Finite Group Databasebeta

← C2×#342×D71 #31994878

Subgroups — C2×#342×D71 #31994878

Order 18176.

4334 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×C22⋊C4×D71C212yesmaximal
2C22×C22⋊C4×D71C211yesmaximal, characteristic
3C2×#17985766C211yesmaximal, characteristic, complemented
4C2×#17985765C212yesmaximal, complemented
5#342×D71C218yesmaximal, direct product factor
6C2×#342×C71C211yesmaximal, characteristic, complemented
7C25×D71C2211yescharacteristic
8C23×C4×D71—22no—
9C23×C4×D71C412yescomplemented
10C22×C22⋊Dic71C2212yes—
11C22×C22⋊Dic71C2211yescharacteristic
12C22×(C2×D71)⋊C4—22no—
13C22×(C2×D71)⋊C4C412yescomplemented
14C2×C22⋊C4×D71—28no—
15C2×C22⋊C4×D71—22no—
16C2×C22⋊C4×D71—24no—
17C2×C22⋊C4×D71—22no—
18C2×C22⋊C4×D71—24no—
19C2×C22⋊C4×D71C4116yescomplemented
20C2×C22⋊C4×D71C414yescomplemented
21C2×C22⋊C4×D71C2218yes—
22C2×C22⋊C4×D71C2214yes—
23#17985766C2214yescomplemented
24#17985765C2218yescomplemented
25C22×C22⋊C4×C71C2211yescharacteristic
26C22×C22⋊C4×C71C2212yes—
27#342×C71C2214yescomplemented
28C24×D71—48no—
29C24×D71C2×C411yescharacteristic
30C24×D71C2×C418yes—
31C24×D71C2×C412yes—
32C24×D71C2×C411yescharacteristic
33C24×D71D412yes—
34C24×D71D414yes—
35C24×D71D414yes—
36C24×D71D412yes—
37C24×D71Q814yes—
38C24×D71C2312yes—
39C23×Dic71—22no—
40C23×Dic71C2×C412yescomplemented
41C22×C4×D71—216no—
42C22×C4×D71—24no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—28no—
46C22×C4×D71—416no—
47C22×C4×D71—48no—
48C22×C4×D71C2×C418yescomplemented
49C2×C22⋊Dic71—22no—
50C2×C22⋊Dic71—22no—
51C2×C22⋊Dic71—24no—
52C2×C22⋊Dic71—22no—
53C2×C22⋊Dic71C2×C412yescomplemented
54C2×C22⋊Dic71C2×C418yescomplemented
55C2×C22⋊Dic71C2312yes—
56C2×C22⋊Dic71C2314yes—
57C2×(C2×D71)⋊C4—24no—
58C2×(C2×D71)⋊C4—24no—
59C2×(C2×D71)⋊C4—216no—
60C2×(C2×D71)⋊C4—28no—
61C2×(C2×D71)⋊C4C2×C418yescomplemented
62C22⋊C4×D71—28no—
63C22⋊C4×D71—28no—
64C22⋊C4×D71—232no—
65C22⋊C4×D71—28no—
66C22⋊C4×D71—28no—
67C22⋊C4×D71—216no—
68C22⋊C4×D71C2×C4132yescomplemented
69C25×C71C2311yescharacteristic
70C23×C4×C71—22no—
71C23×C4×C71C2×C412yescomplemented
72C2×C22⋊C4×C71—22no—
73C2×C22⋊C4×C71—22no—
74C2×C22⋊C4×C71—22no—
75C2×C22⋊C4×C71—24no—
76C2×C22⋊C4×C71C2×C412yescomplemented
77C2×C22⋊C4×C71C2×C418yescomplemented
78C2×C22⋊C4×C71C2312yes—
79C2×C22⋊C4×C71C2314yes—
80C23×D71—28no—
81C23×D71—216no—
82C23×D71—432no—
83C23×D71—48no—
84C23×D71—416no—
85C23×D71—416no—
86C23×D71—416no—
87C23×D71—48no—
88C23×D71—44no—
89C23×D71—48no—
90C23×D71C4214yes—
91C23×D71C4218yes—
92C23×D71C4214yes—
93C23×D71C22⋊C418yes—
94C23×D71C22⋊C412yes—
95C23×D71C22⋊C411yescharacteristic
96C23×D71C22⋊C414yes—
97C23×D71C22⋊C412yes—
98C23×D71C22⋊C4116yes—
99C23×D71C22⋊C414yes—
100C23×D71C22⋊C418yes—
101C23×D71C22⋊C412yes—
102C23×D71C22⋊C411yescharacteristic
103C23×D71C4⋊C414yes—
104C23×D71C4⋊C418yes—
