Finite Group Databasebeta

← C41×#34522 #23810723

Subgroups — C41×#34522 #23810723

Order 10496.

2216 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C41×#562C211yesmaximal, characteristic
2C41×#2382C211yesmaximal, characteristic, complemented
3C41×#2379C212yesmaximal, complemented
4C41×#2368C214yesmaximal, complemented
5C41×#2363C214yesmaximal, complemented
6C41×#2359C214yesmaximal, complemented
7C41×#2335C214yesmaximal, complemented
8C41×#2325C214yesmaximal, complemented
9C41×#2323C214yesmaximal, complemented
10C4×#125×C41C211yesmaximal, characteristic, complemented
11C4×#124×C41C212yesmaximal, complemented
12C41×#562C4116yescomplemented
13C2×#125×C41C2211yescharacteristic
14C2×#124×C41C2212yes—
15C2×#123×C41C2214yes—
16C2×C22⋊Q8×C41C2214yes—
17C2×#121×C41C2214yes—
18C2×C4×Q8×C41C2212yes—
19C2×C4×D4×C41C2212yes—
20C2×C4×D4×C41C2214yes—
21C2×#117×C41C2214yes—
22C41×#391C2212yes—
23C41×#391C2214yes—
24C41×#390C2212yes—
25C41×#390C2214yes—
26C41×#389C2212yes—
27C41×#389C2214yes—
28C41×#389C2214yes—
29C41×#388C2214yes—
30C41×#388C2214yes—
31C41×#388C2214yes—
32C41×#388C2214yes—
33C41×#388C2214yes—
34C41×#388C2214yes—
35C41×#387C2214yes—
36C41×#387C2214yes—
37C41×#387C2214yes—
38C41×#387C2214yes—
39C41×#387C2214yes—
40C41×#386C2214yes—
41C41×#386C2214yes—
42C41×#386C2214yes—
43C41×#385C2214yes—
44C41×#385C2214yes—
45C41×#385C2214yes—
46C41×#382C2212yes—
47C41×#382C2211yescharacteristic
48C41×#380C2214yes—
49C4×C4⋊C4×C41C2212yes—
50C4×C4⋊C4×C41C2214yes—
51C4×C4⋊C4×C41C2214yes—
52C4×C22⋊C4×C41C2214yes—
53C4×C22⋊C4×C41C2214yes—
54C4×C22⋊C4×C41C2214yes—
55C4×C22⋊C4×C41C2214yes—
56C2×#95×C41C2214yes—
57C43×C41C2211yescharacteristic
58C22×Q8×C41C2312yes—
59C22×D4×C41C2314yes—
60C22×D4×C41C2312yes—
61C23×C4×C41C2314yes—
62C23×C4×C41C2314yes—
63C23×C4×C41C2314yes—
64#125×C41C2×C418yescomplemented
65#124×C41C2×C4116yescomplemented
66#123×C41C2×C4132yescomplemented
67C22⋊Q8×C41C2×C4132yescomplemented
68#121×C41C2×C4132yescomplemented
69C4×Q8×C41—44no—
70C4×D4×C41—44no—
71C4×D4×C41—48no—
72#117×C41—48no—
73C2×C4⋊C4×C41C2314yes—
74C2×C4⋊C4×C41C2314yes—
75C2×C4⋊C4×C41C2312yes—
76C2×C4⋊C4×C41C2314yes—
77C2×C4⋊C4×C41C2312yes—
78C2×C4⋊C4×C41C2314yes—
79C2×C4⋊C4×C41C2312yes—
80C2×C4⋊C4×C41C2314yes—
81C2×C4⋊C4×C41C2314yes—
82C2×C4⋊C4×C41C2312yes—
83C2×C22⋊C4×C41C2314yes—
84C2×C22⋊C4×C41C2314yes—
85C2×C22⋊C4×C41C2314yes—
86C2×C22⋊C4×C41C2314yes—
87C2×C22⋊C4×C41C2314yes—
88C2×C22⋊C4×C41C2314yes—
89C2×C22⋊C4×C41C2314yes—
90C2×C22⋊C4×C41C2314yes—
91C2×C22⋊C4×C41C2314yes—
92C2×C42×C41C2314yes—
93C2×C42×C41C2312yes—
94C2×C42×C41C2311yescharacteristic
95C2×C42×C41C2314yes—
96C2×C42×C41C2314yes—
97C2×C42×C41C2311yescharacteristic
98C2×C42×C41C2312yes—
99C2×C42×C41C2314yes—
100C2×C42×C41C2311yescharacteristic
101#95×C41C2314yes—
102#95×C41C2314yes—
103#95×C41C2314yes—
104#95×C41C2314yes—
105#95×C41C2314yes—
106#95×C41C2314yes—
107#95×C41C2314yes—
108#95×C41C2314yes—
109#95×C41C2314yes—
