Finite Group Databasebeta

← C7×#18454 #13195670

Subgroups — C7×#18454 #13195670

Order 1792.

936 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3538C211yesmaximal, characteristic, complemented
2C7×#3194C211yesmaximal, characteristic
3C7×#3187C211yesmaximal, characteristic, complemented
4C7×#2816C211yesmaximal, characteristic, complemented
5C7×#1611C212yesmaximal, complemented
6C7×#1609C212yesmaximal, complemented
7C7×#1599C212yesmaximal, complemented
8C7×#1597C212yesmaximal, complemented
9C7×#1516C211yesmaximal, characteristic, complemented
10C7×#1511C211yesmaximal, characteristic, complemented
11C7×#1472C211yesmaximal, characteristic, complemented
12C7×#563—22no—
13C7×#552—21no—
14C7×#552—21no—
15C7×#551—22no—
16C7×D4×Q8—21no—
17C7×D4×Q8—21no—
18C7×#544—21no—
19C7×#544C2212yes—
20C7×#541—21no—
21C7×#539—21no—
22C7×#539C2212yes—
23C7×#536—21no—
24C7×#525C2211yescharacteristic
25C2×C7×C22⋊Q8C2211yescharacteristic
26C2×C7×C22⋊Q8C2211yescharacteristic
27C2×C7×#121C2211yescharacteristic
28C7×#483C2212yes—
29C7×#483C2211yescharacteristic
30C7×#482—21no—
31C7×#481—21no—
32C7×#480C2211yescharacteristic
33C7×#469—21no—
34C7×#468—21no—
35C7×#465—22no—
36C7×#450C2212yes—
37C7×#450C2211yescharacteristic
38C7×#449—21no—
39C7×#448—21no—
40C7×#447C2211yescharacteristic
41C7×#407C2211yescharacteristic
42C2×C7×C22⋊C8C2211yescharacteristic
43C7×#399—21no—
44C7×#398C2212yes—
45C7×#394—22no—
46C7×#394C2212yes—
47C7×#393—21no—
48C7×#381C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18454C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41D411yescharacteristic
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8D411yescharacteristic
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C2×C7×SD16—21no—
64C2×C7×SD16—22no—
65C2×C7×D8—21no—
66C2×C7×M16—21no—
67C22×C7×C8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8—22no—
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—22no—
75C7×#126—22no—
76C7×#125—21no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—22no—
81C7×#124—22no—
82C7×#123—22no—
83C7×#123—22no—
84C7×#123—22no—
85C7×#123—21no—
86C7×#123—22no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—21no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—21no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8D411yescharacteristic
100C7×C22⋊Q8D412yes—
101C7×C22⋊Q8C2312yes—
102C7×C22⋊Q8C2311yescharacteristic
103C7×#121—21no—
104C7×#121—21no—
105C7×#121—21no—
106C7×#121—21no—
107C7×#121D411yescharacteristic
108C7×#121C2311yescharacteristic
109C7×#120—21no—
110C4×C7×Q8—21no—
111C4×C7×Q8—21no—
112C4×C7×D4—22no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4—22no—
116C7×#117—21no—
117C7×#117—22no—
118C7×#117—22no—
119C7×#117D411yescharacteristic
120C2×C7×C4⋊C4—21no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4D411yescharacteristic
124C2×C7×C4⋊C4C2311yescharacteristic
125C2×C7×C4⋊C4C2312yes—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4—22no—
130C2×C7×C22⋊C4—22no—
131C2×C7×C22⋊C4D412yes—
132C7×#107—21no—
133C7×#106—21no—
134C7×#106—22no—
135C7×#103—21no—
136C7×#103—22no—
137C7×#102—21no—
138C7×#102—21no—
139C7×#102—21no—
140C7×#102—22no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8C2311yescharacteristic