105C23×D71C4⋊C4116yes—
106C23×D71C4⋊C418yes—
107C23×D71C4⋊C418yes—
108C23×D71C4⋊C414yes—
109C23×D71C22×C412yes—
110C23×D71C22×C418yes—
111C23×D71C22×C412yes—
112C23×D71C2×D414yes—
113C23×D71C2×D414yes—
114C23×D71C2×D414yes—
115C23×D71C2×Q814yes—
116C22×Dic71—28no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—24no—
120C22×Dic71—48no—
121C22×Dic71—44no—
122C22×Dic71C22×C414yescomplemented
123C2×C4×D71—232no—
124C2×C4×D71—28no—
125C2×C4×D71—28no—
126C2×C4×D71—464no—
127C2×C4×D71—432no—
128C2×C4×D71—48no—
129C2×C4×D71—416no—
130C2×C4×D71—48no—
131C2×C4×D71—416no—
132C22⋊Dic71—24no—
133C22⋊Dic71—28no—
134C22⋊Dic71—24no—
135C22⋊Dic71—24no—
136C22⋊Dic71C22×C418yescomplemented
137(C2×D71)⋊C4—216no—
138(C2×D71)⋊C4—232no—
139C24×C71—44no—
140C24×C71C22×C411yescharacteristic
141C24×C71C22×C414yes—
142C24×C71C22×C411yescharacteristic
143C24×C71C2×D412yes—
144C24×C71C2×D412yes—
145C24×C71C2×D412yes—
146C24×C71C2×Q812yes—
147C24×C71C2411yescharacteristic
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—28no—
151C22×C4×C71—24no—
152C22×C4×C71—48no—
153C22×C4×C71—44no—
154C22×C4×C71C22×C414yescomplemented
155C22⋊C4×C71—24no—
156C22⋊C4×C71—28no—
157C22⋊C4×C71—24no—
158C22⋊C4×C71—24no—
159C22⋊C4×C71C22×C418yescomplemented
160C23×C71—24no—
161C23×C71—28no—
162C23×C71—42no—
163C23×C71—48no—
164C23×C71—44no—
165C23×C71—44no—
166C23×C71—44no—
167C23×C71—42no—
168C23×C71C2×C4212yes—
169C23×C71C2×C4212yes—
170C23×C71C2×C22⋊C412yes—
171C23×C71C2×C22⋊C411yescharacteristic
172C23×C71C2×C22⋊C414yes—
173C23×C71C2×C22⋊C414yes—
174C23×C71C2×C22⋊C411yescharacteristic
175C23×C71C2×C4⋊C412yes—
176C23×C71C2×C4⋊C414yes—
177C23×C71C2×C4⋊C412yes—
178C23×C71C2×C4⋊C414yes—
179C23×C71C23×C411yescharacteristic
180C23×C71C23×C412yes—
181C23×C71C22×D412yes—
182C23×C71C22×D411yescharacteristic
183C23×C71C22×Q811yescharacteristic
184C22×D71—24no—
185C22×D71—28no—
186C22×D71—232no—
187C22×D71—216no—
188C22×D71—216no—
189C22×D71—232no—
190C22×D71—28no—
191C22×D71—28no—
192C22×D71—216no—
193C22×D71—216no—
194C22×D71—416no—
195C22×D71—432no—
196C22×D71—416no—
197C22×D71—432no—
198C22×D71—416no—
199C22×D71—48no—
200C22×D71—48no—
201C22×D71—48no—
202C22×D71—48no—
203C22×D71—44no—
204C22×D71—416no—
205C22×D71—416no—
206C22×D71—48no—
207C22×D71—44no—
208C22×D71—48no—
209C22×D71#9514yes—
210C22×D71#9518yes—
211C22×D71#9518yes—
212C22×D71#95116yes—
213C22×D71#95116yes—
214C22×D71#9518yes—
215C22×D71#9514yes—
216C22×D71#9918yes—
217C22×D71C2×C4214yes—
218C22×D71C2×C4214yes—
219C22×D71C2×C22⋊C414yes—
220C22×D71C2×C22⋊C412yes—
221C22×D71C2×C22⋊C418yes—
222C22×D71C2×C22⋊C418yes—
223C22×D71C2×C22⋊C412yes—
224C22×D71C2×C4⋊C414yes—
225C22×D71C2×C4⋊C418yes—
226C22×D71C2×C4⋊C418yes—
227C22×D71C2×C4⋊C414yes—
228C2×C4×C71—24no—
229C2×C4×C71—28no—
230C2×C4×C71—48no—
231C2×C4×C71—416no—
232C2×C4×C71—44no—
