Finite Group Databasebeta

← C23×#60032 #19207658

Subgroups — C23×#60032 #19207658

Order 5888.

2212 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2736C211yesmaximal, characteristic, complemented
2C23×#2668C213yesmaximal, complemented
3C23×#2578C216yesmaximal, complemented
4C23×#2564C213yesmaximal, complemented
5C23×#2535C213yesmaximal, complemented
6C23×#2517C213yesmaximal, complemented
7C23×#2463C213yesmaximal, complemented
8C23×#2364C216yesmaximal, complemented
9C4×C23×#120C213yesmaximal, complemented
10C24×C4×C23C2211yescharacteristic
11C2×C23×C22⋊Q8C2213yescomplemented
12C2×C23×#121C2216yes—
13C2×C23×#120C2213yes—
14C2×C4×D4×C23C2216yes—
15C2×C4×D4×C23C2213yes—
16C2×C4×D4×C23C2213yescomplemented
17C23×#400C2213yescomplemented
18C23×#399C2213yes—
19C23×#397C2216yes—
20C23×#397C2213yes—
21C23×#397C2213yescomplemented
22C23×#395C2211yescharacteristic, complemented
23C23×#394C2216yes—
24C23×#393C2213yes—
25C23×#392C2213yes—
26C23×#392C2211yescharacteristic, complemented
27C23×#388C2216yes—
28C23×#388C2216yescomplemented
29C23×#387C2216yescomplemented
30C23×#387C2216yescomplemented
31C23×#387C2216yescomplemented
32C23×#386C2216yes—
33C23×#386C2216yes—
34C23×#386C2216yes—
35C23×#386C2213yes—
36C23×#386C2216yescomplemented
37C23×#385C2213yes—
38C23×#385C2216yes—
39C23×#383C2212yescomplemented
40C23×#382C2213yescomplemented
41C23×#380C2216yescomplemented
42C23×#379C2213yes—
43C4×C22⋊C4×C23C2216yes—
44C4×C22⋊C4×C23C2213yes—
45C4×C22⋊C4×C23C2216yescomplemented
46C2×C23×#95C2213yescomplemented
47C25×C23C2311yescharacteristic
48C22×Q8×C23C2311yescharacteristic
49C22×D4×C23C2316yes—
50C22×D4×C23C2313yes—
51C22×D4×C23C2313yes—
52C23×C4×C23—46no—
53C23×C4×C23C2313yes—
54C23×C4×C23C2313yes—
55C23×C4×C23C2316yes—
56C23×C4×C23C2313yes—
57C23×C22⋊Q8—46no—
58C23×#121—224no—
59C23×#120—212no—
60C4×D4×C23—46no—
61C4×D4×C23—46no—
62C4×D4×C23—412no—
63C2×C4⋊C4×C23C2316yes—
64C2×C4⋊C4×C23C2316yes—
65C2×C4⋊C4×C23C2313yes—
66C2×C4⋊C4×C23C2313yes—
67C2×C4⋊C4×C23C2313yes—
68C2×C22⋊C4×C23C2316yes—
69C2×C22⋊C4×C23C2316yes—
70C2×C22⋊C4×C23C2313yes—
71C2×C22⋊C4×C23C2316yes—
72C2×C22⋊C4×C23C2316yes—
73C2×C22⋊C4×C23C2313yes—
74C2×C22⋊C4×C23C2316yes—
75C2×C22⋊C4×C23C2313yes—
76C2×C22⋊C4×C23C2313yes—
77C2×C22⋊C4×C23C2316yes—
78C2×C42×C23C2313yes—
79C2×C42×C23C2316yes—
80C2×C42×C23C2312yes—
81C23×#95C2316yes—
82C23×#95C2313yes—
83C23×#95C2313yes—
84C23×#95C2316yes—
85C23×#95C2316yes—
86C23×#95C2316yes—
87C23×#95C2316yes—
88C23×#95C2311yescharacteristic
89C23×#95C2313yes—
90C23×#95C2316yes—
91C24×C23—26no—
92C24×C23—43no—
93C24×C23—41no—
94C24×C23C2413yes—
95C24×C23C2413yes—
96C2×Q8×C23—43no—
97C2×D4×C23—212no—
98C2×D4×C23—212no—
99C2×D4×C23—212no—
100C2×D4×C23—43no—
