Finite Group Databasebeta

← C7×#18457 #13195673

Subgroups — C7×#18457 #13195673

Order 1792.

954 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3536C211yesmaximal, characteristic
2C7×#3196C211yesmaximal, characteristic, complemented
3C7×#3187C211yesmaximal, characteristic, complemented
4C7×#2817C211yesmaximal, characteristic, complemented
5C7×#1616C212yesmaximal, complemented
6C7×#1615C212yesmaximal, complemented
7C7×#1604C212yesmaximal, complemented
8C7×#1603C212yesmaximal, complemented
9C7×#1516C211yesmaximal, characteristic, complemented
10C7×#1510C211yesmaximal, characteristic, complemented
11C7×#1472C211yesmaximal, characteristic, complemented
12C7×#562—22no—
13C7×#554—21no—
14C7×#548—21no—
15C7×#547—21no—
16C7×#546—21no—
17C7×#546—22no—
18C7×#543—21no—
19C7×#542—21no—
20C7×#539C2212yes—
21C7×#538—21no—
22C7×#537—21no—
23C7×#537C2212yes—
24C7×#525C2211yescharacteristic
25C2×C7×C22⋊Q8C2211yescharacteristic
26C2×C7×C22⋊Q8C2211yescharacteristic
27C2×C7×#121C2211yescharacteristic
28C7×#485—21no—
29C7×#484C2211yescharacteristic, complemented
30C7×#483—21no—
31C7×#481C2212yes—
32C7×#481C2211yescharacteristic, complemented
33C7×#470—21no—
34C7×#468—21no—
35C7×#465—22no—
36C7×#452—21no—
37C7×#451C2211yescharacteristic, complemented
38C7×#450—21no—
39C7×#448C2212yes—
40C7×#448C2211yescharacteristic, complemented
41C7×#407C2211yescharacteristic
42C2×C7×C22⋊C8C2211yescharacteristic
43C7×#400—21no—
44C7×#400C2212yescomplemented
45C7×#397—22no—
46C7×#395—21no—
47C7×#395C2212yescomplemented
48C7×#381C2211yescharacteristic
49C7×#328C2212yescomplemented
50C7×#327C2212yes—
51C7×#327C2212yes—
52C7×#327C2212yescomplemented
53C7×#324C2212yescomplemented
54C7×#324C2212yescomplemented
55#18457C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—22no—
58C2×C7×#41D411yescharacteristic
59C22×C7×Q8D411yescharacteristic
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4—21no—
62C22×C7×D4D411yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C2×C7×Q16—21no—
65C2×C7×SD16—21no—
66C2×C7×SD16—22no—
67C2×C7×M16—21no—
68C22×C7×C8—21no—
69C7×C4⋊Q8—21no—
70C7×#126—22no—
71C7×#125—22no—
72C7×#124—22no—
73C7×#124—21no—
74C7×#124—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—21no—
79C7×#123—22no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8D411yescharacteristic
96C7×C22⋊Q8C2311yescharacteristic
97C7×#121—21no—
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—22no—
101C7×#121—22no—
102C7×#121—21no—
103C7×#121—21no—
104C7×#121—22no—
105C7×#121D411yescharacteristic
106C7×#121D412yes—
107C7×#121C2312yes—
108C7×#121C2311yescharacteristic
109C7×#120—22no—
110C7×#120—21no—
111C4×C7×Q8—21no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—21no—
116C4×C7×D4—22no—
117C7×#117D411yescharacteristic
118C2×C7×C4⋊C4—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4—22no—
121C2×C7×C4⋊C4D411yescharacteristic
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4C2311yescharacteristic
124C2×C7×C4⋊C4C2312yes—
125C2×C7×C22⋊C4—22no—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4D412yes—
130C7×#107—21no—
131C7×#106—21no—
132C7×#106—22no—
133C7×#103—21no—
134C7×#103—21no—
135C7×#103—21no—
136C7×#103—22no—
137C7×#102—21no—
138C7×#102—22no—
139C7×C22⋊C8—21no—
140C7×C22⋊C8—21no—
141C7×C22⋊C8C2312yes—
142C7×C22⋊C8C2311yescharacteristic
143C7×C22⋊C8C2311yescharacteristic
144C7×#95—21no—
145C7×#95—22no—
