Finite Group Databasebeta

← C2×C71×#1390 #31833444

Subgroups — C2×C71×#1390 #31833444

Order 18176.

1202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#100×C71C211yesmaximal, characteristic
2C22×#99×C71C211yesmaximal, characteristic
3C2×#386×C71C211yesmaximal, characteristic
4C71×#1390C214yesmaximal, direct product factor
5C23×D4×C71C2211yescharacteristic
6C24×C4×C71C411yescharacteristic, complemented
7C22×M16×C71—21no—
8C22×C22⋊C4×C71—21no—
9C22×C22⋊C4×C71—21no—
10C22×C22⋊C4×C71C411yescharacteristic, complemented
11C2×#100×C71—21no—
12C2×#100×C71—22no—
13C2×#100×C71C414yescomplemented
14C2×#100×C71C2212yes—
15C2×#99×C71—21no—
16C2×#99×C71—22no—
17C2×#99×C71C414yescomplemented
18C2×#99×C71C2212yes—
19#386×C71—21no—
20#386×C71—21no—
21#386×C71C412yescomplemented
22#386×C71C2212yes—
23C2×#95×C71—21no—
24C25×C71C2×C411yescharacteristic
25C25×C71D411yescharacteristic
26C22×D4×C71—42no—
27C22×D4×C71—41no—
28C22×D4×C71—41no—
29C22×D4×C71C2×C411yescharacteristic
30C22×D4×C71C2×C412yes—
31C22×D4×C71C2×C412yes—
32C22×D4×C71D411yescharacteristic
33C22×D4×C71D412yes—
34C22×D4×C71D411yescharacteristic
35C22×D4×C71Q812yes—
36C22×D4×C71C2311yescharacteristic
37C23×C4×C71—21no—
38C23×C4×C71—21no—
39C23×C4×C71—21no—
40C23×C4×C71—44no—
41C23×C4×C71—41no—
42C23×C4×C71—41no—
43C23×C4×C71—41no—
44C23×C4×C71C2×C412yescomplemented
45C23×C4×C71D411yescharacteristic
46C2×M16×C71—24no—
47C2×M16×C71—22no—
48C2×M16×C71—42no—
49C2×M16×C71—41no—
50C22×C8×C71—41no—
51C2×C22⋊C4×C71—24no—
52C2×C22⋊C4×C71—22no—
53C2×C22⋊C4×C71—22no—
54C2×C22⋊C4×C71—21no—
55C2×C22⋊C4×C71—22no—
56C2×C22⋊C4×C71—44no—
57C2×C22⋊C4×C71—41no—
58C2×C22⋊C4×C71—42no—
59C2×C22⋊C4×C71—42no—
60C2×C22⋊C4×C71C2×C412yescomplemented
61#100×C71—24no—
62#100×C71—22no—
63#100×C71C2×C414yescomplemented
64#99×C71—24no—
65#99×C71—22no—
66#99×C71C2×C414yescomplemented
67#95×C71—22no—
68#95×C71—22no—
69C24×C71—22no—
70C24×C71—21no—
71C24×C71—21no—
72C24×C71—21no—
73C24×C71—22no—
74C24×C71—21no—
75C24×C71—42no—
76C24×C71—42no—
77C24×C71—42no—
78C24×C71—42no—
79C24×C71C4212yes—
80C24×C71C22⋊C412yes—
81C24×C71C22⋊C411yescharacteristic
82C24×C71C22⋊C411yescharacteristic
83C24×C71C22⋊C411yescharacteristic
84C24×C71C4⋊C412yes—
85C24×C71C4⋊C412yes—
86C24×C71C22×C411yescharacteristic
87C24×C71C2×D411yescharacteristic
88C2×D4×C71—42no—
89C2×D4×C71—42no—
90C2×D4×C71—42no—
91C2×D4×C71—42no—
92C2×D4×C71—41no—
93C2×D4×C71—41no—
94C2×D4×C71—42no—
95C2×D4×C71—42no—
96C2×D4×C71—41no—
97C2×D4×C71—41no—
98C2×D4×C71—41no—
99C2×D4×C71—41no—
100C2×D4×C71—42no—
101C2×D4×C71—42no—
102C2×D4×C71—41no—
103C2×D4×C71—41no—
104C2×D4×C71C4212yes—
105C2×D4×C71C22⋊C412yes—
106C2×D4×C71C22⋊C412yes—
107C2×D4×C71C4⋊C412yes—
108C2×D4×C71C4⋊C412yes—
109C2×D4×C71C4⋊C412yes—
110C2×D4×C71C22×C411yescharacteristic
111C2×D4×C71C22×C411yescharacteristic
112C2×D4×C71C2×D411yescharacteristic