233C2×C4×C71—44no—
234C2×C4×C71—44no—
235C2×Dic71—24no—
236C2×Dic71—28no—
237C2×Dic71—48no—
238C2×Dic71—416no—
239C2×Dic71—44no—
240C2×Dic71—44no—
241C2×Dic71—44no—
242C4×D71—464no—
243C4×D71—416no—
244C4×D71—416no—
245C22×C71—22no—
246C22×C71—22no—
247C22×C71—24no—
248C22×C71—24no—
249C22×C71—28no—
250C22×C71—24no—
251C22×C71—24no—
252C22×C71—28no—
253C22×C71—42no—
254C22×C71—44no—
255C22×C71—42no—
256C22×C71—44no—
257C22×C71—42no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71C2×#9512yes—
261C22×C71C2×#9514yes—
262C22×C71C2×#9512yes—
263C22×C71C2×#9914yes—
264C22×C71C22×C4211yescharacteristic
265C22×C71C22×C22⋊C412yes—
266C22×C71C22×C22⋊C411yescharacteristic
267C22×C71C22×C4⋊C411yescharacteristic
268C22×C71C22×C4⋊C412yes—
269C2×D71—24no—
270C2×D71—28no—
271C2×D71—28no—
272C2×D71—216no—
273C2×D71—216no—
274C2×D71—232no—
275C2×D71—216no—
276C2×D71—28no—
277C2×D71—28no—
278C2×D71—216no—
279C2×D71—416no—
280C2×D71—48no—
281C2×D71—44no—
282C2×D71—48no—
283C2×D71—48no—
284C2×D71—44no—
285C2×D71#342116yesdirect product factor
286C2×D71C2×#9514yes—
287C2×D71C2×#9518yes—
288C2×D71C2×#9514yes—
289C2×D71C2×#9918yes—
290C4×C71—48no—
291C4×C71—44no—
292Dic71—48no—
293Dic71—44no—
294C22×#342—711nomaximal, Sylow
295C2×C71—22no—
296C2×C71—21no—
297C2×C71—22no—
298C2×C71—22no—
299C2×C71—24no—
300C2×C71—21no—
301C2×C71C2×#34214yescomplemented
302C2×C71C22×#9511yescharacteristic
303C2×C71C22×#9912yes—
304D71—24no—
305D71—28no—
306D71C2×#34218yesdirect product factor
307C23×C22⋊C4—712no—
308C23×C22⋊C4—711no—
309C2×#342—718no—
310C2×#342—712no—
311C2×#342—711no—
312C2×#342D7111yescharacteristic, direct product factor
313C71C22×#34211yescharacteristic, complemented, Sylow
314C26—711no—
315C24×C4—712no—
316C24×C4—1422no—
317C22×C22⋊C4—712no—
318C22×C22⋊C4—7116no—
319C22×C22⋊C4—718no—
320C22×C22⋊C4—711no—
321C22×C22⋊C4—712no—
322C22×C22⋊C4—714no—
323C22×C22⋊C4—714no—
324C22×C22⋊C4—1428no—
325C22×C22⋊C4—1424no—
326C22×C22⋊C4—1424no—
327C22×C22⋊C4—1422no—
328C22×C22⋊C4—1422no—
329C22×C22⋊C4—1422no—
330C22×C22⋊C4C2×D7111yescharacteristic
331C22×C22⋊C4C2×D7112yes—
332#342—718no—
333#342—714no—
334#342C2×D7114yesdirect product factor
335C25—2848no—
336C25—711no—
337C25—718no—
338C25—712no—
339C25—712no—
340C25—714no—
341C25—714no—
342C25—714no—
343C25—712no—
344C25—711no—
345C25—712no—
346C25C22×D7111yescharacteristic
347C23×C4—22no—
348C23×C4—28416no—
349C23×C4—2848no—
350C23×C4—712no—
351C23×C4—718no—
352C23×C4—1428no—
353C23×C4—14216no—
354C23×C4—1424no—
355C23×C4—1422no—
356C23×C4—1422no—
357C23×C4—1422no—
358C23×C4C4×D7112yescomplemented
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—718no—
364C2×C22⋊C4—714no—
365C2×C22⋊C4—7132no—
366C2×C22⋊C4—712no—
367C2×C22⋊C4—712no—
368C2×C22⋊C4—718no—
369C2×C22⋊C4—14216no—
370C2×C22⋊C4—1428no—
371C2×C22⋊C4—1428no—
