Finite Group Databasebeta

← C7×#18323 #13195539

Subgroups — C7×#18323 #13195539

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3196C212yesmaximal, complemented
3C7×#2564C211yesmaximal, characteristic, complemented
4C7×#1615C212yesmaximal, complemented
5C7×#1614C211yesmaximal, characteristic, complemented
6C7×#1613C211yesmaximal, characteristic, complemented
7C7×#1603C212yesmaximal, complemented
8C7×#1602C211yesmaximal, characteristic, complemented
9C7×#1601C211yesmaximal, characteristic, complemented
10C7×#1504C212yesmaximal, complemented
11C7×#1465C211yesmaximal, characteristic, complemented
12C7×#543C2211yescharacteristic
13C7×#539—22no—
14C7×#539C2212yes—
15C7×#538—22no—
16C7×#538—21no—
17C7×#534—21no—
18C7×#534C2211yescharacteristic
19C7×#529—22no—
20C7×#529—21no—
21C7×#525—22no—
22C7×#525C2212yes—
23C2×C7×C22⋊Q8C2211yescharacteristic, complemented
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#485—22no—
27C7×#484C2212yescomplemented
28C7×#481C2212yes—
29C7×#470—22no—
30C7×#468—22no—
31C7×#452—22no—
32C7×#451C2212yescomplemented
33C7×#448C2212yes—
34C7×#407C2212yescomplemented
35C7×#400C2211yescharacteristic, complemented
36C7×#399C2211yescharacteristic
37C7×#395C2211yescharacteristic, complemented
38C7×#393C2211yescharacteristic
39C7×#387—21no—
40C7×#386—21no—
41C7×#385—21no—
42C7×#380—21no—
43C2×C7×#95C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yescomplemented
49C7×#323C2211yescharacteristic
50C7×#323C2211yescharacteristic
51#18323C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41D412yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C2×C7×Q16—22no—
61C2×C7×SD16—22no—
62C2×C7×M16—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#125—21no—
67C7×#125—22no—
68C7×#124—22no—
69C7×#124—21no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8D412yes—
83C7×C22⋊Q8C2312yes—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—21no—
88C7×#121D412yes—
89C7×#121C2312yes—
90C7×#120—22no—
91C7×#120—22no—
92C7×#120—22no—
93C4×C7×Q8—21no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—21no—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C7×#117—22no—
103C7×#117D412yes—
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4C2311yescharacteristic
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C22⋊C4—21no—
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4D411yescharacteristic
116C2×C7×C22⋊C4C2311yescharacteristic
117C2×C42×C7—21no—
118C2×C42×C7D411yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×#103—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×C22⋊C8—22no—
126C7×C22⋊C8C2312yes—
127C7×C22⋊C8C2312yes—
128C7×#95—21no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3459C2×C711yescharacteristic
136#3196C2×C712yescomplemented
137#2564C2×C711yescharacteristic, complemented
138#1615C2×C712yescomplemented
139#1614C2×C711yescharacteristic, complemented
140#1613C2×C711yescharacteristic, complemented
141#1603C2×C712yescomplemented
142#1602C2×C711yescharacteristic, complemented
143#1601C2×C711yescharacteristic, complemented
144#1504C2×C712yescomplemented
145#1465C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C24×C7C2×D411yescharacteristic
148C24×C7C2×D411yescharacteristic
149C7×#41—82no—
150C2×C7×Q8—21no—
151C2×C7×Q8—22no—
152C2×C7×Q8—81no—
153C2×C7×Q8C2×D412yes—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—41no—
162C2×C7×D4—42no—
163C2×C7×D4—42no—
164C2×C7×D4C2×D412yes—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—42no—
177C22×C4×C7—42no—
178C22×C4×C7—81no—
179C22×C4×C7—81no—
180C22×C4×C7—82no—
181C22×C4×C7—81no—
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2411yescharacteristic
189C7×Q16—82no—
