Finite Group Databasebeta

← C4×C41×#555 #23810366

Subgroups — C4×C41×#555 #23810366

Order 10496.

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×C41×#555C211yesmaximal, characteristic
2C41×#2372C214yesmaximal, complemented
3C41×#2369C211yesmaximal, characteristic, complemented
4C41×#2363C214yesmaximal, complemented
5C41×#2354C212yesmaximal, complemented
6C41×#2346C214yesmaximal, complemented
7C4×C4⋊Q8×C41C211yesmaximal, characteristic, complemented
8C4×#124×C41C214yesmaximal, complemented
9C4×#121×C41C214yesmaximal, complemented
10C42×D4×C41C212yesmaximal, complemented
11C4×#117×C41C214yesmaximal, complemented
12C41×#555C4116yesdirect product factor
13C2×C4⋊Q8×C41C2211yescharacteristic
14C2×#124×C41C2214yes—
15C2×#121×C41C2214yes—
16C2×C4×Q8×C41C2212yes—
17C2×C4×D4×C41C2212yes—
18C2×C4×D4×C41C2212yes—
19C2×C4×D4×C41C2214yes—
20C2×C4×D4×C41C2212yes—
21C2×#117×C41C2214yes—
22C2×#117×C41C2214yes—
23C22×C42×C41C2214yes—
24C41×#391C2214yes—
25C41×#391C2212yes—
26C41×#390C2214yes—
27C41×#390C2214yes—
28C41×#389C2214yes—
29C41×#389C2214yes—
30C41×#388C2218yes—
31C41×#388C2218yes—
32C41×#388C2218yes—
33C41×#386C2218yes—
34C41×#384C2211yescharacteristic
35C41×#382C2214yes—
36C41×#380C2214yes—
37C4×C4⋊C4×C41C2212yes—
38C4×C4⋊C4×C41C2214yes—
39C4×C4⋊C4×C41C2214yes—
40C4×C4⋊C4×C41C2214yes—
41C4×C4⋊C4×C41C2212yes—
42C4×C4⋊C4×C41C2212yes—
43C4×C22⋊C4×C41C2218yes—
44C4×C22⋊C4×C41C2218yes—
45C4×C22⋊C4×C41C2214yes—
46C4×C22⋊C4×C41C2214yes—
47C4×C22⋊C4×C41C2214yes—
48C41×#376C2218yes—
49C43×C41C2214yes—
50C43×C41C2211yescharacteristic
51C22×Q8×C41C2312yes—
52C22×D4×C41C2312yes—
53C22×D4×C41C2314yes—
54C23×C4×C41C2314yes—
55C23×C4×C41C2314yes—
56C23×C4×C41C2314yes—
57C4⋊Q8×C41C2×C418yescomplemented
58#124×C41C2×C4132yescomplemented
59#121×C41C2×C4132yescomplemented
60C4×Q8×C41—44no—
61C4×D4×C41—28no—
62C4×D4×C41—28no—
63C4×D4×C41—48no—
64C4×D4×C41C2×C4116yescomplemented
65#117×C41—216no—
66#117×C41C2×C4132yescomplemented
67C2×C4⋊C4×C41C2314yes—
68C2×C4⋊C4×C41C2312yes—
69C2×C4⋊C4×C41C2314yes—
70C2×C4⋊C4×C41C2314yes—
71C2×C4⋊C4×C41C2312yes—
72C2×C4⋊C4×C41C2312yes—
73C2×C4⋊C4×C41C2314yes—
74C2×C4⋊C4×C41C2312yes—
75C2×C22⋊C4×C41C2318yes—
76C2×C22⋊C4×C41C2314yes—
77C2×C22⋊C4×C41C2314yes—
78C2×C22⋊C4×C41C2318yes—
79C2×C22⋊C4×C41C2318yes—
80C2×C22⋊C4×C41C2314yes—
81C2×C42×C41—216no—
82C2×C42×C41C2314yes—
83C2×C42×C41C2314yes—
84C2×C42×C41C2318yes—
85C2×C42×C41C2312yes—
86C2×C42×C41C2314yes—
87C2×C42×C41C2314yes—
88C2×C42×C41C2311yescharacteristic
89C2×C42×C41C2311yescharacteristic
90C2×C42×C41C2314yes—
91C2×C42×C41C2314yes—
92C2×C42×C41C2311yescharacteristic
93C2×C42×C41C2312yes—
94#95×C41C2314yes—
95#95×C41C2318yes—
96#95×C41C2318yes—
97#95×C41C2318yes—
98#95×C41C2314yes—
99#95×C41C2314yes—
100C24×C41C2414yes—
101C2×Q8×C41—42no—
102C2×Q8×C41—42no—
103C2×Q8×C41C22×C418yes—
104C2×D4×C41—24no—
105C2×D4×C41—24no—