101C2×D4×C23—43no—
102C2×D4×C23—46no—
103C2×D4×C23—43no—
104C2×D4×C23#41112yes—
105C22×C4×C23—212no—
106C22×C4×C23—26no—
107C22×C4×C23—212no—
108C22×C4×C23—26no—
109C22×C4×C23—43no—
110C22×C4×C23—46no—
111C22×C4×C23—46no—
112C22×C4×C23—46no—
113C22×C4×C23—46no—
114C22×C4×C23—416no—
115C22×C4×C23C2416yes—
116C22×C4×C23C2416yes—
117C22×C4×C23C2413yes—
118C22×C4×C23C2413yes—
119C22×C4×C23C2411yescharacteristic
120C22×C4×C23C2413yes—
121C22×C4×C23C2413yes—
122C4⋊C4×C23—212no—
123C4⋊C4×C23—212no—
124C4⋊C4×C23—43no—
125C4⋊C4×C23—43no—
126C4⋊C4×C23—43no—
127C22⋊C4×C23—26no—
128C22⋊C4×C23—212no—
129C22⋊C4×C23—212no—
130C22⋊C4×C23—212no—
131C22⋊C4×C23—212no—
132C22⋊C4×C23—43no—
133C22⋊C4×C23—43no—
134C22⋊C4×C23—43no—
135C22⋊C4×C23—46no—
136C22⋊C4×C23—46no—
137C42×C23—26no—
138C42×C23—42no—
139C42×C23—46no—
140#60032C2311yesmaximal, characteristic, direct product factor, Sylow
141C23×C23—26no—
142C23×C23—26no—
143C23×C23—26no—
144C23×C23—23no—
145C23×C23—26no—
146C23×C23—23no—
147C23×C23—23no—
148C23×C23—23no—
149C23×C23—412no—
150C23×C23—412no—
151C23×C23—44no—
152C23×C23C2×#4116yes—
153C23×C23C2×#4116yes—
154C23×C23C2511yescharacteristic
155Q8×C23—82no—
156D4×C23—412no—
157D4×C23—412no—
158D4×C23—424no—
159C2×C4×C23—23no—
160C2×C4×C23—26no—
161C2×C4×C23—23no—
162C2×C4×C23—26no—
163C2×C4×C23—26no—
164C2×C4×C23—26no—
165C2×C4×C23—26no—
166C2×C4×C23—23no—
167C2×C4×C23—26no—
168C2×C4×C23—26no—
169C2×C4×C23—26no—
170C2×C4×C23—23no—
171C2×C4×C23—23no—
172C2×C4×C23—26no—
173C2×C4×C23—412no—
174C2×C4×C23—412no—
175C2×C4×C23—812no—
176C2×C4×C23—86no—
177C2×C4×C23C2×#4116yes—
178C2×C4×C23H(5,𝔽2)16yes—
179#2736C2×C2311yescharacteristic, complemented
180#2668C2×C2313yescomplemented
181#2578C2×C2316yescomplemented
182#2564C2×C2313yescomplemented
183#2535C2×C2313yescomplemented
184#2517C2×C2313yescomplemented
185#2463C2×C2313yescomplemented
186#2364C2×C2316yescomplemented
187C4×#120C2×C2313yescomplemented
188C22×C23—212no—
189C22×C23—212no—
190C22×C23—26no—
191C22×C23—26no—
192C22×C23—416no—
193C22×C23—43no—
194C22×C23—43no—
195C22×C23—46no—
196C22×C23—46no—
197C22×C23—43no—
198C22×C23—43no—
199C22×C23C22×#4113yes—
200C22×C23C2×H(5,𝔽2)13yes—
201C22×C23#58511yescharacteristic
202C4×C23—26no—
203C4×C23—26no—
204C4×C23—46no—
205C4×C23—43no—
206C4×C23—46no—
207C4×C23—41no—
208C4×C23—43no—
209C24×C4C22×C2311yescharacteristic
210C2×C22⋊Q8C22×C2313yescomplemented
211C2×#121C22×C2316yes—
212C2×#120C22×C2313yes—
213C2×C4×D4C22×C2313yes—
214C2×C4×D4C22×C2316yes—
215C2×C4×D4C22×C2313yescomplemented
216#400C22×C2313yescomplemented
217#399C22×C2313yes—
218#397C22×C2316yes—
219#397C22×C2313yes—
220#397C22×C2313yescomplemented
221#395C22×C2311yescharacteristic, complemented
