Finite Group Databasebeta

← C7×#18307 #13195523

Subgroups — C7×#18307 #13195523

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3461C211yesmaximal, characteristic, complemented
2C7×#3211C212yesmaximal, complemented
3C7×#2632C211yesmaximal, characteristic, complemented
4C7×#1605C211yesmaximal, characteristic, complemented
5C7×#1604C211yesmaximal, characteristic, complemented
6C7×#1603C212yesmaximal, complemented
7C7×#1599C211yesmaximal, characteristic, complemented
8C7×#1598C211yesmaximal, characteristic
9C7×#1597C212yesmaximal, complemented
10C7×#1521C212yesmaximal, complemented
11C7×#1461C211yesmaximal, characteristic, complemented
12C7×#544C2211yescharacteristic
13C7×#542—22no—
14C7×#541—22no—
15C7×#539—21no—
16C7×#539C2212yes—
17C7×#537C2211yescharacteristic
18C7×#536—21no—
19C7×#533—22no—
20C7×#529—22no—
21C7×#529—21no—
22C7×#529C2212yes—
23C2×C7×#124C2211yescharacteristic
24C4×C7×#41—21no—
25C2×C4×C7×Q8C2211yescharacteristic
26C7×#495—22no—
27C7×#488—22no—
28C7×#464—22no—
29C7×#463—22no—
30C7×#451C2212yes—
31C7×#450C2212yes—
32C7×#448C2212yes—
33C7×#447C2212yes—
34C7×#431C2212yes—
35C7×#399—21no—
36C7×#395C2211yescharacteristic
37C7×#395C2211yescharacteristic
38C7×#394C2211yescharacteristic
39C7×#394C2211yescharacteristic
40C7×#393—21no—
41C7×#388—21no—
42C7×#388—21no—
43C7×#383C2211yescharacteristic
44C7×#328C2212yes—
45C7×#328C2212yes—
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yes—
49C7×#324C2211yescharacteristic
50C7×#324C2211yescharacteristic
51#18307C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C22×C7×Q8D411yescharacteristic
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4C2311yescharacteristic
56C2×C7×Q16—22no—
57C2×C7×SD16—22no—
58C2×C7×SD16—22no—
59C2×C7×D8—22no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#126—22no—
66C7×#125—21no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#125—22no—
70C7×#125D412yes—
71C7×#124—22no—
72C7×#124—21no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124D412yes—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8C2312yes—
87C7×#121—22no—
88C7×#121—21no—
89C7×#121C2312yes—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8—22no—
94C4×C7×Q8—22no—
95C4×C7×Q8D412yes—
96C4×C7×D4—21no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4D412yes—
102C7×#117—22no—
103C7×#117—22no—
104C7×#117—21no—
105C7×#117—22no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4D411yescharacteristic
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4—21no—
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C42×C7—22no—
118C2×C42×C7D411yescharacteristic
119C2×C42×C7D411yescharacteristic
120C2×C42×C7C2311yescharacteristic
121C7×C4⋊C8—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×#102—22no—
126C7×C22⋊C8C2312yes—
127C7×C22⋊C8C2312yes—
128C4×C7×C8—22no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3461C2×C711yescharacteristic, complemented
136#3211C2×C712yescomplemented
137#2632C2×C711yescharacteristic, complemented
138#1605C2×C711yescharacteristic, complemented
139#1604C2×C711yescharacteristic, complemented
140#1603C2×C712yescomplemented
141#1599C2×C711yescharacteristic, complemented
142#1598C2×C711yescharacteristic
143#1597C2×C712yescomplemented
144#1521C2×C712yescomplemented
145#1461C2×C711yescharacteristic, complemented
146C24×C7—21no—
147C7×#41—42no—
148C7×#41—42no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—22no—
152C2×C7×Q8—22no—
