Finite Group Databasebeta

← C4×#89660 #13360968

Subgroups — C4×#89660 #13360968

Order 1792.

888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89660C211yesmaximal, characteristic
2C4×#6460C211yesmaximal, characteristic
3C4×#6455C211yesmaximal, characteristic
4C4×C7×#107C211yesmaximal, characteristic, complemented
5#11728387C211yesmaximal, characteristic
6#11728378C211yesmaximal, characteristic, complemented
7#11728363C211yesmaximal, characteristic, complemented
8C4×#6390C211yesmaximal, characteristic, complemented
9C4×(C2×D7)⋊C8C211yesmaximal, characteristic
10C4×#6382C211yesmaximal, characteristic, complemented
11C4×#6384C211yesmaximal, characteristic, complemented
12#11726633C211yesmaximal, characteristic
13#11726627C211yesmaximal, characteristic
14#11726571C211yesmaximal, characteristic, complemented
15#11726547C211yesmaximal, characteristic, complemented
16C2×#6460C2211yescharacteristic
17C2×#6455C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×C42×D7C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×#107C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C4⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C8C2211yescharacteristic
27C2×(C2×D7)⋊C8C2211yescharacteristic
28C2×#6390C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30#89759C2211yescharacteristic
31#89757C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C4⋊C4×Dic7C2211yescharacteristic
34#89746C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C4×(C2×D7)⋊C4—21no—
40C4×(C2×D7)⋊C4—21no—
41C4×C4⋊Dic7—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43C42×Dic7—21no—
44C2×C4×C7⋊C8C2211yescharacteristic
45#89660C418yesdirect product factor
46#89440—21no—
47#89438—21no—
48#89425—21no—
49#89420—21no—
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60C4×#485—71nomaximal, Sylow
61C22×C4×D7D411yescharacteristic
62C22×C4×D7D411yescharacteristic
63C22×C4×D7C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—21no—
84C2×C4×Dic7—21no—
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7D411yescharacteristic
91C2×C4×Dic7D411yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95#6460—22no—
96#6460C2×C414yes—
97#6455—22no—
98#6455—22no—
99#6455—22no—
100#6455C2×C414yes—
101C42×D7—22no—
102C42×D7—22no—
103C4×Dic14—22no—
104C4×Dic14—22no—
105C7×#107—22no—
106C7×#107C2×C414yescomplemented
107C4×C7×C8—22no—
108C7×#95—21no—
109C7×#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117(C2×D7)⋊C8—22no—
118(C2×D7)⋊C8C2×C414yes—
119#6390—22no—
120#6390C2×C414yescomplemented
121#6384—22no—
122#6384C2×C414yescomplemented
123#6382—22no—
124#6382C2×C414yescomplemented
125C4×C7⋊C8—22no—
126C2×#485—71no—
127C4×C22⋊Q8—71no—
128C4×#117—71no—
129#1928—71no—
130#1927—71no—
131#1919—71no—
132#1918—71no—
133#1873—71no—
134#1854—71no—
135#1825—71no—
136C4×#107D711yescharacteristic, complemented
137C4×#106—71no—
138C4×#103—71no—
139C4×#103—71no—
140C4×C22⋊C8—71no—
141C23×D7C2×D411yescharacteristic
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7C2×D411yescharacteristic
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C2×C4×D7—21no—
157C2×C4×D7—21no—
158C2×C4×D7—21no—
159C2×C4×D7—21no—
160C2×C4×D7—22no—
161C2×C4×D7—22no—
162C2×C4×D7—22no—
163C2×C4×D7—22no—
164C2×C4×D7—22no—
165C2×C4×D7—41no—
166C2×C4×D7C22×C412yes—
167C2×C4×D7#4112yes—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14C22×C412yes—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C22×C412yes—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4C22×C412yes—
186C7×C4⋊C4C22×C412yes—
187C7×C4⋊C4#4112yes—
188C42×C7—42no—
189C42×C7—42no—
