Finite Group Databasebeta

← C2×C31×#1603 #21474858

Subgroups — C2×C31×#1603 #21474858

Order 7936.

1232 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#539C211yesmaximal, characteristic
2C2×C31×#451C211yesmaximal, characteristic, complemented
3C2×C31×#448C211yesmaximal, characteristic
4C2×C31×#395C211yesmaximal, characteristic, complemented
5C31×#1603C218yesmaximal, direct product factor
6C2×C31×#328C211yesmaximal, characteristic, complemented
7C2×C31×#327C211yesmaximal, characteristic
8C2×C31×#324C211yesmaximal, characteristic, complemented
9C23×Q8×C31C2211yescharacteristic
10C22×Q16×C31—21no—
11C22×SD16×C31—21no—
12C31×#539—22no—
13C31×#539—22no—
14C31×#539—22no—
15C31×#539C2214yes—
16C2×C31×#126—21no—
17C2×C31×#125—21no—
18C2×C31×#123—21no—
19C2×C31×#123—21no—
20C2×C31×C22⋊Q8—21no—
21C2×C31×C22⋊Q8C2211yescharacteristic
22C2×C31×#121C2211yescharacteristic
23C2×C4×D4×C31—21no—
24C22×C4⋊C4×C31—21no—
25C22×C4⋊C4×C31C2211yescharacteristic
26C31×#451—22no—
27C31×#451C2214yescomplemented
28C31×#448—22no—
29C31×#448C2214yes—
30C2×C31×#103—21no—
31C2×C31×#103—21no—
32C2×C31×C22⋊C8C2211yescharacteristic
33C2×C31×C22⋊C8C2211yescharacteristic
34C31×#395—22no—
35C31×#395C2214yescomplemented
36C2×C31×#95—21no—
37C2×C31×#95C2211yescharacteristic
38C31×#328C2214yescomplemented
39C31×#327C2214yes—
40C31×#324C2214yescomplemented
41C22×Q8×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31—21no—
44C22×Q8×C31—21no—
45C22×Q8×C31—21no—
46C22×Q8×C31—44no—
47C22×Q8×C31D411yescharacteristic
48C22×Q8×C31C2312yes—
49C22×D4×C31—21no—
50C22×D4×C31D411yescharacteristic
51C23×C4×C31—21no—
52C23×C4×C31—21no—
53C23×C4×C31—21no—
54C23×C4×C31D411yescharacteristic
55C23×C4×C31D411yescharacteristic
56C23×C4×C31C2311yescharacteristic
57C2×Q16×C31—24no—
58C2×Q16×C31—44no—
59C2×SD16×C31—24no—
60C2×SD16×C31—44no—
61C22×C8×C31—21no—
62C22×C8×C31—21no—
63C31×#126—24no—
64C31×#126—24no—
65C31×#125—24no—
66C31×#125—24no—
67C31×#123—24no—
68C31×#123—24no—
69C31×#123—24no—
70C31×#123—24no—
71C31×C22⋊Q8—24no—
72C31×C22⋊Q8—24no—
73C31×C22⋊Q8—21no—
74C31×C22⋊Q8—21no—
75C31×C22⋊Q8D412yes—
76C31×C22⋊Q8C2312yes—
77C31×#121—21no—
78C31×#121—21no—
79C31×#121D412yes—
80C31×#121C2312yes—
81C4×D4×C31—24no—
82C4×D4×C31—24no—
83C2×C4⋊C4×C31—21no—
84C2×C4⋊C4×C31—21no—
85C2×C4⋊C4×C31—21no—
86C2×C4⋊C4×C31—21no—
87C2×C4⋊C4×C31—21no—
88C2×C4⋊C4×C31—21no—
89C2×C4⋊C4×C31—21no—
90C2×C4⋊C4×C31—21no—
91C2×C4⋊C4×C31—24no—
92C2×C4⋊C4×C31—44no—
93C2×C4⋊C4×C31D412yes—
94C2×C4⋊C4×C31D411yescharacteristic
95C2×C4⋊C4×C31D411yescharacteristic
96C2×C4⋊C4×C31C2312yes—
97C2×C22⋊C4×C31—21no—
98C2×C22⋊C4×C31—21no—
99C2×C22⋊C4×C31—21no—
100C2×C22⋊C4×C31—21no—
101C2×C22⋊C4×C31—21no—
102C2×C42×C31—21no—
103C31×#103—24no—
104C31×#103—24no—
105C31×C22⋊C8—21no—
106C31×C22⋊C8—21no—
107C31×C22⋊C8C2312yes—
108C31×C22⋊C8C2312yes—
109C31×#95—21no—
110C31×#95—24no—
111C31×#95C2312yes—
112C24×C31—21no—
113C24×C31C2×D411yescharacteristic
114C2×Q8×C31—21no—
115C2×Q8×C31—24no—
116C2×Q8×C31—24no—
117C2×Q8×C31—22no—
