Finite Group Databasebeta

← C2×C31×#2123 #21490105

Subgroups — C2×C31×#2123 #21490105

Order 7936.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C31×#536C211yesmaximal, characteristic
2C2×C31×#413C211yesmaximal, characteristic
3C2×C31×#410C211yesmaximal, characteristic
4C31×#2123C218yesmaximal, direct product factor
5C2×C31×#352C214yesmaximal, complemented
6C22×#41×C31C2211yescharacteristic
7C23×Q8×C31C411yescharacteristic, complemented
8C22×M16×C31—21no—
9C31×#536—22no—
10C31×#536—22no—
11C31×#536C414yescomplemented
12C31×#536C2214yes—
13C2×C31×C4⋊Q8—21no—
14C2×C31×#124—21no—
15C2×C31×#123C412yescomplemented
16C2×C31×#123C2212yes—
17C2×C31×C22⋊Q8—21no—
18C2×C31×C22⋊Q8—21no—
19C2×C31×#117—21no—
20C2×C31×#117C411yescharacteristic, complemented
21C31×#413—22no—
22C31×#413C2214yes—
23C2×C31×#101—21no—
24C2×C31×#100C2212yes—
25C31×#410—22no—
26C31×#410C2214yes—
27C2×C31×#99—21no—
28C2×C31×#99C2212yes—
29C31×#352C22116yescomplemented
30C2×#41×C31—41no—
31C2×#41×C31—41no—
32C2×#41×C31—42no—
33C2×#41×C31C2×C412yes—
34C2×#41×C31D412yes—
35C2×#41×C31D412yes—
36C2×#41×C31C2312yes—
37C22×Q8×C31—21no—
38C22×Q8×C31—22no—
39C22×Q8×C31—21no—
40C22×Q8×C31—42no—
41C22×Q8×C31—41no—
42C22×Q8×C31—41no—
43C22×Q8×C31C2×C414yescomplemented
44C22×Q8×C31D411yescharacteristic
45C22×D4×C31C2×C411yescharacteristic
46C22×D4×C31D411yescharacteristic
47C22×D4×C31C2311yescharacteristic
48C23×C4×C31C2×C411yescharacteristic
49C23×C4×C31C2×C412yes—
50C23×C4×C31D411yescharacteristic
51C23×C4×C31D411yescharacteristic
52C2×M16×C31—22no—
53C2×M16×C31—24no—
54C2×M16×C31—41no—
55C2×M16×C31—42no—
56C22×C8×C31—41no—
57C31×C4⋊Q8—24no—
58C31×C4⋊Q8—24no—
59C31×#124—24no—
60C31×#124—24no—
61C31×#123—24no—
62C31×#123—22no—
63C31×#123—22no—
64C31×#123C2×C414yes—
65C31×#123C2×C418yescomplemented
66C31×#123C2314yes—
67C31×C22⋊Q8—24no—
68C31×C22⋊Q8—24no—
69C31×C22⋊Q8—24no—
70C31×C22⋊Q8—24no—
71C31×#117—22no—
72C31×#117—24no—
73C31×#117—42no—
74C31×#117C2×C414yescomplemented
75C2×C4⋊C4×C31—22no—
76C2×C4⋊C4×C31—21no—
77C2×C4⋊C4×C31—22no—
78C2×C4⋊C4×C31—41no—
79C2×C22⋊C4×C31—22no—
80C2×C22⋊C4×C31—22no—
81C2×C22⋊C4×C31—22no—
82C2×C22⋊C4×C31C2×C412yes—
83C2×C42×C31—21no—
84C2×C42×C31—41no—
85C31×#101—24no—
86C31×#100—22no—
87C31×#100C2314yes—
88C31×#99—24no—
89C31×#99—22no—
90C31×#99C2314yes—
91C24×C31C22⋊C411yescharacteristic
92C24×C31C22×C411yescharacteristic
93C24×C31C2×D411yescharacteristic
94#41×C31—42no—
95#41×C31—42no—
96#41×C31—42no—
97#41×C31—42no—
98#41×C31—42no—
99#41×C31—42no—
100#41×C31—42no—
101#41×C31—42no—
102C2×Q8×C31—24no—
103C2×Q8×C31—24no—
104C2×Q8×C31—41no—
105C2×Q8×C31—41no—
106C2×Q8×C31—42no—
107C2×Q8×C31—42no—
108C2×Q8×C31—44no—
109C2×Q8×C31—44no—
110C2×Q8×C31—44no—
111C2×Q8×C31—44no—
112C2×Q8×C31—42no—
113C2×Q8×C31—41no—
114C2×Q8×C31—41no—
115C2×Q8×C31C22⋊C412yes—
116C2×Q8×C31C2×D412yes—
117C2×D4×C31—41no—
118C2×D4×C31—41no—
119C2×D4×C31—41no—
120C2×D4×C31—41no—
121C2×D4×C31—41no—
122C2×D4×C31—41no—
