Finite Group Databasebeta

← C23×#36692 #19184318

Subgroups — C23×#36692 #19184318

Order 5888.

1716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2837C214yesmaximal, complemented
2C23×#2836C214yesmaximal, complemented
3C23×#2829C214yesmaximal, complemented
4C23×#2828C214yesmaximal, complemented
5C23×#2402C211yesmaximal, characteristic, complemented
6C23×#2397C211yesmaximal, characteristic, complemented
7C23×#2395C214yesmaximal, complemented
8C23×#2392C214yesmaximal, complemented
9C23×#2392C212yesmaximal, complemented
10C23×#2298C211yesmaximal, characteristic, complemented
11C23×#2293C212yesmaximal, complemented
12C2×C4×D4×C23C2212yes—
13C2×C4×D4×C23C2214yes—
14C2×C4×D4×C23C2214yes—
15C2×C23×#117C2211yescharacteristic
16C2×C23×#117C2214yes—
17C23×#400C2214yes—
18C23×#400C2214yes—
19C23×#399C2214yes—
20C23×#399C2214yes—
21C23×#398C2214yes—
22C23×#398C2212yes—
23C23×#398C2214yes—
24C23×#398C2214yes—
25C23×#397C2214yes—
26C23×#397C2214yes—
27C23×#397C2214yes—
28C23×#397C2214yes—
29C23×#397C2212yes—
30C23×#396C2214yes—
31C23×#396C2214yes—
32C23×#396C2214yes—
33C23×#394C2214yes—
34C23×#394C2214yes—
35C23×#394C2214yes—
36C23×#394C2214yes—
37C23×#394C2214yes—
38C23×#394C2214yes—
39C23×#394C2212yes—
40C23×#392C2214yes—
41C23×#392C2212yes—
42C23×#392C2214yes—
43C23×#390C2214yes—
44C23×#389C2214yes—
45C23×#388C2214yes—
46C23×#388C2214yes—
47C23×#388C2214yes—
48C23×#387C2214yes—
49C4×C4⋊C4×C23C2212yes—
50C4×C4⋊C4×C23C2214yes—
51C4×C4⋊C4×C23C2212yes—
52C4×C22⋊C4×C23C2214yes—
53C4×C22⋊C4×C23C2212yes—
54C4×C22⋊C4×C23C2214yes—
55C23×#376C2211yescharacteristic
56C43×C23C2211yescharacteristic
57C22×D4×C23C2312yes—
58C22×D4×C23C2314yes—
59C22×D4×C23C2314yes—
60C23×C4×C23C2311yescharacteristic
61C23×C4×C23C2314yes—
62C4×D4×C23—28no—
63C4×D4×C23—48no—
64C4×D4×C23—48no—
65C23×#117—24no—
66C23×#117—48no—
67C2×C4⋊C4×C23C2312yes—
68C2×C4⋊C4×C23C2312yes—
69C2×C4⋊C4×C23C2314yes—
70C2×C4⋊C4×C23C2314yes—
71C2×C4⋊C4×C23C2314yes—
72C2×C4⋊C4×C23C2314yes—
73C2×C4⋊C4×C23C2314yes—
74C2×C4⋊C4×C23C2314yes—
75C2×C4⋊C4×C23C2314yes—
76C2×C4⋊C4×C23C2314yes—
77C2×C22⋊C4×C23C2312yes—
78C2×C22⋊C4×C23C2314yes—
79C2×C22⋊C4×C23C2314yes—
80C2×C22⋊C4×C23C2314yes—
81C2×C22⋊C4×C23C2314yes—
82C2×C22⋊C4×C23C2314yes—
83C2×C22⋊C4×C23C2314yes—
84C2×C22⋊C4×C23C2314yes—
85C2×C22⋊C4×C23C2314yes—
86C2×C22⋊C4×C23C2314yes—
87C2×C22⋊C4×C23C2314yes—
88C2×C22⋊C4×C23C2314yes—
89C2×C22⋊C4×C23C2314yes—
90C2×C42×C23C2312yes—
91C2×C42×C23C2312yes—
92C2×C42×C23C2311yescharacteristic
93C2×C42×C23C2314yes—
94C2×C42×C23C2314yes—