110#95×C41C2314yes—
111#95×C41C2314yes—
112C24×C41C2414yes—
113C2×Q8×C41—42no—
114C2×Q8×C41—42no—
115C2×Q8×C41C22×C418yes—
116C2×D4×C41—42no—
117C2×D4×C41—44no—
118C2×D4×C41—44no—
119C2×D4×C41—42no—
120C2×D4×C41C22×C4116yes—
121C2×D4×C41C22×C418yes—
122C22×C4×C41—28no—
123C22×C4×C41—44no—
124C22×C4×C41—44no—
125C22×C4×C41—44no—
126C22×C4×C41—44no—
127C22×C4×C41—44no—
128C22×C4×C41—44no—
129C22×C4×C41—44no—
130C22×C4×C41C22×C4116yes—
131C22×C4×C41C2414yes—
132C22×C4×C41C2414yes—
133C22×C4×C41C2411yescharacteristic
134C22×C4×C41C2414yes—
135C22×C4×C41C2411yescharacteristic
136C22×C4×C41C2412yes—
137C22×C4×C41C2414yes—
138C22×C4×C41C2414yes—
139C22×C4×C41C2412yes—
140C22×C4×C41C2411yescharacteristic
141C4⋊C4×C41—28no—
142C4⋊C4×C41—42no—
143C4⋊C4×C41—44no—
144C4⋊C4×C41—44no—
145C4⋊C4×C41—42no—
146C4⋊C4×C41—44no—
147C4⋊C4×C41—42no—
148C4⋊C4×C41C22×C4116yes—
149C4⋊C4×C41C22×C4116yes—
150C4⋊C4×C41C22×C418yes—
151C22⋊C4×C41—28no—
152C22⋊C4×C41—28no—
153C22⋊C4×C41—44no—
154C22⋊C4×C41—44no—
155C22⋊C4×C41—44no—
156C22⋊C4×C41—44no—
157C22⋊C4×C41C22×C4116yes—
158C22⋊C4×C41C22×C4116yes—
159C22⋊C4×C41C22×C4116yes—
160C42×C41—28no—
161C42×C41—24no—
162C42×C41—22no—
163C42×C41—42no—
164C42×C41—44no—
165C42×C41—41no—
166C42×C41—44no—
167C42×C41—44no—
168C42×C41C22×C414yes—
169C23×C41—24no—
170C23×C41—24no—
171C23×C41—24no—
172C23×C41—24no—
173C23×C41—24no—
174C23×C41—24no—
175C23×C41C23×C418yes—
176C23×C41C2511yescharacteristic
177Q8×C41—48no—
178D4×C41—416no—
179D4×C41—48no—
180C2×C4×C41—24no—
181C2×C4×C41—22no—
182C2×C4×C41—24no—
183C2×C4×C41—21no—
184C2×C4×C41—22no—
185C2×C4×C41—21no—
186C2×C4×C41—22no—
187C2×C4×C41—22no—
188C2×C4×C41—24no—
189C2×C4×C41—22no—
190C2×C4×C41—24no—
191C2×C4×C41—22no—
192C2×C4×C41—24no—
193C2×C4×C41—24no—
194C2×C4×C41—24no—
195C2×C4×C41—24no—
196C2×C4×C41—21no—
197C2×C4×C41—24no—
198C2×C4×C41—24no—
199C2×C4×C41—24no—
200C2×C4×C41—24no—
201C2×C4×C41—24no—
202C2×C4×C41—416no—
203C2×C4×C41—88no—
204C2×C4×C41—88no—
205C2×C4×C41C23×C412yes—
206C2×C4×C41C23×C414yes—
207C2×C4×C41C23×C418yes—
208C2×C4×C41C23×C418yes—
209C2×C4×C41H(5,𝔽2)14yes—
210C2×C4×C412−1+412yes—
211#34522C4111yesmaximal, characteristic, direct product factor, Sylow
212C22×C41—28no—
213C22×C41—28no—
214C22×C41—28no—
215C22×C41—44no—
216C22×C41—44no—
217C22×C41—44no—
218C22×C41—44no—
219C22×C41C24×C411yescharacteristic
220C22×C41C2×H(5,𝔽2)12yes—
221C22×C41C2×2−1+411yescharacteristic
222C22×C41#58512yes—
223C22×C41#58511yescharacteristic
224C4×C41—28no—
225C4×C41—28no—
226C4×C41—24no—
227C4×C41—22no—
228C4×C41—24no—
229C4×C41—22no—
230C4×C41—44no—
231C4×C41—44no—
232C4×C41—41no—
233C4×C41—44no—
234C2×#562C2×C4111yescharacteristic
235#2382C2×C4111yescharacteristic, complemented
236#2379C2×C4112yescomplemented
237#2368C2×C4114yescomplemented
238#2363C2×C4114yescomplemented
239#2359C2×C4114yescomplemented
240#2335C2×C4114yescomplemented
241#2325C2×C4114yescomplemented
