Finite Group Databasebeta

← C2×C31×#1854 #21477601

Subgroups — C2×C31×#1854 #21477601

Order 7936.

1662 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C31×#117C211yesmaximal, characteristic
2C22×C31×#107C211yesmaximal, characteristic, complemented
3C22×C31×#106C211yesmaximal, characteristic, complemented
4C22×C31×C22⋊C8C211yesmaximal, characteristic
5C2×C31×#380C211yesmaximal, characteristic
6C31×#1854C218yesmaximal, direct product factor
7C2×C31×#340C211yesmaximal, characteristic, complemented
8C2×C31×#340C211yesmaximal, characteristic, complemented
9C24×C4×C31C2211yescharacteristic
10C23×C8×C31C2211yescharacteristic
11C23×C8×C31C2211yescharacteristic
12C2×C31×#117—22no—
13C2×C31×#117—24no—
14C2×C31×#117—24no—
15C2×C31×#117C2214yes—
16C22×C4⋊C4×C31C2211yescharacteristic
17C22×C4⋊C4×C31C2211yescharacteristic
18C22×C4⋊C4×C31C2211yescharacteristic
19C22×C4⋊C4×C31C2211yescharacteristic
20C22×C22⋊C4×C31—21no—
21C22×C22⋊C4×C31—21no—
22C22×C42×C31—21no—
23C2×C31×#107—24no—
24C2×C31×#107C2214yescomplemented
25C2×C31×#106—24no—
26C2×C31×#106C2214yescomplemented
27C2×C31×C22⋊C8C418yescomplemented
28C2×C31×C22⋊C8C2214yes—
29C31×#380—22no—
30C31×#380C2214yes—
31C2×C31×#95—21no—
32C31×#340C2214yescomplemented
33C31×#340C2214yescomplemented
34C25×C31D411yescharacteristic
35C23×C4×C31—22no—
36C23×C4×C31—22no—
37C23×C4×C31—22no—
38C23×C4×C31—22no—
39C23×C4×C31—21no—
40C23×C4×C31—21no—
41C23×C4×C31—21no—
42C23×C4×C31—21no—
43C23×C4×C31C2×C414yes—
44C23×C4×C31D412yes—
45C23×C4×C31D411yescharacteristic
46C23×C4×C31Q814yes—
47C23×C4×C31C2312yes—
48C23×C4×C31C2311yescharacteristic
49C22×C8×C31—22no—
50C22×C8×C31—22no—
51C22×C8×C31—22no—
52C22×C8×C31—22no—
53C22×C8×C31C2×C414yes—
54C22×C8×C31C2×C414yes—
55C22×C8×C31C2312yes—
56C22×C8×C31C2312yes—
57C31×#117—216no—
58C31×#117—28no—
59C31×#117—28no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31—21no—
62C2×C4⋊C4×C31—22no—
63C2×C4⋊C4×C31—22no—
64C2×C4⋊C4×C31—22no—
65C2×C4⋊C4×C31—22no—
66C2×C4⋊C4×C31—22no—
67C2×C4⋊C4×C31—22no—
68C2×C4⋊C4×C31—21no—
69C2×C4⋊C4×C31—21no—
70C2×C4⋊C4×C31D414yes—
71C2×C4⋊C4×C31D414yes—
72C2×C4⋊C4×C31C2312yes—
73C2×C4⋊C4×C31C2312yes—
74C2×C4⋊C4×C31C2312yes—
75C2×C4⋊C4×C31C2312yes—
76C2×C22⋊C4×C31—28no—
77C2×C22⋊C4×C31—24no—
78C2×C22⋊C4×C31—24no—
79C2×C22⋊C4×C31—44no—
80C2×C42×C31—28no—
81C2×C42×C31—24no—
82C31×#107—28no—
83C31×#106—28no—
84C31×C22⋊C8C2×C4116yescomplemented
85C31×#95—24no—
86C24×C31—22no—
87C24×C31—22no—
88C24×C31—22no—
89C24×C31—22no—
90C24×C31—22no—
91C24×C31—22no—
92C24×C31C4⋊C414yes—
93C24×C31C2×D412yes—
94C24×C31C2×D411yescharacteristic
95C22×C4×C31—22no—
96C22×C4×C31—22no—
97C22×C4×C31—28no—
98C22×C4×C31—28no—
99C22×C4×C31—24no—
100C22×C4×C31—28no—
101C22×C4×C31—28no—
102C22×C4×C31—24no—
103C22×C4×C31—22no—
104C22×C4×C31—22no—
105C22×C4×C31—22no—
106C22×C4×C31—22no—
107C22×C4×C31—24no—
108C22×C4×C31—24no—
109C22×C4×C31—22no—
110C22×C4×C31—22no—
111C22×C4×C31—22no—
112C22×C4×C31—22no—
113C22×C4×C31—42no—
114C22×C4×C31—42no—
115C22×C4×C31—42no—
116C22×C4×C31—42no—
117C22×C4×C31—42no—
118C22×C4×C31—42no—
119C22×C4×C31—42no—
120C22×C4×C31—42no—