153C2×C7×Q8—22no—
154C2×C7×Q8—41no—
155C2×C7×Q8—81no—
156C2×C7×Q8C2×D412yes—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—21no—
160C2×C7×D4—22no—
161C2×C7×D4—81no—
162C2×C7×D4C2×D412yes—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—21no—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2411yescharacteristic
176C7×Q16—82no—
177C7×SD16—82no—
178C7×SD16—82no—
179C7×D8—82no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—81no—
198C7×C4⋊C4—81no—
199C7×C4⋊C4C2×D412yes—
200C7×C4⋊C4C2×D412yes—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—42no—
211C7×C22⋊C4—41no—
212C7×C22⋊C4—41no—
213C7×C22⋊C4—42no—
214C7×C22⋊C4—42no—
215C7×C22⋊C4—81no—
216C7×C22⋊C4—81no—
217C42×C7—22no—
218C42×C7—22no—
219C42×C7—21no—
220C42×C7—22no—
221C42×C7—21no—
222C42×C7—21no—
223C42×C7—21no—
224C42×C7C2×D412yes—
225C42×C7C2×D412yes—
226#544C22×C711yescharacteristic
227#542—22no—
228#541—22no—
229#539—21no—
230#539C22×C712yes—
231#537C22×C711yescharacteristic
232#536—21no—
233#533—22no—
234#529—22no—
235#529—21no—
236#529C22×C712yes—
237C2×#124C22×C711yescharacteristic
238C4×#41—21no—
239C2×C4×Q8C22×C711yescharacteristic
240#495—22no—
241#488—22no—
242#464—22no—
243#463—22no—
244#451C22×C712yes—
245#450C22×C712yes—
246#448C22×C712yes—
247#447C22×C712yes—
248#431C22×C712yes—
249#399—21no—
250#395C22×C711yescharacteristic
251#395C22×C711yescharacteristic
252#394C22×C711yescharacteristic
253#394C22×C711yescharacteristic
254#393—21no—
255#388—21no—
256#388—21no—
257#383C22×C711yescharacteristic
258#328C22×C712yes—
259#328C22×C712yes—
260#327C22×C712yes—
261#327C22×C712yes—
262#324C22×C712yes—
263#324C22×C711yescharacteristic
264#324C22×C711yescharacteristic
265C23×C7—22no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7C22×D411yescharacteristic
274C7×Q8—44no—
275C7×Q8—42no—
276C7×Q8—42no—
277C7×Q8—42no—
278C7×Q8—42no—
279C7×Q8—81no—
280C7×D4—42no—
281C7×D4—42no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—42no—
286C7×D4—81no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—44no—
302C2×C4×C7—44no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
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—82no—
315C2×C4×C7#12011yescharacteristic
316C2×C4×C7#12011yescharacteristic
317C2×C4×C7#12012yes—
318C2×C4×C7#13512yes—
319C2×C4×C7C22×D412yes—
320C7×C8—42no—
321C7×C8—42no—
322C2×#41—21no—
323C22×Q8C7×D411yescharacteristic
324C22×Q8C23×C711yescharacteristic
325C22×D4C23×C711yescharacteristic
326C2×Q16—22no—
327C2×SD16—22no—
328C2×SD16—22no—
329C2×D8—22no—
330C4⋊Q8—22no—
331C4⋊Q8—22no—
332#126—22no—
333#126—22no—
334#126—22no—
335#126—22no—
336#125—21no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125C7×D412yes—
341#124—22no—
342#124—21no—
343#124—22no—
344#124—22no—
345#124C7×D412yes—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350C22⋊Q8—22no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8C23×C712yes—
357#121—22no—
358#121—21no—
359#121C23×C712yes—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8C7×D412yes—
366C4×D4—21no—
367C4×D4—21no—
368C4×D4—21no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4C7×D412yes—
372#117—22no—
373#117—22no—
374#117—21no—
375#117—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C7×D411yescharacteristic
379C2×C4⋊C4C23×C711yescharacteristic
380C2×C4⋊C4C23×C711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C42—22no—
388C2×C42C7×D411yescharacteristic
