Finite Group Databasebeta

← C4×#555×C131 #45720049

Subgroups — C4×#555×C131 #45720049

Order 33536.

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×#555×C131C211yesmaximal, characteristic
2#2372×C131C214yesmaximal, complemented
3#2369×C131C211yesmaximal, characteristic, complemented
4#2363×C131C214yesmaximal, complemented
5#2354×C131C212yesmaximal, complemented
6#2346×C131C214yesmaximal, complemented
7C4×C4⋊Q8×C131C211yesmaximal, characteristic, complemented
8C4×#124×C131C214yesmaximal, complemented
9C4×#121×C131C214yesmaximal, complemented
10C42×D4×C131C212yesmaximal, complemented
11C4×#117×C131C214yesmaximal, complemented
12#555×C131C4116yesdirect product factor
13C2×C4⋊Q8×C131C2211yescharacteristic
14C2×#124×C131C2214yes—
15C2×#121×C131C2214yes—
16C2×C4×Q8×C131C2212yes—
17C2×C4×D4×C131C2212yes—
18C2×C4×D4×C131C2212yes—
19C2×C4×D4×C131C2214yes—
20C2×C4×D4×C131C2212yes—
21C2×#117×C131C2214yes—
22C2×#117×C131C2214yes—
23C22×C42×C131C2214yes—
24#391×C131C2214yes—
25#391×C131C2212yes—
26#390×C131C2214yes—
27#390×C131C2214yes—
28#389×C131C2214yes—
29#389×C131C2214yes—
30#388×C131C2218yes—
31#388×C131C2218yes—
32#388×C131C2218yes—
33#386×C131C2218yes—
34#384×C131C2211yescharacteristic
35#382×C131C2214yes—
36#380×C131C2214yes—
37C4×C4⋊C4×C131C2212yes—
38C4×C4⋊C4×C131C2214yes—
39C4×C4⋊C4×C131C2214yes—
40C4×C4⋊C4×C131C2214yes—
41C4×C4⋊C4×C131C2212yes—
42C4×C4⋊C4×C131C2212yes—
43C4×C22⋊C4×C131C2218yes—
44C4×C22⋊C4×C131C2218yes—
45C4×C22⋊C4×C131C2214yes—
46C4×C22⋊C4×C131C2214yes—
47C4×C22⋊C4×C131C2214yes—
48#376×C131C2218yes—
49C43×C131C2214yes—
50C43×C131C2211yescharacteristic
51C22×Q8×C131C2312yes—
52C22×D4×C131C2312yes—
53C22×D4×C131C2314yes—
54C23×C4×C131C2314yes—
55C23×C4×C131C2314yes—
56C23×C4×C131C2314yes—
57C4⋊Q8×C131C2×C418yescomplemented
58#124×C131C2×C4132yescomplemented
59#121×C131C2×C4132yescomplemented
60C4×Q8×C131—44no—
61C4×D4×C131—28no—
62C4×D4×C131—28no—
63C4×D4×C131—48no—
64C4×D4×C131C2×C4116yescomplemented
65#117×C131—216no—
66#117×C131C2×C4132yescomplemented
67C2×C4⋊C4×C131C2312yes—
68C2×C4⋊C4×C131C2314yes—
69C2×C4⋊C4×C131C2312yes—
70C2×C4⋊C4×C131C2314yes—
71C2×C4⋊C4×C131C2314yes—
72C2×C4⋊C4×C131C2314yes—
73C2×C4⋊C4×C131C2312yes—
74C2×C4⋊C4×C131C2312yes—
75C2×C22⋊C4×C131C2314yes—
76C2×C22⋊C4×C131C2314yes—
77C2×C22⋊C4×C131C2318yes—
78C2×C22⋊C4×C131C2318yes—
79C2×C22⋊C4×C131C2318yes—
80C2×C22⋊C4×C131C2314yes—
81C2×C42×C131—216no—
82C2×C42×C131C2314yes—
83C2×C42×C131C2318yes—
84C2×C42×C131C2312yes—
85C2×C42×C131C2314yes—
86C2×C42×C131C2314yes—
87C2×C42×C131C2314yes—
88C2×C42×C131C2311yescharacteristic
89C2×C42×C131C2314yes—
90C2×C42×C131C2311yescharacteristic
91C2×C42×C131C2314yes—
92C2×C42×C131C2311yescharacteristic
93C2×C42×C131C2312yes—
94#95×C131C2314yes—
95#95×C131C2318yes—
96#95×C131C2314yes—
97#95×C131C2318yes—
98#95×C131C2318yes—
99#95×C131C2314yes—
100C24×C131C2414yes—
101C2×Q8×C131—42no—
102C2×Q8×C131—42no—
103C2×Q8×C131C22×C418yes—
104C2×D4×C131—24no—
105C2×D4×C131—24no—