113C2×D4×C71C2×Q811yescharacteristic
114C22×C4×C71—22no—
115C22×C4×C71—21no—
116C22×C4×C71—21no—
117C22×C4×C71—24no—
118C22×C4×C71—22no—
119C22×C4×C71—22no—
120C22×C4×C71—44no—
121C22×C4×C71—44no—
122C22×C4×C71—42no—
123C22×C4×C71—42no—
124C22×C4×C71—48no—
125C22×C4×C71—44no—
126C22×C4×C71—41no—
127C22×C4×C71—41no—
128C22×C4×C71—42no—
129C22×C4×C71—44no—
130C22×C4×C71—44no—
131C22×C4×C71—42no—
132C22×C4×C71—42no—
133C22×C4×C71—42no—
134C22×C4×C71—42no—
135C22×C4×C71—82no—
136C22×C4×C71—82no—
137C22×C4×C71C22⋊C411yescharacteristic
138C22×C4×C71C22⋊C412yes—
139C22×C4×C71C4⋊C412yes—
140C22×C4×C71C2×D411yescharacteristic
141M16×C71—24no—
142M16×C71—44no—
143M16×C71—42no—
144C2×C8×C71—44no—
145C2×C8×C71—42no—
146C22⋊C4×C71—24no—
147C22⋊C4×C71—44no—
148C22⋊C4×C71—44no—
149C22⋊C4×C71—44no—
150C22⋊C4×C71—42no—
151C22⋊C4×C71—42no—
152C22⋊C4×C71—42no—
153C23×C71—22no—
154C23×C71—22no—
155C23×C71—22no—
156C23×C71—22no—
157C23×C71—22no—
158C23×C71—22no—
159C23×C71—22no—
160C23×C71—22no—
161C23×C71—21no—
162C23×C71—21no—
163C23×C71—21no—
164C23×C71—21no—
165C23×C71—21no—
166C23×C71—21no—
167C23×C71—22no—
168C23×C71—22no—
169C23×C71—21no—
170C23×C71—21no—
171C23×C71—41no—
172C23×C71—41no—
173C23×C71—44no—
174C23×C71—41no—
175C23×C71—44no—
176C23×C71—41no—
177C23×C71—42no—
178C23×C71—42no—
179C23×C71—48no—
180C23×C71—41no—
181C23×C71—41no—
182C23×C71—41no—
183C23×C71—44no—
184C23×C71—44no—
185C23×C71—41no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71—81no—
191C23×C71—81no—
192C23×C71—84no—
193C23×C71#9512yes—
194C23×C71#9512yes—
195C23×C71#9512yes—
196C23×C71#9911yescharacteristic
197C23×C71#9912yes—
198C23×C71#9911yescharacteristic
199C23×C71C2×C4211yescharacteristic
200C23×C71C2×C22⋊C411yescharacteristic
201C23×C71C2×C22⋊C411yescharacteristic
202C23×C71C2×C4⋊C411yescharacteristic
203C23×C71C2×C4⋊C411yescharacteristic
204D4×C71—42no—
205D4×C71—42no—
206D4×C71—42no—
207D4×C71—42no—
208D4×C71—42no—
209D4×C71—42no—
210D4×C71—41no—
211D4×C71—41no—
212D4×C71—41no—
213D4×C71—41no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—24no—
223C2×C4×C71—42no—
224C2×C4×C71—44no—
225C2×C4×C71—48no—
226C2×C4×C71—44no—
227C2×C4×C71—44no—
228C2×C4×C71—42no—
229C2×C4×C71—44no—
230C2×C4×C71—44no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—84no—
235C2×C4×C71—82no—
236C2×C4×C71—82no—
237C2×C4×C71—88no—
238C2×C4×C71#9512yes—
239C2×C4×C71C2×C22⋊C411yescharacteristic
240C2×C4×C71C2×C4⋊C411yescharacteristic
241C8×C71—44no—
242C22×C71—21no—
243C22×C71—21no—
244C22×C71—21no—
245C22×C71—21no—
246C22×C71—24no—
247C22×C71—24no—
248C22×C71—21no—
249C22×C71—22no—
250C22×C71—22no—
251C22×C71—21no—
252C22×C71—21no—
253C22×C71—21no—
254C22×C71—42no—
255C22×C71—42no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—42no—
261C22×C71—42no—
262C22×C71—44no—
263C22×C71—44no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—41no—
271C22×C71—41no—