389C2×C42C7×D411yescharacteristic
390C2×C42C23×C711yescharacteristic
391C4⋊C8—22no—
392#103—22no—
393#103—22no—
394#102—22no—
395#102—22no—
396C22⋊C8C23×C712yes—
397C22⋊C8C23×C712yes—
398C4×C8—22no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95C23×C711yescharacteristic
404#95C23×C711yescharacteristic
405C22×C7—21no—
406C22×C7—21no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—81no—
411C22×C7—81no—
412C22×C7—81no—
413C22×C7—81no—
414C22×C7—81no—
415C22×C7—81no—
416C22×C7—81no—
417C22×C7C2×#12011yescharacteristic
418C22×C7C2×#13511yescharacteristic
419C22×C7#57511yescharacteristic
420C4×C7—22no—
421C4×C7—22no—
422C4×C7—21no—
423C4×C7—21no—
424C4×C7—22no—
425C4×C7—41no—
426C4×C7—42no—
427C4×C7—42no—
428C4×C7—41no—
429C4×C7—41no—
430C4×C7—42no—
431C4×C7—81no—
432C4×C7—81no—
433C4×C7—81no—
434C24—21no—
435#41—42no—
436#41—42no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—41no—
443C2×Q8—81no—
444C2×Q8C2×C7×D412yes—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—81no—
450C2×D4C2×C7×D412yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4C2×C7×D411yescharacteristic
458C22×C4C2×C7×D411yescharacteristic
459C22×C4C2×C7×D411yescharacteristic
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D411yescharacteristic
462C22×C4C2×C7×D411yescharacteristic
463C22×C4C24×C711yescharacteristic
464Q16—82no—
465SD16—82no—
466SD16—82no—
467D8—82no—
468C2×C8—22no—
469C2×C8—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4C2×C7×D412yes—
488C4⋊C4C2×C7×D412yes—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42C2×C7×D412yes—
513C42C2×C7×D412yes—
514C2×C7—21no—
515C2×C7—21no—
516C2×C7—82no—
517C2×C7—161no—
518C2×C7#300211yescharacteristic
519C2×C7#302011yescharacteristic
520C2×C7#302711yescharacteristic
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C22×C7×D411yescharacteristic
530Q8—44no—
531Q8—42no—
532Q8—42no—
533Q8—42no—
534Q8—42no—
535Q8—81no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—81no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—82no—
571C2×C4C7×#12011yescharacteristic
572C2×C4C7×#12011yescharacteristic
573C2×C4C7×#12012yes—
574C2×C4C7×#13512yes—
575C2×C4C22×C7×D412yes—
576C8—42no—
577C8—42no—
578C7#1830711yescharacteristic, direct product factor, Sylow
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C7×#12011yescharacteristic
592C22C2×C7×#13511yescharacteristic
593C22C7×#57511yescharacteristic
594C4—22no—
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—81no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—161no—
612C2C7×#300211yescharacteristic
613C2C7×#302011yescharacteristic
614C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#1598C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3461C2semidirect
2. C7×#3211C2semidirect
3. C7×#2632C2semidirect
4. C7×#1605C2semidirect
5. C7×#1604C2semidirect
6. C7×#1603C2semidirect
7. C7×#1599C2semidirect
9. C7×#1597C2semidirect
10. C7×#1521C2semidirect
11. C7×#1461C2semidirect
51. #18307C7direct (N × H)
135. #3461C2×C7semidirect
136. #3211C2×C7semidirect
137. #2632C2×C7semidirect
138. #1605C2×C7semidirect
139. #1604C2×C7semidirect
140. #1603C2×C7semidirect
141. #1599C2×C7semidirect
143. #1597C2×C7semidirect
144. #1521C2×C7semidirect
145. #1461C2×C7semidirect
578. C7#18307direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34612
C7×#32112
C7×#26322
C7×#16052
C7×#16042
C7×#16032
C7×#15992
C7×#15982
C7×#15972
C7×#15212
C7×#14612
#183077