106C2×D4×C131—44no—
107C2×D4×C131—44no—
108C2×D4×C131C22×C418yes—
109C2×D4×C131C22×C4116yes—
110C22×C4×C131—28no—
111C22×C4×C131—28no—
112C22×C4×C131—28no—
113C22×C4×C131—28no—
114C22×C4×C131—28no—
115C22×C4×C131—28no—
116C22×C4×C131—28no—
117C22×C4×C131—44no—
118C22×C4×C131C22×C4116yes—
119C22×C4×C131C2414yes—
120C22×C4×C131C2411yescharacteristic
121C22×C4×C131C2414yes—
122C22×C4×C131C2414yes—
123C22×C4×C131C2411yescharacteristic
124C22×C4×C131C2414yes—
125C22×C4×C131C2412yes—
126C22×C4×C131C2414yes—
127C22×C4×C131C2412yes—
128C22×C4×C131C2411yescharacteristic
129C4⋊C4×C131—28no—
130C4⋊C4×C131—24no—
131C4⋊C4×C131—44no—
132C4⋊C4×C131—44no—
133C4⋊C4×C131—42no—
134C4⋊C4×C131C22×C418yes—
135C4⋊C4×C131C22×C4116yes—
136C4⋊C4×C131#4118yes—
137C22⋊C4×C131—216no—
138C22⋊C4×C131—28no—
139C22⋊C4×C131—28no—
140C22⋊C4×C131—48no—
141C22⋊C4×C131C22×C4132yes—
142C22⋊C4×C131C22×C4116yes—
143C42×C131—28no—
144C42×C131—28no—
145C42×C131—28no—
146C42×C131—216no—
147C42×C131—24no—
148C42×C131—28no—
149C42×C131—22no—
150C42×C131—44no—
151C42×C131—41no—
152C42×C131C22×C4116yes—
153C42×C131C22×C414yes—
154C42×C131#4118yes—
155C23×C131—24no—
156C23×C131—24no—
157C23×C131—24no—
158C23×C131—24no—
159C23×C131—24no—
160C23×C131—24no—
161C23×C131C23×C418yes—
162C23×C131C2511yescharacteristic
163Q8×C131—48no—
164D4×C131—216no—
165D4×C131—416no—
166C2×C4×C131—24no—
167C2×C4×C131—24no—
168C2×C4×C131—24no—
169C2×C4×C131—22no—
170C2×C4×C131—24no—
171C2×C4×C131—21no—
172C2×C4×C131—24no—
173C2×C4×C131—22no—
174C2×C4×C131—24no—
175C2×C4×C131—21no—
176C2×C4×C131—232no—
177C2×C4×C131—24no—
178C2×C4×C131—22no—
179C2×C4×C131—28no—
180C2×C4×C131—24no—
181C2×C4×C131—24no—
182C2×C4×C131—21no—
183C2×C4×C131—24no—
184C2×C4×C131—416no—
185C2×C4×C131—416no—
186C2×C4×C131C23×C418yes—
187C2×C4×C131C23×C412yes—
188C2×C4×C131C23×C418yes—
189C2×C4×C131C23×C414yes—
190C2×C4×C131C2×#4114yes—
191C2×C4×C131C2×#4118yes—
192C2×C4×C131C2×#4114yes—
193C2×C4×C131C2×#4114yes—
194C2×C4×C131C2×#4114yes—
195C2×C4×C131H(5,𝔽2)12yes—
196C22×C131—28no—
197C22×C131—28no—
198C22×C131—28no—
199C22×C131—44no—
200C22×C131—44no—
201C22×C131—44no—
202C22×C131—44no—
203C22×C131C24×C411yescharacteristic
204C22×C131C22×#4112yes—
205C22×C131C22×#4112yes—
206C22×C131C2×H(5,𝔽2)11yescharacteristic
207C22×C131#58511yescharacteristic
208C4×C131—28no—
209C4×C131—28no—
210C4×C131—22no—
211C4×C131—24no—
212C4×C131—28no—
213C4×C131—44no—
214C4×C131—41no—
215C4×C131—44no—
216C4×C131C4×#4118yes—
217C4×C131#55514yesdirect product factor
218C2×C131—48no—
219C2×C131C2×C4×#4112yes—
220C2×C131C4×H(5,𝔽2)11yescharacteristic
221C2×C131C2×#55511yescharacteristic
222C2×C131#347512yes—
223C2×C131#348311yescharacteristic
224C4×#555C13111yesmaximal, characteristic, direct product factor, Sylow
225C131C4×#55511yescharacteristic, direct product factor, Sylow
226C2×#555C2×C13111yescharacteristic
227#2372C2×C13114yescomplemented
228#2369C2×C13111yescharacteristic, complemented
229#2363C2×C13114yescomplemented
