Finite Group Databasebeta

← C7×#18791 #13196007

Subgroups — C7×#18791 #13196007

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3480C211yesmaximal, characteristic
2C7×#3206C211yesmaximal, characteristic, complemented
3C7×#3079C211yesmaximal, characteristic, complemented
4C2×C7×#477C211yesmaximal, characteristic, complemented
5C7×#1638C212yesmaximal, complemented
6C7×#1636C212yesmaximal, complemented
7C7×#1631C212yesmaximal, complemented
8C7×#1630C212yesmaximal, complemented
9C7×#1547C211yesmaximal, characteristic, complemented
10C7×#1487C211yesmaximal, characteristic, complemented
11C7×#1477C211yesmaximal, characteristic, complemented
12C7×#559—21no—
13C7×#559C2211yescharacteristic
14C7×#557C2211yescharacteristic
15C7×#556—21no—
16C7×#556C2211yescharacteristic
17C7×#553—22no—
18C7×#553—21no—
19C7×#550C2211yescharacteristic
20C7×#548—21no—
21C7×#532—21no—
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529—21no—
25C7×#529—22no—
26C4×C7×#41—21no—
27C4×C7×#41—21no—
28C4×C7×#41C2211yescharacteristic
29C2×C4×C7×Q8C2211yescharacteristic
30C7×#492C2212yes—
31C7×#487C2212yes—
32C7×#477—21no—
33C7×#477—21no—
34C7×#477C2212yes—
35C7×#477C2212yes—
36C7×#477C2212yes—
37C7×#476C2212yes—
38C7×#475—21no—
39C7×#474—21no—
40C7×#461—22no—
41C7×#448—22no—
42C7×#431C2211yescharacteristic
43C7×#428—21no—
44C2×C7×#107—21no—
45C7×#423C2211yescharacteristic
46C2×C7×C4⋊C8C2211yescharacteristic
47C7×#417—21no—
48C2×C7×#103—21no—
49C7×#332C2212yes—
50C7×#331C2212yes—
51C7×Q8⋊C8C2212yes—
52C7×Q8⋊C8C2212yes—
53C7×Q8⋊C8C2212yes—
54C7×D4⋊C8C2212yes—
55#18791C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—21no—
58C2×C7×#41D411yescharacteristic
59C22×C7×Q8—21no—
60C22×C7×Q8D411yescharacteristic
61C2×C7×SD16—24no—
62C2×C7×M16—21no—
63C22×C7×C8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8C2312yes—
69C7×C4⋊Q8C2311yescharacteristic
70C7×#127—21no—
71C7×#127C2311yescharacteristic
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—22no—
75C7×#125—22no—
76C7×#125—21no—
77C7×#124—22no—
78C7×#124—21no—
79C7×#124—21no—
80C7×#124—21no—
81C7×#124—21no—
82C7×#123—22no—
83C7×#123—22no—
84C7×#123—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—21no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—21no—
91C7×#121—21no—
92C7×#121D412yes—
93C4×C7×Q8—21no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×Q8—21no—
98C4×C7×Q8D411yescharacteristic
99C4×C7×Q8D411yescharacteristic
100C4×C7×Q8D411yescharacteristic
101C4×C7×Q8C2311yescharacteristic
102C4×C7×Q8C2311yescharacteristic
103C4×C7×Q8C2311yescharacteristic
104C4×C7×D4—22no—
105C4×C7×D4—22no—
106C4×C7×D4—22no—
107C4×C7×D4—22no—
108C4×C7×D4—22no—
109C4×C7×D4—21no—
110C4×C7×D4—21no—
111C4×C7×D4—21no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—21no—
115C4×C7×D4D411yescharacteristic
116C4×C7×D4D412yes—
117C4×C7×D4C2311yescharacteristic
118C7×#117—22no—
119C7×#117—22no—
120C7×#117—21no—
121C7×#117—21no—
122C7×#117D411yescharacteristic
123C2×C7×C4⋊C4—21no—
124C2×C7×C4⋊C4—21no—
125C2×C7×C4⋊C4—21no—
126C2×C7×C4⋊C4D411yescharacteristic
127C2×C42×C7—22no—
128C2×C42×C7—21no—
129C2×C42×C7—21no—
130C2×C42×C7C2311yescharacteristic
131C7×#107—22no—
132C7×#107—22no—