123C2×D4×C31C22⋊C412yes—
124C2×D4×C31C22⋊C412yes—
125C2×D4×C31C22×C411yescharacteristic
126C2×D4×C31C22×C412yes—
127C2×D4×C31C2×D411yescharacteristic
128C2×D4×C31C2×D412yes—
129C2×D4×C31C2×D411yescharacteristic
130C2×D4×C31C2411yescharacteristic
131C22×C4×C31—21no—
132C22×C4×C31—21no—
133C22×C4×C31—22no—
134C22×C4×C31—21no—
135C22×C4×C31—22no—
136C22×C4×C31—21no—
137C22×C4×C31—22no—
138C22×C4×C31—21no—
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—22no—
143C22×C4×C31—42no—
144C22×C4×C31—42no—
145C22×C4×C31—42no—
146C22×C4×C31C22⋊C411yescharacteristic
147C22×C4×C31C22⋊C411yescharacteristic
148C22×C4×C31C22⋊C412yes—
149C22×C4×C31C22⋊C412yes—
150C22×C4×C31C22⋊C412yes—
151C22×C4×C31C22⋊C411yescharacteristic
152C22×C4×C31C22×C412yes—
153C22×C4×C31C22×C414yes—
154C22×C4×C31C2×D412yes—
155C22×C4×C31C2×D411yescharacteristic
156C22×C4×C31C2×D412yes—
157M16×C31—24no—
158M16×C31—42no—
159M16×C31—44no—
160C2×C8×C31—42no—
161C2×C8×C31—44no—
162C4⋊C4×C31—24no—
163C4⋊C4×C31—24no—
164C4⋊C4×C31—28no—
165C4⋊C4×C31—24no—
166C4⋊C4×C31—44no—
167C22⋊C4×C31—24no—
168C22⋊C4×C31—28no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—24no—
171C22⋊C4×C31—24no—
172C22⋊C4×C31—42no—
173C22⋊C4×C31C22×C414yes—
174C42×C31—24no—
175C42×C31—44no—
176C2×#2123C3111yesmaximal, characteristic, direct product factor, Sylow
177C23×C31—21no—
178C23×C31—21no—
179C23×C31—21no—
180C23×C31—21no—
181C23×C31—21no—
182C23×C31—21no—
183C23×C31—42no—
184C23×C31—42no—
185C23×C31—42no—
186C23×C31C2×C22⋊C411yescharacteristic
187C23×C31C2×C22⋊C411yescharacteristic
188C23×C31C2×C22⋊C412yes—
189C23×C31C2×C22⋊C411yescharacteristic
190C23×C31C23×C411yescharacteristic
191C23×C31C22×D411yescharacteristic
192Q8×C31—42no—
193Q8×C31—42no—
194Q8×C31—48no—
195Q8×C31—44no—
196Q8×C31—44no—
197D4×C31—42no—
198D4×C31—42no—
199D4×C31—42no—
200D4×C31—41no—
201D4×C31—41no—
202D4×C31—41no—
203D4×C31—42no—
204D4×C31—41no—
205C2×C4×C31—21no—
206C2×C4×C31—21no—
207C2×C4×C31—21no—
208C2×C4×C31—21no—
209C2×C4×C31—21no—
210C2×C4×C31—21no—
211C2×C4×C31—22no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—22no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—22no—
218C2×C4×C31—22no—
219C2×C4×C31—22no—
220C2×C4×C31—22no—
221C2×C4×C31—22no—
222C2×C4×C31—22no—
223C2×C4×C31—24no—
224C2×C4×C31—24no—
225C2×C4×C31—44no—
226C2×C4×C31—44no—
227C2×C4×C31—42no—
228C2×C4×C31—42no—
229C2×C4×C31—44no—
230C2×C4×C31—44no—
231C2×C4×C31—84no—
232C2×C4×C31#9912yes—
233C2×C4×C31#9911yescharacteristic
234C2×C4×C31#9911yescharacteristic
235C2×C4×C31C2×C22⋊C412yes—
236C2×C4×C31C2×C22⋊C412yes—
237C2×C4×C31C2×C22⋊C411yescharacteristic
238C2×C4×C31C2×C22⋊C412yes—
239C2×C4×C31C22×D411yescharacteristic
240C8×C31—44no—
241C2×#536C2×C3111yescharacteristic
242C2×#413C2×C3111yescharacteristic
243C2×#410C2×C3111yescharacteristic
244#2123C2×C3118yesdirect product factor
245C2×#352C2×C3114yescomplemented
246C22×C31—22no—
247C22×C31—21no—
248C22×C31—21no—
249C22×C31—21no—
