Finite Group Databasebeta

← #13360952

Subgroups — #13360952

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic7×#121C211yesmaximal, characteristic
2C2×#89807C211yesmaximal, characteristic
3Dic7×#117C211yesmaximal, characteristic, complemented
4C7×#1875C211yesmaximal, characteristic
5#11729330C211yesmaximal, characteristic, complemented
6#11727928C211yesmaximal, characteristic, complemented
7#11727927C211yesmaximal, characteristic, complemented
8Dic7×#102C211yesmaximal, characteristic, complemented
9Dic7×#102C211yesmaximal, characteristic, complemented
10#11727680C211yesmaximal, characteristic, complemented
11Dic7×C22⋊C8C211yesmaximal, characteristic, complemented
12#11726575C211yesmaximal, characteristic, complemented
13#11726573C211yesmaximal, characteristic, complemented
14#11726539C211yesmaximal, characteristic, complemented
15#11726538C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×D4×Dic7—21no—
19C2×D4×Dic7C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C7×#102C2211yescharacteristic
24C2×C7×C22⋊C8C2211yescharacteristic
25#89993C2211yescharacteristic
26#89992—21no—
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29#89984C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89807C418yescomplemented
34#89757C2211yescharacteristic
35C4⋊C4×Dic7C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89723—21no—
45C22⋊C4×Dic7—21no—
46C22⋊C4×Dic7—21no—
47C22⋊C4×Dic7—21no—
48#89704—21no—
49C42×Dic7—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60C4×#482—71nomaximal, Sylow
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C23×Dic7D411yescharacteristic
66C23×Dic7D411yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C7×#121—22no—
70C7×#121C2×C414yes—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C42×C7—21no—
79C2×C22⋊Dic7—21no—
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C22⋊Dic7—21no—
83C2×C22⋊Dic7—21no—
84D4×Dic7—22no—
85D4×Dic7—22no—
86D4×Dic7—44no—
87#6489—22no—
88#6489C2×C414yescomplemented
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4×Dic7—21no—
95C2×C4×Dic7—22no—
96C2×C4×Dic7—22no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7D411yescharacteristic
103C2×C4×Dic7D411yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105C2×C4×Dic7C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C7×#102—22no—
109C7×#102—22no—
110C7×C22⋊C8—22no—
111#6406C2×C414yescomplemented
112#6406C2×C414yescomplemented
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C22⋊(C7⋊C8)C2×C414yescomplemented
121C8×Dic7—22no—
122C8×Dic7—22no—
123#6382—22no—
124#6382C2×C414yescomplemented
125#6381—22no—
126#6381C2×C414yescomplemented
127C2×#482—71no—
128C4×#121—71no—
129C4×#117—71no—
130#1926—71no—
131#1925—71no—
132#1921—71no—
133#1920—71no—
134#1875D711yescharacteristic
135#1854—71no—
136#1825—71no—
137C4×#107—71no—
138C4×#106—71no—
139C4×#102—71no—
140C4×#102—71no—
141C4×C22⋊C8—71no—
142C24×C7—21no—
143C24×C7C2×D411yescharacteristic
144C2×C7×D4—24no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—42no—
151C2×C7×D4—42no—
152C2×C7×D4C22×C412yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7C22×C412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7#4112yes—
163C22×C4×C7C2411yescharacteristic
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—22no—
167C22×Dic7—22no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—41no—
174C22×Dic7—42no—
175C22×Dic7—42no—
176C22×Dic7—42no—
177C22×Dic7C2×D411yescharacteristic
178C22×Dic7C2×D411yescharacteristic
179C22×Dic7C2×D411yescharacteristic
180C22×Dic7C2×D411yescharacteristic
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—21no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4C22×C412yes—
191C7×C4⋊C4#4112yes—
192C7×C4⋊C4#4112yes—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C42×C7—24no—
196C22⋊Dic7—24no—
197C22⋊Dic7—42no—
198C22⋊Dic7—42no—
199C22⋊Dic7—42no—
200C22⋊Dic7—42no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—42no—