121C22×C4×C31—42no—
122C22×C4×C31—42no—
123C22×C4×C31C4⋊C414yes—
124C22×C4×C31C4⋊C418yes—
125C22×C4×C31C22×C414yes—
126C22×C4×C31C22×C412yes—
127C22×C4×C31C2×D412yes—
128C22×C4×C31C2×D411yescharacteristic
129C22×C4×C31C2×D412yes—
130C22×C4×C31C2×D412yes—
131C22×C4×C31C2×Q814yes—
132C22×C4×C31C2×Q812yes—
133C22×C4×C31#4112yes—
134C22×C4×C31#4112yes—
135C22×C4×C31C2411yescharacteristic
136C2×C8×C31—28no—
137C2×C8×C31—28no—
138C2×C8×C31—22no—
139C2×C8×C31—22no—
140C2×C8×C31—22no—
141C2×C8×C31—22no—
142C2×C8×C31C22×C414yes—
143C2×C8×C31C22×C414yes—
144C4⋊C4×C31—24no—
145C4⋊C4×C31—24no—
146C4⋊C4×C31—22no—
147C4⋊C4×C31—22no—
148C4⋊C4×C31—22no—
149C4⋊C4×C31—22no—
150C4⋊C4×C31—22no—
151C4⋊C4×C31—22no—
152C4⋊C4×C31—42no—
153C4⋊C4×C31—42no—
154C4⋊C4×C31C2×D414yes—
155C4⋊C4×C31C2×D414yes—
156C22⋊C4×C31—216no—
157C22⋊C4×C31—48no—
158C42×C31—216no—
159C2×#1854C3111yesmaximal, characteristic, direct product factor, Sylow
160C23×C31—22no—
161C23×C31—22no—
162C23×C31—22no—
163C23×C31—22no—
164C23×C31—22no—
165C23×C31—22no—
166C23×C31—28no—
167C23×C31—28no—
168C23×C31—28no—
169C23×C31—44no—
170C23×C31—44no—
171C23×C31—44no—
172C23×C31—44no—
173C23×C31C2×C4⋊C414yes—
174C23×C31C2×C4⋊C412yes—
175C23×C31#12012yes—
176C23×C31#12312yes—
177C23×C31#13512yes—
178C23×C31#13512yes—
179C23×C31AGL(1,ℤ/8)12yes—
180C23×C31#13712yes—
181C23×C31C22×D411yescharacteristic
182C2×C4×C31—22no—
183C2×C4×C31—22no—
184C2×C4×C31—28no—
185C2×C4×C31—24no—
186C2×C4×C31—24no—
187C2×C4×C31—24no—
188C2×C4×C31—24no—
189C2×C4×C31—216no—
190C2×C4×C31—24no—
191C2×C4×C31—24no—
192C2×C4×C31—42no—
193C2×C4×C31—42no—
194C2×C4×C31—42no—
195C2×C4×C31—42no—
196C2×C4×C31—48no—
197C2×C4×C31—48no—
198C2×C4×C31—42no—
199C2×C4×C31—42no—
200C2×C4×C31—42no—
201C2×C4×C31—42no—
202C2×C4×C31—42no—
203C2×C4×C31—42no—
204C2×C4×C31—84no—
205C2×C4×C31—84no—
206C2×C4×C31C2×C4⋊C414yes—
207C2×C4×C31C2×C4⋊C412yes—
208C2×C4×C31C4×D414yes—
209C2×C4×C31C4×D414yes—
210C2×C4×C31#12012yes—
211C2×C4×C31C22⋊Q814yes—
212C2×C4×C31C22⋊Q814yes—
213C2×C4×C31#12312yes—
214C2×C4×C31C23×C411yescharacteristic
215C2×C4×C31C22×D411yescharacteristic
216C2×C4×C31C22×D411yescharacteristic
217C2×C4×C31C22×Q811yescharacteristic
218C2×C4×C31C2×#4111yescharacteristic
219C2×C4×C31C2×#4111yescharacteristic
220C8×C31—24no—
221C8×C31—24no—
222C22×#117C2×C3111yescharacteristic
223C22×#107C2×C3111yescharacteristic, complemented
224C22×#106C2×C3111yescharacteristic, complemented
225C22×C22⋊C8C2×C3111yescharacteristic
226C2×#380C2×C3111yescharacteristic
227#1854C2×C3118yesdirect product factor
228C2×#340C2×C3111yescharacteristic, complemented
229C2×#340C2×C3111yescharacteristic, complemented
230C22×C31—24no—
231C22×C31—24no—
232C22×C31—24no—
233C22×C31—42no—
234C22×C31—42no—
235C22×C31—416no—
236C22×C31—42no—
237C22×C31—42no—
238C22×C31#38514yes—
239C22×C31#42714yes—
240C22×C31#42814yes—
241C22×C31#44914yes—
242C22×C31#45214yes—
243C22×C31#48214yes—
244C22×C31#48514yes—
245C22×C31C22×C4⋊C411yescharacteristic
246C22×C31C2×#12011yescharacteristic
247C22×C31C2×#12311yescharacteristic
248C22×C31C2×#13511yescharacteristic
249C22×C31C2×#13511yescharacteristic
