Finite Group Databasebeta

← C7×#18277 #13195493

Subgroups — C7×#18277 #13195493

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic
2C7×#3112C211yesmaximal, characteristic, complemented
3C7×#3108C211yesmaximal, characteristic, complemented
4C7×#2584C211yesmaximal, characteristic, complemented
5C7×#1610C212yesmaximal, complemented
6C7×#1609C212yesmaximal, complemented
7C7×#1604C212yesmaximal, complemented
8C7×#1603C212yesmaximal, complemented
9C7×#1509C211yesmaximal, characteristic, complemented
10C7×#1502C211yesmaximal, characteristic, complemented
11C7×#1470C211yesmaximal, characteristic, complemented
12C7×#563—21no—
13C7×#562—21no—
14C7×#554—21no—
15C7×#553—21no—
16C7×#553—21no—
17C7×#547—22no—
18C7×#547—21no—
19C7×#544—21no—
20C7×#540—21no—
21C7×#539—22no—
22C7×#539C2212yes—
23C7×#537C2212yes—
24C7×#532C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C2×C4×C7×Q8C2211yescharacteristic
28C7×#483C2211yescharacteristic
29C7×#480C2212yes—
30C7×#480C2211yescharacteristic
31C7×#479—21no—
32C7×#477—21no—
33C7×#464—21no—
34C7×#460—21no—
35C7×#451C2211yescharacteristic
36C7×#448C2212yes—
37C7×#448C2211yescharacteristic
38C7×#440—22no—
39C4×C7×Q16—21no—
40C4×C7×SD16—21no—
41C7×#432C2211yescharacteristic
42C7×#431C2211yescharacteristic
43C7×#398C2212yes—
44C7×#395C2212yes—
45C7×#389—21no—
46C7×#389—22no—
47C7×#389—21no—
48C4×C7×C4⋊C4C2211yescharacteristic
49C7×#328C2212yes—
50C7×#327C2212yes—
51C7×#327C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18277C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41—21no—
57C2×C7×#41—22no—
58C22×C7×Q8C2311yescharacteristic
59C2×C7×Q16—21no—
60C2×C7×SD16—21no—
61C2×C7×SD16—22no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D411yescharacteristic
65C7×#127D411yescharacteristic
66C7×#126—22no—
67C7×#126—22no—
68C7×#126—21no—
69C7×#125—21no—
70C7×#125—22no—
71C7×#125—21no—
72C7×#125—21no—
73C7×#125—21no—
74C7×#125—21no—
75C7×#125D411yescharacteristic
76C7×#124—21no—
77C7×#124D411yescharacteristic
78C7×#123—22no—
79C7×#123—22no—
80C7×#123—22no—
81C7×#123—22no—
82C7×#123—22no—
83C7×#123—21no—
84C7×#123—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8C2311yescharacteristic
92C7×#121—21no—
93C7×#121—22no—
94C7×#121—21no—
95C7×#121—22no—
96C7×#121—22no—
97C7×#121—21no—
98C7×#121—22no—
99C7×#121C2311yescharacteristic
100C7×#121C2312yes—
101C4×C7×Q8—21no—
102C4×C7×Q8—21no—
103C4×C7×Q8—21no—
104C4×C7×Q8—21no—
105C4×C7×Q8D411yescharacteristic
106C4×C7×D4—22no—
107C4×C7×D4—21no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—22no—
111C4×C7×D4—21no—
112C4×C7×D4D411yescharacteristic
113C4×C7×D4D412yes—
114C7×#117—21no—
115C7×#117—21no—
116C2×C7×C4⋊C4—22no—
117C2×C7×C4⋊C4—21no—
118C2×C7×C4⋊C4—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4—22no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4D412yes—
124C2×C7×C4⋊C4D411yescharacteristic
125C2×C7×C4⋊C4C2311yescharacteristic
126C2×C7×C4⋊C4C2312yes—
127C2×C42×C7—21no—
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×#103—21no—
138C7×#103—21no—
139C7×#102—21no—
140C7×#102—22no—
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#3559C2×C711yescharacteristic
150#3112C2×C711yescharacteristic, complemented
151#3108C2×C711yescharacteristic, complemented
152#2584C2×C711yescharacteristic, complemented
153#1610C2×C712yescomplemented
154#1609C2×C712yescomplemented
155#1604C2×C712yescomplemented
156#1603C2×C712yescomplemented
157#1509C2×C711yescharacteristic, complemented
158#1502C2×C711yescharacteristic, complemented
159#1470C2×C711yescharacteristic, complemented
160C7×#41—81no—
161C7×#41—81no—
162C7×#41—82no—
163C7×#41—82no—
164C2×C7×Q8—22no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—42no—
169C2×C7×Q8C2×D411yescharacteristic
170C2×C7×D4—21no—
171C2×C7×D4—22no—
172C2×C7×D4—22no—
173C2×C7×D4—22no—
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4C2×D411yescharacteristic
177C2×C7×D4C2×D412yes—
178C22×C4×C7—22no—
179C22×C4×C7—21no—
180C22×C4×C7—22no—
181C22×C4×C7—22no—
182C22×C4×C7—22no—
183C22×C4×C7—21no—