208C4⋊Dic7C22×C412yes—
209C4⋊Dic7C22×C412yes—
210C4×Dic7—24no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7—22no—
214C4×Dic7—22no—
215C4×Dic7—42no—
216C4×Dic7—42no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C4×Dic7—42no—
220C4×Dic7#4112yes—
221C4×Dic7#4112yes—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8C22×C412yes—
227C2×C7⋊C8C22×C412yes—
228C2×#121C2×D711yescharacteristic
229C2×C4×D4—71no—
230C2×C4×D4—141no—
231C2×#117—71no—
232C2×#117C2×D711yescharacteristic
233C22×C42—71no—
234#482—78no—
235C2×#107—71no—
236C2×#107—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102C2×D711yescharacteristic
242C2×#102C2×D711yescharacteristic
243C2×C22⋊C8—71no—
244C2×C22⋊C8C2×D711yescharacteristic
245C2×C4×C8—71no—
246C2×C4×C8—71no—
247#390—71no—
248#389—71no—
249#388—141no—
250#386—141no—
251#380—71no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—71no—
255C4×C4⋊C4—71no—
256C4×C4⋊C4—71no—
257C4×C22⋊C4—141no—
258C4×C22⋊C4—141no—
259C4×C22⋊C4—141no—
260#376—141no—
261C43—141no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340C2×D711yescharacteristic
269#340C2×D711yescharacteristic
270#336—71no—
271#336—71no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—22no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7C4×D412yes—
286C23×C7C22×D411yescharacteristic
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—48no—
290C7×D4—41no—
291C7×D4—41no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7C4×D412yes—
308C2×C4×C7C23×C411yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—44no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—44no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—84no—
336C2×Dic7#12312yes—
337C2×Dic7#12312yes—
338C2×Dic7#13512yes—
339C2×Dic7#13512yes—
340C2×Dic7AGL(1,ℤ/8)12yes—
341C7×C8—41no—
342C7×C8—41no—
343C7⋊C8—22no—
344C7⋊C8—22no—
345C22×D4—21no—
346C22×D4C22×D711yescharacteristic
347C23×C4—71no—
348C23×C4—71no—
349C23×C4—141no—
350C23×C4C22×D711yescharacteristic
351C22×C8—71no—
352C22×C8—71no—
353C22×C8C22×D711yescharacteristic
354C22×C8C22×D711yescharacteristic
355#121—22no—
356#121C4×D714yes—
357C4×D4—142no—
358C4×D4—142no—
359C4×D4—284no—
360#117—22no—
361#117—22no—
362#117—74no—
363#117—142no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—141no—
375C2×C22⋊C4—141no—
376C2×C22⋊C4—141no—
377C2×C22⋊C4—141no—
378C2×C22⋊C4—141no—
379C2×C42—21no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—142no—
385C2×C42—142no—
386C2×C42—141no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42—141no—
390C2×C42—141no—
391C2×C42—141no—
392#107—74no—
393#107—142no—
394#106—74no—
395#106—142no—
396#102—22no—
397#102—22no—
398#102—74no—
399#102—74no—
400C22⋊C8—22no—
401C22⋊C8—74no—
402C4×C8—142no—
403C4×C8—142no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95—141no—
410#95—141no—
411C22×C7—22no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—41no—
415C22×C7—41no—
416C22×C7—41no—
417C22×C7—41no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—81no—
422C22×C7—81no—
423C22×C7—81no—
424C22×C7—81no—
425C22×C7C2×C4×D411yescharacteristic
426C22×C7C2×#12311yescharacteristic
427C22×C7#52511yescharacteristic
428C22×C7C2×#13511yescharacteristic
429C22×C7C2×#13511yescharacteristic
430C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
431C22×C7#57511yescharacteristic
432C4×C7—22no—
433C4×C7—42no—
434C4×C7—42no—
435C4×C7—42no—
436C4×C7C4×#4111yescharacteristic
437C4×C7C4×#4111yescharacteristic
438C4×C7#52911yescharacteristic
439C4×C7#52911yescharacteristic
440Dic7—22no—
441Dic7—22no—
442Dic7—41no—
443Dic7—42no—
444Dic7—42no—
445Dic7—81no—
446Dic7—81no—
447Dic7#48214yescomplemented
448C24—21no—
449C24D4×D711yescharacteristic
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4C2×C4×D712yes—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—71no—
467C22×C4—71no—
468C22×C4—71no—
469C22×C4—71no—
470C22×C4—142no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—142no—