95C2×C42×C23C2311yescharacteristic
96C23×#95C2314yes—
97C23×#95C2314yes—
98C23×#95C2314yes—
99C23×#95C2314yes—
100C23×#95C2314yes—
101C23×#95C2314yes—
102C23×#95C2314yes—
103C23×#95C2312yes—
104C23×#95C2312yes—
105C23×#95C2314yes—
106C23×#95C2314yes—
107C24×C23C2411yescharacteristic
108C24×C23C2414yes—
109C2×D4×C23—24no—
110C2×D4×C23—44no—
111C2×D4×C23—44no—
112C2×D4×C23—44no—
113C2×D4×C23—44no—
114C2×D4×C23—44no—
115C2×D4×C23—44no—
116C2×D4×C23—44no—
117C22×C4×C23—22no—
118C22×C4×C23—28no—
119C22×C4×C23—28no—
120C22×C4×C23—24no—
121C22×C4×C23—28no—
122C22×C4×C23C2414yes—
123C22×C4×C23C2414yes—
124C22×C4×C23C2412yes—
125C22×C4×C23C2414yes—
126C22×C4×C23C2414yes—
127C22×C4×C23C2414yes—
128C22×C4×C23C2411yescharacteristic
129C22×C4×C23C2412yes—
130C22×C4×C23C2411yescharacteristic
131C4⋊C4×C23—24no—
132C4⋊C4×C23—24no—
133C4⋊C4×C23—44no—
134C4⋊C4×C23—44no—
135C4⋊C4×C23—44no—
136C4⋊C4×C23—44no—
137C4⋊C4×C23—44no—
138C4⋊C4×C23—44no—
139C4⋊C4×C23—44no—
140C4⋊C4×C23—44no—
141C22⋊C4×C23—24no—
142C22⋊C4×C23—28no—
143C22⋊C4×C23—44no—
144C22⋊C4×C23—44no—
145C22⋊C4×C23—44no—
146C22⋊C4×C23—44no—
147C22⋊C4×C23—44no—
148C22⋊C4×C23—44no—
149C22⋊C4×C23—44no—
150C22⋊C4×C23—44no—
151C22⋊C4×C23—44no—
152C22⋊C4×C23—44no—
153C22⋊C4×C23—44no—
154C42×C23—22no—
155C42×C23—24no—
156C42×C23—28no—
157C42×C23—24no—
158C42×C23—44no—
159C42×C23#4114yes—
160#36692C2311yesmaximal, characteristic, direct product factor, Sylow
161C23×C23—21no—
162C23×C23—24no—
163C23×C23—24no—
164C23×C23—22no—
165C23×C23—24no—
166C23×C23—24no—
167C23×C23—24no—
168C23×C23—24no—
169C23×C23—24no—
170C23×C23—24no—
171C23×C23C2511yescharacteristic
172D4×C23—48no—
173D4×C23—416no—
174D4×C23—416no—
175C2×C4×C23—24no—
176C2×C4×C23—22no—
177C2×C4×C23—24no—
178C2×C4×C23—21no—
179C2×C4×C23—24no—
180C2×C4×C23—24no—
181C2×C4×C23—24no—
182C2×C4×C23—24no—
183C2×C4×C23—24no—
184C2×C4×C23—22no—
185C2×C4×C23—24no—
186C2×C4×C23—24no—
187C2×C4×C23—24no—
188C2×C4×C23—24no—
189C2×C4×C23—24no—
190C2×C4×C23—24no—
191C2×C4×C23—24no—
192C2×C4×C23—24no—
193C2×C4×C23—24no—
194C2×C4×C23—24no—
195C2×C4×C23—44no—
196C2×C4×C23—416no—
197C2×C4×C23C2×#4114yes—
198C2×C4×C23C2×#4112yes—
199C2×C4×C23H(5,𝔽2)14yes—
200#2837C2×C2314yescomplemented
201#2836C2×C2314yescomplemented
202#2829C2×C2314yescomplemented
203#2828C2×C2314yescomplemented
204#2402C2×C2311yescharacteristic, complemented
205#2397C2×C2311yescharacteristic, complemented
206#2395C2×C2314yescomplemented
207#2392C2×C2312yescomplemented
208#2392C2×C2314yescomplemented
209#2298C2×C2311yescharacteristic, complemented