222#394C22×C2316yes—
223#393C22×C2313yes—
224#392C22×C2313yes—
225#392C22×C2311yescharacteristic, complemented
226#388C22×C2316yes—
227#388C22×C2316yescomplemented
228#387C22×C2316yescomplemented
229#387C22×C2316yescomplemented
230#387C22×C2316yescomplemented
231#386C22×C2316yes—
232#386C22×C2316yes—
233#386C22×C2316yes—
234#386C22×C2313yes—
235#386C22×C2316yescomplemented
236#385C22×C2316yes—
237#385C22×C2313yes—
238#383C22×C2312yescomplemented
239#382C22×C2313yescomplemented
240#380C22×C2316yescomplemented
241#379C22×C2313yes—
242C4×C22⋊C4C22×C2313yes—
243C4×C22⋊C4C22×C2316yes—
244C4×C22⋊C4C22×C2316yescomplemented
245C2×#95C22×C2313yescomplemented
246C2×C23—46no—
247C2×C23—46no—
248C2×C23#348113yes—
249C2×C23#350513yes—
250C2×C23#353211yescharacteristic
251C25C23×C2311yescharacteristic
252C22×Q8C23×C2311yescharacteristic
253C22×D4C23×C2316yes—
254C22×D4C23×C2313yes—
255C22×D4C23×C2313yes—
256C23×C4—46no—
257C23×C4C23×C2313yes—
258C23×C4C23×C2313yes—
259C23×C4C23×C2313yes—
260C23×C4C23×C2316yes—
261C22⋊Q8—46no—
262#121—224no—
263#120—212no—
264C4×D4—46no—
265C4×D4—46no—
266C4×D4—412no—
267C2×C4⋊C4C23×C2316yes—
268C2×C4⋊C4C23×C2316yes—
269C2×C4⋊C4C23×C2313yes—
270C2×C4⋊C4C23×C2313yes—
271C2×C4⋊C4C23×C2313yes—
272C2×C22⋊C4C23×C2316yes—
273C2×C22⋊C4C23×C2316yes—
274C2×C22⋊C4C23×C2313yes—
275C2×C22⋊C4C23×C2316yes—
276C2×C22⋊C4C23×C2316yes—
277C2×C22⋊C4C23×C2316yes—
278C2×C22⋊C4C23×C2313yes—
279C2×C22⋊C4C23×C2313yes—
280C2×C22⋊C4C23×C2316yes—
281C2×C22⋊C4C23×C2313yes—
282C2×C42C23×C2313yes—
283C2×C42C23×C2316yes—
284C2×C42C23×C2312yes—
285#95C23×C2316yes—
286#95C23×C2313yes—
287#95C23×C2313yes—
288#95C23×C2316yes—
289#95C23×C2316yes—
290#95C23×C2316yes—
291#95C23×C2316yes—
292#95C23×C2311yescharacteristic
293#95C23×C2313yes—
294#95C23×C2316yes—
295C23#6003211yescharacteristic, direct product factor, Sylow
296C24—26no—
297C24—43no—
298C24—41no—
299C24C24×C2313yes—
300C24C24×C2313yes—
301C2×Q8—43no—
302C2×D4—212no—
303C2×D4—212no—
304C2×D4—212no—
305C2×D4—43no—
306C2×D4—43no—
307C2×D4—46no—
308C2×D4—43no—
309C2×D4#41×C23112yes—
310C22×C4—212no—
311C22×C4—26no—
312C22×C4—212no—
313C22×C4—26no—
314C22×C4—43no—
315C22×C4—46no—
316C22×C4—46no—
317C22×C4—46no—
318C22×C4—46no—
319C22×C4—416no—
320C22×C4C24×C2316yes—
321C22×C4C24×C2316yes—
322C22×C4C24×C2313yes—
323C22×C4C24×C2311yescharacteristic
324C22×C4C24×C2313yes—
325C22×C4C24×C2313yes—
326C22×C4C24×C2313yes—
327C4⋊C4—212no—
328C4⋊C4—212no—
329C4⋊C4—43no—
330C4⋊C4—43no—
331C4⋊C4—43no—
332C22⋊C4—26no—
333C22⋊C4—212no—
334C22⋊C4—212no—
335C22⋊C4—212no—
336C22⋊C4—212no—
337C22⋊C4—43no—
338C22⋊C4—43no—
339C22⋊C4—46no—
340C22⋊C4—46no—
341C22⋊C4—43no—
342C42—26no—
343C42—46no—
344C42—42no—
345C23—26no—