146C7×#95—22no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3536C2×C711yescharacteristic
150#3196C2×C711yescharacteristic, complemented
151#3187C2×C711yescharacteristic, complemented
152#2817C2×C711yescharacteristic, complemented
153#1616C2×C712yescomplemented
154#1615C2×C712yescomplemented
155#1604C2×C712yescomplemented
156#1603C2×C712yescomplemented
157#1516C2×C711yescharacteristic, complemented
158#1510C2×C711yescharacteristic, complemented
159#1472C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C24×C7C2×D411yescharacteristic
162C7×#41—84no—
163C7×#41—81no—
164C7×#41—82no—
165C2×C7×Q8—22no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—21no—
169C2×C7×Q8—21no—
170C2×C7×Q8—81no—
171C2×C7×Q8—81no—
172C2×C7×Q8C2×D411yescharacteristic
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4—22no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—22no—
180C2×C7×D4—21no—
181C2×C7×D4—81no—
182C2×C7×D4—82no—
183C2×C7×D4C2×D411yescharacteristic
184C2×C7×D4C2×D412yes—
185C22×C4×C7—22no—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—21no—
189C22×C4×C7—21no—
190C22×C4×C7—22no—
191C22×C4×C7—21no—
192C22×C4×C7—21no—
193C22×C4×C7—81no—
194C22×C4×C7C2×D412yes—
195C22×C4×C7C2×D412yes—
196C22×C4×C7C2×D411yescharacteristic
197C22×C4×C7C2×D411yescharacteristic
198C22×C4×C7C2×D411yescharacteristic
199C22×C4×C7C2×D411yescharacteristic
200C22×C4×C7C2×D411yescharacteristic
201C22×C4×C7C2411yescharacteristic
202C7×Q16—81no—
203C7×SD16—82no—
204C7×SD16—81no—
205C7×M16—81no—
206C2×C7×C8—22no—
207C2×C7×C8—21no—
208C2×C7×C8—21no—
209C2×C7×C8—81no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—21no—
218C7×C4⋊C4—22no—
219C7×C4⋊C4—22no—
220C7×C4⋊C4—21no—
221C7×C4⋊C4—21no—
222C7×C4⋊C4—21no—
223C7×C4⋊C4—41no—
224C7×C4⋊C4—41no—
225C7×C4⋊C4—82no—
226C7×C4⋊C4C2×D412yes—
227C7×C4⋊C4C2×D411yescharacteristic
228C7×C4⋊C4C2×D411yescharacteristic
229C7×C22⋊C4—22no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—21no—
238C7×C22⋊C4—21no—
239C7×C22⋊C4—22no—
240C7×C22⋊C4—21no—
241C7×C22⋊C4—22no—
242C7×C22⋊C4—21no—
243C7×C22⋊C4—41no—
244C7×C22⋊C4—41no—
245C7×C22⋊C4—81no—
246C42×C7—22no—
247C42×C7—21no—
248#562—22no—
249#554—21no—
250#548—21no—
251#547—21no—
252#546—21no—
253#546—22no—
254#543—21no—
255#542—21no—
256#539C22×C712yes—
257#538—21no—
258#537—21no—
259#537C22×C712yes—
260#525C22×C711yescharacteristic
261C2×C22⋊Q8C22×C711yescharacteristic
262C2×C22⋊Q8C22×C711yescharacteristic
263C2×#121C22×C711yescharacteristic
264#485—21no—
265#484C22×C711yescharacteristic, complemented
266#483—21no—
267#481C22×C712yes—
268#481C22×C711yescharacteristic, complemented
269#470—21no—
270#468—21no—
271#465—22no—
272#452—21no—
273#451C22×C711yescharacteristic, complemented
274#450—21no—
275#448C22×C712yes—
276#448C22×C711yescharacteristic, complemented
277#407C22×C711yescharacteristic
278C2×C22⋊C8C22×C711yescharacteristic
279#400—21no—
280#400C22×C712yescomplemented
281#397—22no—
282#395—21no—
283#395C22×C712yescomplemented
284#381C22×C711yescharacteristic
285#328C22×C712yescomplemented
286#327C22×C712yes—
287#327C22×C712yes—
288#327C22×C712yescomplemented
289#324C22×C712yescomplemented
290#324C22×C712yescomplemented
291C23×C7—22no—
292C23×C7—22no—
293C23×C7—21no—
294C23×C7—41no—
295C23×C7—41no—
296C23×C7—42no—
297C23×C7—81no—
298C23×C7—81no—
299C23×C7#12011yescharacteristic
300C23×C7#12011yescharacteristic
301C23×C7C22×D411yescharacteristic
302C7×Q8—41no—
303C7×Q8—42no—
304C7×Q8—42no—