106C2×D4×C41—44no—
107C2×D4×C41—44no—
108C2×D4×C41C22×C418yes—
109C2×D4×C41C22×C4116yes—
110C22×C4×C41—28no—
111C22×C4×C41—28no—
112C22×C4×C41—28no—
113C22×C4×C41—28no—
114C22×C4×C41—28no—
115C22×C4×C41—28no—
116C22×C4×C41—28no—
117C22×C4×C41—44no—
118C22×C4×C41C22×C4116yes—
119C22×C4×C41C2414yes—
120C22×C4×C41C2411yescharacteristic
121C22×C4×C41C2414yes—
122C22×C4×C41C2414yes—
123C22×C4×C41C2411yescharacteristic
124C22×C4×C41C2414yes—
125C22×C4×C41C2412yes—
126C22×C4×C41C2414yes—
127C22×C4×C41C2412yes—
128C22×C4×C41C2411yescharacteristic
129C4⋊C4×C41—28no—
130C4⋊C4×C41—24no—
131C4⋊C4×C41—44no—
132C4⋊C4×C41—44no—
133C4⋊C4×C41—42no—
134C4⋊C4×C41C22×C418yes—
135C4⋊C4×C41C22×C4116yes—
136C4⋊C4×C41#4118yes—
137C22⋊C4×C41—216no—
138C22⋊C4×C41—28no—
139C22⋊C4×C41—28no—
140C22⋊C4×C41—48no—
141C22⋊C4×C41C22×C4132yes—
142C22⋊C4×C41C22×C4116yes—
143C42×C41—28no—
144C42×C41—28no—
145C42×C41—28no—
146C42×C41—216no—
147C42×C41—24no—
148C42×C41—28no—
149C42×C41—22no—
150C42×C41—44no—
151C42×C41—41no—
152C42×C41C22×C4116yes—
153C42×C41C22×C414yes—
154C42×C41#4118yes—
155C23×C41—24no—
156C23×C41—24no—
157C23×C41—24no—
158C23×C41—24no—
159C23×C41—24no—
160C23×C41—24no—
161C23×C41C23×C418yes—
162C23×C41C2511yescharacteristic
163Q8×C41—48no—
164D4×C41—216no—
165D4×C41—416no—
166C2×C4×C41—24no—
167C2×C4×C41—24no—
168C2×C4×C41—24no—
169C2×C4×C41—22no—
170C2×C4×C41—24no—
171C2×C4×C41—21no—
172C2×C4×C41—24no—
173C2×C4×C41—22no—
174C2×C4×C41—24no—
175C2×C4×C41—21no—
176C2×C4×C41—232no—
177C2×C4×C41—24no—
178C2×C4×C41—22no—
179C2×C4×C41—28no—
180C2×C4×C41—24no—
181C2×C4×C41—24no—
182C2×C4×C41—21no—
183C2×C4×C41—24no—
184C2×C4×C41—416no—
185C2×C4×C41—416no—
186C2×C4×C41C23×C418yes—
187C2×C4×C41C23×C412yes—
188C2×C4×C41C23×C418yes—
189C2×C4×C41C23×C414yes—
190C2×C4×C41C2×#4114yes—
191C2×C4×C41C2×#4118yes—
192C2×C4×C41C2×#4114yes—
193C2×C4×C41C2×#4114yes—
194C2×C4×C41C2×#4114yes—
195C2×C4×C41H(5,𝔽2)12yes—
196C4×#555C4111yesmaximal, characteristic, direct product factor, Sylow
197C22×C41—28no—
198C22×C41—28no—
199C22×C41—28no—
200C22×C41—44no—
201C22×C41—44no—
202C22×C41—44no—
203C22×C41—44no—
204C22×C41C24×C411yescharacteristic
205C22×C41C22×#4112yes—
206C22×C41C22×#4112yes—
207C22×C41C2×H(5,𝔽2)11yescharacteristic
208C22×C41#58511yescharacteristic
209C4×C41—28no—
210C4×C41—28no—
211C4×C41—22no—
212C4×C41—24no—
213C4×C41—28no—
214C4×C41—44no—
215C4×C41—41no—
216C4×C41—44no—
217C4×C41C4×#4118yes—
218C4×C41#55514yesdirect product factor
219C2×#555C2×C4111yescharacteristic
220#2372C2×C4114yescomplemented
221#2369C2×C4111yescharacteristic, complemented
222#2363C2×C4114yescomplemented
223#2354C2×C4112yescomplemented
224#2346C2×C4114yescomplemented
225C4×C4⋊Q8C2×C4111yescharacteristic, complemented
226C4×#124C2×C4114yescomplemented
227C4×#121C2×C4114yescomplemented
228C42×D4C2×C4112yescomplemented
229C4×#117C2×C4114yescomplemented
