Finite Group Databasebeta

← C4×#562×C131 #45720236

Subgroups — C4×#562×C131 #45720236

Order 33536.

2248 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×#562×C131C211yesmaximal, characteristic
2#2382×C131C211yesmaximal, characteristic, complemented
3#2379×C131C212yesmaximal, complemented
4#2378×C131C214yesmaximal, complemented
5#2377×C131C214yesmaximal, complemented
6#2376×C131C214yesmaximal, complemented
7C4×#125×C131C211yesmaximal, characteristic, complemented
8C4×#124×C131C212yesmaximal, complemented
9C4×C22⋊Q8×C131C214yesmaximal, complemented
10C4×#123×C131C214yesmaximal, complemented
11C4×#121×C131C214yesmaximal, complemented
12#562×C131C4116yesdirect product factor
13C2×#125×C131C2211yescharacteristic
14C2×#124×C131C2212yes—
15C2×#123×C131C2214yes—
16C2×C22⋊Q8×C131C2214yes—
17C2×#121×C131C2214yes—
18C2×C4×Q8×C131C2212yes—
19C2×C4×D4×C131C2214yes—
20C2×C4×D4×C131C2212yes—
21C22×C42×C131C2214yes—
22#391×C131C2214yes—
23#391×C131C2212yes—
24#390×C131C2214yes—
25#390×C131C2212yes—
26#389×C131C2214yes—
27#389×C131C2214yes—
28#389×C131C2212yes—
29#388×C131C2214yes—
30#388×C131C2214yes—
31#388×C131C2214yes—
32#388×C131C2214yes—
33#388×C131C2214yes—
34#388×C131C2214yes—
35#387×C131C2214yes—
36#387×C131C2214yes—
37#387×C131C2214yes—
38#387×C131C2214yes—
39#387×C131C2214yes—
40#386×C131C2214yes—
41#386×C131C2214yes—
42#386×C131C2214yes—
43#385×C131C2214yes—
44#385×C131C2214yes—
45#385×C131C2214yes—
46#382×C131C2212yes—
47#382×C131C2211yescharacteristic
48#381×C131C2214yes—
49#380×C131C2214yes—
50#380×C131C2214yes—
51C4×C4⋊C4×C131C2214yes—
52C4×C4⋊C4×C131C2212yes—
53C4×C4⋊C4×C131C2214yes—
54C4×C22⋊C4×C131C2214yes—
55C4×C22⋊C4×C131C2214yes—
56C4×C22⋊C4×C131C2214yes—
57C43×C131C2211yescharacteristic
58C22×Q8×C131C2312yes—
59C22×D4×C131C2314yes—
60C22×D4×C131C2312yes—
61C23×C4×C131C2314yes—
62C23×C4×C131C2314yes—
63C23×C4×C131C2314yes—
64#125×C131C2×C418yescomplemented
65#124×C131C2×C4116yescomplemented
66#123×C131C2×C4132yescomplemented
67C22⋊Q8×C131C2×C4132yescomplemented
68#121×C131C2×C4132yescomplemented
69C4×Q8×C131—44no—
70C4×D4×C131—48no—
71C4×D4×C131—44no—
72C2×C4⋊C4×C131C2312yes—
73C2×C4⋊C4×C131C2314yes—
74C2×C4⋊C4×C131C2312yes—
75C2×C4⋊C4×C131C2314yes—
76C2×C4⋊C4×C131C2314yes—
77C2×C4⋊C4×C131C2314yes—
78C2×C4⋊C4×C131C2312yes—
79C2×C4⋊C4×C131C2314yes—
80C2×C4⋊C4×C131C2314yes—
81C2×C4⋊C4×C131C2312yes—
82C2×C22⋊C4×C131C2314yes—
83C2×C22⋊C4×C131C2314yes—
84C2×C22⋊C4×C131C2314yes—
85C2×C22⋊C4×C131C2314yes—
86C2×C22⋊C4×C131C2314yes—
87C2×C22⋊C4×C131C2314yes—
88C2×C22⋊C4×C131C2314yes—
89C2×C22⋊C4×C131C2314yes—
90C2×C22⋊C4×C131C2314yes—
91C2×C42×C131—48no—
92C2×C42×C131C2314yes—
93C2×C42×C131C2314yes—
94C2×C42×C131C2312yes—
95C2×C42×C131C2311yescharacteristic
96C2×C42×C131C2311yescharacteristic
97C2×C42×C131C2314yes—
98C2×C42×C131C2312yes—
99C2×C42×C131C2314yes—
100C2×C42×C131C2311yescharacteristic
101#95×C131C2314yes—
102#95×C131C2314yes—
103#95×C131C2314yes—
104#95×C131C2314yes—
105#95×C131C2314yes—
106#95×C131C2314yes—
107#95×C131C2314yes—
108#95×C131C2314yes—
109#95×C131C2314yes—
110#95×C131C2314yes—