190C42×C7#4112yes—
191C42×C7#4112yes—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—24no—
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—42no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—24no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—42no—
202C4⋊Dic7C22×C412yes—
203C4⋊Dic7#4112yes—
204Dic7⋊C4—42no—
205C4×Dic7—24no—
206C4×Dic7—22no—
207C4×Dic7—22no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—24no—
211C4×Dic7—42no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C4×Dic7—42no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C22×C412yes—
221C2×C22⋊Q8—71no—
222C2×C4×Q8—71no—
223C2×#117—71no—
224C2×#117—71no—
225C22×C42—71no—
226#485—78no—
227C2×#107C2×D711yescharacteristic
228C2×#107C2×D711yescharacteristic
229C2×#106—71no—
230C2×#106—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#103—71no—
235C2×C22⋊C8—71no—
236C2×C22⋊C8—71no—
237C2×C4×C8—71no—
238C2×C4×C8C2×D711yescharacteristic
239#391—71no—
240#389—71no—
241#387—141no—
242#385—141no—
243#380—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4—141no—
248C4×C4⋊C4C2×D711yescharacteristic
249C4×C4⋊C4C2×D711yescharacteristic
250C4×C22⋊C4—141no—
251C4×C22⋊C4—141no—
252C4×C22⋊C4—141no—
253#376—141no—
254C43—141no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340C2×D711yescharacteristic
262#340C2×D711yescharacteristic
263#336—71no—
264#336—71no—
265C23×C7C22×D411yescharacteristic
266C22×D7—21no—
267C22×D7—21no—
268C22×D7—21no—
269C22×D7—21no—
270C22×D7—21no—
271C22×D7—21no—
272C22×D7C4×D412yes—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7#12312yes—
288C2×C4×C7#12312yes—
289C2×C4×C7#13512yes—
290C2×C4×C7#13712yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—21no—
304C2×Dic7—22no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7C4×D412yes—
317C2×Dic7#13512yes—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—22no—
322C4×D7—44no—
323C4×D7—44no—
324Dic14—22no—
325Dic14—22no—
326Dic14—41no—
327Dic14—41no—
328C7×C8—22no—
329C7×C8—41no—
330C7⋊C8—22no—
331C7⋊C8—41no—
332C22×Q8—71no—
333C23×C4—71no—
334C23×C4—71no—
335C23×C4—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8C22×D711yescharacteristic
339C22×C8C22×D711yescharacteristic
340C22⋊Q8—74no—
341C22⋊Q8—142no—
342C4×Q8—142no—
343C4×Q8—142no—
344#117—74no—
345#117—142no—
346#117—142no—
347#117—142no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—141no—
368C2×C42—141no—
369C2×C42—142no—
370C2×C42—142no—
371C2×C42—141no—
372C2×C42—141no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42C22×D711yescharacteristic
376#107—22no—
377#107C4×D714yescomplemented
378#106—74no—
379#106—142no—
380#103—74no—
381#103—74no—
382#103—142no—
383#103—142no—
384C22⋊C8—74no—
385C22⋊C8—142no—
386C4×C8—22no—
387C4×C8—142no—
388#95—21no—
389#95—21no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397C22×C7C2×C4×D411yescharacteristic
398C22×C7C2×#12311yescharacteristic
399C22×C7#52511yescharacteristic
400C22×C7C2×#13511yescharacteristic
401C22×C7C2×#13511yescharacteristic
402C22×C7C2×#13711yescharacteristic
403C22×C7#57511yescharacteristic
404C2×D7—22no—
405C2×D7—22no—
406C2×D7—22no—
407C2×D7—41no—
408C2×D7—41no—
409C2×D7—41no—
410C2×D7—41no—
411C4×C7—22no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—81no—
416C4×C7#48514yescomplemented
417C4×C7C4×#4111yescharacteristic
418C4×C7C4×#4111yescharacteristic
419C4×C7#52911yescharacteristic
420C4×C7#52911yescharacteristic
421Dic7—22no—
422Dic7—22no—
423Dic7—41no—
424Dic7—42no—
425Dic7—42no—
426Dic7—42no—
427Dic7—81no—
428C24—71no—
429C2×Q8—72no—
430C2×Q8—141no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—72no—