210#2293C2×C2312yescomplemented
211C22×C23—41no—
212C22×C23—44no—
213C22×C23—44no—
214C22×C23—42no—
215C22×C23—44no—
216C22×C23—44no—
217C22×C23—44no—
218C22×C23—44no—
219C22×C23—44no—
220C22×C23—44no—
221C22×C23C22×#4111yescharacteristic
222C22×C23C2×H(5,𝔽2)12yes—
223C22×C23#58514yes—
224C4×C23—24no—
225C4×C23—44no—
226C4×C23—41no—
227C4×C23—42no—
228C4×C23—44no—
229C4×C23—44no—
230C4×C23—44no—
231C4×C23—44no—
232C4×C23#54014yes—
233C2×C4×D4C22×C2312yes—
234C2×C4×D4C22×C2314yes—
235C2×C4×D4C22×C2314yes—
236C2×#117C22×C2311yescharacteristic
237C2×#117C22×C2314yes—
238#400C22×C2314yes—
239#400C22×C2314yes—
240#399C22×C2314yes—
241#399C22×C2314yes—
242#398C22×C2314yes—
243#398C22×C2314yes—
244#398C22×C2312yes—
245#398C22×C2314yes—
246#397C22×C2314yes—
247#397C22×C2314yes—
248#397C22×C2314yes—
249#397C22×C2314yes—
250#397C22×C2312yes—
251#396C22×C2314yes—
252#396C22×C2314yes—
253#396C22×C2314yes—
254#394C22×C2314yes—
255#394C22×C2314yes—
256#394C22×C2314yes—
257#394C22×C2314yes—
258#394C22×C2314yes—
259#394C22×C2314yes—
260#394C22×C2312yes—
261#392C22×C2312yes—
262#392C22×C2314yes—
263#392C22×C2314yes—
264#390C22×C2314yes—
265#389C22×C2314yes—
266#388C22×C2314yes—
267#388C22×C2314yes—
268#388C22×C2314yes—
269#387C22×C2314yes—
270C4×C4⋊C4C22×C2314yes—
271C4×C4⋊C4C22×C2312yes—
272C4×C4⋊C4C22×C2312yes—
273C4×C22⋊C4C22×C2312yes—
274C4×C22⋊C4C22×C2314yes—
275C4×C22⋊C4C22×C2314yes—
276#376C22×C2311yescharacteristic
277C43C22×C2311yescharacteristic
278C2×C23—84no—
279C2×C23—81no—
280C2×C23C2×#54011yescharacteristic
281C2×C23#345512yes—
282C2×C23#352914yes—
283C22×D4C23×C2312yes—
284C22×D4C23×C2314yes—
285C22×D4C23×C2314yes—
286C23×C4C23×C2311yescharacteristic
287C23×C4C23×C2314yes—
288C4×D4—28no—
289C4×D4—48no—
290C4×D4—48no—
291#117—24no—
292#117—48no—
293C2×C4⋊C4C23×C2312yes—
294C2×C4⋊C4C23×C2312yes—
295C2×C4⋊C4C23×C2314yes—
296C2×C4⋊C4C23×C2314yes—
297C2×C4⋊C4C23×C2314yes—
298C2×C4⋊C4C23×C2314yes—
299C2×C4⋊C4C23×C2314yes—
300C2×C4⋊C4C23×C2314yes—
301C2×C4⋊C4C23×C2314yes—
302C2×C4⋊C4C23×C2314yes—
303C2×C22⋊C4C23×C2312yes—
304C2×C22⋊C4C23×C2314yes—
305C2×C22⋊C4C23×C2314yes—
306C2×C22⋊C4C23×C2314yes—
307C2×C22⋊C4C23×C2314yes—
308C2×C22⋊C4C23×C2314yes—
309C2×C22⋊C4C23×C2314yes—
310C2×C22⋊C4C23×C2314yes—
311C2×C22⋊C4C23×C2314yes—
312C2×C22⋊C4C23×C2314yes—
313C2×C22⋊C4C23×C2314yes—
314C2×C22⋊C4C23×C2314yes—
315C2×C22⋊C4C23×C2314yes—
316C2×C42C23×C2312yes—
317C2×C42C23×C2314yes—
318C2×C42C23×C2311yescharacteristic
319C2×C42C23×C2312yes—
320C2×C42C23×C2314yes—