111#95×C131C2314yes—
112C24×C131C2414yes—
113C2×Q8×C131—42no—
114C2×Q8×C131—42no—
115C2×Q8×C131C22×C418yes—
116C2×D4×C131—44no—
117C2×D4×C131—42no—
118C2×D4×C131—42no—
119C2×D4×C131—44no—
120C2×D4×C131C22×C4116yes—
121C2×D4×C131C22×C418yes—
122C22×C4×C131—28no—
123C22×C4×C131—28no—
124C22×C4×C131—28no—
125C22×C4×C131—44no—
126C22×C4×C131—44no—
127C22×C4×C131—44no—
128C22×C4×C131—44no—
129C22×C4×C131—44no—
130C22×C4×C131C22×C4116yes—
131C22×C4×C131C2411yescharacteristic
132C22×C4×C131C2414yes—
133C22×C4×C131C2414yes—
134C22×C4×C131C2414yes—
135C22×C4×C131C2412yes—
136C22×C4×C131C2411yescharacteristic
137C22×C4×C131C2412yes—
138C22×C4×C131C2414yes—
139C22×C4×C131C2414yes—
140C22×C4×C131C2411yescharacteristic
141C4⋊C4×C131—42no—
142C4⋊C4×C131—44no—
143C4⋊C4×C131—44no—
144C4⋊C4×C131—44no—
145C4⋊C4×C131—42no—
146C4⋊C4×C131—44no—
147C4⋊C4×C131—42no—
148C4⋊C4×C131C22×C4116yes—
149C4⋊C4×C131C22×C4116yes—
150C4⋊C4×C131C22×C418yes—
151C22⋊C4×C131—44no—
152C22⋊C4×C131—44no—
153C22⋊C4×C131—44no—
154C22⋊C4×C131—44no—
155C22⋊C4×C131—44no—
156C22⋊C4×C131—44no—
157C22⋊C4×C131C22×C4116yes—
158C22⋊C4×C131C22×C4116yes—
159C22⋊C4×C131C22×C4116yes—
160C42×C131—28no—
161C42×C131—28no—
162C42×C131—24no—
163C42×C131—22no—
164C42×C131—44no—
165C42×C131—44no—
166C42×C131—41no—
167C42×C131—42no—
168C42×C131C22×C414yes—
169C23×C131—24no—
170C23×C131—24no—
171C23×C131—24no—
172C23×C131—24no—
173C23×C131—24no—
174C23×C131—24no—
175C23×C131C23×C418yes—
176C23×C131C2511yescharacteristic
177Q8×C131—48no—
178D4×C131—416no—
179D4×C131—48no—
180C2×C4×C131—24no—
181C2×C4×C131—24no—
182C2×C4×C131—22no—
183C2×C4×C131—21no—
184C2×C4×C131—21no—
185C2×C4×C131—22no—
186C2×C4×C131—24no—
187C2×C4×C131—24no—
188C2×C4×C131—24no—
189C2×C4×C131—24no—
190C2×C4×C131—22no—
191C2×C4×C131—24no—
192C2×C4×C131—24no—
193C2×C4×C131—21no—
194C2×C4×C131—24no—
195C2×C4×C131—22no—
196C2×C4×C131—22no—
197C2×C4×C131—22no—
198C2×C4×C131—24no—
199C2×C4×C131—24no—
200C2×C4×C131—24no—
201C2×C4×C131—24no—
202C2×C4×C131—416no—
203C2×C4×C131—416no—
204C2×C4×C131—416no—
205C2×C4×C131C23×C412yes—
206C2×C4×C131C23×C414yes—
207C2×C4×C131C23×C418yes—
208C2×C4×C131C23×C418yes—
209C2×C4×C131H(5,𝔽2)14yes—
210C2×C4×C1312−1+412yes—
211C22×C131—28no—
212C22×C131—28no—
213C22×C131—28no—
214C22×C131—44no—
215C22×C131—44no—
216C22×C131—44no—
217C22×C131—44no—
218C22×C131C24×C411yescharacteristic
219C22×C131C2×H(5,𝔽2)12yes—
220C22×C131C2×2−1+411yescharacteristic
221C22×C131#58512yes—
222C22×C131#58511yescharacteristic
223C4×C131—28no—
224C4×C131—28no—
225C4×C131—24no—
226C4×C131—22no—
227C4×C131—44no—
228C4×C131—42no—
229C4×C131—44no—
230C4×C131—44no—
231C4×C131—41no—
232C4×C131#56214yesdirect product factor
233C2×C131—48no—
234C2×C131C4×H(5,𝔽2)12yes—
235C2×C131C4×2−1+411yescharacteristic
236C2×C131C2×#56211yescharacteristic
237C2×C131#352912yes—
238C2×C131#353111yescharacteristic
239C4×#562C13111yesmaximal, characteristic, direct product factor, Sylow
240C131C4×#56211yescharacteristic, direct product factor, Sylow
241C2×#562C2×C13111yescharacteristic
242#2382C2×C13111yescharacteristic, complemented