118C2×Q8×C31—22no—
119C2×Q8×C31—44no—
120C2×Q8×C31—42no—
121C2×Q8×C31—42no—
122C2×Q8×C31—44no—
123C2×Q8×C31—42no—
124C2×Q8×C31—44no—
125C2×Q8×C31—44no—
126C2×Q8×C31—41no—
127C2×Q8×C31—41no—
128C2×Q8×C31C2×D412yes—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—21no—
132C2×D4×C31—42no—
133C2×D4×C31—42no—
134C2×D4×C31—42no—
135C2×D4×C31C2×D412yes—
136C22×C4×C31—24no—
137C22×C4×C31—24no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—21no—
141C22×C4×C31—21no—
142C22×C4×C31—21no—
143C22×C4×C31—21no—
144C22×C4×C31—22no—
145C22×C4×C31—22no—
146C22×C4×C31—22no—
147C22×C4×C31—22no—
148C22×C4×C31—24no—
149C22×C4×C31—21no—
150C22×C4×C31—42no—
151C22×C4×C31—42no—
152C22×C4×C31—42no—
153C22×C4×C31—41no—
154C22×C4×C31—41no—
155C22×C4×C31—42no—
156C22×C4×C31—42no—
157C22×C4×C31—82no—
158C22×C4×C31C2×D412yes—
159C22×C4×C31C2×D411yescharacteristic
160C22×C4×C31C2×D411yescharacteristic
161C22×C4×C31C2×D412yes—
162C22×C4×C31C2×D411yescharacteristic
163C22×C4×C31C2×D411yescharacteristic
164C22×C4×C31C2×D411yescharacteristic
165C22×C4×C31C2411yescharacteristic
166Q16×C31—48no—
167SD16×C31—48no—
168C2×C8×C31—22no—
169C2×C8×C31—22no—
170C2×C8×C31—42no—
171C2×C8×C31—42no—
172C4⋊C4×C31—24no—
173C4⋊C4×C31—24no—
174C4⋊C4×C31—24no—
175C4⋊C4×C31—22no—
176C4⋊C4×C31—22no—
177C4⋊C4×C31—22no—
178C4⋊C4×C31—21no—
179C4⋊C4×C31—21no—
180C4⋊C4×C31—22no—
181C4⋊C4×C31—22no—
182C4⋊C4×C31—22no—
183C4⋊C4×C31—48no—
184C4⋊C4×C31C2×D412yes—
185C4⋊C4×C31C2×D412yes—
186C22⋊C4×C31—24no—
187C22⋊C4×C31—24no—
188C22⋊C4×C31—24no—
189C22⋊C4×C31—22no—
190C22⋊C4×C31—22no—
191C22⋊C4×C31—22no—
192C22⋊C4×C31—22no—
193C42×C31—24no—
194C2×#1603C3111yesmaximal, characteristic, direct product factor, Sylow
195C23×C31—22no—
196C23×C31—22no—
197C23×C31—22no—
198C23×C31—42no—
199C23×C31—42no—
200C23×C31—42no—
201C23×C31#12011yescharacteristic
202C23×C31#12011yescharacteristic
203C23×C31#13512yes—
204C23×C31#13712yes—
205C23×C31C22×D411yescharacteristic
206Q8×C31—24no—
207Q8×C31—24no—
208Q8×C31—42no—
209Q8×C31—42no—
210Q8×C31—42no—
211Q8×C31—44no—
212Q8×C31—44no—
213Q8×C31—81no—
214D4×C31—42no—
215D4×C31—42no—
216D4×C31—42no—
217D4×C31—44no—
218D4×C31—81no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—24no—
222C2×C4×C31—24no—
223C2×C4×C31—22no—
224C2×C4×C31—22no—
225C2×C4×C31—22no—
226C2×C4×C31—22no—
227C2×C4×C31—22no—
228C2×C4×C31—22no—
229C2×C4×C31—22no—
230C2×C4×C31—41no—
231C2×C4×C31—41no—
232C2×C4×C31—41no—
233C2×C4×C31—41no—
234C2×C4×C31—42no—
235C2×C4×C31—48no—
236C2×C4×C31—42no—
237C2×C4×C31—44no—
238C2×C4×C31—42no—
239C2×C4×C31—42no—
240C2×C4×C31—42no—
241C2×C4×C31—42no—
242C2×C4×C31—42no—
243C2×C4×C31—42no—
244C2×C4×C31—44no—
245C2×C4×C31—42no—
246C2×C4×C31—42no—
247C2×C4×C31—42no—
248C2×C4×C31—81no—
249C2×C4×C31—84no—
250C2×C4×C31#12011yescharacteristic
251C2×C4×C31#12011yescharacteristic
252C2×C4×C31C22×D411yescharacteristic
253C2×C4×C31C22×D411yescharacteristic
254C8×C31—42no—
255C8×C31—42no—
256C2×#539C2×C3111yescharacteristic
257C2×#451C2×C3111yescharacteristic, complemented
