
221C4×C23—28no—
222C4×C23—22no—
223C4×C23—24no—
224C4×C23—28no—
225C4×C23—44no—
226C4×C23—41no—
227C4×C23—44no—
228C4×C23C4×#4118yes—
229C4×C23#55514yesdirect product factor
230#555C4×C23116yesdirect product factor
231C2×C4⋊Q8C22×C2311yescharacteristic
232C2×#124C22×C2314yes—
233C2×#121C22×C2314yes—
234C2×C4×Q8C22×C2312yes—
235C2×C4×D4C22×C2312yes—
236C2×C4×D4C22×C2312yes—
237C2×C4×D4C22×C2314yes—
238C2×C4×D4C22×C2312yes—
239C2×#117C22×C2314yes—
240C2×#117C22×C2314yes—
241C22×C42C22×C2314yes—
242#391C22×C2312yes—
243#391C22×C2314yes—
244#390C22×C2314yes—
245#390C22×C2314yes—
246#389C22×C2314yes—
247#389C22×C2314yes—
248#388C22×C2318yes—
249#388C22×C2318yes—
250#388C22×C2318yes—
251#386C22×C2318yes—
252#384C22×C2311yescharacteristic
253#382C22×C2314yes—
254#380C22×C2314yes—
255C4×C4⋊C4C22×C2314yes—
256C4×C4⋊C4C22×C2312yes—
257C4×C4⋊C4C22×C2314yes—
258C4×C4⋊C4C22×C2312yes—
259C4×C4⋊C4C22×C2314yes—
260C4×C4⋊C4C22×C2312yes—
261C4×C22⋊C4C22×C2314yes—
262C4×C22⋊C4C22×C2318yes—
263C4×C22⋊C4C22×C2314yes—
264C4×C22⋊C4C22×C2318yes—
265C4×C22⋊C4C22×C2314yes—
266#376C22×C2318yes—
267C43C22×C2311yescharacteristic
268C43C22×C2314yes—
269C2×C23—48no—
270C2×C23C2×C4×#4112yes—
271C2×C23C4×H(5,𝔽2)11yescharacteristic
272C2×C23C2×#55511yescharacteristic
273C2×C23#347512yes—
274C2×C23#348311yescharacteristic
275C22×Q8C23×C2312yes—
276C22×D4C23×C2312yes—
277C22×D4C23×C2314yes—
278C23×C4C23×C2314yes—
279C23×C4C23×C2314yes—
280C23×C4C23×C2314yes—
281C4⋊Q8C2×C4×C2318yescomplemented
282#124C2×C4×C23132yescomplemented
283#121C2×C4×C23132yescomplemented
284C4×Q8—44no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—48no—
288C4×D4C2×C4×C23116yescomplemented
289#117—216no—
290#117C2×C4×C23132yescomplemented
291C2×C4⋊C4C23×C2314yes—
292C2×C4⋊C4C23×C2312yes—
293C2×C4⋊C4C23×C2314yes—
294C2×C4⋊C4C23×C2314yes—
295C2×C4⋊C4C23×C2312yes—
296C2×C4⋊C4C23×C2312yes—
297C2×C4⋊C4C23×C2314yes—
298C2×C4⋊C4C23×C2312yes—
299C2×C22⋊C4C23×C2318yes—
300C2×C22⋊C4C23×C2314yes—
301C2×C22⋊C4C23×C2314yes—
302C2×C22⋊C4C23×C2318yes—
303C2×C22⋊C4C23×C2318yes—
304C2×C22⋊C4C23×C2314yes—
305C2×C42—216no—
306C2×C42C23×C2318yes—
307C2×C42C23×C2314yes—
308C2×C42C23×C2312yes—
309C2×C42C23×C2311yescharacteristic
310C2×C42C23×C2314yes—
311C2×C42C23×C2314yes—
312C2×C42C23×C2314yes—
313C2×C42C23×C2311yescharacteristic
314C2×C42C23×C2314yes—
315C2×C42C23×C2314yes—
316C2×C42C23×C2311yescharacteristic
317C2×C42C23×C2312yes—
318#95C23×C2314yes—
319#95C23×C2318yes—
320#95C23×C2318yes—
321#95C23×C2318yes—
322#95C23×C2314yes—
323#95C23×C2314yes—
324C23C4×#55511yescharacteristic, direct product factor, Sylow
325C24C24×C2314yes—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C22×C4×C2318yes—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4C22×C4×C2318yes—
334C2×D4C22×C4×C23116yes—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—44no—
343C22×C4C22×C4×C23116yes—
344C22×C4C24×C2314yes—
345C22×C4C24×C2311yescharacteristic
346C22×C4C24×C2314yes—
347C22×C4C24×C2314yes—
