Finite Group Databasebeta

← C7×#18278 #13195494

Subgroups — C7×#18278 #13195494

Order 1792.

910 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic, complemented
2C7×#3110C211yesmaximal, characteristic, complemented
3C7×#3105C211yesmaximal, characteristic
4C7×#2562C211yesmaximal, characteristic, complemented
5C7×#1617C212yesmaximal, complemented
6C7×#1615C212yesmaximal, complemented
7C7×#1599C212yesmaximal, complemented
8C7×#1597C212yesmaximal
9C7×#1509C211yesmaximal, characteristic, complemented
10C7×#1503C211yesmaximal, characteristic, complemented
11C7×#1470C211yesmaximal, characteristic, complemented
12C7×#564—21no—
13C7×#563—21no—
14C7×#557—21no—
15C7×#554—21no—
16C7×#552—21no—
17C7×#552—21no—
18C7×#552—22no—
19C7×#544—21no—
20C7×#544C2212yes—
21C7×#541—21no—
22C7×#541—22no—
23C7×#539C2212yes—
24C7×#533C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C2×C4×C7×D4C2211yescharacteristic
28C7×#484C2212yes—
29C7×#484C2211yescharacteristic
30C7×#481C2211yescharacteristic
31C7×#478—21no—
32C7×#474—21no—
33C7×#463—21no—
34C7×#461—21no—
35C7×#450C2212yes—
36C7×#450C2211yescharacteristic
37C7×#447C2211yescharacteristic
38C7×#440—22no—
39C4×C7×SD16—21no—
40C4×C7×D8—21no—
41C7×#432C2211yescharacteristic
42C7×#431C2211yescharacteristic
43C7×#400C2212yes—
44C7×#394C2212yes—
45C7×#388—22no—
46C7×#380—21no—
47C7×#380—21no—
48C4×C7×C4⋊C4C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18278C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7—21no—
58C2×C7×SD16—21no—
59C2×C7×SD16—22no—
60C2×C7×D8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8D411yescharacteristic
66C7×#126—22no—
67C7×#126—22no—
68C7×#126—22no—
69C7×#126—22no—
70C7×#126—21no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#125—21no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#125—21no—
81C7×#125—21no—
82C7×#125—21no—
83C7×#125—21no—
84C7×#125—21no—
85C7×#125D411yescharacteristic
86C7×#124D411yescharacteristic
87C7×#123—22no—
88C7×#123—21no—
89C7×#123—22no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8C2311yescharacteristic
98C7×C22⋊Q8C2312yes—
99C7×#121C2311yescharacteristic
100C4×C7×Q8—21no—
101C4×C7×Q8—22no—
102C4×C7×Q8—21no—
103C4×C7×Q8—21no—
104C4×C7×Q8D411yescharacteristic
105C4×C7×Q8D412yes—
106C4×C7×D4—21no—
107C4×C7×D4—22no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—21no—
111C4×C7×D4—21no—
112C4×C7×D4D411yescharacteristic
113C7×#117—22no—
114C7×#117—21no—
115C7×#117—21no—
116C7×#117—22no—
117C7×#117—21no—
118C2×C7×C4⋊C4—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4D411yescharacteristic
121C2×C7×C4⋊C4D411yescharacteristic
122C2×C7×C4⋊C4D412yes—
123C2×C7×C4⋊C4C2311yescharacteristic
124C2×C7×C4⋊C4C2312yes—
125C2×C7×C22⋊C4—21no—
126C2×C7×C22⋊C4—22no—
127C2×C7×C22⋊C4—22no—
128C2×C42×C7—21no—
129C2×C42×C7C2311yescharacteristic
130C7×#107—22no—
131C7×#107—21no—
132C7×#106—21no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8—21no—
135C7×#103—21no—
136C7×#103—22no—
137C7×#102—21no—
138C7×#102—22no—
139C7×#102—21no—
140C7×#102—21no—
141C7×C22⋊C8C2311yescharacteristic
142C7×C22⋊C8C2312yes—
143C7×C22⋊C8C2311yescharacteristic
144C7×#97—21no—
145C4×C7×C8—21no—
146C7×#95—22no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3563C2×C711yescharacteristic, complemented
150#3110C2×C711yescharacteristic, complemented
151#3105C2×C711yescharacteristic
152#2562C2×C711yescharacteristic, complemented
153#1617C2×C712yescomplemented
154#1615C2×C712yescomplemented
155#1599C2×C712yescomplemented
156#1597C2×C712yes—
157#1509C2×C711yescharacteristic, complemented
158#1503C2×C711yescharacteristic, complemented
159#1470C2×C711yescharacteristic, complemented
160C24×C7—21no—
161C2×C7×Q8—22no—
162C2×C7×Q8—21no—
163C2×C7×Q8C2×D412yes—
164C2×C7×Q8C2×D411yescharacteristic
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—42no—
169C2×C7×D4C2×D411yescharacteristic