184C22×C4×C7—22no—
185C22×C4×C7—21no—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D412yes—
190C22×C4×C7C2411yescharacteristic
191C7×Q16—42no—
192C7×SD16—44no—
193C7×SD16—42no—
194C2×C7×C8—21no—
195C2×C7×C8—21no—
196C2×C7×C8—22no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
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—21no—
215C7×C4⋊C4—21no—
216C7×C4⋊C4—21no—
217C7×C4⋊C4—44no—
218C7×C4⋊C4—41no—
219C7×C4⋊C4—41no—
220C7×C4⋊C4—82no—
221C7×C4⋊C4C2×D411yescharacteristic
222C7×C4⋊C4C2×D412yes—
223C7×C4⋊C4C2×D411yescharacteristic
224C7×C22⋊C4—22no—
225C7×C22⋊C4—22no—
226C7×C22⋊C4—22no—
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—21no—
230C7×C22⋊C4—21no—
231C7×C22⋊C4—21no—
232C7×C22⋊C4—21no—
233C7×C22⋊C4—22no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C42×C7—22no—
237C42×C7—21no—
238C42×C7—41no—
239C42×C7—41no—
240C42×C7—81no—
241C42×C7C2×D411yescharacteristic
242C42×C7C2×D411yescharacteristic
243C42×C7C2×D411yescharacteristic
244C42×C7C2×D411yescharacteristic
245#563—21no—
246#562—21no—
247#554—21no—
248#553—21no—
249#553—21no—
250#547—22no—
251#547—21no—
252#544—21no—
253#540—21no—
254#539—22no—
255#539C22×C712yes—
256#537C22×C712yes—
257#532C22×C711yescharacteristic
258#529C22×C711yescharacteristic
259C2×#125C22×C711yescharacteristic
260C2×C4×Q8C22×C711yescharacteristic
261#483C22×C711yescharacteristic
262#480C22×C711yescharacteristic
263#480C22×C712yes—
264#479—21no—
265#477—21no—
266#464—21no—
267#460—21no—
268#451C22×C711yescharacteristic
269#448C22×C711yescharacteristic
270#448C22×C712yes—
271#440—22no—
272C4×Q16—21no—
273C4×SD16—21no—
274#432C22×C711yescharacteristic
275#431C22×C711yescharacteristic
276#398C22×C712yes—
277#395C22×C712yes—
278#389—21no—
279#389—22no—
280#389—21no—
281C4×C4⋊C4C22×C711yescharacteristic
282#328C22×C712yes—
283#327C22×C712yes—
284#327C22×C712yes—
285#327C22×C712yes—
286#324C22×C712yes—
287#324C22×C712yes—
288C23×C7—21no—
289C23×C7—22no—
290C23×C7C22×D411yescharacteristic
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—41no—
294C7×Q8—41no—
295C7×Q8—41no—
296C7×Q8—81no—
297C7×Q8—82no—
298C7×D4—42no—
299C7×D4—41no—
300C7×D4—42no—
301C7×D4—41no—
302C7×D4—42no—
303C7×D4—41no—
304C7×D4—42no—
305C7×D4—81no—
306C7×D4—82no—
307C7×D4—82no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—24no—
312C2×C4×C7—22no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—22no—
316C2×C4×C7—22no—
317C2×C4×C7—21no—
318C2×C4×C7—22no—
319C2×C4×C7—22no—
320C2×C4×C7—22no—
321C2×C4×C7—41no—
322C2×C4×C7—42no—
323C2×C4×C7—41no—
324C2×C4×C7—41no—
325C2×C4×C7—42no—
326C2×C4×C7—42no—
327C2×C4×C7—42no—
328C2×C4×C7—41no—
329C2×C4×C7—41no—
330C2×C4×C7—41no—
331C2×C4×C7—82no—
332C2×C4×C7—82no—
333C2×C4×C7—82no—
334C2×C4×C7—82no—
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7#12011yescharacteristic
338C2×C4×C7#12011yescharacteristic
339C2×C4×C7C22×D411yescharacteristic
340C2×C4×C7C22×D411yescharacteristic
341C7×C8—42no—
342C7×C8—41no—
343C7×C8—41no—
344C2×#41—21no—
345C2×#41—22no—
346C22×Q8C23×C711yescharacteristic
347C2×Q16—21no—
348C2×SD16—21no—
349C2×SD16—22no—
350C4⋊Q8—21no—
351C4⋊Q8—22no—
352C4⋊Q8C7×D411yescharacteristic
353#127C7×D411yescharacteristic
354#126—22no—
355#126—22no—
356#126—21no—
357#125—21no—
358#125—22no—
359#125—21no—
360#125—21no—
361#125—21no—
362#125—21no—
363#125C7×D411yescharacteristic
364#124—21no—
365#124C7×D411yescharacteristic
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371#123—21no—
372#123—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8C23×C711yescharacteristic
380#121—21no—
381#121—22no—
382#121—21no—
383#121—22no—
384#121—22no—
385#121—21no—
386#121—22no—
387#121C23×C711yescharacteristic
388#121C23×C712yes—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8C7×D411yescharacteristic
394C4×D4—22no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—22no—
399C4×D4—21no—
400C4×D4C7×D411yescharacteristic
401C4×D4C7×D412yes—
402#117—21no—
403#117—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4C7×D411yescharacteristic
