Finite Group Databasebeta

← C7×#18291 #13195507

Subgroups — C7×#18291 #13195507

Order 1792.

984 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3190C212yesmaximal, complemented
3C7×#2638C211yesmaximal, characteristic, complemented
4C7×#1609C212yesmaximal, complemented
5C7×#1608C211yesmaximal, characteristic, complemented
6C7×#1607C211yesmaximal, characteristic, complemented
7C7×#1603C212yesmaximal, complemented
8C7×#1602C211yesmaximal, characteristic, complemented
9C7×#1601C211yesmaximal, characteristic, complemented
10C7×#1516C212yesmaximal, complemented
11C7×#1459C211yesmaximal, 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×#483C2212yescomplemented
28C7×#480C2212yes—
29C7×#467—22no—
30C7×#465—22no—
31C7×#452—22no—
32C7×#451C2212yescomplemented
33C7×#448C2212yes—
34C2×C7×C22⋊C8C2212yescomplemented
35C7×#398C2211yescharacteristic, complemented
36C7×#397—21no—
37C7×#397C2211yescharacteristic
38C7×#395C2211yescharacteristic, complemented
39C7×#394—21no—
40C7×#393C2211yescharacteristic
41C7×#387—21no—
42C7×#385—21no—
43C7×#381C2211yescharacteristic, complemented
44C7×#328C2212yescomplemented
45C7×#328C2212yescomplemented
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yescomplemented
49C7×#323C2211yescharacteristic
50C7×#323C2211yescharacteristic
51#18291C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41D412yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C2×C7×Q16—22no—
60C2×C7×SD16—22no—
61C22×C7×C8—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#125—21no—
66C7×#125—22no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—21no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8D412yes—
82C7×C22⋊Q8C2312yes—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—21no—
87C7×#121D412yes—
88C7×#121C2312yes—
89C7×#120—22no—
90C7×#120—22no—
91C7×#120—22no—
92C4×C7×Q8—21no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—21no—
96C4×C7×D4—22no—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C7×#117—22no—
102C7×#117D412yes—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4C2311yescharacteristic
108C2×C7×C4⋊C4C2311yescharacteristic
109C2×C7×C22⋊C4—21no—
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#3190C2×C712yescomplemented
137#2638C2×C711yescharacteristic, complemented
138#1609C2×C712yescomplemented
139#1608C2×C711yescharacteristic, complemented
140#1607C2×C711yescharacteristic, complemented
141#1603C2×C712yescomplemented
142#1602C2×C711yescharacteristic, complemented
143#1601C2×C711yescharacteristic, complemented
144#1516C2×C712yescomplemented
145#1459C2×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—22no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—81no—
177C22×C4×C7—81no—
178C22×C4×C7—82no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D412yes—
184C22×C4×C7C2×D412yes—
185C22×C4×C7C2411yescharacteristic
186C7×Q16—82no—
187C7×SD16—82no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C2×C7×C8—82no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—42no—
202C7×C4⋊C4—41no—
203C7×C4⋊C4—41no—
204C7×C4⋊C4—81no—
205C7×C4⋊C4C2×D412yes—
206C7×C4⋊C4C2×D412yes—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
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—42no—
220C7×C22⋊C4—42no—
221C7×C22⋊C4—81no—
222C7×C22⋊C4—81no—
223C42×C7—22no—
224C42×C7—22no—
225C42×C7—21no—
226C42×C7—21no—
227#543C22×C711yescharacteristic
228#539—22no—
229#539C22×C712yes—
230#538—22no—
231#538—21no—
232#534—21no—
233#534C22×C711yescharacteristic
234#529—22no—
235#529—21no—
236#525—22no—
237#525C22×C712yes—
238C2×C22⋊Q8C22×C711yescharacteristic, complemented
239C2×C4×D4—21no—
240C2×C4×D4C22×C711yescharacteristic
241#485—22no—
242#483C22×C712yescomplemented
243#480C22×C712yes—
244#467—22no—
245#465—22no—
246#452—22no—
247#451C22×C712yescomplemented
248#448C22×C712yes—
249C2×C22⋊C8C22×C712yescomplemented
250#398C22×C711yescharacteristic, complemented
251#397—21no—
252#397C22×C711yescharacteristic
253#395C22×C711yescharacteristic, complemented
254#394—21no—
255#393C22×C711yescharacteristic
256#387—21no—
257#385—21no—
258#381C22×C711yescharacteristic, complemented
259#328C22×C712yescomplemented
260#328C22×C712yescomplemented
261#327C22×C712yes—
262#327C22×C712yes—
263#324C22×C712yescomplemented
264#323C22×C711yescharacteristic
265#323C22×C711yescharacteristic
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—22no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—42no—
276C23×C7—81no—
277C23×C7—81no—
278C23×C7#12012yes—
279C23×C7#13512yes—
280C23×C7C22×D411yescharacteristic
281C7×Q8—42no—
282C7×Q8—42no—
283C7×Q8—42no—
284C7×Q8—81no—
285C7×D4—42no—
286C7×D4—44no—
287C7×D4—44no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—82no—
291C7×D4—82no—
292C7×D4—81no—
293C2×C4×C7—21no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—22no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—42no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—44no—
307C2×C4×C7—42no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
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—81no—
323C2×C4×C7—81no—
324C2×C4×C7—82no—
