Finite Group Databasebeta

← Dic7×#397 #13275617

Subgroups — Dic7×#397 #13275617

Order 1792.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#397C211yesmaximal, characteristic
2C2×C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic7C212yesmaximal
4C2×C22⋊C4×Dic7C211yesmaximal, characteristic
5#11729323C211yesmaximal, characteristic, complemented
6#11729302C212yesmaximal
7#11729300C211yesmaximal, characteristic
8#11727170C212yesmaximal, complemented
9#11727166C211yesmaximal, characteristic, complemented
10Dic7×#95C212yesmaximal, complemented
11Dic7×#95C211yesmaximal, characteristic, complemented
12C24×Dic7C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C22⋊Dic7C2212yes—
17C22×C22⋊Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×C4×Dic7C2212yes—
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C7×#397C418yescomplemented
24C2×C7×#95C2211yescharacteristic
25C2×C7×#95C2212yes—
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2211yescharacteristic
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2211yescharacteristic
33C2×C7⋊#95C2211yescharacteristic
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36C4⋊C4×Dic7—24no—
37C22⋊C4×Dic7—28no—
38C22⋊C4×Dic7—24no—
39C4×#397—71nomaximal, Sylow
40C25×C7C2311yescharacteristic
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×Dic7—22no—
46C23×Dic7—22no—
47C23×Dic7—24no—
48C23×Dic7—24no—
49C23×Dic7D414yes—
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C23×Dic7C2312yes—
53C23×Dic7C2311yescharacteristic
54C23×Dic7C2312yes—
55C23×Dic7C2311yescharacteristic
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4C2×C414yescomplemented
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—22no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4C2×C418yes—
64C2×C7×C22⋊C4C2×C414yes—
65C2×C22⋊Dic7—24no—
66C2×C22⋊Dic7—22no—
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7—24no—
69C2×C22⋊Dic7—22no—
70C2×C22⋊Dic7—24no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7—24no—
73C2×C4×Dic7—24no—
74C2×C4×Dic7—24no—
75C2×C4×Dic7—24no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—24no—
78C2×C4×Dic7—24no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7—24no—
81C2×C4×Dic7—24no—
82C2×C4×Dic7D414yes—
83C7×#95C2×C418yescomplemented
84C7×#95C2×C414yescomplemented
85C7⋊#95—22no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88C7⋊#95—24no—
89C7⋊#95—24no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C2×#397D711yescharacteristic
96C2×C4×C4⋊C4—71no—
97C2×C4×C22⋊C4—72no—
98C2×C4×C22⋊C4—71no—
99#1445—71no—
100#1443—72no—
101#1442—71no—
102#1440—72no—
103#1439—71no—
104C4×#95—72no—
105C4×#95—71no—
106C24×C7—21no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—21no—
110C24×C7—21no—
111C24×C7—21no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7C22×C412yes—
115C24×C7C2×D412yes—
116C24×C7#4112yes—
117C24×C7C2411yescharacteristic
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7C22×C414yes—
137C22×C4×C7C22×C412yes—
138C22×C4×C7C22×C414yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C2×D412yes—
141C22×C4×C7#4112yes—
142C22×Dic7—21no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—21no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—28no—
167C22×Dic7—216no—
168C22×Dic7—44no—
169C22×Dic7—44no—
170C22×Dic7—48no—
171C22×Dic7C2×D412yes—
172C22×Dic7C2×D412yes—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D412yes—
175C22×Dic7#4112yes—
176C22×Dic7#4114yes—
177C22×Dic7#4114yes—
178C22×Dic7#4112yes—
179C22×Dic7#4114yes—
180C22×Dic7#4114yes—
181C7×C4⋊C4—28no—
182C7×C22⋊C4—216no—
183C7×C22⋊C4—28no—
184C22⋊Dic7—28no—
185C22⋊Dic7—216no—
186C4⋊Dic7—28no—
187C4×Dic7—28no—
188C4×Dic7—48no—
189C4×Dic7—48no—
190C4×Dic7—44no—
191C24×C4—71no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4C2×D711yescharacteristic
194C22×C22⋊C4—72no—
195C22×C22⋊C4—71no—
196C22×C22⋊C4C2×D711yescharacteristic
197C22×C22⋊C4C2×D712yes—
198C22×C42—71no—
199C22×C42—71no—
200C22×C42—72no—