144C7×C22⋊C8C2312yes—
145C7×C22⋊C8C2311yescharacteristic
146C7×#95—21no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3538C2×C711yescharacteristic, complemented
150#3194C2×C711yescharacteristic
151#3187C2×C711yescharacteristic, complemented
152#2816C2×C711yescharacteristic, complemented
153#1611C2×C712yescomplemented
154#1609C2×C712yescomplemented
155#1599C2×C712yescomplemented
156#1597C2×C712yescomplemented
157#1516C2×C711yescharacteristic, complemented
158#1511C2×C711yescharacteristic, complemented
159#1472C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C24×C7C2×D411yescharacteristic
162C7×#41—81no—
163C2×C7×Q8—22no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—81no—
168C2×C7×Q8—82no—
169C2×C7×Q8—81no—
170C2×C7×Q8C2×D412yes—
171C2×C7×Q8C2×D411yescharacteristic
172C2×C7×D4—21no—
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4—21no—
177C2×C7×D4—41no—
178C2×C7×D4—41no—
179C2×C7×D4—81no—
180C2×C7×D4—82no—
181C2×C7×D4C2×D411yescharacteristic
182C22×C4×C7—21no—
183C22×C4×C7—22no—
184C22×C4×C7—21no—
185C22×C4×C7—21no—
186C22×C4×C7—22no—
187C22×C4×C7—21no—
188C22×C4×C7—81no—
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2×D412yes—
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C22×C4×C7C2×D411yescharacteristic
195C22×C4×C7C2×D411yescharacteristic
196C22×C4×C7C2411yescharacteristic
197C7×SD16—82no—
198C7×SD16—81no—
199C7×D8—81no—
200C7×M16—81no—
201C2×C7×C8—21no—
202C2×C7×C8—22no—
203C2×C7×C8—21no—
204C2×C7×C8—81no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
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—22no—
221C7×C4⋊C4—21no—
222C7×C4⋊C4—21no—
223C7×C4⋊C4—21no—
224C7×C4⋊C4—82no—
225C7×C4⋊C4C2×D412yes—
226C7×C4⋊C4C2×D411yescharacteristic
227C7×C4⋊C4C2×D411yescharacteristic
228C7×C22⋊C4—22no—
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—21no—
236C7×C22⋊C4—22no—
237C7×C22⋊C4—21no—
238C7×C22⋊C4—21no—
239C7×C22⋊C4—21no—
240C7×C22⋊C4—41no—
241C7×C22⋊C4—41no—
242C7×C22⋊C4—82no—
243C7×C22⋊C4—82no—
244C7×C22⋊C4—81no—
245C42×C7—22no—
246C42×C7—22no—
247C42×C7—22no—
248C42×C7—21no—
249#563—22no—
250#552—21no—
251#552—21no—
252#551—22no—
253D4×Q8—21no—
254D4×Q8—21no—
255#544—21no—
256#544C22×C712yes—
257#541—21no—
258#539—21no—
259#539C22×C712yes—
260#536—21no—
261#525C22×C711yescharacteristic
262C2×C22⋊Q8C22×C711yescharacteristic
263C2×C22⋊Q8C22×C711yescharacteristic
264C2×#121C22×C711yescharacteristic
265#483C22×C712yes—
266#483C22×C711yescharacteristic
267#482—21no—
268#481—21no—
269#480C22×C711yescharacteristic
270#469—21no—
271#468—21no—
272#465—22no—
273#450C22×C712yes—
274#450C22×C711yescharacteristic
275#449—21no—
276#448—21no—
277#447C22×C711yescharacteristic
278#407C22×C711yescharacteristic
279C2×C22⋊C8C22×C711yescharacteristic
280#399—21no—
281#398C22×C712yes—
282#394—22no—
283#394C22×C712yes—
284#393—21no—
285#381C22×C711yescharacteristic
286#328C22×C712yes—
287#328C22×C712yes—
288#328C22×C712yes—
289#327C22×C712yes—
290#324C22×C712yes—
291#324C22×C712yes—
292C23×C7—22no—
293C23×C7—21no—
294C23×C7—41no—
295C23×C7—41no—
296C23×C7—42no—
297C23×C7—42no—
298C23×C7—41no—
299C23×C7—42no—
300C23×C7#12011yescharacteristic