372C2×C22⋊C4—14232no—
373C2×C22⋊C4—1428no—
374C2×C22⋊C4—1428no—
375C2×C22⋊C4—1428no—
376C2×C22⋊C4—1424no—
377C2×C22⋊C4—14216no—
378C2×C22⋊C4—1424no—
379C2×C22⋊C4—1424no—
380C2×C22⋊C4—1422no—
381C2×C22⋊C4—1422no—
382C2×C22⋊C4—1422no—
383C2×C22⋊C4C4×D7112yescomplemented
384C2×C22⋊C4C4×D7118yescomplemented
385C2×C22⋊C4C22×D7112yes—
386C2×C22⋊C4C22×D7114yes—
387C24—44no—
388C24—2844no—
389C24—2848no—
390C24—28432no—
391C24—2848no—
392C24—28416no—
393C24—28416no—
394C24—28416no—
395C24—2848no—
396C24—712no—
397C24—718no—
398C24—714no—
399C24—714no—
400C24—714no—
401C24—714no—
402C24—718no—
403C24—712no—
404C24—714no—
405C24—718no—
406C24—7116no—
407C24—714no—
408C24—718no—
409C24—718no—
410C24—7116no—
411C24—711no—
412C24—712no—
413C24—714no—
414C24—714no—
415C24—714no—
416C24—718no—
417C24—712no—
418C24—712no—
419C24—711no—
420C24—718no—
421C24—714no—
422C24—1428no—
423C24—14216no—
424C24C2×C4×D7111yescharacteristic
425C24C2×C4×D7111yescharacteristic
426C24C2×C4×D7114yes—
427C24D4×D7112yes—
428C24D4×D7112yes—
429C24D4×D7112yes—
430C24Q8×D7112yes—
431C24C23×D7111yescharacteristic
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—2848no—
439C22×C4—28416no—
440C22×C4—28464no—
441C22×C4—28432no—
442C22×C4—2848no—
443C22×C4—28416no—
444C22×C4—2848no—
445C22×C4—2844no—
446C22×C4—714no—
447C22×C4—14232no—
448C22×C4—1428no—
449C22×C4—1428no—
450C22×C4—1424no—
451C22×C4—1428no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4C2×C4×D7114yescomplemented
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—28no—
459C22⋊C4—718no—
460C22⋊C4—14216no—
461C22⋊C4—14232no—
462C22⋊C4—1424no—
463C22⋊C4—1424no—
464C22⋊C4—1428no—
465C22⋊C4—1424no—
466C22⋊C4C2×C4×D7118yescomplemented
467C23—24no—
468C23—28no—
469C23—48no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—28416no—
476C23—2848no—
477C23—2848no—
478C23—28416no—
479C23—2844no—
480C23—2848no—
481C23—2844no—
482C23—2848no—
483C23—28416no—
484C23—2848no—
485C23—2848no—
486C23—28416no—
487C23—28432no—
488C23—28416no—
489C23—28432no—
490C23—718no—
491C23—7116no—
492C23—714no—
493C23—718no—
494C23—714no—
495C23—718no—
496C23—714no—
497C23—712no—
498C23—718no—
499C23—7116no—
500C23—718no—
501C23—718no—
502C23—714no—
503C23—714no—
504C23—714no—
505C23—718no—
506C23—712no—
507C23—718no—
508C23—714no—
509C23—14232no—
510C23—1428no—
511C23—14216no—
512C23—1428no—
513C23—14216no—
514C23—14216no—
515C23—14232no—
516C23—14216no—
517C23—1424no—
518C23—1428no—
519C23C42×D7112yes—
520C23C42×D7112yes—
521C23C22⋊C4×D7112yes—
522C23C22⋊C4×D7114yes—
523C23C22⋊C4×D7111yescharacteristic
524C23C22⋊C4×D7114yes—
525C23C22⋊C4×D7111yescharacteristic
526C23C4⋊C4×D7112yes—
527C23C4⋊C4×D7114yes—
528C23C4⋊C4×D7114yes—
529C23C4⋊C4×D7112yes—
530C23C22×C4×D7111yescharacteristic
531C23C22×C4×D7112yes—
532C23C2×D4×D7112yes—
533C23C2×D4×D7111yescharacteristic