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—42no—
175C22×C4×C7—42no—
176C22×C4×C7—41no—
177C22×C4×C7—41no—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2411yescharacteristic
183C7×SD16—42no—
184C7×SD16—44no—
185C7×D8—42no—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8—22no—
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—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—22no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—22no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—42no—
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D411yescharacteristic
219C7×C4⋊C4C2×D412yes—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—44no—
230C7×C22⋊C4—41no—
231C7×C22⋊C4—41no—
232C7×C22⋊C4—82no—
233C42×C7—22no—
234C42×C7—22no—
235C42×C7—22no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7—81no—
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243#564—21no—
244#563—21no—
245#557—21no—
246#554—21no—
247#552—21no—
248#552—21no—
249#552—22no—
250#544—21no—
251#544C22×C712yes—
252#541—21no—
253#541—22no—
254#539C22×C712yes—
255#533C22×C711yescharacteristic
256#529C22×C711yescharacteristic
257C2×#125C22×C711yescharacteristic
258C2×C4×D4C22×C711yescharacteristic
259#484C22×C711yescharacteristic
260#484C22×C712yes—
261#481C22×C711yescharacteristic
262#478—21no—
263#474—21no—
264#463—21no—
265#461—21no—
266#450C22×C711yescharacteristic
267#450C22×C712yes—
268#447C22×C711yescharacteristic
269#440—22no—
270C4×SD16—21no—
271C4×D8—21no—
272#432C22×C711yescharacteristic
273#431C22×C711yescharacteristic
274#400C22×C712yes—
275#394C22×C712yes—
276#388—22no—
277#380—21no—
278#380—21no—
279C4×C4⋊C4C22×C711yescharacteristic
280#328C22×C712yes—
281#328C22×C712yes—
282#328C22×C712yes—
283#327C22×C712yes—
284#324C22×C712yes—
285#324C22×C712yes—
286C23×C7—22no—
287C23×C7—21no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7—41no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7—41no—
294C23×C7C22×D411yescharacteristic
295C7×Q8—41no—
296C7×Q8—42no—
297C7×Q8—42no—
298C7×Q8—82no—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—41no—
302C7×D4—41no—
303C7×D4—41no—
304C7×D4—41no—
305C7×D4—41no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—22no—
313C2×C4×C7—21no—
314C2×C4×C7—22no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—22no—
318C2×C4×C7—41no—
319C2×C4×C7—42no—
320C2×C4×C7—42no—
321C2×C4×C7—42no—
322C2×C4×C7—42no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—41no—
326C2×C4×C7—42no—
327C2×C4×C7—82no—
328C2×C4×C7—82no—
329C2×C4×C7—82no—
330C2×C4×C7#12011yescharacteristic
331C2×C4×C7#12011yescharacteristic
332C2×C4×C7#12011yescharacteristic
333C2×C4×C7#12011yescharacteristic
334C2×C4×C7C22×D411yescharacteristic
335C2×C4×C7C22×D411yescharacteristic
336C7×C8—41no—
337C7×C8—41no—
338C7×C8—42no—
339C22×D4C23×C711yescharacteristic
340C23×C4—21no—
341C2×SD16—21no—
342C2×SD16—22no—
343C2×D8—21no—
344C4⋊Q8—21no—
345C4⋊Q8—22no—
346C4⋊Q8—22no—
347C4⋊Q8C7×D411yescharacteristic
348C4⋊Q8C7×D411yescharacteristic
349#126—22no—
350#126—22no—
351#126—22no—
352#126—22no—
353#126—21no—
354#125—22no—
355#125—22no—
356#125—22no—
357#125—22no—
358#125—21no—
359#125—22no—
360#125—22no—
361#125—22no—
362#125—22no—
363#125—21no—
364#125—21no—
365#125—21no—
366#125—21no—
367#125—21no—
368#125C7×D411yescharacteristic
369#124C7×D411yescharacteristic
370#123—22no—
371#123—21no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—22no—
380C22⋊Q8C23×C711yescharacteristic
381C22⋊Q8C23×C712yes—
382#121C23×C711yescharacteristic
383C4×Q8—21no—
384C4×Q8—22no—
385C4×Q8—21no—
386C4×Q8—21no—
387C4×Q8C7×D411yescharacteristic
388C4×Q8C7×D412yes—
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C7×D411yescharacteristic
396#117—22no—
397#117—21no—
398#117—21no—
399#117—22no—
400#117—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4C7×D411yescharacteristic