133C7×#107—22no—
134C7×#106—22no—
135C7×C4⋊C8—21no—
136C7×C4⋊C8—21no—
137C7×C4⋊C8—21no—
138C7×C4⋊C8C2312yes—
139C7×C4⋊C8C2312yes—
140C7×#103—22no—
141C7×#103—22no—
142C7×#103—22no—
143C7×#103—24no—
144C7×#102—22no—
145C7×C22⋊C8—21no—
146C4×C7×C8C2312yes—
147#3480C2×C711yescharacteristic
148#3206C2×C711yescharacteristic, complemented
149#3079C2×C711yescharacteristic, complemented
150C2×#477C2×C711yescharacteristic, complemented
151#1638C2×C712yescomplemented
152#1636C2×C712yescomplemented
153#1631C2×C712yescomplemented
154#1630C2×C712yescomplemented
155#1547C2×C711yescharacteristic, complemented
156#1487C2×C711yescharacteristic, complemented
157#1477C2×C711yescharacteristic, complemented
158C7×#41—22no—
159C7×#41—22no—
160C7×#41—42no—
161C7×#41—42no—
162C7×#41—42no—
163C7×#41—42no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—22no—
167C2×C7×Q8—22no—
168C2×C7×Q8—41no—
169C2×C7×Q8—41no—
170C2×C7×Q8—82no—
171C2×C7×Q8C2×D411yescharacteristic
172C2×C7×Q8C2×D411yescharacteristic
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×D4—22no—
175C2×C7×D4—21no—
176C2×C7×D4—22no—
177C2×C7×D4—22no—
178C2×C7×D4—21no—
179C2×C7×D4—21no—
180C2×C7×D4C2×D411yescharacteristic
181C2×C7×D4C2×D412yes—
182C22×C4×C7—22no—
183C22×C4×C7—21no—
184C22×C4×C7—22no—
185C22×C4×C7—21no—
186C22×C4×C7—21no—
187C22×C4×C7—22no—
188C22×C4×C7—21no—
189C22×C4×C7—21no—
190C22×C4×C7—41no—
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D411yescharacteristic
194C7×SD16—84no—
195C7×M16—42no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C2×C7×C8—22no—
199C2×C7×C8—42no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—22no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4—41no—
216C7×C4⋊C4C2×D412yes—
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C4⋊C4C2×D411yescharacteristic
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—21no—
232C42×C7—22no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—21no—
236C42×C7—21no—
237C42×C7—21no—
238C42×C7—21no—
239C42×C7—21no—
240C42×C7—21no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2411yescharacteristic
245#559—21no—
246#559C22×C711yescharacteristic
247#557C22×C711yescharacteristic
248#556—21no—
249#556C22×C711yescharacteristic
250#553—22no—
251#553—21no—
252#550C22×C711yescharacteristic
253#548—21no—
254#532—21no—
255#532C22×C711yescharacteristic
256C2×C4⋊Q8C22×C711yescharacteristic
257#529—21no—
258#529—22no—
259C4×#41—21no—
260C4×#41—21no—
261C4×#41C22×C711yescharacteristic
262C2×C4×Q8C22×C711yescharacteristic
263#492C22×C712yes—
264#487C22×C712yes—
265#477—21no—
266#477—21no—
267#477C22×C712yes—
268#477C22×C712yes—
269#477C22×C712yes—
270#476C22×C712yes—
271#475—21no—
272#474—21no—
273#461—22no—
274#448—22no—
275#431C22×C711yescharacteristic
276#428—21no—
277C2×#107—21no—
278#423C22×C711yescharacteristic
279C2×C4⋊C8C22×C711yescharacteristic
280#417—21no—
281C2×#103—21no—
282#332C22×C712yes—
283#331C22×C712yes—
284Q8⋊C8C22×C712yes—
285Q8⋊C8C22×C712yes—
286Q8⋊C8C22×C712yes—
287D4⋊C8C22×C712yes—
288C23×C7—21no—
289C23×C7—22no—
290C23×C7#12011yescharacteristic
291C7×Q8—21no—
292C7×Q8—21no—
293C7×Q8—21no—
294C7×Q8—21no—
295C7×Q8—21no—
296C7×Q8—42no—