272C22×C71—44no—
273C22×C71—44no—
274C22×C71—44no—
275C22×C71—44no—
276C22×C71—81no—
277C22×C71—81no—
278C22×C71—82no—
279C22×C71—82no—
280C22×C71#34212yes—
281C22×C71#34212yes—
282C22×C71AΣL(1,𝔽16)12yes—
283C22×C71#35212yes—
284C22×C71C2×#9511yescharacteristic
285C22×C71C2×#9911yescharacteristic
286C22×C71C2×#9911yescharacteristic
287C4×C71—21no—
288C4×C71—22no—
289C4×C71—21no—
290C4×C71—44no—
291C4×C71—82no—
292C4×C71—84no—
293C2×#1390C7111yesmaximal, characteristic, direct product factor, Sylow
294C2×C71—21no—
295C2×C71—21no—
296C2×C71—22no—
297C2×C71—41no—
298C2×C71—42no—
299C2×C71—41no—
300C2×C71—41no—
301C2×C71—42no—
302C2×C71—41no—
303C2×C71#139014yesdirect product factor
304C2×C71C2×#34211yescharacteristic
305C2×C71C2×AΣL(1,𝔽16)11yescharacteristic
306C2×C71C2×#35211yescharacteristic
307C22×#100C2×C7111yescharacteristic
308C22×#99C2×C7111yescharacteristic
309C2×#386C2×C7111yescharacteristic
310#1390C2×C7114yesdirect product factor
311C71C2×#139011yescharacteristic, direct product factor, Sylow
312C23×D4C22×C7111yescharacteristic
313C24×C4C4×C7111yescharacteristic, complemented
314C22×M16—21no—
315C22×C22⋊C4—21no—
316C22×C22⋊C4—21no—
317C22×C22⋊C4C4×C7111yescharacteristic, complemented
318C2×#100—21no—
319C2×#100—22no—
320C2×#100C4×C7114yescomplemented
321C2×#100C22×C7112yes—
322C2×#99—21no—
323C2×#99—22no—
324C2×#99C4×C7114yescomplemented
325C2×#99C22×C7112yes—
326#386—21no—
327#386—21no—
328#386C4×C7112yescomplemented
329#386C22×C7112yes—
330C2×#95—21no—
331C25C2×C4×C7111yescharacteristic
332C25D4×C7111yescharacteristic
333C22×D4—41no—
334C22×D4—42no—
335C22×D4—41no—
336C22×D4C2×C4×C7111yescharacteristic
337C22×D4C2×C4×C7112yes—
338C22×D4C2×C4×C7112yes—
339C22×D4D4×C7111yescharacteristic
340C22×D4D4×C7112yes—
341C22×D4D4×C7111yescharacteristic
342C22×D4Q8×C7112yes—
343C22×D4C23×C7111yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—44no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4C2×C4×C7112yescomplemented
352C23×C4D4×C7111yescharacteristic
353C2×M16—24no—
354C2×M16—22no—
355C2×M16—41no—
356C2×M16—42no—
357C22×C8—41no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—44no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—41no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4C2×C4×C7112yescomplemented
368#100—24no—
369#100—22no—
370#100C2×C4×C7114yescomplemented
371#99—24no—
372#99—22no—
373#99C2×C4×C7114yescomplemented
374#95—22no—
375#95—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C42×C7112yes—
387C24C22⋊C4×C7111yescharacteristic
388C24C22⋊C4×C7111yescharacteristic
389C24C22⋊C4×C7112yes—
390C24C22⋊C4×C7111yescharacteristic
391C24C4⋊C4×C7112yes—
392C24C4⋊C4×C7112yes—
393C24C22×C4×C7111yescharacteristic
394C24C2×D4×C7111yescharacteristic
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C42×C7112yes—
412C2×D4C22⋊C4×C7112yes—
413C2×D4C22⋊C4×C7112yes—