348C22×C4C24×C2311yescharacteristic
349C22×C4C24×C2314yes—
350C22×C4C24×C2312yes—
351C22×C4C24×C2314yes—
352C22×C4C24×C2312yes—
353C22×C4C24×C2311yescharacteristic
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4C22×C4×C2318yes—
360C4⋊C4C22×C4×C23116yes—
361C4⋊C4#41×C2318yes—
362C22⋊C4—216no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—48no—
366C22⋊C4C22×C4×C23132yes—
367C22⋊C4C22×C4×C23116yes—
368C42—28no—
369C42—28no—
370C42—28no—
371C42—216no—
372C42—24no—
373C42—28no—
374C42—22no—
375C42—44no—
376C42—41no—
377C42C22×C4×C23116yes—
378C42C22×C4×C2314yes—
379C42#41×C2318yes—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23C23×C4×C2318yes—
387C23C25×C2311yescharacteristic
388Q8—48no—
389D4—216no—
390D4—416no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—232no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—416no—
410C2×C4—416no—
411C2×C4C23×C4×C2318yes—
412C2×C4C23×C4×C2312yes—
413C2×C4C23×C4×C2318yes—
414C2×C4C23×C4×C2314yes—
415C2×C4C2×#41×C2314yes—
416C2×C4C2×#41×C2318yes—
417C2×C4C2×#41×C2314yes—
418C2×C4C2×#41×C2314yes—
419C2×C4C2×#41×C2314yes—
420C2×C4C23×H(5,𝔽2)12yes—
421C22—28no—
422C22—28no—
423C22—28no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22C24×C4×C2311yescharacteristic
429C22C22×#41×C2312yes—
430C22C22×#41×C2312yes—
431C22C2×C23×H(5,𝔽2)11yescharacteristic
432C22C23×#58511yescharacteristic
433C4—28no—
434C4—28no—
435C4—22no—
436C4—24no—
437C4—28no—
438C4—44no—
439C4—41no—
440C4—44no—
441C4C4×#41×C2318yes—
442C4C23×#55514yesdirect product factor
443C2—48no—
444C2C2×C4×#41×C2312yes—
445C2C4×C23×H(5,𝔽2)11yescharacteristic
446C2C2×C23×#55511yescharacteristic
447C2C23×#347512yes—
448C2C23×#348311yescharacteristic
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC2×C23×#555C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2372C2semidirect
3. C23×#2369C2semidirect
4. C23×#2363C2semidirect
5. C23×#2354C2semidirect
6. C23×#2346C2semidirect
7. C4×C23×C4⋊Q8C2semidirect
8. C4×C23×#124C2semidirect
9. C4×C23×#121C2semidirect
10. C42×D4×C23C2semidirect
11. C4×C23×#117C2semidirect
12. C23×#555C4direct (N × H)
57. C23×C4⋊Q8C2×C4semidirect
58. C23×#124C2×C4semidirect
59. C23×#121C2×C4semidirect
64. C4×D4×C23C2×C4semidirect
66. C23×#117C2×C4semidirect
155. C4×#555C23direct (N × H)
198. #2372C2×C23semidirect
199. #2369C2×C23semidirect
200. #2363C2×C23semidirect
201. #2354C2×C23semidirect
202. #2346C2×C23semidirect
203. C4×C4⋊Q8C2×C23semidirect
204. C4×#124C2×C23semidirect
205. C4×#121C2×C23semidirect
206. C42×D4C2×C23semidirect
207. C4×#117C2×C23semidirect
229. C4×C23#555direct (N × H)
230. #555C4×C23direct (N × H)
281. C4⋊Q8C2×C4×C23semidirect
282. #124C2×C4×C23semidirect
283. #121C2×C4×C23semidirect
288. C4×D4C2×C4×C23semidirect
290. #117C2×C4×C23semidirect
324. C23C4×#555direct (N × H)
442. C4C23×#555direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5552
C23×#23722
C23×#23692
C23×#23632
C23×#23542
C23×#23462
C4×C23×C4⋊Q82
C4×C23×#1242
C4×C23×#1212
C42×D4×C232
C4×C23×#1172
C4×#55523