346C23—26no—
347C23—26no—
348C23—23no—
349C23—26no—
350C23—23no—
351C23—23no—
352C23—23no—
353C23—412no—
354C23—412no—
355C23—44no—
356C23C2×#41×C2316yes—
357C23C2×#41×C2316yes—
358C23C25×C2311yescharacteristic
359Q8—82no—
360D4—412no—
361D4—412no—
362D4—424no—
363C2×C4—23no—
364C2×C4—26no—
365C2×C4—23no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—26no—
369C2×C4—26no—
370C2×C4—23no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—23no—
375C2×C4—23no—
376C2×C4—26no—
377C2×C4—412no—
378C2×C4—412no—
379C2×C4—812no—
380C2×C4—86no—
381C2×C4C2×#41×C2316yes—
382C2×C4C23×H(5,𝔽2)16yes—
383C22—212no—
384C22—212no—
385C22—26no—
386C22—26no—
387C22—416no—
388C22—43no—
389C22—43no—
390C22—46no—
391C22—46no—
392C22—43no—
393C22—43no—
394C22C22×#41×C2313yes—
395C22C2×C23×H(5,𝔽2)13yes—
396C22C23×#58511yescharacteristic
397C4—26no—
398C4—26no—
399C4—46no—
400C4—43no—
401C4—46no—
402C4—41no—
403C4—43no—
404C2—46no—
405C2—46no—
406C2C23×#348113yes—
407C2C23×#350513yes—
408C2C23×#353211yescharacteristic
subgroupG/H
center, socleC23×C23C25
commutator, Frattini, cosocleC23C25×C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2736C2semidirect
2. C23×#2668C2semidirect
3. C23×#2578C2semidirect
4. C23×#2564C2semidirect
5. C23×#2535C2semidirect
6. C23×#2517C2semidirect
7. C23×#2463C2semidirect
8. C23×#2364C2semidirect
9. C4×C23×#120C2semidirect
11. C2×C23×C22⋊Q8C22semidirect
16. C2×C4×D4×C23C22semidirect
17. C23×#400C22semidirect
21. C23×#397C22semidirect
22. C23×#395C22semidirect
26. C23×#392C22semidirect
28. C23×#388C22semidirect
29. C23×#387C22semidirect
30. C23×#387C22semidirect
31. C23×#387C22semidirect
36. C23×#386C22semidirect
39. C23×#383C22semidirect
40. C23×#382C22semidirect
41. C23×#380C22semidirect
45. C4×C22⋊C4×C23C22semidirect
46. C2×C23×#95C22semidirect
140. #60032C23direct (N × H)
179. #2736C2×C23semidirect
180. #2668C2×C23semidirect
181. #2578C2×C23semidirect
182. #2564C2×C23semidirect
183. #2535C2×C23semidirect
184. #2517C2×C23semidirect
185. #2463C2×C23semidirect
186. #2364C2×C23semidirect
187. C4×#120C2×C23semidirect
210. C2×C22⋊Q8C22×C23semidirect
215. C2×C4×D4C22×C23semidirect
216. #400C22×C23semidirect
220. #397C22×C23semidirect
221. #395C22×C23semidirect
225. #392C22×C23semidirect
227. #388C22×C23semidirect
228. #387C22×C23semidirect
229. #387C22×C23semidirect
230. #387C22×C23semidirect
235. #386C22×C23semidirect
238. #383C22×C23semidirect
239. #382C22×C23semidirect
240. #380C22×C23semidirect
244. C4×C22⋊C4C22×C23semidirect
245. C2×#95C22×C23semidirect
295. C23#60032direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27362
C23×#26682
C23×#25782
C23×#25642
C23×#25352
C23×#25172
C23×#24632
C23×#23642
C4×C23×#1202
#6003223