242#2323C2×C4114yescomplemented
243C4×#125C2×C4111yescharacteristic, complemented
244C4×#124C2×C4112yescomplemented
245C2×C41—48no—
246C2×C41C4×H(5,𝔽2)12yes—
247C2×C41C4×2−1+411yescharacteristic
248C2×C41#354512yes—
249C2×C41#354811yescharacteristic
250C2×C41#355311yescharacteristic
251#562C4×C41116yescomplemented
252C2×#125C22×C4111yescharacteristic
253C2×#124C22×C4112yes—
254C2×#123C22×C4114yes—
255C2×C22⋊Q8C22×C4114yes—
256C2×#121C22×C4114yes—
257C2×C4×Q8C22×C4112yes—
258C2×C4×D4C22×C4112yes—
259C2×C4×D4C22×C4114yes—
260C2×#117C22×C4114yes—
261#391C22×C4114yes—
262#391C22×C4112yes—
263#390C22×C4114yes—
264#390C22×C4112yes—
265#389C22×C4112yes—
266#389C22×C4114yes—
267#389C22×C4114yes—
268#388C22×C4114yes—
269#388C22×C4114yes—
270#388C22×C4114yes—
271#388C22×C4114yes—
272#388C22×C4114yes—
273#388C22×C4114yes—
274#387C22×C4114yes—
275#387C22×C4114yes—
276#387C22×C4114yes—
277#387C22×C4114yes—
278#387C22×C4114yes—
279#386C22×C4114yes—
280#386C22×C4114yes—
281#386C22×C4114yes—
282#385C22×C4114yes—
283#385C22×C4114yes—
284#385C22×C4114yes—
285#382C22×C4111yescharacteristic
286#382C22×C4112yes—
287#380C22×C4114yes—
288C4×C4⋊C4C22×C4112yes—
289C4×C4⋊C4C22×C4114yes—
290C4×C4⋊C4C22×C4114yes—
291C4×C22⋊C4C22×C4114yes—
292C4×C22⋊C4C22×C4114yes—
293C4×C22⋊C4C22×C4114yes—
294C4×C22⋊C4C22×C4114yes—
295C2×#95C22×C4114yes—
296C43C22×C4111yescharacteristic
297C41#3452211yescharacteristic, direct product factor, Sylow
298C22×Q8C23×C4112yes—
299C22×D4C23×C4114yes—
300C22×D4C23×C4112yes—
301C23×C4C23×C4114yes—
302C23×C4C23×C4114yes—
303C23×C4C23×C4114yes—
304#125C2×C4×C4118yescomplemented
305#124C2×C4×C41116yescomplemented
306#123C2×C4×C41132yescomplemented
307C22⋊Q8C2×C4×C41132yescomplemented
308#121C2×C4×C41132yescomplemented
309C4×Q8—44no—
310C4×D4—44no—
311C4×D4—48no—
312#117—48no—
313C2×C4⋊C4C23×C4114yes—
314C2×C4⋊C4C23×C4114yes—
315C2×C4⋊C4C23×C4112yes—
316C2×C4⋊C4C23×C4114yes—
317C2×C4⋊C4C23×C4112yes—
318C2×C4⋊C4C23×C4114yes—
319C2×C4⋊C4C23×C4112yes—
320C2×C4⋊C4C23×C4114yes—
321C2×C4⋊C4C23×C4114yes—
322C2×C4⋊C4C23×C4112yes—
323C2×C22⋊C4C23×C4114yes—
324C2×C22⋊C4C23×C4114yes—
325C2×C22⋊C4C23×C4114yes—
326C2×C22⋊C4C23×C4114yes—
327C2×C22⋊C4C23×C4114yes—
328C2×C22⋊C4C23×C4114yes—
329C2×C22⋊C4C23×C4114yes—
330C2×C22⋊C4C23×C4114yes—
331C2×C22⋊C4C23×C4114yes—
332C2×C42C23×C4114yes—
333C2×C42C23×C4112yes—
334C2×C42C23×C4111yescharacteristic
335C2×C42C23×C4114yes—
336C2×C42C23×C4114yes—
337C2×C42C23×C4111yescharacteristic
338C2×C42C23×C4112yes—
339C2×C42C23×C4114yes—
340C2×C42C23×C4111yescharacteristic
341#95C23×C4114yes—
342#95C23×C4114yes—
343#95C23×C4114yes—
344#95C23×C4114yes—
345#95C23×C4114yes—
346#95C23×C4114yes—
347#95C23×C4114yes—
348#95C23×C4114yes—
349#95C23×C4114yes—
350#95C23×C4114yes—
351#95C23×C4114yes—
352C24C24×C4114yes—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C22×C4×C4118yes—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4C22×C4×C41116yes—
361C2×D4C22×C4×C4118yes—