305C7×Q8—41no—
306C7×Q8—41no—
307C7×Q8—81no—
308C7×Q8—82no—
309C7×D4—42no—
310C7×D4—41no—
311C7×D4—42no—
312C7×D4—42no—
313C7×D4—44no—
314C7×D4—42no—
315C7×D4—82no—
316C7×D4—81no—
317C7×D4—82no—
318C7×D4—82no—
319C2×C4×C7—21no—
320C2×C4×C7—22no—
321C2×C4×C7—21no—
322C2×C4×C7—22no—
323C2×C4×C7—21no—
324C2×C4×C7—22no—
325C2×C4×C7—24no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—42no—
333C2×C4×C7—42no—
334C2×C4×C7—42no—
335C2×C4×C7—42no—
336C2×C4×C7—42no—
337C2×C4×C7—42no—
338C2×C4×C7—41no—
339C2×C4×C7—41no—
340C2×C4×C7—42no—
341C2×C4×C7—81no—
342C2×C4×C7—82no—
343C2×C4×C7—82no—
344C2×C4×C7—82no—
345C2×C4×C7#12011yescharacteristic
346C2×C4×C7#12011yescharacteristic
347C2×C4×C7C22×D411yescharacteristic
348C2×C4×C7C22×D411yescharacteristic
349C7×C8—42no—
350C7×C8—41no—
351C7×C8—41no—
352C2×#41—21no—
353C2×#41—22no—
354C2×#41C7×D411yescharacteristic
355C22×Q8C7×D411yescharacteristic
356C22×Q8C23×C711yescharacteristic
357C22×D4—21no—
358C22×D4C7×D411yescharacteristic
359C23×C4C23×C711yescharacteristic
360C2×Q16—21no—
361C2×SD16—22no—
362C2×SD16—21no—
363C2×M16—21no—
364C22×C8—21no—
365C4⋊Q8—21no—
366#126—22no—
367#125—22no—
368#124—22no—
369#124—21no—
370#124—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374#123—21no—
375#123—22no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8C7×D411yescharacteristic
392C22⋊Q8C23×C711yescharacteristic
393#121—21no—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—22no—
398#121—21no—
399#121—21no—
400#121—22no—
401#121C7×D412yes—
402#121C7×D411yescharacteristic
403#121C23×C712yes—
404#121C23×C711yescharacteristic
405#120—22no—
406#120—21no—
407C4×Q8—21no—
408C4×D4—22no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—21no—
412C4×D4—22no—
413#117C7×D411yescharacteristic
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4C7×D411yescharacteristic
418C2×C4⋊C4C7×D411yescharacteristic
419C2×C4⋊C4C23×C711yescharacteristic
420C2×C4⋊C4C23×C712yes—
421C2×C22⋊C4—22no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4C7×D412yes—
426#107—21no—
427#106—22no—
428#106—21no—
429#103—21no—
430#103—22no—
431#103—21no—
432#103—21no—
433#102—22no—
434#102—21no—
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8C23×C711yescharacteristic
438C22⋊C8C23×C711yescharacteristic
439C22⋊C8C23×C712yes—
440#95—21no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95C23×C712yes—
445C22×C7—21no—
446C22×C7—21no—
447C22×C7—21no—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—21no—
451C22×C7—21no—
452C22×C7—21no—
453C22×C7—41no—
454C22×C7—42no—
455C22×C7—41no—
456C22×C7—42no—
457C22×C7—41no—
458C22×C7—42no—
459C22×C7—82no—
460C22×C7—162no—
461C22×C7C2×#12011yescharacteristic
462C22×C7#57712yes—
463C4×C7—21no—
464C4×C7—21no—
465C4×C7—21no—
466C4×C7—21no—
467C4×C7—42no—
468C4×C7—42no—
469C4×C7—41no—
470C4×C7—41no—
471C4×C7—41no—
472C4×C7—42no—
473C4×C7—82no—
474C4×C7—81no—
475C4×C7—81no—
476C24—21no—
477C24C2×C7×D411yescharacteristic
478#41—84no—
479#41—81no—
480#41—82no—
481C2×Q8—22no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—81no—
487C2×Q8—81no—
488C2×Q8C2×C7×D411yescharacteristic
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—22no—
493C2×D4—22no—
494C2×D4—21no—
495C2×D4—22no—
496C2×D4—21no—
497C2×D4—81no—
498C2×D4—82no—
499C2×D4C2×C7×D411yescharacteristic
500C2×D4C2×C7×D412yes—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—22no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—81no—