474C22×C4—142no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—282no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4—281no—
483C22×C4C2×C4×D712yes—
484C22×C4D4×D711yescharacteristic
485C22×C4#115912yes—
486C22×C4C23×D711yescharacteristic
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—72no—
494C2×C8—72no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—72no—
503C4⋊C4—72no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—282no—
511C4⋊C4C2×C4×D712yes—
512C4⋊C4#115912yes—
513C4⋊C4#115912yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—144no—
517C22⋊C4—282no—
518C22⋊C4—282no—
519C22⋊C4—282no—
520C22⋊C4—282no—
521C42—24no—
522C42—72no—
523C42—72no—
524C42—142no—
525C42—142no—
526C42—144no—
527C42—141no—
528C42—141no—
529C42—282no—
530C42—282no—
531C42—282no—
532C42—282no—
533C42—282no—
534C2×C7—42no—
535C2×C7—82no—
536C2×C7C4×#12311yescharacteristic
537C2×C7C4×#13511yescharacteristic
538C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
539C2×C7C2×#48211yescharacteristic
540C2×C7#309111yescharacteristic
541C2×C7#310211yescharacteristic
542C2×C7#310611yescharacteristic
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23(C22⋊D7)⋊C412yes—
557C23C2×D4×D711yescharacteristic
558D4—22no—
559D4—22no—
560D4—48no—
561D4—41no—
562D4—41no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—72no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—72no—
582C2×C4—72no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—141no—
586C2×C4—141no—
587C2×C4—142no—
588C2×C4—142no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—284no—
593C2×C4—284no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—564no—
605C2×C4(C22⋊D7)⋊C412yes—
606C2×C4C22×C4×D711yescharacteristic
607C2×C4C2×#115911yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4C2×#115911yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4D7×#4111yescharacteristic
612C8—41no—
613C8—41no—
614C8—142no—
615C8—142no—
616C7C4×#48211yescharacteristic, complemented, Sylow
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×(C22⋊D7)⋊C411yescharacteristic
632C22#9027711yescharacteristic
633C22D7×#12311yescharacteristic
634C22D7×#13511yescharacteristic
635C22D7×#13511yescharacteristic
636C22D7×AGL(1,ℤ/8)11yescharacteristic
637C22#9046411yescharacteristic
638C4—22no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—74no—
643C4—142no—
644C4—142no—
645C4—282no—
646C4—281no—
647C4—282no—
648C4—561no—
649C4—561no—
650C4C4×#115911yescharacteristic
651C4C4×#115911yescharacteristic
652C4#9029111yescharacteristic
653C4#9029111yescharacteristic
654C2—42no—
655C2—82no—
656C2#1173210611yescharacteristic
657C2#1173626311yescharacteristic
658C2#1173635911yescharacteristic
659C2#1173751311yescharacteristic
660C2#1173751411yescharacteristic
661C2D7×#48211yescharacteristic
662C2#1173881611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1875C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
cosocleC22×C4×C7C24
hypercenter#1875D7
hypocenterC7C4×#482
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1875

Complements

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

NH ≅ G/Ntype
3. Dic7×#117C2semidirect
5. #11729330C2semidirect
6. #11727928C2semidirect
7. #11727927C2semidirect
8. Dic7×#102C2semidirect
9. Dic7×#102C2semidirect
10. #11727680C2semidirect
11. Dic7×C22⋊C8C2semidirect
12. #11726575C2semidirect
13. #11726573C2semidirect
14. #11726539C2semidirect
15. #11726538C2semidirect
33. #89807C4semidirect
88. #6489C2×C4semidirect
111. #6406C2×C4semidirect
112. #6406C2×C4semidirect
120. C22⋊(C7⋊C8)C2×C4semidirect
124. #6382C2×C4semidirect
126. #6381C2×C4semidirect
447. Dic7#482semidirect
616. C7C4×#482semidirect

Maximal subgroups

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

subgroupindex
Dic7×#1212
C2×#898072
Dic7×#1172
C7×#18752
#117293302
#117279282
#117279272
Dic7×#1022
#117276802
Dic7×C22⋊C82
#117265752
#117265732
#117265392
#117265382
C4×#4827