258C2×#448C2×C3111yescharacteristic
259C2×#395C2×C3111yescharacteristic, complemented
260#1603C2×C3118yesdirect product factor
261C2×#328C2×C3111yescharacteristic, complemented
262C2×#327C2×C3111yescharacteristic
263C2×#324C2×C3111yescharacteristic, complemented
264C22×C31—24no—
265C22×C31—24no—
266C22×C31—21no—
267C22×C31—21no—
268C22×C31—21no—
269C22×C31—21no—
270C22×C31—42no—
271C22×C31—42no—
272C22×C31—42no—
273C22×C31—84no—
274C22×C31#45212yes—
275C22×C31#45212yes—
276C22×C31#45612yes—
277C22×C31B(4,𝔽2)12yes—
278C22×C31C2×#12011yescharacteristic
279C22×C31C2×#13511yescharacteristic
280C22×C31C2×#13711yescharacteristic
281C4×C31—21no—
282C4×C31—21no—
283C4×C31—21no—
284C4×C31—21no—
285C4×C31—44no—
286C4×C31—42no—
287C4×C31—42no—
288C4×C31—42no—
289C4×C31—44no—
290C4×C31—42no—
291C4×C31—42no—
292C23×Q8C22×C3111yescharacteristic
293C22×Q16—21no—
294C22×SD16—21no—
295#539—22no—
296#539—22no—
297#539—22no—
298#539C22×C3114yes—
299C2×#126—21no—
300C2×#125—21no—
301C2×#123—21no—
302C2×#123—21no—
303C2×C22⋊Q8—21no—
304C2×C22⋊Q8C22×C3111yescharacteristic
305C2×#121C22×C3111yescharacteristic
306C2×C4×D4—21no—
307C22×C4⋊C4—21no—
308C22×C4⋊C4C22×C3111yescharacteristic
309#451—22no—
310#451C22×C3114yescomplemented
311#448—22no—
312#448C22×C3114yes—
313C2×#103—21no—
314C2×#103—21no—
315C2×C22⋊C8C22×C3111yescharacteristic
316C2×C22⋊C8C22×C3111yescharacteristic
317#395—22no—
318#395C22×C3114yescomplemented
319C2×#95—21no—
320C2×#95C22×C3111yescharacteristic
321#328C22×C3114yescomplemented
322#327C22×C3114yes—
323#324C22×C3114yescomplemented
324C2×C31—21no—
325C2×C31—21no—
326C2×C31—22no—
327C2×C31—82no—
328C2×C31#160314yesdirect product factor
329C2×C31C2×#45211yescharacteristic
330C2×C31C2×#45611yescharacteristic
331C2×C31C2×B(4,𝔽2)11yescharacteristic
332C22×Q8—21no—
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8—44no—
338C22×Q8D4×C3111yescharacteristic
339C22×Q8C23×C3112yes—
340C22×D4—21no—
341C22×D4D4×C3111yescharacteristic
342C23×C4—21no—
343C23×C4—21no—
344C23×C4—21no—
345C23×C4D4×C3111yescharacteristic
346C23×C4D4×C3111yescharacteristic
347C23×C4C23×C3111yescharacteristic
348C2×Q16—24no—
349C2×Q16—44no—
350C2×SD16—24no—
351C2×SD16—44no—
352C22×C8—21no—
353C22×C8—21no—
354#126—24no—
355#126—24no—
356#125—24no—
357#125—24no—
358#123—24no—
359#123—24no—
360#123—24no—
361#123—24no—
362C22⋊Q8—24no—
363C22⋊Q8—24no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8D4×C3112yes—
367C22⋊Q8C23×C3112yes—
368#121—21no—
369#121—21no—
370#121D4×C3112yes—
371#121C23×C3112yes—
372C4×D4—24no—
373C4×D4—24no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—44no—
384C2×C4⋊C4D4×C3112yes—
385C2×C4⋊C4D4×C3111yescharacteristic
386C2×C4⋊C4D4×C3111yescharacteristic
387C2×C4⋊C4C23×C3112yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394#103—24no—
395#103—24no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8C23×C3112yes—
399C22⋊C8C23×C3112yes—
400#95—24no—
401#95—21no—
402#95C23×C3112yes—
403C31C2×#160311yescharacteristic, direct product factor, Sylow