301C23×C7#12011yescharacteristic
302C23×C7C22×D411yescharacteristic
303C7×Q8—42no—
304C7×Q8—41no—
305C7×Q8—42no—
306C7×Q8—42no—
307C7×Q8—81no—
308C7×Q8—82no—
309C7×D4—41no—
310C7×D4—41no—
311C7×D4—41no—
312C7×D4—42no—
313C7×D4—42no—
314C7×D4—42no—
315C7×D4—81no—
316C7×D4—82no—
317C7×D4—82no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—22no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—22no—
324C2×C4×C7—22no—
325C2×C4×C7—22no—
326C2×C4×C7—22no—
327C2×C4×C7—41no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—41no—
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—42no—
340C2×C4×C7—42no—
341C2×C4×C7—81no—
342C2×C4×C7—82no—
343C2×C4×C7#12011yescharacteristic
344C2×C4×C7#12011yescharacteristic
345C2×C4×C7C22×D411yescharacteristic
346C2×C4×C7C22×D411yescharacteristic
347C7×C8—42no—
348C7×C8—41no—
349C7×C8—41no—
350C2×#41C7×D411yescharacteristic
351C22×Q8—21no—
352C22×Q8C7×D411yescharacteristic
353C22×Q8C7×D411yescharacteristic
354C22×D4—21no—
355C22×D4C23×C711yescharacteristic
356C23×C4C23×C711yescharacteristic
357C2×SD16—22no—
358C2×SD16—21no—
359C2×D8—21no—
360C2×M16—21no—
361C22×C8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—22no—
364C4⋊Q8—22no—
365C4⋊Q8—22no—
366#126—22no—
367#126—22no—
368#126—22no—
369#126—22no—
370#125—21no—
371#125—22no—
372#125—22no—
373#125—22no—
374#125—22no—
375#124—22no—
376#123—22no—
377#123—22no—
378#123—22no—
379#123—21no—
380#123—22no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—22no—
393C22⋊Q8C7×D412yes—
394C22⋊Q8C7×D411yescharacteristic
395C22⋊Q8C23×C712yes—
396C22⋊Q8C23×C711yescharacteristic
397#121—21no—
398#121—21no—
399#121—21no—
400#121—21no—
401#121C7×D411yescharacteristic
402#121C23×C711yescharacteristic
403#120—21no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×D4—22no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—22no—
410#117—21no—
411#117—22no—
412#117—22no—
413#117C7×D411yescharacteristic
414C2×C4⋊C4—21no—
415C2×C4⋊C4—22no—
416C2×C4⋊C4C7×D411yescharacteristic
417C2×C4⋊C4C7×D411yescharacteristic
418C2×C4⋊C4C23×C711yescharacteristic
419C2×C4⋊C4C23×C712yes—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—22no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4C7×D412yes—
426#107—21no—
427#106—22no—
428#106—21no—
429#103—22no—
430#103—21no—
431#102—21no—
432#102—21no—
433#102—22no—
434#102—21no—
435C22⋊C8—21no—
436C22⋊C8—21no—
437C22⋊C8C23×C712yes—
438C22⋊C8C23×C711yescharacteristic
439C22⋊C8C23×C711yescharacteristic
440#95—21no—
441#95—22no—
442#95C23×C712yes—
443C22×C7—21no—
444C22×C7—21no—
445C22×C7—21no—
446C22×C7—21no—
447C22×C7—21no—
448C22×C7—21no—
449C22×C7—21no—
450C22×C7—21no—
451C22×C7—41no—
452C22×C7—41no—
453C22×C7—41no—
454C22×C7—82no—
455C22×C7—84no—
456C22×C7—81no—
457C22×C7—81no—
458C22×C7—81no—
459C22×C7C2×#12011yescharacteristic
460C22×C7#57712yes—
461C4×C7—21no—
462C4×C7—21no—
463C4×C7—21no—
464C4×C7—21no—
465C4×C7—42no—
466C4×C7—41no—
467C4×C7—42no—
468C4×C7—41no—
469C4×C7—41no—
470C4×C7—42no—
471C4×C7—42no—
472C4×C7—82no—
473C4×C7—81no—
474C24—21no—
475C24C2×C7×D411yescharacteristic
476#41—81no—
477C2×Q8—22no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—81no—
482C2×Q8—82no—