190C7×SD16—82no—
191C7×M16—82no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—42no—
205C7×C4⋊C4—41no—
206C7×C4⋊C4—41no—
207C7×C4⋊C4—81no—
208C7×C4⋊C4C2×D412yes—
209C7×C4⋊C4C2×D412yes—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—42no—
223C7×C22⋊C4—42no—
224C7×C22⋊C4—81no—
225C42×C7—22no—
226C42×C7—22no—
227C42×C7—21no—
228C42×C7—21no—
229#543C22×C711yescharacteristic
230#539—22no—
231#539C22×C712yes—
232#538—22no—
233#538—21no—
234#534—21no—
235#534C22×C711yescharacteristic
236#529—22no—
237#529—21no—
238#525—22no—
239#525C22×C712yes—
240C2×C22⋊Q8C22×C711yescharacteristic, complemented
241C2×C4×D4—21no—
242C2×C4×D4C22×C711yescharacteristic
243#485—22no—
244#484C22×C712yescomplemented
245#481C22×C712yes—
246#470—22no—
247#468—22no—
248#452—22no—
249#451C22×C712yescomplemented
250#448C22×C712yes—
251#407C22×C712yescomplemented
252#400C22×C711yescharacteristic, complemented
253#399C22×C711yescharacteristic
254#395C22×C711yescharacteristic, complemented
255#393C22×C711yescharacteristic
256#387—21no—
257#386—21no—
258#385—21no—
259#380—21no—
260C2×#95C22×C711yescharacteristic, complemented
261#328C22×C712yescomplemented
262#328C22×C712yescomplemented
263#327C22×C712yes—
264#327C22×C712yes—
265#324C22×C712yescomplemented
266#323C22×C711yescharacteristic
267#323C22×C711yescharacteristic
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—22no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—42no—
278C23×C7—81no—
279C23×C7—81no—
280C23×C7#12012yes—
281C23×C7#13712yes—
282C23×C7C22×D411yescharacteristic
283C7×Q8—42no—
284C7×Q8—42no—
285C7×Q8—42no—
286C7×Q8—81no—
287C7×D4—42no—
288C7×D4—44no—
289C7×D4—44no—
290C7×D4—42no—
291C7×D4—42no—
292C7×D4—82no—
293C7×D4—82no—
294C7×D4—81no—
295C2×C4×C7—22no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—42no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—41no—
308C2×C4×C7—42no—
309C2×C4×C7—44no—
310C2×C4×C7—41no—
311C2×C4×C7—42no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—41no—
324C2×C4×C7—81no—
325C2×C4×C7—81no—
326C2×C4×C7—82no—
327C2×C4×C7—81no—
328C2×C4×C7—84no—
329C2×C4×C7#12011yescharacteristic
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7C22×D412yes—
332C7×C8—42no—
333C7×C8—42no—
334C2×#41C7×D412yes—
335C22×Q8C23×C711yescharacteristic
336C22×D4—21no—
337C22×D4C7×D411yescharacteristic
338C23×C4—21no—
339C23×C4—21no—
340C23×C4C7×D411yescharacteristic
341C23×C4C23×C711yescharacteristic
342C2×Q16—22no—
343C2×SD16—22no—
344C2×M16—22no—
345#126—22no—
346#126—22no—
347#126—22no—
348#125—21no—
349#125—22no—
350#124—22no—
351#124—21no—
352#123—22no—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—21no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8C7×D412yes—
365C22⋊Q8C23×C712yes—
366#121—22no—
367#121—22no—
368#121—22no—
369#121—21no—
370#121C7×D412yes—
371#121C23×C712yes—
372#120—22no—
373#120—22no—
374#120—22no—
375C4×Q8—21no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—22no—
384#117—22no—
385#117C7×D412yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4C23×C711yescharacteristic
391C2×C4⋊C4C23×C711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4C7×D411yescharacteristic
398C2×C22⋊C4C23×C711yescharacteristic
399C2×C42—21no—
400C2×C42C7×D411yescharacteristic
401#107—22no—
402#106—22no—
403#103—22no—
404#103—22no—
405#103—22no—
406#102—22no—
407C22⋊C8—22no—
408C22⋊C8C23×C712yes—
409C22⋊C8C23×C712yes—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95C23×C711yescharacteristic
416#95C23×C711yescharacteristic
417C22×C7—21no—
418C22×C7—21no—
419C22×C7—22no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—22no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—41no—