297C7×Q8—44no—
298C7×Q8—41no—
299C7×Q8—41no—
300C7×Q8—82no—
301C7×Q8#13712yes—
302C7×D4—21no—
303C7×D4—21no—
304C7×D4—42no—
305C7×D4—44no—
306C7×D4—42no—
307C7×D4—44no—
308C7×D4—41no—
309C7×D4—41no—
310C7×D4—41no—
311C7×D4—41no—
312C7×D4—82no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—21no—
317C2×C4×C7—21no—
318C2×C4×C7—21no—
319C2×C4×C7—21no—
320C2×C4×C7—21no—
321C2×C4×C7—22no—
322C2×C4×C7—21no—
323C2×C4×C7—41no—
324C2×C4×C7—42no—
325C2×C4×C7—44no—
326C2×C4×C7—41no—
327C2×C4×C7—41no—
328C2×C4×C7—42no—
329C2×C4×C7—44no—
330C2×C4×C7—42no—
331C2×C4×C7—81no—
332C2×C4×C7—81no—
333C2×C4×C7—82no—
334C2×C4×C7—82no—
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7#12011yescharacteristic
338C2×C4×C7#13512yes—
339C2×C4×C7#13712yes—
340C2×C4×C7C22×D411yescharacteristic
341C2×C4×C7C22×D411yescharacteristic
342C2×C4×C7C22×D411yescharacteristic
343C7×C8—42no—
344C7×C8—42no—
345C7×C8—42no—
346C2×#41—21no—
347C2×#41—21no—
348C2×#41C7×D411yescharacteristic
349C22×Q8—21no—
350C22×Q8C7×D411yescharacteristic
351C2×SD16—24no—
352C2×M16—21no—
353C22×C8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8—21no—
357C4⋊Q8—21no—
358C4⋊Q8C23×C712yes—
359C4⋊Q8C23×C711yescharacteristic
360#127—21no—
361#127C23×C711yescharacteristic
362#126—22no—
363#126—22no—
364#126—22no—
365#125—22no—
366#125—21no—
367#124—22no—
368#124—21no—
369#124—21no—
370#124—21no—
371#124—21no—
372#123—22no—
373#123—22no—
374#123—22no—
375C22⋊Q8—22no—
376C22⋊Q8—21no—
377#121—22no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121—21no—
382#121C7×D412yes—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8C7×D411yescharacteristic
389C4×Q8C7×D411yescharacteristic
390C4×Q8C7×D411yescharacteristic
391C4×Q8C23×C711yescharacteristic
392C4×Q8C23×C711yescharacteristic
393C4×Q8C23×C711yescharacteristic
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—21no—
404C4×D4—21no—
405C4×D4C7×D411yescharacteristic
406C4×D4C7×D412yes—
407C4×D4C23×C711yescharacteristic
408#117—22no—
409#117—22no—
410#117—21no—
411#117—21no—
412#117C7×D411yescharacteristic
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4C7×D411yescharacteristic
417C2×C42—22no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42C23×C711yescharacteristic
421#107—22no—
422#107—22no—
423#107—22no—
424#106—22no—
425C4⋊C8—21no—
426C4⋊C8—21no—
427C4⋊C8—21no—
428C4⋊C8C23×C712yes—
429C4⋊C8C23×C712yes—
430#103—22no—
431#103—22no—
432#103—22no—
433#103—24no—
434#102—22no—
435C22⋊C8—21no—
436C4×C8C23×C712yes—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—41no—
440C22×C7—42no—
441C22×C7—42no—
442C22×C7—41no—
443C22×C7—42no—
444C22×C7—41no—
445C22×C7#45212yes—
446C22×C7C2×#12011yescharacteristic
447C4×C7—21no—
448C4×C7—22no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—41no—
452C4×C7—42no—
453C4×C7—41no—
454C4×C7—41no—
455C4×C7—41no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7—82no—
459C4×C7#45212yes—
460C4×C7C2×#13511yescharacteristic
461C4×C7C2×#13711yescharacteristic
462C4×C7C2×#13711yescharacteristic
463C4×C7#57511yescharacteristic
464#41—22no—
465#41—22no—
466#41—42no—
467#41—42no—
468#41—42no—
469#41—42no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—41no—