250C22×C31C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C31C2×#13711yescharacteristic
252C4×C31—24no—
253C4×C31—44no—
254C4×C31—44no—
255C4×C31—82no—
256C4×C31—82no—
257C4×C31#38514yes—
258C4×C31C2×C4×D411yescharacteristic
259C4×C31C2×C4×D411yescharacteristic
260C4×C31C2×C22⋊Q811yescharacteristic
261C4×C31C2×C22⋊Q811yescharacteristic
262C24×C4C22×C3111yescharacteristic
263C23×C8C22×C3111yescharacteristic
264C23×C8C22×C3111yescharacteristic
265C2×#117—22no—
266C2×#117—24no—
267C2×#117—24no—
268C2×#117C22×C3114yes—
269C22×C4⋊C4C22×C3111yescharacteristic
270C22×C4⋊C4C22×C3111yescharacteristic
271C22×C4⋊C4C22×C3111yescharacteristic
272C22×C4⋊C4C22×C3111yescharacteristic
273C22×C22⋊C4—21no—
274C22×C22⋊C4—21no—
275C22×C42—21no—
276C2×#107—24no—
277C2×#107C22×C3114yescomplemented
278C2×#106—24no—
279C2×#106C22×C3114yescomplemented
280C2×C22⋊C8C4×C3118yescomplemented
281C2×C22⋊C8C22×C3114yes—
282#380—22no—
283#380C22×C3114yes—
284C2×#95—21no—
285#340C22×C3114yescomplemented
286#340C22×C3114yescomplemented
287C2×C31—44no—
288C2×C31#185418yesdirect product factor
289C2×C31C2×#38511yescharacteristic
290C2×C31C2×#42711yescharacteristic
291C2×C31C2×#42811yescharacteristic
292C2×C31C2×#44911yescharacteristic
293C2×C31C2×#45211yescharacteristic
294C2×C31C2×#48211yescharacteristic
295C2×C31C2×#48511yescharacteristic
296C25D4×C3111yescharacteristic
297C23×C4—22no—
298C23×C4—22no—
299C23×C4—22no—
300C23×C4—22no—
301C23×C4—21no—
302C23×C4—21no—
303C23×C4—21no—
304C23×C4—21no—
305C23×C4C2×C4×C3114yes—
306C23×C4D4×C3112yes—
307C23×C4D4×C3111yescharacteristic
308C23×C4Q8×C3114yes—
309C23×C4C23×C3112yes—
310C23×C4C23×C3111yescharacteristic
311C22×C8—22no—
312C22×C8—22no—
313C22×C8—22no—
314C22×C8—22no—
315C22×C8C2×C4×C3114yes—
316C22×C8C2×C4×C3114yes—
317C22×C8C23×C3112yes—
318C22×C8C23×C3112yes—
319#117—216no—
320#117—28no—
321#117—28no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4D4×C3114yes—
333C2×C4⋊C4D4×C3114yes—
334C2×C4⋊C4C23×C3112yes—
335C2×C4⋊C4C23×C3112yes—
336C2×C4⋊C4C23×C3112yes—
337C2×C4⋊C4C23×C3112yes—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—44no—
342C2×C42—28no—
343C2×C42—24no—
344#107—28no—
345#106—28no—
346C22⋊C8C2×C4×C31116yescomplemented
347#95—24no—
348C31C2×#185411yescharacteristic, direct product factor, Sylow
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C4⋊C4×C3114yes—
356C24C2×D4×C3112yes—
357C24C2×D4×C3111yescharacteristic
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4C4⋊C4×C3114yes—
387C22×C4C4⋊C4×C3118yes—
388C22×C4C22×C4×C3114yes—
389C22×C4C22×C4×C3112yes—
390C22×C4C2×D4×C3112yes—
391C22×C4C2×D4×C3111yescharacteristic
392C22×C4C2×D4×C3112yes—
393C22×C4C2×D4×C3112yes—
394C22×C4C2×Q8×C3114yes—
395C22×C4C2×Q8×C3112yes—
396C22×C4#41×C3112yes—
397C22×C4#41×C3112yes—
398C22×C4C24×C3111yescharacteristic
399C2×C8—28no—
400C2×C8—28no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8C22×C4×C3114yes—
406C2×C8C22×C4×C3114yes—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4C2×D4×C3114yes—
418C4⋊C4C2×D4×C3114yes—
419C22⋊C4—216no—
420C22⋊C4—48no—
421C42—216no—
422C23—28no—