510C22×C4C2×C7×D412yes—
511C22×C4C2×C7×D412yes—
512C22×C4C2×C7×D411yescharacteristic
513C22×C4C2×C7×D411yescharacteristic
514C22×C4C2×C7×D411yescharacteristic
515C22×C4C2×C7×D411yescharacteristic
516C22×C4C2×C7×D411yescharacteristic
517C22×C4C24×C711yescharacteristic
518Q16—81no—
519SD16—82no—
520SD16—81no—
521M16—81no—
522C2×C8—22no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—81no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—82no—
542C4⋊C4C2×C7×D412yes—
543C4⋊C4C2×C7×D411yescharacteristic
544C4⋊C4C2×C7×D411yescharacteristic
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—21no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—21no—
554C22⋊C4—21no—
555C22⋊C4—22no—
556C22⋊C4—21no—
557C22⋊C4—22no—
558C22⋊C4—21no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—81no—
562C42—22no—
563C42—21no—
564C2×C7—22no—
565C2×C7—41no—
566C2×C7—41no—
567C2×C7—81no—
568C2×C7—82no—
569C2×C7#300911yescharacteristic
570C2×C7#302011yescharacteristic
571C2×C7#302611yescharacteristic
572C23—22no—
573C23—22no—
574C23—21no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—81no—
579C23—81no—
580C23C7×#12011yescharacteristic
581C23C7×#12011yescharacteristic
582C23C22×C7×D411yescharacteristic
583Q8—41no—
584Q8—42no—
585Q8—42no—
586Q8—41no—
587Q8—41no—
588Q8—81no—
589Q8—82no—
590D4—42no—
591D4—41no—
592D4—42no—
593D4—42no—
594D4—44no—
595D4—42no—
596D4—82no—
597D4—81no—
598D4—82no—
599D4—82no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—81no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4C7×#12011yescharacteristic
627C2×C4C7×#12011yescharacteristic
628C2×C4C22×C7×D411yescharacteristic
629C2×C4C22×C7×D411yescharacteristic
630C8—42no—
631C8—41no—
632C8—41no—
633C7#1845711yescharacteristic, direct product factor, Sylow
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—42no—
648C22—82no—
649C22—162no—
650C22C2×C7×#12011yescharacteristic
651C22C7×#57712yes—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—82no—
663C4—81no—
664C4—81no—
665C2—22no—
666C2—41no—
667C2—41no—
668C2—82no—
669C2—81no—
670C2C7×#300911yescharacteristic
671C2C7×#302011yescharacteristic
672C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3196C2semidirect
3. C7×#3187C2semidirect
4. C7×#2817C2semidirect
5. C7×#1616C2semidirect
6. C7×#1615C2semidirect
7. C7×#1604C2semidirect
8. C7×#1603C2semidirect
9. C7×#1516C2semidirect
10. C7×#1510C2semidirect
11. C7×#1472C2semidirect
29. C7×#484C22semidirect
32. C7×#481C22semidirect
37. C7×#451C22semidirect
40. C7×#448C22semidirect
44. C7×#400C22semidirect
47. C7×#395C22semidirect
49. C7×#328C22semidirect
52. C7×#327C22semidirect
53. C7×#324C22semidirect
54. C7×#324C22semidirect
55. #18457C7direct (N × H)
150. #3196C2×C7semidirect
151. #3187C2×C7semidirect
152. #2817C2×C7semidirect
153. #1616C2×C7semidirect
154. #1615C2×C7semidirect
155. #1604C2×C7semidirect
156. #1603C2×C7semidirect
157. #1516C2×C7semidirect
158. #1510C2×C7semidirect
159. #1472C2×C7semidirect
265. #484C22×C7semidirect
268. #481C22×C7semidirect
273. #451C22×C7semidirect
276. #448C22×C7semidirect
280. #400C22×C7semidirect
283. #395C22×C7semidirect
285. #328C22×C7semidirect
288. #327C22×C7semidirect
289. #324C22×C7semidirect
290. #324C22×C7semidirect
633. C7#18457direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35362
C7×#31962
C7×#31872
C7×#28172
C7×#16162
C7×#16152
C7×#16042
C7×#16032
C7×#15162
C7×#15102
C7×#14722
#184577