534C23C2×Q8×D7111yescharacteristic
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—416no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—28416no—
543C2×C4—28464no—
544C2×C4—28416no—
545C2×C4—28416no—
546C2×C4—2848no—
547C2×C4—2844no—
548C2×C4—2844no—
549C2×C4—2844no—
550C2×C4—1428no—
551C2×C4—1424no—
552C22—28no—
553C22—24no—
554C22—24no—
555C22—28no—
556C22—22no—
557C22—24no—
558C22—22no—
559C22—24no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—44no—
565C22—42no—
566C22—44no—
567C22—2848no—
568C22—28416no—
569C22—2848no—
570C22—2844no—
571C22—2844no—
572C22—2848no—
573C22—7116no—
574C22—718no—
575C22—714no—
576C22—718no—
577C22—714no—
578C22—14216no—
579C22—14232no—
580C22—14216no—
581C22—1428no—
582C22—1428no—
583C22—14216no—
584C22—14216no—
585C22—1428no—
586C22—1424no—
587C22—1428no—
588C22#95×D7112yes—
589C22#95×D7114yes—
590C22#95×D7112yes—
591C22#99×D7114yes—
592C22C2×C42×D7111yescharacteristic
593C22C2×C22⋊C4×D7112yes—
594C22C2×C22⋊C4×D7111yescharacteristic
595C22C2×C4⋊C4×D7111yescharacteristic
596C22C2×C4⋊C4×D7112yes—
597C4—48no—
598C4—44no—
599C4—2848no—
600C4—2844no—
601C2—22no—
602C2—24no—
603C2—22no—
604C2—21no—
605C2—21no—
606C2—22no—
607C2—718no—
608C2—1428no—
609C2—1424no—
610C2#342×D7114yesdirect product factor
611C2C2×#95×D7111yescharacteristic
612C2C2×#99×D7112yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C71C22×C42
FrattiniC24C23×D71
FittingC2×#342×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC25×D71C22
cosocleC24×C71C24
hypercenterC2×#342D71
hypocenterC71C22×#342
LCSG ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. C2×#17985766C2semidirect
4. C2×#17985765C2semidirect
5. #342×D71C2direct (N × H)
6. C2×#342×C71C2semidirect
9. C23×C4×D71C4semidirect
13. C22×(C2×D71)⋊C4C4semidirect
19. C2×C22⋊C4×D71C4semidirect
20. C2×C22⋊C4×D71C4semidirect
23. #17985766C22semidirect
24. #17985765C22semidirect
27. #342×C71C22semidirect
40. C23×Dic71C2×C4semidirect
48. C22×C4×D71C2×C4semidirect
53. C2×C22⋊Dic71C2×C4semidirect
54. C2×C22⋊Dic71C2×C4semidirect
61. C2×(C2×D71)⋊C4C2×C4semidirect
68. C22⋊C4×D71C2×C4semidirect
71. C23×C4×C71C2×C4semidirect
76. C2×C22⋊C4×C71C2×C4semidirect
77. C2×C22⋊C4×C71C2×C4semidirect
122. C22×Dic71C22×C4semidirect
136. C22⋊Dic71C22×C4semidirect
154. C22×C4×C71C22×C4semidirect
159. C22⋊C4×C71C22×C4semidirect
285. C2×D71#342direct (N × H)
301. C2×C71C2×#342semidirect
306. D71C2×#342direct (N × H)
312. C2×#342D71direct (N × H)
313. C71C22×#342semidirect
334. #342C2×D71direct (N × H)
358. C23×C4C4×D71semidirect
383. C2×C22⋊C4C4×D71semidirect
384. C2×C22⋊C4C4×D71semidirect
454. C22×C4C2×C4×D71semidirect
466. C22⋊C4C2×C4×D71semidirect
610. C2#342×D71direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×D712
C2×#179857662
C2×#179857652
#342×D712
C2×#342×C712
C22×#34271