230C2×C41—48no—
231C2×C41C2×C4×#4112yes—
232C2×C41C4×H(5,𝔽2)11yescharacteristic
233C2×C41C2×#55511yescharacteristic
234C2×C41#347512yes—
235C2×C41#348311yescharacteristic
236#555C4×C41116yesdirect product factor
237C2×C4⋊Q8C22×C4111yescharacteristic
238C2×#124C22×C4114yes—
239C2×#121C22×C4114yes—
240C2×C4×Q8C22×C4112yes—
241C2×C4×D4C22×C4112yes—
242C2×C4×D4C22×C4112yes—
243C2×C4×D4C22×C4114yes—
244C2×C4×D4C22×C4112yes—
245C2×#117C22×C4114yes—
246C2×#117C22×C4114yes—
247C22×C42C22×C4114yes—
248#391C22×C4112yes—
249#391C22×C4114yes—
250#390C22×C4114yes—
251#390C22×C4114yes—
252#389C22×C4114yes—
253#389C22×C4114yes—
254#388C22×C4118yes—
255#388C22×C4118yes—
256#388C22×C4118yes—
257#386C22×C4118yes—
258#384C22×C4111yescharacteristic
259#382C22×C4114yes—
260#380C22×C4114yes—
261C4×C4⋊C4C22×C4114yes—
262C4×C4⋊C4C22×C4112yes—
263C4×C4⋊C4C22×C4114yes—
264C4×C4⋊C4C22×C4112yes—
265C4×C4⋊C4C22×C4114yes—
266C4×C4⋊C4C22×C4112yes—
267C4×C22⋊C4C22×C4114yes—
268C4×C22⋊C4C22×C4118yes—
269C4×C22⋊C4C22×C4114yes—
270C4×C22⋊C4C22×C4118yes—
271C4×C22⋊C4C22×C4114yes—
272#376C22×C4118yes—
273C43C22×C4111yescharacteristic
274C43C22×C4114yes—
275C41C4×#55511yescharacteristic, direct product factor, Sylow
276C22×Q8C23×C4112yes—
277C22×D4C23×C4112yes—
278C22×D4C23×C4114yes—
279C23×C4C23×C4114yes—
280C23×C4C23×C4114yes—
281C23×C4C23×C4114yes—
282C4⋊Q8C2×C4×C4118yescomplemented
283#124C2×C4×C41132yescomplemented
284#121C2×C4×C41132yescomplemented
285C4×Q8—44no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—48no—
289C4×D4C2×C4×C41116yescomplemented
290#117—216no—
291#117C2×C4×C41132yescomplemented
292C2×C4⋊C4C23×C4114yes—
293C2×C4⋊C4C23×C4112yes—
294C2×C4⋊C4C23×C4114yes—
295C2×C4⋊C4C23×C4114yes—
296C2×C4⋊C4C23×C4112yes—
297C2×C4⋊C4C23×C4112yes—
298C2×C4⋊C4C23×C4114yes—
299C2×C4⋊C4C23×C4112yes—
300C2×C22⋊C4C23×C4118yes—
301C2×C22⋊C4C23×C4114yes—
302C2×C22⋊C4C23×C4114yes—
303C2×C22⋊C4C23×C4118yes—
304C2×C22⋊C4C23×C4118yes—
305C2×C22⋊C4C23×C4114yes—
306C2×C42—216no—
307C2×C42C23×C4118yes—
308C2×C42C23×C4114yes—
309C2×C42C23×C4112yes—
310C2×C42C23×C4111yescharacteristic
311C2×C42C23×C4114yes—
312C2×C42C23×C4114yes—
313C2×C42C23×C4114yes—
314C2×C42C23×C4111yescharacteristic
315C2×C42C23×C4114yes—
316C2×C42C23×C4114yes—
317C2×C42C23×C4111yescharacteristic
318C2×C42C23×C4112yes—
319#95C23×C4114yes—
320#95C23×C4118yes—
321#95C23×C4118yes—
322#95C23×C4118yes—
323#95C23×C4114yes—
324#95C23×C4114yes—
325C24C24×C4114yes—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C22×C4×C4118yes—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4C22×C4×C4118yes—
334C2×D4C22×C4×C41116yes—
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×C41116yes—
344C22×C4C24×C4114yes—
345C22×C4C24×C4111yescharacteristic
346C22×C4C24×C4114yes—
347C22×C4C24×C4114yes—
348C22×C4C24×C4111yescharacteristic
349C22×C4C24×C4114yes—
350C22×C4C24×C4112yes—