483C2×Q8—81no—
484C2×Q8C2×C7×D412yes—
485C2×Q8C2×C7×D411yescharacteristic
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—41no—
492C2×D4—41no—
493C2×D4—81no—
494C2×D4—82no—
495C2×D4C2×C7×D411yescharacteristic
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—21no—
502C22×C4—81no—
503C22×C4C2×C7×D412yes—
504C22×C4C2×C7×D412yes—
505C22×C4C2×C7×D411yescharacteristic
506C22×C4C2×C7×D411yescharacteristic
507C22×C4C2×C7×D411yescharacteristic
508C22×C4C2×C7×D411yescharacteristic
509C22×C4C2×C7×D411yescharacteristic
510C22×C4C24×C711yescharacteristic
511SD16—82no—
512SD16—81no—
513D8—81no—
514M16—81no—
515C2×C8—21no—
516C2×C8—22no—
517C2×C8—21no—
518C2×C8—81no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—82no—
539C4⋊C4C2×C7×D412yes—
540C4⋊C4C2×C7×D411yescharacteristic
541C4⋊C4C2×C7×D411yescharacteristic
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—21no—
550C22⋊C4—22no—
551C22⋊C4—21no—
552C22⋊C4—21no—
553C22⋊C4—21no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—81no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—21no—
563C2×C7—22no—
564C2×C7—41no—
565C2×C7—41no—
566C2×C7—81no—
567C2×C7—161no—
568C2×C7#301011yescharacteristic
569C2×C7#302011yescharacteristic
570C2×C7#302611yescharacteristic
571C23—22no—
572C23—21no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23—42no—
577C23—41no—
578C23—42no—
579C23C7×#12011yescharacteristic
580C23C7×#12011yescharacteristic
581C23C22×C7×D411yescharacteristic
582Q8—42no—
583Q8—41no—
584Q8—42no—
585Q8—42no—
586Q8—81no—
587Q8—82no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—81no—
595D4—82no—
596D4—82no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—81no—
621C2×C4—82no—
622C2×C4C7×#12011yescharacteristic
623C2×C4C7×#12011yescharacteristic
624C2×C4C22×C7×D411yescharacteristic
625C2×C4C22×C7×D411yescharacteristic
626C8—42no—
627C8—41no—
628C8—41no—
629C7#1845411yescharacteristic, direct product factor, Sylow
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—82no—
642C22—84no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22C2×C7×#12011yescharacteristic
647C22C7×#57712yes—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—41no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—81no—
661C2—22no—
662C2—41no—
663C2—41no—
664C2—81no—
665C2—161no—
666C2C7×#301011yescharacteristic
667C2C7×#302011yescharacteristic
668C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3194C2
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
1. C7×#3538C2semidirect
3. C7×#3187C2semidirect
4. C7×#2816C2semidirect
5. C7×#1611C2semidirect
6. C7×#1609C2semidirect
7. C7×#1599C2semidirect
8. C7×#1597C2semidirect
9. C7×#1516C2semidirect
10. C7×#1511C2semidirect
11. C7×#1472C2semidirect
55. #18454C7direct (N × H)
149. #3538C2×C7semidirect
151. #3187C2×C7semidirect
152. #2816C2×C7semidirect
153. #1611C2×C7semidirect
154. #1609C2×C7semidirect
155. #1599C2×C7semidirect
156. #1597C2×C7semidirect
157. #1516C2×C7semidirect
158. #1511C2×C7semidirect
159. #1472C2×C7semidirect
629. C7#18454direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35382
C7×#31942
C7×#31872
C7×#28162
C7×#16112
C7×#16092
C7×#15992
C7×#15972
C7×#15162
C7×#15112
C7×#14722
#184547