423C23—28no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—28no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23C2×C4⋊C4×C3114yes—
436C23C2×C4⋊C4×C3112yes—
437C23C31×#12012yes—
438C23C31×#12312yes—
439C23C31×#13512yes—
440C23C31×#13512yes—
441C23C31×AGL(1,ℤ/8)12yes—
442C23C31×#13712yes—
443C23C22×D4×C3111yescharacteristic
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—216no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4C2×C4⋊C4×C3114yes—
469C2×C4C2×C4⋊C4×C3112yes—
470C2×C4C4×D4×C3114yes—
471C2×C4C4×D4×C3114yes—
472C2×C4C31×#12012yes—
473C2×C4C31×C22⋊Q814yes—
474C2×C4C31×C22⋊Q814yes—
475C2×C4C31×#12312yes—
476C2×C4C23×C4×C3111yescharacteristic
477C2×C4C22×D4×C3111yescharacteristic
478C2×C4C22×D4×C3111yescharacteristic
479C2×C4C22×Q8×C3111yescharacteristic
480C2×C4C2×#41×C3111yescharacteristic
481C2×C4C2×#41×C3111yescharacteristic
482C8—24no—
483C8—24no—
484C22—24no—
485C22—24no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—416no—
490C22—42no—
491C22—42no—
492C22C31×#38514yes—
493C22C31×#42714yes—
494C22C31×#42814yes—
495C22C31×#44914yes—
496C22C31×#45214yes—
497C22C31×#48214yes—
498C22C31×#48514yes—
499C22C22×C4⋊C4×C3111yescharacteristic
500C22C2×C31×#12011yescharacteristic
501C22C2×C31×#12311yescharacteristic
502C22C2×C31×#13511yescharacteristic
503C22C2×C31×#13511yescharacteristic
504C22C2×C31×AGL(1,ℤ/8)11yescharacteristic
505C22C2×C31×#13711yescharacteristic
506C4—24no—
507C4—44no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4C31×#38514yes—
512C4C2×C4×D4×C3111yescharacteristic
513C4C2×C4×D4×C3111yescharacteristic
514C4C2×C31×C22⋊Q811yescharacteristic
515C4C2×C31×C22⋊Q811yescharacteristic
516C2—44no—
517C2C31×#185418yesdirect product factor
518C2C2×C31×#38511yescharacteristic
519C2C2×C31×#42711yescharacteristic
520C2C2×C31×#42811yescharacteristic
521C2C2×C31×#44911yescharacteristic
522C2C2×C31×#45211yescharacteristic
523C2C2×C31×#48211yescharacteristic
524C2C2×C31×#48511yescharacteristic
subgroupG/H
center, socleC24×C31C2×D4
commutatorC2×C4C23×C4×C31
Frattini, cosocleC22×C4C24×C31
prime radicalC25×C31D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24×C31 ◁ C24×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C31×#107C2semidirect
3. C22×C31×#106C2semidirect
6. C31×#1854C2direct (N × H)
7. C2×C31×#340C2semidirect
8. C2×C31×#340C2semidirect
24. C2×C31×#107C22semidirect
26. C2×C31×#106C22semidirect
27. C2×C31×C22⋊C8C4semidirect
32. C31×#340C22semidirect
33. C31×#340C22semidirect
84. C31×C22⋊C8C2×C4semidirect
159. C2×#1854C31direct (N × H)
223. C22×#107C2×C31semidirect
224. C22×#106C2×C31semidirect
227. #1854C2×C31direct (N × H)
228. C2×#340C2×C31semidirect
229. C2×#340C2×C31semidirect
277. C2×#107C22×C31semidirect
279. C2×#106C22×C31semidirect
280. C2×C22⋊C8C4×C31semidirect
285. #340C22×C31semidirect
286. #340C22×C31semidirect
288. C2×C31#1854direct (N × H)
346. C22⋊C8C2×C4×C31semidirect
348. C31C2×#1854direct (N × H)
517. C2C31×#1854direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C31×#1172
C22×C31×#1072
C22×C31×#1062
C22×C31×C22⋊C82
C2×C31×#3802
C31×#18542
C2×C31×#3402
C2×#185431