475C2×Q8—41no—
476C2×Q8—82no—
477C2×Q8C2×C7×D411yescharacteristic
478C2×Q8C2×C7×D411yescharacteristic
479C2×Q8C2×C7×D411yescharacteristic
480C2×D4—22no—
481C2×D4—21no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C7×D411yescharacteristic
487C2×D4C2×C7×D412yes—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—41no—
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D411yescharacteristic
500SD16—84no—
501M16—42no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—42no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—41no—
521C4⋊C4—41no—
522C4⋊C4C2×C7×D412yes—
523C4⋊C4C2×C7×D411yescharacteristic
524C4⋊C4C2×C7×D411yescharacteristic
525C4⋊C4C2×C7×D411yescharacteristic
526C4⋊C4C2×C7×D411yescharacteristic
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—21no—
534C22⋊C4—22no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42C2×C7×D411yescharacteristic
548C42C2×C7×D411yescharacteristic
549C42C2×C7×D411yescharacteristic
550C42C24×C711yescharacteristic
551C2×C7—22no—
552C2×C7—81no—
553C2×C7—82no—
554C2×C7C2×#45211yescharacteristic
555C2×C7#300711yescharacteristic
556C2×C7#302111yescharacteristic
557C23—21no—
558C23—22no—
559C23C7×#12011yescharacteristic
560Q8—21no—
561Q8—21no—
562Q8—21no—
563Q8—21no—
564Q8—21no—
565Q8—42no—
566Q8—44no—
567Q8—41no—
568Q8—41no—
569Q8—82no—
570Q8C7×#13712yes—
571D4—21no—
572D4—21no—
573D4—42no—
574D4—44no—
575D4—42no—
576D4—44no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4C7×#12011yescharacteristic
605C2×C4C7×#12011yescharacteristic
606C2×C4C7×#12011yescharacteristic
607C2×C4C7×#13512yes—
608C2×C4C7×#13712yes—
609C2×C4C22×C7×D411yescharacteristic
610C2×C4C22×C7×D411yescharacteristic
611C2×C4C22×C7×D411yescharacteristic
612C8—42no—
613C8—42no—
614C8—42no—
615C7#1879111yescharacteristic, direct product factor, Sylow
616C22—21no—
617C22—21no—
618C22—41no—
619C22—42no—
620C22—42no—
621C22—41no—
622C22—42no—
623C22—41no—
624C22C7×#45212yes—
625C22C2×C7×#12011yescharacteristic
626C4—21no—
627C4—22no—
628C4—21no—
629C4—21no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—82no—
638C4C7×#45212yes—
639C4C2×C7×#13511yescharacteristic
640C4C2×C7×#13711yescharacteristic
641C4C2×C7×#13711yescharacteristic
642C4C7×#57511yescharacteristic
643C2—22no—
644C2—81no—
645C2—82no—
646C2C2×C7×#45211yescharacteristic
647C2C7×#300711yescharacteristic
648C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3480C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3206C2semidirect
3. C7×#3079C2semidirect
4. C2×C7×#477C2semidirect
5. C7×#1638C2semidirect
6. C7×#1636C2semidirect
7. C7×#1631C2semidirect
8. C7×#1630C2semidirect
9. C7×#1547C2semidirect
10. C7×#1487C2semidirect
11. C7×#1477C2semidirect
55. #18791C7direct (N × H)
148. #3206C2×C7semidirect
149. #3079C2×C7semidirect
150. C2×#477C2×C7semidirect
151. #1638C2×C7semidirect
152. #1636C2×C7semidirect
153. #1631C2×C7semidirect
154. #1630C2×C7semidirect
155. #1547C2×C7semidirect
156. #1487C2×C7semidirect
157. #1477C2×C7semidirect
615. C7#18791direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34802
C7×#32062
C7×#30792
C2×C7×#4772
C7×#16382
C7×#16362
C7×#16312
C7×#16302
C7×#15472
C7×#14872
C7×#14772
#187917