201C22×C42—72no—
202#397Dic718yesdirect product factor
203C4×C4⋊C4—144no—
204C4×C22⋊C4—144no—
205C4×C22⋊C4—148no—
206C2×#95—72no—
207C2×#95—71no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95—71no—
214C2×#95—71no—
215C2×#95—72no—
216C2×#95C2×D711yescharacteristic
217C2×#95C2×D712yes—
218C23×C7—22no—
219C23×C7—22no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—22no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—24no—
228C23×C7—24no—
229C23×C7—22no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—41no—
240C23×C7C4×D414yes—
241C23×C7C23×C411yescharacteristic
242C23×C7C22×D411yescharacteristic
243C23×C7C22×D411yescharacteristic
244C23×C7C2×#4111yescharacteristic
245C23×C7C2×#4111yescharacteristic
246C23×C7C2×#4111yescharacteristic
247C23×C7C2×#4112yes—
248C23×C7C2×#4112yes—
249C23×C7C2×#4112yes—
250C23×C7C2×#4112yes—
251C23×C7C2×#4111yescharacteristic
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7C4×D414yes—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—22no—
289C2×Dic7—42no—
290C2×Dic7—42no—
291C2×Dic7—41no—
292C2×Dic7—42no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—42no—
297C2×Dic7—42no—
298C2×Dic7—42no—
299C2×Dic7—42no—
300C2×Dic7—42no—
301C2×Dic7—42no—
302C2×Dic7—42no—
303C2×Dic7—42no—
304C2×Dic7—42no—
305C2×Dic7—416no—
306C2×Dic7#12114yes—
307C2×Dic7#12314yes—
308C2×Dic7#12318yes—
309C2×Dic7#12414yes—
310C2×Dic7#12618yes—
311C25C22×D711yescharacteristic
312C23×C4—71no—
313C23×C4—74no—
314C23×C4—72no—
315C23×C4—71no—
316C23×C4—72no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—142no—
320C23×C4—142no—
321C23×C4—144no—
322C23×C4—144no—
323C23×C4C22×D712yes—
324C23×C4C22×D711yescharacteristic
325C23×C4C22×D712yes—
326C23×C4C22×D711yescharacteristic
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—144no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4C2×Dic714yescomplemented
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—144no—
337C2×C22⋊C4—142no—
338C2×C22⋊C4—144no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—144no—
341C2×C22⋊C4—142no—
342C2×C22⋊C4C2×Dic718yes—
343C2×C22⋊C4C2×Dic714yes—
344C2×C42—74no—
345C2×C42—144no—
346C2×C42—142no—
347C2×C42—144no—
348C2×C42—144no—
349C2×C42—144no—
350C2×C42—144no—
351C2×C42—144no—
352C2×C42—144no—
353C2×C42—144no—
354#95—144no—
355#95—144no—
356#95—144no—
357#95—144no—
358#95—144no—
359#95—144no—
360#95—142no—
361#95—144no—
362#95—142no—
363#95—142no—
364#95C2×Dic718yescomplemented
365#95C2×Dic714yescomplemented
366C22×C7—24no—
367C22×C7—28no—
368C22×C7—44no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—44no—
373C22×C7—42no—
374C22×C7—44no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7C2×C4×D411yescharacteristic
378C22×C7C2×C4×D411yescharacteristic
379C22×C7C4×#4111yescharacteristic
380C22×C7C4×#4112yes—
381C22×C7C4×#4112yes—
382C22×C7C2×#12111yescharacteristic
383C22×C7C2×#12311yescharacteristic
384C22×C7C2×#12312yes—
385C22×C7#52511yescharacteristic
386C22×C7#52512yes—
387C22×C7#52511yescharacteristic
388C22×C7C2×#12411yescharacteristic
389C22×C7C2×#12612yes—
390C22×C7#52912yes—
391C22×C7#52912yes—
392C22×C7#52912yes—
393C22×C7#52912yes—
394C22×C7#52911yescharacteristic
395C22×C7#52912yes—
396C22×C7#52912yes—
397C22×C7#52912yes—
398C22×C7#53211yescharacteristic
399C22×C7#53211yescharacteristic
400C4×C7—24no—
401C4×C7—44no—
402C4×C7—44no—
403C4×C7—42no—
404Dic7—42no—
405Dic7—44no—
406Dic7—42no—
407Dic7—44no—
408Dic7—42no—
409Dic7#39718yesdirect product factor
410C24—21no—
411C24—21no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—21no—
416C24—22no—
417C24—22no—
418C24D4×D712yes—
419C24#115912yes—
420C24C22×Dic712yes—
421C24C23×D711yescharacteristic
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—72no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—74no—
444C22×C4—74no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—74no—
448C22×C4—72no—
449C22×C4—74no—
450C22×C4—148no—