321C2×C42C23×C2311yescharacteristic
322#95C23×C2314yes—
323#95C23×C2314yes—
324#95C23×C2314yes—
325#95C23×C2314yes—
326#95C23×C2314yes—
327#95C23×C2314yes—
328#95C23×C2314yes—
329#95C23×C2312yes—
330#95C23×C2312yes—
331#95C23×C2314yes—
332#95C23×C2314yes—
333C23#3669211yescharacteristic, direct product factor, Sylow
334C24C24×C2311yescharacteristic
335C24C24×C2314yes—
336C2×D4—24no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4C24×C2312yes—
350C22×C4C24×C2314yes—
351C22×C4C24×C2314yes—
352C22×C4C24×C2314yes—
353C22×C4C24×C2314yes—
354C22×C4C24×C2314yes—
355C22×C4C24×C2311yescharacteristic
356C22×C4C24×C2312yes—
357C22×C4C24×C2311yescharacteristic
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C22⋊C4—24no—
369C22⋊C4—28no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C42—22no—
382C42—24no—
383C42—28no—
384C42—24no—
385C42—44no—
386C42#41×C2314yes—
387C23—21no—
388C23—24no—
389C23—24no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23C25×C2311yescharacteristic
398D4—48no—
399D4—416no—
400D4—416no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—44no—
422C2×C4—416no—
423C2×C4C2×#41×C2314yes—
424C2×C4C2×#41×C2312yes—
425C2×C4C23×H(5,𝔽2)14yes—
426C22—41no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—44no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22C22×#41×C2311yescharacteristic
437C22C2×C23×H(5,𝔽2)12yes—
438C22C23×#58514yes—
439C4—24no—
440C4—44no—
441C4—41no—
442C4—42no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4C23×#54014yes—
448C2—84no—
449C2—81no—
450C2C2×C23×#54011yescharacteristic
451C2C23×#345512yes—
452C2C23×#352914yes—
subgroupG/H
centerC22×C4×C23C24
commutator, Frattini, cosocleC23C25×C23
socleC23×C23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2837C2semidirect
2. C23×#2836C2semidirect
3. C23×#2829C2semidirect
4. C23×#2828C2semidirect
5. C23×#2402C2semidirect
6. C23×#2397C2semidirect
7. C23×#2395C2semidirect
8. C23×#2392C2semidirect
9. C23×#2392C2semidirect
10. C23×#2298C2semidirect
11. C23×#2293C2semidirect
160. #36692C23direct (N × H)
200. #2837C2×C23semidirect
201. #2836C2×C23semidirect
202. #2829C2×C23semidirect
203. #2828C2×C23semidirect
204. #2402C2×C23semidirect
205. #2397C2×C23semidirect
206. #2395C2×C23semidirect
207. #2392C2×C23semidirect
208. #2392C2×C23semidirect
209. #2298C2×C23semidirect
210. #2293C2×C23semidirect
333. C23#36692direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28372
C23×#28362
C23×#28292
C23×#28282
C23×#24022
C23×#23972
C23×#23952
C23×#23922
C23×#22982
C23×#22932
#3669223