442C22×C4—72no—
443C22×C4—71no—
444C22×C4—142no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—141no—
457C22×C4—281no—
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4C23×D711yescharacteristic
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—72no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8C2×C4×D712yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—72no—
479C4⋊C4—72no—
480C4⋊C4—144no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C4⋊C4C2×C4×D712yes—
488C4⋊C4C2×C4×D712yes—
489C4⋊C4#115712yes—
490C22⋊C4—144no—
491C22⋊C4—144no—
492C22⋊C4—144no—
493C22⋊C4—282no—
494C42—42no—
495C42—42no—
496C42—144no—
497C42—144no—
498C42—142no—
499C42—142no—
500C42—141no—
501C42—141no—
502C42—282no—
503C42—282no—
504C42—282no—
505C42—282no—
506C42#115712yes—
507C42#116112yes—
508C2×C7C4×#12311yescharacteristic
509C2×C7C4×#13511yescharacteristic
510C2×C7C4×#13711yescharacteristic
511C2×C7C2×#48511yescharacteristic
512C2×C7#309111yescharacteristic
513C2×C7#310211yescharacteristic
514C2×C7#310711yescharacteristic
515D7—42no—
516C23—72no—
517C23—141no—
518C23—141no—
519C23—141no—
520C23—141no—
521C23—141no—
522C23—141no—
523C23C2×D4×D711yescharacteristic
524Q8—142no—
525Q8—142no—
526Q8—281no—
527Q8—281no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—284no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—284no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4#645712yes—
570C2×C4#645712yes—
571C2×C4#647512yes—
572C2×C4#648112yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4C2×#115711yescharacteristic
576C2×C4C2×#116111yescharacteristic
577C2×C4D7×#4111yescharacteristic
578C2×C4D7×#4111yescharacteristic
579C8—22no—
580C8—41no—
581C8—142no—
582C8—281no—
583C7C4×#48511yescharacteristic, complemented, Sylow
584C22—142no—
585C22—142no—
586C22—142no—
587C22—281no—
588C22—281no—
589C22—281no—
590C22—281no—
591C22C2×#645711yescharacteristic
592C22C4×D4×D711yescharacteristic
593C22#9023811yescharacteristic
594C22C2×#647511yescharacteristic
595C22C2×#648111yescharacteristic
596C22D7×#13511yescharacteristic
597C22#9045911yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—142no—
604C4—142no—
605C4—282no—
606C4—281no—
607C4—282no—
608C4—282no—
609C4—561no—
610C4#8966014yesdirect product factor
611C4C4×#115711yescharacteristic
612C4#9021911yescharacteristic
613C4C4×#116111yescharacteristic
614C4#9027111yescharacteristic
615C2—282no—
616C2C4×#645711yescharacteristic
617C2C2×#8966011yescharacteristic
618C2#1173663011yescharacteristic
619C2C4×#647511yescharacteristic
620C2#1173703611yescharacteristic
621C2C4×#648111yescharacteristic
622C2#1173719211yescharacteristic
subgroupG/H
centerC22×C4D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#107C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenterC4×#107D7
hypocenterC7C4×#485
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
4. C4×C7×#107C2semidirect
6. #11728378C2semidirect
7. #11728363C2semidirect
8. C4×#6390C2semidirect
10. C4×#6382C2semidirect
11. C4×#6384C2semidirect
14. #11726571C2semidirect
15. #11726547C2semidirect
45. #89660C4direct (N × H)
106. C7×#107C2×C4semidirect
120. #6390C2×C4semidirect
122. #6384C2×C4semidirect
124. #6382C2×C4semidirect
136. C4×#107D7semidirect
377. #107C4×D7semidirect
416. C4×C7#485semidirect
583. C7C4×#485semidirect
610. C4#89660direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#896602
C4×#64602
C4×#64552
C4×C7×#1072
#117283872
#117283782
#117283632
C4×#63902
C4×(C2×D7)⋊C82
C4×#63822
C4×#63842
#117266332
#117266272
#117265712
#117265472
C4×#4857