451C22×C4—1416no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—142no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—142no—
461C22×C4—142no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—142no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—141no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—141no—
474C22×C4—142no—
475C22×C4—142no—
476C22×C4—288no—
477C22×C4—284no—
478C22×C4—284no—
479C22×C4D4×D712yes—
480C22×C4#115912yes—
481C22×C4C22×Dic714yes—
482C22×C4C22×Dic712yes—
483C22×C4C22×Dic714yes—
484C22×C4C22×Dic712yes—
485C4⋊C4—28no—
486C4⋊C4—148no—
487C22⋊C4—216no—
488C22⋊C4—28no—
489C22⋊C4—148no—
490C22⋊C4—1416no—
491C42—148no—
492C42—288no—
493C42—288no—
494C42—284no—
495C2×C7—44no—
496C2×C7C4×#12111yescharacteristic
497C2×C7C4×#12311yescharacteristic
498C2×C7C4×#12312yes—
499C2×C7C4×#12411yescharacteristic
500C2×C7C4×#12612yes—
501C2×C7C2×#39711yescharacteristic
502C2×C7#239211yescharacteristic
503C2×C7#239511yescharacteristic
504C2×C7#239512yes—
505C2×C7#239811yescharacteristic
506C2×C7#240112yes—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23D4×Dic714yes—
530C23C2×D4×D711yescharacteristic
531C23C2×D4×D711yescharacteristic
532C23C2×#115911yescharacteristic
533C23C2×#115911yescharacteristic
534C23D7×#4112yes—
535C23D7×#4112yes—
536C23D7×#4111yescharacteristic
537C23D7×#4112yes—
538C23D7×#4111yescharacteristic
539C23D7×#4112yes—
540C23C23×Dic711yescharacteristic
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—78no—
565C2×C4—74no—
566C2×C4—74no—
567C2×C4—78no—
568C2×C4—74no—
569C2×C4—144no—
570C2×C4—144no—
571C2×C4—144no—
572C2×C4—144no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—144no—
578C2×C4—144no—
579C2×C4—144no—
580C2×C4—144no—
581C2×C4—142no—
582C2×C4—2816no—
583C2×C4—282no—
584C2×C4—282no—
585C2×C4—282no—
586C2×C4—282no—
587C2×C4—282no—
588C2×C4—282no—
589C2×C4—282no—
590C2×C4—282no—
591C2×C4—282no—
592C2×C4—282no—
593C2×C4—281no—
594C2×C4—282no—
595C2×C4—281no—
596C2×C4—282no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4D4×Dic714yes—
600C7C4×#39711yescharacteristic, complemented, Sylow
601C22—28no—
602C22—24no—
603C22—42no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—42no—
612C22#9028711yescharacteristic
613C22#9029111yescharacteristic
614C22D7×#12111yescharacteristic
615C22#9029711yescharacteristic
616C22#9034012yes—
617C22#9034012yes—
618C22#9034012yes—
619C22D7×#12311yescharacteristic
620C22D7×#12312yes—
621C22#9034712yes—
622C22#9034711yescharacteristic
623C22#9035911yescharacteristic
624C22D7×#12411yescharacteristic
625C22#9037112yes—
626C22D7×#12612yes—
627C22#9039712yes—
628C22#9039712yes—
629C22#9039712yes—
630C22C2×D4×Dic711yescharacteristic
631C22C2×D4×Dic711yescharacteristic
632C22#41×Dic711yescharacteristic
633C22#41×Dic712yes—
634C22#41×Dic712yes—
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—78no—
640C4—282no—
641C4—284no—
642C4—282no—
643C4—284no—
644C4—282no—
645C2—44no—
646C2#1173490211yescharacteristic
647C2#1173490311yescharacteristic
648C2#1173490412yes—
649C2#1173490511yescharacteristic
650C2#1173490612yes—
651C2D7×#39711yescharacteristic
652C2Dic7×#12111yescharacteristic
653C2Dic7×#12311yescharacteristic
654C2Dic7×#12312yes—
655C2Dic7×#12411yescharacteristic
656C2Dic7×#12612yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#397C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#397D7
hypocenterC7C4×#397
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic7C2semidirect
5. #11729323C2semidirect
8. #11727170C2semidirect
9. #11727166C2semidirect
10. Dic7×#95C2semidirect
11. Dic7×#95C2semidirect
23. C7×#397C4semidirect
58. C2×C7×C4⋊C4C2×C4semidirect
83. C7×#95C2×C4semidirect
84. C7×#95C2×C4semidirect
202. #397Dic7direct (N × H)
331. C2×C4⋊C4C2×Dic7semidirect
364. #95C2×Dic7semidirect
365. #95C2×Dic7semidirect
409. Dic7#397direct (N × H)
600. C7C4×#397semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3972
C2×C4⋊C4×Dic72
C2×C22⋊C4×Dic72
#117293232
#117293022
#117293002
#117271702
#117271662
Dic7×#952
C4×#3977