325C2×C4×C7—81no—
326C2×C4×C7#12011yescharacteristic
327C2×C4×C7#12011yescharacteristic
328C2×C4×C7C22×D412yes—
329C7×C8—42no—
330C7×C8—42no—
331C2×#41C7×D412yes—
332C22×Q8C23×C711yescharacteristic
333C22×D4—21no—
334C22×D4C7×D411yescharacteristic
335C23×C4—21no—
336C23×C4C7×D411yescharacteristic
337C23×C4C23×C711yescharacteristic
338C2×Q16—22no—
339C2×SD16—22no—
340C22×C8—22no—
341#126—22no—
342#126—22no—
343#126—22no—
344#125—21no—
345#125—22no—
346#124—22no—
347#124—21no—
348#123—22no—
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8C7×D412yes—
361C22⋊Q8C23×C712yes—
362#121—22no—
363#121—22no—
364#121—22no—
365#121—21no—
366#121C7×D412yes—
367#121C23×C712yes—
368#120—22no—
369#120—22no—
370#120—22no—
371C4×Q8—21no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380#117—22no—
381#117C7×D412yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4C23×C711yescharacteristic
387C2×C4⋊C4C23×C711yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4C7×D411yescharacteristic
395C2×C22⋊C4C23×C711yescharacteristic
396C2×C42—21no—
397C2×C42C7×D411yescharacteristic
398#107—22no—
399#106—22no—
400#103—22no—
401#103—22no—
402#103—22no—
403#102—22no—
404C22⋊C8—22no—
405C22⋊C8C23×C712yes—
406C22⋊C8C23×C712yes—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95C23×C711yescharacteristic
413#95C23×C711yescharacteristic
414C22×C7—21no—
415C22×C7—21no—
416C22×C7—22no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—42no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7—84no—
430C22×C7C2×#12011yescharacteristic
431C22×C7C2×#13511yescharacteristic
432C22×C7#57511yescharacteristic
433C4×C7—22no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—41no—
437C4×C7—42no—
438C4×C7—41no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—81no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7—81no—
446C24—21no—
447C24C2×C7×D411yescharacteristic
448C24C2×C7×D411yescharacteristic
449#41—82no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—81no—
453C2×Q8C2×C7×D412yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—41no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4C2×C7×D412yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—82no—
479C22×C4C2×C7×D411yescharacteristic
480C22×C4C2×C7×D411yescharacteristic
481C22×C4C2×C7×D411yescharacteristic
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D412yes—
484C22×C4C2×C7×D412yes—
485C22×C4C24×C711yescharacteristic
486Q16—82no—
487SD16—82no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—82no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—81no—
505C4⋊C4C2×C7×D412yes—
506C4⋊C4C2×C7×D412yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C42—22no—
524C42—22no—
525C42—21no—
526C42—21no—
527C2×C7—21no—
528C2×C7—21no—
529C2×C7—42no—
530C2×C7—42no—
531C2×C7—82no—
532C2×C7#300211yescharacteristic
533C2×C7#302211yescharacteristic
534C2×C7#302611yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—81no—
546C23—81no—
547C23C7×#12012yes—
548C23C7×#13512yes—
549C23C22×C7×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—81no—
554D4—42no—
555D4—44no—
556D4—44no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—82no—
561D4—81no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—81no—
595C2×C4C7×#12011yescharacteristic
596C2×C4C7×#12011yescharacteristic
597C2×C4C22×C7×D412yes—
598C8—42no—
599C8—42no—
600C7#1829111yescharacteristic, direct product factor, Sylow
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—84no—
617C22C2×C7×#12011yescharacteristic
618C22C2×C7×#13511yescharacteristic
619C22C7×#57511yescharacteristic
620C4—22no—
621C4—21no—
622C4—21no—
623C4—41no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—21no—
634C2—21no—
635C2—42no—
636C2—42no—
637C2—82no—
638C2C7×#300211yescharacteristic
639C2C7×#302211yescharacteristic
640C2C7×#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×#3190C2semidirect
3. C7×#2638C2semidirect
4. C7×#1609C2semidirect
5. C7×#1608C2semidirect
6. C7×#1607C2semidirect
7. C7×#1603C2semidirect
8. C7×#1602C2semidirect
9. C7×#1601C2semidirect
10. C7×#1516C2semidirect
11. C7×#1459C2semidirect
23. C2×C7×C22⋊Q8C22semidirect
27. C7×#483C22semidirect
32. C7×#451C22semidirect
34. C2×C7×C22⋊C8C22semidirect
35. C7×#398C22semidirect
38. C7×#395C22semidirect
43. C7×#381C22semidirect
44. C7×#328C22semidirect
45. C7×#328C22semidirect
48. C7×#324C22semidirect
51. #18291C7direct (N × H)
136. #3190C2×C7semidirect
137. #2638C2×C7semidirect
138. #1609C2×C7semidirect
139. #1608C2×C7semidirect
140. #1607C2×C7semidirect
141. #1603C2×C7semidirect
142. #1602C2×C7semidirect
143. #1601C2×C7semidirect
144. #1516C2×C7semidirect
145. #1459C2×C7semidirect
238. C2×C22⋊Q8C22×C7semidirect
242. #483C22×C7semidirect
247. #451C22×C7semidirect
249. C2×C22⋊C8C22×C7semidirect
250. #398C22×C7semidirect
253. #395C22×C7semidirect
258. #381C22×C7semidirect
259. #328C22×C7semidirect
260. #328C22×C7semidirect
263. #324C22×C7semidirect
600. C7#18291direct (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×#31902
C7×#26382
C7×#16092
C7×#16082
C7×#16072
C7×#16032
C7×#16022
C7×#16012
C7×#15162
C7×#14592
#182917