414C2×D4C4⋊C4×C7112yes—
415C2×D4C4⋊C4×C7112yes—
416C2×D4C4⋊C4×C7112yes—
417C2×D4C22×C4×C7111yescharacteristic
418C2×D4C22×C4×C7111yescharacteristic
419C2×D4C2×D4×C7111yescharacteristic
420C2×D4C2×Q8×C7111yescharacteristic
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22⋊C4×C7111yescharacteristic
445C22×C4C22⋊C4×C7112yes—
446C22×C4C4⋊C4×C7112yes—
447C22×C4C2×D4×C7111yescharacteristic
448M16—24no—
449M16—42no—
450M16—44no—
451C2×C8—42no—
452C2×C8—44no—
453C22⋊C4—24no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—44no—
481C23—41no—
482C23—44no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—81no—
498C23—81no—
499C23—84no—
500C23#95×C7112yes—
501C23#95×C7112yes—
502C23#95×C7112yes—
503C23#99×C7111yescharacteristic
504C23#99×C7112yes—
505C23#99×C7111yescharacteristic
506C23C2×C42×C7111yescharacteristic
507C23C2×C22⋊C4×C7111yescharacteristic
508C23C2×C22⋊C4×C7111yescharacteristic
509C23C2×C4⋊C4×C7111yescharacteristic
510C23C2×C4⋊C4×C7111yescharacteristic
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—88no—
545C2×C4#95×C7112yes—
546C2×C4C2×C22⋊C4×C7111yescharacteristic
547C2×C4C2×C4⋊C4×C7111yescharacteristic
548C8—44no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—24no—
554C22—24no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22#342×C7112yes—
588C22#342×C7112yes—
589C22AΣL(1,𝔽16)×C7112yes—
590C22#352×C7112yes—
591C22C2×#95×C7111yescharacteristic
592C22C2×#99×C7111yescharacteristic
593C22C2×#99×C7111yescharacteristic
594C4—21no—
595C4—22no—
596C4—21no—
597C4—44no—
598C4—82no—
599C4—84no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—42no—
605C2—41no—
606C2—41no—
607C2—42no—
608C2—41no—
609C2C71×#139014yesdirect product factor
610C2C2×#342×C7111yescharacteristic
611C2C2×AΣL(1,𝔽16)×C7111yescharacteristic
612C2C2×#352×C7111yescharacteristic
subgroupG/H
center, socleC23×C71#99
commutatorC23C2×C42×C71
Frattini, cosocleC22×D4C23×C71
prime radicalC23×D4×C71C22
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C71 ◁ C24×C71 ◁ C23×D4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
4. C71×#1390C2direct (N × H)
6. C24×C4×C71C4semidirect
10. C22×C22⋊C4×C71C4semidirect
13. C2×#100×C71C4semidirect
17. C2×#99×C71C4semidirect
21. #386×C71C4semidirect
44. C23×C4×C71C2×C4semidirect
60. C2×C22⋊C4×C71C2×C4semidirect
63. #100×C71C2×C4semidirect
66. #99×C71C2×C4semidirect
293. C2×#1390C71direct (N × H)
303. C2×C71#1390direct (N × H)
310. #1390C2×C71direct (N × H)
311. C71C2×#1390direct (N × H)
313. C24×C4C4×C71semidirect
317. C22×C22⋊C4C4×C71semidirect
320. C2×#100C4×C71semidirect
324. C2×#99C4×C71semidirect
328. #386C4×C71semidirect
351. C23×C4C2×C4×C71semidirect
367. C2×C22⋊C4C2×C4×C71semidirect
370. #100C2×C4×C71semidirect
373. #99C2×C4×C71semidirect
609. C2C71×#1390direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#100×C712
C22×#99×C712
C2×#386×C712
C71×#13902
C2×#139071