243#2379C2×C13112yescomplemented
244#2378C2×C13114yescomplemented
245#2377C2×C13114yescomplemented
246#2376C2×C13114yescomplemented
247C4×#125C2×C13111yescharacteristic, complemented
248C4×#124C2×C13112yescomplemented
249C4×C22⋊Q8C2×C13114yescomplemented
250C4×#123C2×C13114yescomplemented
251C4×#121C2×C13114yescomplemented
252#562C4×C131116yesdirect product factor
253C2×#125C22×C13111yescharacteristic
254C2×#124C22×C13112yes—
255C2×#123C22×C13114yes—
256C2×C22⋊Q8C22×C13114yes—
257C2×#121C22×C13114yes—
258C2×C4×Q8C22×C13112yes—
259C2×C4×D4C22×C13114yes—
260C2×C4×D4C22×C13112yes—
261C22×C42C22×C13114yes—
262#391C22×C13112yes—
263#391C22×C13114yes—
264#390C22×C13114yes—
265#390C22×C13112yes—
266#389C22×C13112yes—
267#389C22×C13114yes—
268#389C22×C13114yes—
269#388C22×C13114yes—
270#388C22×C13114yes—
271#388C22×C13114yes—
272#388C22×C13114yes—
273#388C22×C13114yes—
274#388C22×C13114yes—
275#387C22×C13114yes—
276#387C22×C13114yes—
277#387C22×C13114yes—
278#387C22×C13114yes—
279#387C22×C13114yes—
280#386C22×C13114yes—
281#386C22×C13114yes—
282#386C22×C13114yes—
283#385C22×C13114yes—
284#385C22×C13114yes—
285#385C22×C13114yes—
286#382C22×C13111yescharacteristic
287#382C22×C13112yes—
288#381C22×C13114yes—
289#380C22×C13114yes—
290#380C22×C13114yes—
291C4×C4⋊C4C22×C13114yes—
292C4×C4⋊C4C22×C13112yes—
293C4×C4⋊C4C22×C13114yes—
294C4×C22⋊C4C22×C13114yes—
295C4×C22⋊C4C22×C13114yes—
296C4×C22⋊C4C22×C13114yes—
297C43C22×C13111yescharacteristic
298C22×Q8C23×C13112yes—
299C22×D4C23×C13114yes—
300C22×D4C23×C13112yes—
301C23×C4C23×C13114yes—
302C23×C4C23×C13114yes—
303C23×C4C23×C13114yes—
304#125C2×C4×C13118yescomplemented
305#124C2×C4×C131116yescomplemented
306#123C2×C4×C131132yescomplemented
307C22⋊Q8C2×C4×C131132yescomplemented
308#121C2×C4×C131132yescomplemented
309C4×Q8—44no—
310C4×D4—48no—
311C4×D4—44no—
312C2×C4⋊C4C23×C13114yes—
313C2×C4⋊C4C23×C13112yes—
314C2×C4⋊C4C23×C13114yes—
315C2×C4⋊C4C23×C13112yes—
316C2×C4⋊C4C23×C13114yes—
317C2×C4⋊C4C23×C13114yes—
318C2×C4⋊C4C23×C13112yes—
319C2×C4⋊C4C23×C13114yes—
320C2×C4⋊C4C23×C13114yes—
321C2×C4⋊C4C23×C13112yes—
322C2×C22⋊C4C23×C13114yes—
323C2×C22⋊C4C23×C13114yes—
324C2×C22⋊C4C23×C13114yes—
325C2×C22⋊C4C23×C13114yes—
326C2×C22⋊C4C23×C13114yes—
327C2×C22⋊C4C23×C13114yes—
328C2×C22⋊C4C23×C13114yes—
329C2×C22⋊C4C23×C13114yes—
330C2×C22⋊C4C23×C13114yes—
331C2×C42—48no—
332C2×C42C23×C13112yes—
333C2×C42C23×C13114yes—
334C2×C42C23×C13114yes—
335C2×C42C23×C13111yescharacteristic
336C2×C42C23×C13112yes—
337C2×C42C23×C13111yescharacteristic
338C2×C42C23×C13114yes—
339C2×C42C23×C13114yes—
340C2×C42C23×C13111yescharacteristic
341#95C23×C13114yes—
342#95C23×C13114yes—
343#95C23×C13114yes—
344#95C23×C13114yes—
345#95C23×C13114yes—
346#95C23×C13114yes—
347#95C23×C13114yes—
348#95C23×C13114yes—
349#95C23×C13114yes—
350#95C23×C13114yes—
351#95C23×C13114yes—
352C24C24×C13114yes—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8C22×C4×C13118yes—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4C22×C4×C131116yes—
361C2×D4C22×C4×C13118yes—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4C22×C4×C131116yes—