362C22×C4—28no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×C4×C41116yes—
371C22×C4C24×C4114yes—
372C22×C4C24×C4114yes—
373C22×C4C24×C4111yescharacteristic
374C22×C4C24×C4114yes—
375C22×C4C24×C4111yescharacteristic
376C22×C4C24×C4112yes—
377C22×C4C24×C4114yes—
378C22×C4C24×C4114yes—
379C22×C4C24×C4112yes—
380C22×C4C24×C4111yescharacteristic
381C4⋊C4—28no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4C22×C4×C41116yes—
389C4⋊C4C22×C4×C41116yes—
390C4⋊C4C22×C4×C4118yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4C22×C4×C41116yes—
398C22⋊C4C22×C4×C41116yes—
399C22⋊C4C22×C4×C41116yes—
400C42—28no—
401C42—24no—
402C42—22no—
403C42—44no—
404C42—44no—
405C42—42no—
406C42—44no—
407C42—41no—
408C42C22×C4×C4114yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23C23×C4×C4118yes—
416C23C25×C4111yescharacteristic
417Q8—48no—
418D4—416no—
419D4—48no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4C23×C4×C4112yes—
446C2×C4C23×C4×C4114yes—
447C2×C4C23×C4×C4118yes—
448C2×C4C23×C4×C4118yes—
449C2×C4H(5,𝔽2)×C4114yes—
450C2×C42−1+4×C4112yes—
451C22—28no—
452C22—28no—
453C22—28no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C24×C4×C4111yescharacteristic
459C22C2×H(5,𝔽2)×C4112yes—
460C22C2×2−1+4×C4111yescharacteristic
461C22C41×#58512yes—
462C22C41×#58511yescharacteristic
463C4—22no—
464C4—28no—
465C4—28no—
466C4—24no—
467C4—22no—
468C4—24no—
469C4—41no—
470C4—44no—
471C4—44no—
472C4—44no—
473C2—48no—
474C2C4×H(5,𝔽2)×C4112yes—
475C2C4×2−1+4×C4111yescharacteristic
476C2C41×#354512yes—
477C2C41×#354811yescharacteristic
478C2C41×#355311yescharacteristic
subgroupG/H
center, socleC23×C41C25
commutatorC22C24×C4×C41
Frattini, cosocleC23C25×C41
prime radicalC2×C41×#562C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2382C2semidirect
3. C41×#2379C2semidirect
4. C41×#2368C2semidirect
5. C41×#2363C2semidirect
6. C41×#2359C2semidirect
7. C41×#2335C2semidirect
8. C41×#2325C2semidirect
9. C41×#2323C2semidirect
10. C4×#125×C41C2semidirect
11. C4×#124×C41C2semidirect
12. C41×#562C4semidirect
64. #125×C41C2×C4semidirect
65. #124×C41C2×C4semidirect
66. #123×C41C2×C4semidirect
67. C22⋊Q8×C41C2×C4semidirect
68. #121×C41C2×C4semidirect
211. #34522C41direct (N × H)
235. #2382C2×C41semidirect
236. #2379C2×C41semidirect
237. #2368C2×C41semidirect
238. #2363C2×C41semidirect
239. #2359C2×C41semidirect
240. #2335C2×C41semidirect
241. #2325C2×C41semidirect
242. #2323C2×C41semidirect
243. C4×#125C2×C41semidirect
244. C4×#124C2×C41semidirect
251. #562C4×C41semidirect
297. C41#34522direct (N × H)
304. #125C2×C4×C41semidirect
305. #124C2×C4×C41semidirect
306. #123C2×C4×C41semidirect
307. C22⋊Q8C2×C4×C41semidirect
308. #121C2×C4×C41semidirect

Maximal subgroups

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

subgroupindex
C2×C41×#5622
C41×#23822
C41×#23792
C41×#23682
C41×#23632
C41×#23592
C41×#23352
C41×#23252
C41×#23232
C4×#125×C412
C4×#124×C412
#3452241