250C22×C31—22no—
251C22×C31—21no—
252C22×C31—41no—
253C22×C31—42no—
254C22×C31—41no—
255C22×C31—41no—
256C22×C31—41no—
257C22×C31—42no—
258C22×C31—42no—
259C22×C31—42no—
260C22×C31—42no—
261C22×C31—81no—
262C22×C31—81no—
263C22×C31C2×#9911yescharacteristic
264C22×C31C2×#9911yescharacteristic
265C22×C31C22×C22⋊C411yescharacteristic
266C4×C31—21no—
267C4×C31—21no—
268C4×C31—22no—
269C4×C31—22no—
270C4×C31—44no—
271C4×C31—44no—
272C4×C31—84no—
273C4×C31C2×#9912yes—
274C4×C31C2×#9912yes—
275C22×#41C22×C3111yescharacteristic
276C23×Q8C4×C3111yescharacteristic, complemented
277C22×M16—21no—
278#536—22no—
279#536—22no—
280#536C4×C3114yescomplemented
281#536C22×C3114yes—
282C2×C4⋊Q8—21no—
283C2×#124—21no—
284C2×#123C4×C3112yescomplemented
285C2×#123C22×C3112yes—
286C2×C22⋊Q8—21no—
287C2×C22⋊Q8—21no—
288C2×#117—21no—
289C2×#117C4×C3111yescharacteristic, complemented
290#413—22no—
291#413C22×C3114yes—
292C2×#101—21no—
293C2×#100C22×C3112yes—
294#410—22no—
295#410C22×C3114yes—
296C2×#99—21no—
297C2×#99C22×C3112yes—
298#352C22×C31116yescomplemented
299C2×C31—21no—
300C2×C31—21no—
301C2×C31—41no—
302C2×C31—41no—
303C2×C31—41no—
304C2×C31—41no—
305C2×C31—42no—
306C2×C31#212312yesdirect product factor
307C2×C31C22×#9911yescharacteristic
308C2×#41—41no—
309C2×#41—41no—
310C2×#41—42no—
311C2×#41C2×C4×C3112yes—
312C2×#41D4×C3112yes—
313C2×#41D4×C3112yes—
314C2×#41C23×C3112yes—
315C22×Q8—21no—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8—42no—
319C22×Q8—41no—
320C22×Q8—41no—
321C22×Q8C2×C4×C3114yescomplemented
322C22×Q8D4×C3111yescharacteristic
323C22×D4C2×C4×C3111yescharacteristic
324C22×D4D4×C3111yescharacteristic
325C22×D4C23×C3111yescharacteristic
326C23×C4C2×C4×C3111yescharacteristic
327C23×C4C2×C4×C3112yes—
328C23×C4D4×C3111yescharacteristic
329C23×C4D4×C3111yescharacteristic
330C2×M16—22no—
331C2×M16—24no—
332C2×M16—41no—
333C2×M16—42no—
334C22×C8—41no—
335C4⋊Q8—24no—
336C4⋊Q8—24no—
337#124—24no—
338#124—24no—
339#123—24no—
340#123—22no—
341#123—22no—
342#123C2×C4×C3114yes—
343#123C2×C4×C3118yescomplemented
344#123C23×C3114yes—
345C22⋊Q8—24no—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349#117—22no—
350#117—24no—
351#117—42no—
352#117C2×C4×C3114yescomplemented
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—41no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4C2×C4×C3112yes—
361C2×C42—21no—
362C2×C42—41no—
363#101—24no—
364#100—22no—
365#100C23×C3114yes—
366#99—24no—
367#99—22no—
368#99C23×C3114yes—
369C31C2×#212311yescharacteristic, direct product factor, Sylow
370C24C22⋊C4×C3111yescharacteristic
371C24C22×C4×C3111yescharacteristic
372C24C2×D4×C3111yescharacteristic
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8C22⋊C4×C3112yes—
395C2×Q8C2×D4×C3112yes—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4C22⋊C4×C3112yes—
403C2×D4C22⋊C4×C3112yes—
404C2×D4C22×C4×C3111yescharacteristic
405C2×D4C22×C4×C3112yes—
406C2×D4C2×D4×C3111yescharacteristic