371C22×C4C24×C13111yescharacteristic
372C22×C4C24×C13114yes—
373C22×C4C24×C13114yes—
374C22×C4C24×C13114yes—
375C22×C4C24×C13112yes—
376C22×C4C24×C13111yescharacteristic
377C22×C4C24×C13112yes—
378C22×C4C24×C13114yes—
379C22×C4C24×C13114yes—
380C22×C4C24×C13111yescharacteristic
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4C22×C4×C131116yes—
389C4⋊C4C22×C4×C131116yes—
390C4⋊C4C22×C4×C13118yes—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4C22×C4×C131116yes—
398C22⋊C4C22×C4×C131116yes—
399C22⋊C4C22×C4×C131116yes—
400C42—28no—
401C42—28no—
402C42—24no—
403C42—22no—
404C42—44no—
405C42—44no—
406C42—41no—
407C42—42no—
408C42C22×C4×C13114yes—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23C23×C4×C13118yes—
416C23C25×C13111yescharacteristic
417Q8—48no—
418D4—416no—
419D4—48no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—416no—
444C2×C4—416no—
445C2×C4C23×C4×C13112yes—
446C2×C4C23×C4×C13114yes—
447C2×C4C23×C4×C13118yes—
448C2×C4C23×C4×C13118yes—
449C2×C4H(5,𝔽2)×C13114yes—
450C2×C42−1+4×C13112yes—
451C22—28no—
452C22—28no—
453C22—28no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22C24×C4×C13111yescharacteristic
459C22C2×H(5,𝔽2)×C13112yes—
460C22C2×2−1+4×C13111yescharacteristic
461C22#585×C13111yescharacteristic
462C22#585×C13112yes—
463C4—28no—
464C4—28no—
465C4—24no—
466C4—22no—
467C4—44no—
468C4—44no—
469C4—44no—
470C4—42no—
471C4—41no—
472C4#562×C13114yesdirect product factor
473C2—48no—
474C2C4×H(5,𝔽2)×C13112yes—
475C2C4×2−1+4×C13111yescharacteristic
476C2C2×#562×C13111yescharacteristic
477C2#3529×C13112yes—
478C2#3531×C13111yescharacteristic
subgroupG/H
centerC22×C4×C131C24
commutatorC22C24×C4×C131
Frattini, cosocleC23C25×C131
socleC23×C131C25
prime radicalC2×#562×C131C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2382×C131C2semidirect
3. #2379×C131C2semidirect
4. #2378×C131C2semidirect
5. #2377×C131C2semidirect
6. #2376×C131C2semidirect
7. C4×#125×C131C2semidirect
8. C4×#124×C131C2semidirect
9. C4×C22⋊Q8×C131C2semidirect
10. C4×#123×C131C2semidirect
11. C4×#121×C131C2semidirect
12. #562×C131C4direct (N × H)
64. #125×C131C2×C4semidirect
65. #124×C131C2×C4semidirect
66. #123×C131C2×C4semidirect
67. C22⋊Q8×C131C2×C4semidirect
68. #121×C131C2×C4semidirect
232. C4×C131#562direct (N × H)
239. C4×#562C131direct (N × H)
240. C131C4×#562direct (N × H)
242. #2382C2×C131semidirect
243. #2379C2×C131semidirect
244. #2378C2×C131semidirect
245. #2377C2×C131semidirect
246. #2376C2×C131semidirect
247. C4×#125C2×C131semidirect
248. C4×#124C2×C131semidirect
249. C4×C22⋊Q8C2×C131semidirect
250. C4×#123C2×C131semidirect
251. C4×#121C2×C131semidirect
252. #562C4×C131direct (N × H)
304. #125C2×C4×C131semidirect
305. #124C2×C4×C131semidirect
306. #123C2×C4×C131semidirect
307. C22⋊Q8C2×C4×C131semidirect
308. #121C2×C4×C131semidirect
472. C4#562×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#562×C1312
#2382×C1312
#2379×C1312
#2378×C1312
#2377×C1312
#2376×C1312
C4×#125×C1312
C4×#124×C1312
C4×C22⋊Q8×C1312
C4×#123×C1312
C4×#121×C1312
C4×#562131