411C2×C4⋊C4C7×D412yes—
412C2×C4⋊C4C7×D411yescharacteristic
413C2×C4⋊C4C23×C711yescharacteristic
414C2×C4⋊C4C23×C712yes—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C23×C711yescharacteristic
418#107—21no—
419#107—22no—
420#106—21no—
421C4⋊C8—21no—
422C4⋊C8—21no—
423#103—21no—
424#103—21no—
425#103—21no—
426#103—22no—
427#102—21no—
428#102—22no—
429C22⋊C8C23×C711yescharacteristic
430C22⋊C8C23×C711yescharacteristic
431C22⋊C8C23×C712yes—
432#97—21no—
433C4×C8—21no—
434#95—22no—
435#95—22no—
436#95C23×C712yes—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—42no—
440C22×C7—41no—
441C22×C7—41no—
442C22×C7—42no—
443C22×C7—42no—
444C22×C7—41no—
445C22×C7C2×#12011yescharacteristic
446C22×C7#57611yescharacteristic
447C22×C7#57711yescharacteristic
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—44no—
454C4×C7—42no—
455C4×C7—42no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—42no—
460C4×C7—41no—
461C4×C7—81no—
462C4×C7#45612yes—
463#41—81no—
464#41—81no—
465#41—82no—
466#41—82no—
467C2×Q8—22no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—42no—
472C2×Q8C2×C7×D411yescharacteristic
473C2×D4—21no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4C2×C7×D411yescharacteristic
480C2×D4C2×C7×D412yes—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4C2×C7×D412yes—
490C22×C4C2×C7×D411yescharacteristic
491C22×C4C2×C7×D411yescharacteristic
492C22×C4C2×C7×D412yes—
493C22×C4C24×C711yescharacteristic
494Q16—42no—
495SD16—44no—
496SD16—42no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—44no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—82no—
524C4⋊C4C2×C7×D411yescharacteristic
525C4⋊C4C2×C7×D412yes—
526C4⋊C4C2×C7×D411yescharacteristic
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—22no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C42—22no—
540C42—21no—
541C42—41no—
542C42—41no—
543C42—81no—
544C42C2×C7×D411yescharacteristic
545C42C2×C7×D411yescharacteristic
546C42C2×C7×D411yescharacteristic
547C42C2×C7×D411yescharacteristic
548C2×C7—21no—
549C2×C7—21no—
550C2×C7—81no—
551C2×C7—82no—
552C2×C7#301211yescharacteristic
553C2×C7C2×#45611yescharacteristic
554C2×C7#302511yescharacteristic
555C23—21no—
556C23—22no—
557C23C22×C7×D411yescharacteristic
558Q8—22no—
559Q8—22no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—81no—
564Q8—82no—
565D4—42no—
566D4—41no—
567D4—42no—
568D4—41no—
569D4—42no—
570D4—41no—
571D4—42no—
572D4—81no—
573D4—82no—
574D4—82no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4C7×#12011yescharacteristic
603C2×C4C7×#12011yescharacteristic
604C2×C4C7×#12011yescharacteristic
605C2×C4C7×#12011yescharacteristic
606C2×C4C22×C7×D411yescharacteristic
607C2×C4C22×C7×D411yescharacteristic
608C8—42no—
609C8—41no—
610C8—41no—
611C7#1827711yescharacteristic, direct product factor, Sylow
612C22—21no—
613C22—21no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22C2×C7×#12011yescharacteristic
621C22C7×#57611yescharacteristic
622C22C7×#57711yescharacteristic
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—81no—
637C4C7×#45612yes—
638C2—21no—
639C2—21no—
640C2—81no—
641C2—82no—
642C2C7×#301211yescharacteristic
643C2C2×C7×#45611yescharacteristic
644C2C7×#302511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3559C2
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
2. C7×#3112C2semidirect
3. C7×#3108C2semidirect
4. C7×#2584C2semidirect
5. C7×#1610C2semidirect
6. C7×#1609C2semidirect
7. C7×#1604C2semidirect
8. C7×#1603C2semidirect
9. C7×#1509C2semidirect
10. C7×#1502C2semidirect
11. C7×#1470C2semidirect
55. #18277C7direct (N × H)
150. #3112C2×C7semidirect
151. #3108C2×C7semidirect
152. #2584C2×C7semidirect
153. #1610C2×C7semidirect
154. #1609C2×C7semidirect
155. #1604C2×C7semidirect
156. #1603C2×C7semidirect
157. #1509C2×C7semidirect
158. #1502C2×C7semidirect
159. #1470C2×C7semidirect
611. C7#18277direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#31122
C7×#31082
C7×#25842
C7×#16102
C7×#16092
C7×#16042
C7×#16032
C7×#15092
C7×#15022
C7×#14702
#182777