230#2354C2×C13112yescomplemented
231#2346C2×C13114yescomplemented
232C4×C4⋊Q8C2×C13111yescharacteristic, complemented
233C4×#124C2×C13114yescomplemented
234C4×#121C2×C13114yescomplemented
235C42×D4C2×C13112yescomplemented
236C4×#117C2×C13114yescomplemented
237#555C4×C131116yesdirect product factor
238C2×C4⋊Q8C22×C13111yescharacteristic
239C2×#124C22×C13114yes—
240C2×#121C22×C13114yes—
241C2×C4×Q8C22×C13112yes—
242C2×C4×D4C22×C13112yes—
243C2×C4×D4C22×C13112yes—
244C2×C4×D4C22×C13114yes—
245C2×C4×D4C22×C13112yes—
246C2×#117C22×C13114yes—
247C2×#117C22×C13114yes—
248C22×C42C22×C13114yes—
249#391C22×C13112yes—
250#391C22×C13114yes—
251#390C22×C13114yes—
252#390C22×C13114yes—
253#389C22×C13114yes—
254#389C22×C13114yes—
255#388C22×C13118yes—
256#388C22×C13118yes—
257#388C22×C13118yes—
258#386C22×C13118yes—
259#384C22×C13111yescharacteristic
260#382C22×C13114yes—
261#380C22×C13114yes—
262C4×C4⋊C4C22×C13114yes—
263C4×C4⋊C4C22×C13112yes—
264C4×C4⋊C4C22×C13114yes—
265C4×C4⋊C4C22×C13112yes—
266C4×C4⋊C4C22×C13114yes—
267C4×C4⋊C4C22×C13112yes—
268C4×C22⋊C4C22×C13114yes—
269C4×C22⋊C4C22×C13118yes—
270C4×C22⋊C4C22×C13114yes—
271C4×C22⋊C4C22×C13118yes—
272C4×C22⋊C4C22×C13114yes—
273#376C22×C13118yes—
274C43C22×C13111yescharacteristic
275C43C22×C13114yes—
276C22×Q8C23×C13112yes—
277C22×D4C23×C13112yes—
278C22×D4C23×C13114yes—
279C23×C4C23×C13114yes—
280C23×C4C23×C13114yes—
281C23×C4C23×C13114yes—
282C4⋊Q8C2×C4×C13118yescomplemented
283#124C2×C4×C131132yescomplemented
284#121C2×C4×C131132yescomplemented
285C4×Q8—44no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—48no—
289C4×D4C2×C4×C131116yescomplemented
290#117—216no—
291#117C2×C4×C131132yescomplemented
292C2×C4⋊C4C23×C13114yes—
293C2×C4⋊C4C23×C13112yes—
294C2×C4⋊C4C23×C13114yes—
295C2×C4⋊C4C23×C13114yes—
296C2×C4⋊C4C23×C13112yes—
297C2×C4⋊C4C23×C13112yes—
298C2×C4⋊C4C23×C13114yes—
299C2×C4⋊C4C23×C13112yes—
300C2×C22⋊C4C23×C13118yes—
301C2×C22⋊C4C23×C13114yes—
302C2×C22⋊C4C23×C13114yes—
303C2×C22⋊C4C23×C13118yes—
304C2×C22⋊C4C23×C13118yes—
305C2×C22⋊C4C23×C13114yes—
306C2×C42—216no—
307C2×C42C23×C13118yes—
308C2×C42C23×C13114yes—
309C2×C42C23×C13112yes—
310C2×C42C23×C13111yescharacteristic
311C2×C42C23×C13114yes—
312C2×C42C23×C13114yes—
313C2×C42C23×C13114yes—
314C2×C42C23×C13111yescharacteristic
315C2×C42C23×C13114yes—
316C2×C42C23×C13114yes—
317C2×C42C23×C13111yescharacteristic
318C2×C42C23×C13112yes—
319#95C23×C13114yes—
320#95C23×C13118yes—
321#95C23×C13118yes—
322#95C23×C13118yes—
323#95C23×C13114yes—
324#95C23×C13114yes—
325C24C24×C13114yes—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8C22×C4×C13118yes—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4C22×C4×C13118yes—
334C2×D4C22×C4×C131116yes—
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×C131116yes—
344C22×C4C24×C13114yes—
345C22×C4C24×C13111yescharacteristic
346C22×C4C24×C13114yes—
347C22×C4C24×C13114yes—
348C22×C4C24×C13111yescharacteristic
349C22×C4C24×C13114yes—