404C24—21no—
405C24C2×D4×C3111yescharacteristic
406C2×Q8—21no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—44no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8C2×D4×C3112yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×D4×C3112yes—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4C2×D4×C3112yes—
451C22×C4C2×D4×C3111yescharacteristic
452C22×C4C2×D4×C3111yescharacteristic
453C22×C4C2×D4×C3112yes—
454C22×C4C2×D4×C3111yescharacteristic
455C22×C4C2×D4×C3111yescharacteristic
456C22×C4C2×D4×C3111yescharacteristic
457C22×C4C24×C3111yescharacteristic
458Q16—48no—
459SD16—48no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—42no—
463C2×C8—42no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—48no—
476C4⋊C4C2×D4×C3112yes—
477C4⋊C4C2×D4×C3112yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C42—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C31×#12011yescharacteristic
493C23C31×#12011yescharacteristic
494C23C31×#13512yes—
495C23C31×#13712yes—
496C23C22×D4×C3111yescharacteristic
497Q8—24no—
498Q8—24no—
499Q8—42no—
500Q8—42no—
501Q8—42no—
502Q8—44no—
503Q8—44no—
504Q8—81no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—81no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—81no—
540C2×C4—84no—
541C2×C4C31×#12011yescharacteristic
542C2×C4C31×#12011yescharacteristic
543C2×C4C22×D4×C3111yescharacteristic
544C2×C4C22×D4×C3111yescharacteristic
545C8—42no—
546C8—42no—
547C22—24no—
548C22—24no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22C31×#45212yes—
558C22C31×#45212yes—
559C22C31×#45612yes—
560C22C31×B(4,𝔽2)12yes—
561C22C2×C31×#12011yescharacteristic
562C22C2×C31×#13511yescharacteristic
563C22C2×C31×#13711yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—44no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—42no—
575C2—21no—
576C2—22no—
577C2—21no—
578C2—82no—
579C2C31×#160314yesdirect product factor
580C2C2×C31×#45211yescharacteristic
581C2C2×C31×#45611yescharacteristic
582C2C2×C31×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23×C31#120
commutator, Frattini, cosocleC22×C4C24×C31
prime radicalC2×C31×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C31 ◁ C23×C4×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C31×#451C2semidirect
4. C2×C31×#395C2semidirect
5. C31×#1603C2direct (N × H)
6. C2×C31×#328C2semidirect
8. C2×C31×#324C2semidirect
27. C31×#451C22semidirect
35. C31×#395C22semidirect
38. C31×#328C22semidirect
40. C31×#324C22semidirect
194. C2×#1603C31direct (N × H)
257. C2×#451C2×C31semidirect
259. C2×#395C2×C31semidirect
260. #1603C2×C31direct (N × H)
261. C2×#328C2×C31semidirect
263. C2×#324C2×C31semidirect
310. #451C22×C31semidirect
318. #395C22×C31semidirect
321. #328C22×C31semidirect
323. #324C22×C31semidirect
328. C2×C31#1603direct (N × H)
403. C31C2×#1603direct (N × H)
579. C2C31×#1603direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5392
C2×C31×#4512
C2×C31×#4482
C2×C31×#3952
C31×#16032
C2×C31×#3282
C2×C31×#3272
C2×C31×#3242
C2×#160331