404C2×C4⋊C4C7×D412yes—
405C2×C4⋊C4C7×D411yescharacteristic
406C2×C4⋊C4C23×C711yescharacteristic
407C2×C4⋊C4C23×C712yes—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—22no—
411C2×C42—21no—
412C2×C42C23×C711yescharacteristic
413#107—21no—
414#107—22no—
415#106—21no—
416C4⋊C8—21no—
417C4⋊C8—21no—
418#103—21no—
419#103—22no—
420#102—21no—
421#102—21no—
422#102—21no—
423#102—22no—
424C22⋊C8C23×C712yes—
425C22⋊C8C23×C711yescharacteristic
426C22⋊C8C23×C711yescharacteristic
427#97—21no—
428C4×C8—21no—
429#95—22no—
430#95—22no—
431#95C23×C712yes—
432C22×C7—21no—
433C22×C7—21no—
434C22×C7—42no—
435C22×C7—42no—
436C22×C7—41no—
437C22×C7—41no—
438C22×C7—41no—
439C22×C7—42no—
440C22×C7—81no—
441C22×C7—81no—
442C22×C7—81no—
443C22×C7—81no—
444C22×C7C2×#12011yescharacteristic
445C22×C7#57711yescharacteristic
446C22×C7#57811yescharacteristic
447C4×C7—21no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—42no—
453C4×C7—41no—
454C4×C7—41no—
455C4×C7—42no—
456C4×C7—42no—
457C4×C7—42no—
458C4×C7—41no—
459C4×C7—82no—
460C4×C7—81no—
461C4×C7#45412yes—
462C24—21no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8C2×C7×D412yes—
466C2×Q8C2×C7×D411yescharacteristic
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—42no—
471C2×D4C2×C7×D411yescharacteristic
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4C2×C7×D412yes—
481C22×C4C2×C7×D411yescharacteristic
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D412yes—
484C22×C4C24×C711yescharacteristic
485SD16—42no—
486SD16—44no—
487D8—42no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—42no—
519C4⋊C4C2×C7×D411yescharacteristic
520C4⋊C4C2×C7×D411yescharacteristic
521C4⋊C4C2×C7×D412yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—44no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—81no—
541C42C2×C7×D411yescharacteristic
542C42C2×C7×D411yescharacteristic
543C42C2×C7×D411yescharacteristic
544C42C2×C7×D411yescharacteristic
545C2×C7—21no—
546C2×C7—21no—
547C2×C7—81no—
548C2×C7—82no—
549C2×C7#301311yescharacteristic
550C2×C7C2×#45411yescharacteristic
551C2×C7#302511yescharacteristic
552C23—22no—
553C23—21no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C22×C7×D411yescharacteristic
561Q8—41no—
562Q8—42no—
563Q8—42no—
564Q8—82no—
565D4—22no—
566D4—22no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4C7×#12011yescharacteristic
597C2×C4C7×#12011yescharacteristic
598C2×C4C7×#12011yescharacteristic
599C2×C4C7×#12011yescharacteristic
600C2×C4C22×C7×D411yescharacteristic
601C2×C4C22×C7×D411yescharacteristic
602C8—41no—
603C8—41no—
604C8—42no—
605C7#1827811yescharacteristic, direct product factor, Sylow
606C22—21no—
607C22—21no—
608C22—42no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×C7×#12011yescharacteristic
619C22C7×#57711yescharacteristic
620C22C7×#57811yescharacteristic
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—82no—
634C4—81no—
635C4C7×#45412yes—
636C2—21no—
637C2—21no—
638C2—82no—
639C2—81no—
640C2C7×#301311yescharacteristic
641C2C2×C7×#45411yescharacteristic
642C2C7×#302511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#447C22
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×#3563C2semidirect
2. C7×#3110C2semidirect
4. C7×#2562C2semidirect
5. C7×#1617C2semidirect
6. C7×#1615C2semidirect
7. C7×#1599C2semidirect
9. C7×#1509C2semidirect
10. C7×#1503C2semidirect
11. C7×#1470C2semidirect
55. #18278C7direct (N × H)
149. #3563C2×C7semidirect
150. #3110C2×C7semidirect
152. #2562C2×C7semidirect
153. #1617C2×C7semidirect
154. #1615C2×C7semidirect
155. #1599C2×C7semidirect
157. #1509C2×C7semidirect
158. #1503C2×C7semidirect
159. #1470C2×C7semidirect
605. C7#18278direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#31102
C7×#31052
C7×#25622
C7×#16172
C7×#16152
C7×#15992
C7×#15972
C7×#15092
C7×#15032
C7×#14702
#182787