350C22×C4C24×C13112yes—
351C22×C4C24×C13114yes—
352C22×C4C24×C13112yes—
353C22×C4C24×C13111yescharacteristic
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4C22×C4×C13118yes—
360C4⋊C4C22×C4×C131116yes—
361C4⋊C4#41×C13118yes—
362C22⋊C4—216no—
363C22⋊C4—28no—
364C22⋊C4—28no—
365C22⋊C4—48no—
366C22⋊C4C22×C4×C131132yes—
367C22⋊C4C22×C4×C131116yes—
368C42—28no—
369C42—28no—
370C42—28no—
371C42—216no—
372C42—24no—
373C42—28no—
374C42—22no—
375C42—44no—
376C42—41no—
377C42C22×C4×C131116yes—
378C42C22×C4×C13114yes—
379C42#41×C13118yes—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23C23×C4×C13118yes—
387C23C25×C13111yescharacteristic
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×C13118yes—
412C2×C4C23×C4×C13112yes—
413C2×C4C23×C4×C13118yes—
414C2×C4C23×C4×C13114yes—
415C2×C4C2×#41×C13114yes—
416C2×C4C2×#41×C13118yes—
417C2×C4C2×#41×C13114yes—
418C2×C4C2×#41×C13114yes—
419C2×C4C2×#41×C13114yes—
420C2×C4H(5,𝔽2)×C13112yes—
421C22—28no—
422C22—28no—
423C22—28no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—44no—
428C22C24×C4×C13111yescharacteristic
429C22C22×#41×C13112yes—
430C22C22×#41×C13112yes—
431C22C2×H(5,𝔽2)×C13111yescharacteristic
432C22#585×C13111yescharacteristic
433C4—28no—
434C4—28no—
435C4—22no—
436C4—24no—
437C4—28no—
438C4—44no—
439C4—41no—
440C4—44no—
441C4C4×#41×C13118yes—
442C4#555×C13114yesdirect product factor
443C2—48no—
444C2C2×C4×#41×C13112yes—
445C2C4×H(5,𝔽2)×C13111yescharacteristic
446C2C2×#555×C13111yescharacteristic
447C2#3475×C13112yes—
448C2#3483×C13111yescharacteristic
subgroupG/H
centerC22×C4×C131C24
commutatorC22C24×C4×C131
Frattini, cosocleC23C25×C131
socleC23×C131C25
prime radicalC2×#555×C131C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2372×C131C2semidirect
3. #2369×C131C2semidirect
4. #2363×C131C2semidirect
5. #2354×C131C2semidirect
6. #2346×C131C2semidirect
7. C4×C4⋊Q8×C131C2semidirect
8. C4×#124×C131C2semidirect
9. C4×#121×C131C2semidirect
10. C42×D4×C131C2semidirect
11. C4×#117×C131C2semidirect
12. #555×C131C4direct (N × H)
57. C4⋊Q8×C131C2×C4semidirect
58. #124×C131C2×C4semidirect
59. #121×C131C2×C4semidirect
64. C4×D4×C131C2×C4semidirect
66. #117×C131C2×C4semidirect
217. C4×C131#555direct (N × H)
224. C4×#555C131direct (N × H)
225. C131C4×#555direct (N × H)
227. #2372C2×C131semidirect
228. #2369C2×C131semidirect
229. #2363C2×C131semidirect
230. #2354C2×C131semidirect
231. #2346C2×C131semidirect
232. C4×C4⋊Q8C2×C131semidirect
233. C4×#124C2×C131semidirect
234. C4×#121C2×C131semidirect
235. C42×D4C2×C131semidirect
236. C4×#117C2×C131semidirect
237. #555C4×C131direct (N × H)
282. C4⋊Q8C2×C4×C131semidirect
283. #124C2×C4×C131semidirect
284. #121C2×C4×C131semidirect
289. C4×D4C2×C4×C131semidirect
291. #117C2×C4×C131semidirect
442. C4#555×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#555×C1312
#2372×C1312
#2369×C1312
#2363×C1312
#2354×C1312
#2346×C1312
C4×C4⋊Q8×C1312
C4×#124×C1312
C4×#121×C1312
C42×D4×C1312
C4×#117×C1312
C4×#555131