427C22×C7—41no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—81no—
432C22×C7—84no—
433C22×C7C2×#12011yescharacteristic
434C22×C7C2×#13711yescharacteristic
435C22×C7#57511yescharacteristic
436C4×C7—22no—
437C4×C7—21no—
438C4×C7—21no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—41no—
443C4×C7—42no—
444C4×C7—41no—
445C4×C7—42no—
446C4×C7—81no—
447C4×C7—81no—
448C4×C7—81no—
449C24—21no—
450C24C2×C7×D411yescharacteristic
451C24C2×C7×D411yescharacteristic
452#41—82no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—81no—
456C2×Q8C2×C7×D412yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4C2×C7×D412yes—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4—82no—
484C22×C4—81no—
485C22×C4C2×C7×D411yescharacteristic
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C2×C7×D411yescharacteristic
488C22×C4C2×C7×D411yescharacteristic
489C22×C4C2×C7×D412yes—
490C22×C4C2×C7×D412yes—
491C22×C4C24×C711yescharacteristic
492Q16—82no—
493SD16—82no—
494M16—82no—
495C2×C8—22no—
496C2×C8—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—42no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—81no—
511C4⋊C4C2×C7×D412yes—
512C4⋊C4C2×C7×D412yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—21no—
532C2×C7—21no—
533C2×C7—21no—
534C2×C7—42no—
535C2×C7—42no—
536C2×C7—82no—
537C2×C7#300711yescharacteristic
538C2×C7#301811yescharacteristic
539C2×C7#302611yescharacteristic
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—81no—
551C23—81no—
552C23C7×#12012yes—
553C23C7×#13712yes—
554C23C22×C7×D411yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—81no—
559D4—42no—
560D4—44no—
561D4—44no—
562D4—42no—
563D4—42no—
564D4—82no—
565D4—82no—
566D4—81no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—81no—
600C2×C4—84no—
601C2×C4C7×#12011yescharacteristic
602C2×C4C7×#12011yescharacteristic
603C2×C4C22×C7×D412yes—
604C8—42no—
605C8—42no—
606C7#1832311yescharacteristic, direct product factor, Sylow
607C22—21no—
608C22—21no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—84no—
623C22C2×C7×#12011yescharacteristic
624C22C2×C7×#13711yescharacteristic
625C22C7×#57511yescharacteristic
626C4—22no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—81no—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—42no—
643C2—82no—
644C2C7×#300711yescharacteristic
645C2C7×#301811yescharacteristic
646C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3459C2
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×#2564C2semidirect
4. C7×#1615C2semidirect
5. C7×#1614C2semidirect
6. C7×#1613C2semidirect
7. C7×#1603C2semidirect
8. C7×#1602C2semidirect
9. C7×#1601C2semidirect
10. C7×#1504C2semidirect
11. C7×#1465C2semidirect
23. C2×C7×C22⋊Q8C22semidirect
27. C7×#484C22semidirect
32. C7×#451C22semidirect
34. C7×#407C22semidirect
35. C7×#400C22semidirect
37. C7×#395C22semidirect
43. C2×C7×#95C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
48. C7×#324C22semidirect
51. #18323C7direct (N × H)
136. #3196C2×C7semidirect
137. #2564C2×C7semidirect
138. #1615C2×C7semidirect
139. #1614C2×C7semidirect
140. #1613C2×C7semidirect
141. #1603C2×C7semidirect
142. #1602C2×C7semidirect
143. #1601C2×C7semidirect
144. #1504C2×C7semidirect
145. #1465C2×C7semidirect
240. C2×C22⋊Q8C22×C7semidirect
244. #484C22×C7semidirect
249. #451C22×C7semidirect
251. #407C22×C7semidirect
252. #400C22×C7semidirect
254. #395C22×C7semidirect
260. C2×#95C22×C7semidirect
261. #328C22×C7semidirect
262. #328C22×C7semidirect
265. #324C22×C7semidirect
606. C7#18323direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#31962
C7×#25642
C7×#16152
C7×#16142
C7×#16132
C7×#16032
C7×#16022
C7×#16012
C7×#15042
C7×#14652
#183237