351C22×C4C24×C4114yes—
352C22×C4C24×C4112yes—
353C22×C4C24×C4111yescharacteristic
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4C22×C4×C4118yes—
360C4⋊C4C22×C4×C41116yes—
361C4⋊C4#41×C4118yes—
362C22⋊C4—216no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—48no—
366C22⋊C4C22×C4×C41132yes—
367C22⋊C4C22×C4×C41116yes—
368C42—28no—
369C42—28no—
370C42—28no—
371C42—216no—
372C42—24no—
373C42—28no—
374C42—22no—
375C42—44no—
376C42—41no—
377C42C22×C4×C41116yes—
378C42C22×C4×C4114yes—
379C42#41×C4118yes—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23C23×C4×C4118yes—
387C23C25×C4111yescharacteristic
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×C4118yes—
412C2×C4C23×C4×C4112yes—
413C2×C4C23×C4×C4118yes—
414C2×C4C23×C4×C4114yes—
415C2×C4C2×#41×C4114yes—
416C2×C4C2×#41×C4118yes—
417C2×C4C2×#41×C4114yes—
418C2×C4C2×#41×C4114yes—
419C2×C4C2×#41×C4114yes—
420C2×C4H(5,𝔽2)×C4112yes—
421C22—28no—
422C22—28no—
423C22—28no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22C24×C4×C4111yescharacteristic
429C22C22×#41×C4112yes—
430C22C22×#41×C4112yes—
431C22C2×H(5,𝔽2)×C4111yescharacteristic
432C22C41×#58511yescharacteristic
433C4—28no—
434C4—28no—
435C4—22no—
436C4—24no—
437C4—28no—
438C4—44no—
439C4—41no—
440C4—44no—
441C4C4×#41×C4118yes—
442C4C41×#55514yesdirect product factor
443C2—48no—
444C2C2×C4×#41×C4112yes—
445C2C4×H(5,𝔽2)×C4111yescharacteristic
446C2C2×C41×#55511yescharacteristic
447C2C41×#347512yes—
448C2C41×#348311yescharacteristic
subgroupG/H
centerC22×C4×C41C24
commutatorC22C24×C4×C41
Frattini, cosocleC23C25×C41
socleC23×C41C25
prime radicalC2×C41×#555C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2372C2semidirect
3. C41×#2369C2semidirect
4. C41×#2363C2semidirect
5. C41×#2354C2semidirect
6. C41×#2346C2semidirect
7. C4×C4⋊Q8×C41C2semidirect
8. C4×#124×C41C2semidirect
9. C4×#121×C41C2semidirect
10. C42×D4×C41C2semidirect
11. C4×#117×C41C2semidirect
12. C41×#555C4direct (N × H)
57. C4⋊Q8×C41C2×C4semidirect
58. #124×C41C2×C4semidirect
59. #121×C41C2×C4semidirect
64. C4×D4×C41C2×C4semidirect
66. #117×C41C2×C4semidirect
196. C4×#555C41direct (N × H)
218. C4×C41#555direct (N × H)
220. #2372C2×C41semidirect
221. #2369C2×C41semidirect
222. #2363C2×C41semidirect
223. #2354C2×C41semidirect
224. #2346C2×C41semidirect
225. C4×C4⋊Q8C2×C41semidirect
226. C4×#124C2×C41semidirect
227. C4×#121C2×C41semidirect
228. C42×D4C2×C41semidirect
229. C4×#117C2×C41semidirect
236. #555C4×C41direct (N × H)
275. C41C4×#555direct (N × H)
282. C4⋊Q8C2×C4×C41semidirect
283. #124C2×C4×C41semidirect
284. #121C2×C4×C41semidirect
289. C4×D4C2×C4×C41semidirect
291. #117C2×C4×C41semidirect
442. C4C41×#555direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C41×#5552
C41×#23722
C41×#23692
C41×#23632
C41×#23542
C41×#23462
C4×C4⋊Q8×C412
C4×#124×C412
C4×#121×C412
C42×D4×C412
C4×#117×C412
C4×#55541