407C2×D4C2×D4×C3112yes—
408C2×D4C2×D4×C3111yescharacteristic
409C2×D4C24×C3111yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4C22⋊C4×C3111yescharacteristic
426C22×C4C22⋊C4×C3111yescharacteristic
427C22×C4C22⋊C4×C3112yes—
428C22×C4C22⋊C4×C3112yes—
429C22×C4C22⋊C4×C3112yes—
430C22×C4C22⋊C4×C3111yescharacteristic
431C22×C4C22×C4×C3112yes—
432C22×C4C22×C4×C3114yes—
433C22×C4C2×D4×C3112yes—
434C22×C4C2×D4×C3111yescharacteristic
435C22×C4C2×D4×C3112yes—
436M16—24no—
437M16—42no—
438M16—44no—
439C2×C8—42no—
440C2×C8—44no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—24no—
445C4⋊C4—44no—
446C22⋊C4—24no—
447C22⋊C4—28no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4C22×C4×C3114yes—
453C42—24no—
454C42—44no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×C22⋊C4×C3111yescharacteristic
465C23C2×C22⋊C4×C3111yescharacteristic
466C23C2×C22⋊C4×C3112yes—
467C23C2×C22⋊C4×C3111yescharacteristic
468C23C23×C4×C3111yescharacteristic
469C23C22×D4×C3111yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—48no—
473Q8—44no—
474Q8—44no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—42no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—84no—
510C2×C4C31×#9912yes—
511C2×C4C31×#9911yescharacteristic
512C2×C4C31×#9911yescharacteristic
513C2×C4C2×C22⋊C4×C3112yes—
514C2×C4C2×C22⋊C4×C3112yes—
515C2×C4C2×C22⋊C4×C3111yescharacteristic
516C2×C4C2×C22⋊C4×C3112yes—
517C2×C4C22×D4×C3111yescharacteristic
518C8—44no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22C2×C31×#9911yescharacteristic
537C22C2×C31×#9911yescharacteristic
538C22C22×C22⋊C4×C3111yescharacteristic
539C4—21no—
540C4—22no—
541C4—21no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—84no—
546C4C2×C31×#9912yes—
547C4C2×C31×#9912yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2C31×#212312yesdirect product factor
556C2C22×C31×#9911yescharacteristic
subgroupG/H
center, socleC22×C31C2×#99
commutatorC23C23×C4×C31
Frattini, cosocleC2×D4C24×C31
prime radicalC22×#41×C31C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C22×C4×C31 ◁ C22×#41×C31 ◁ G

Complements

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

NH ≅ G/Ntype
4. C31×#2123C2direct (N × H)
5. C2×C31×#352C2semidirect
7. C23×Q8×C31C4semidirect
11. C31×#536C4semidirect
15. C2×C31×#123C4semidirect
20. C2×C31×#117C4semidirect
29. C31×#352C22semidirect
43. C22×Q8×C31C2×C4semidirect
65. C31×#123C2×C4semidirect
74. C31×#117C2×C4semidirect
176. C2×#2123C31direct (N × H)
244. #2123C2×C31direct (N × H)
245. C2×#352C2×C31semidirect
276. C23×Q8C4×C31semidirect
280. #536C4×C31semidirect
284. C2×#123C4×C31semidirect
289. C2×#117C4×C31semidirect
298. #352C22×C31semidirect
306. C2×C31#2123direct (N × H)
321. C22×Q8C2×C4×C31semidirect
343. #123C2×C4×C31semidirect
352. #117C2×C4×C31semidirect
369. C31C2×#2123direct (N × H)
555. C2C31×#2123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5362
C2×C31×#4132
C2×C31×#4102
C31×#21232
C2×C31×#3522
C2×#212331