Finite Group Databasebeta

← C7×#18455 #13195671

Subgroups — C7×#18455 #13195671

Order 1792.

894 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×#3538C211yesmaximal, characteristic
2C7×#3196C211yesmaximal, characteristic
3C7×#3189C211yesmaximal, characteristic, complemented
4C7×#2817C211yesmaximal, characteristic, complemented
5C7×#1617C212yesmaximal, complemented
6C7×#1615C212yesmaximal, complemented
7C7×#1605C212yesmaximal, complemented
8C7×#1603C212yesmaximal, complemented
9C7×#1516C211yesmaximal, characteristic
10C7×#1511C211yesmaximal, characteristic, complemented
11C7×#1472C211yesmaximal, characteristic, complemented
12C7×#563—22no—
13C7×#552—21no—
14C7×#552—21no—
15C7×#551—22no—
16C7×D4×Q8—21no—
17C7×D4×Q8—21no—
18C7×#544—21no—
19C7×#544C2212yes—
20C7×#541—21no—
21C7×#539—21no—
22C7×#539C2212yes—
23C7×#536—21no—
24C7×#525C2211yescharacteristic
25C2×C7×C22⋊Q8C2211yescharacteristic
26C2×C7×C22⋊Q8C2211yescharacteristic
27C2×C7×C22⋊Q8C2211yescharacteristic
28C7×#485—21no—
29C7×#484—21no—
30C7×#484C2212yes—
31C7×#484C2211yescharacteristic
32C7×#481C2211yescharacteristic
33C7×#470—21no—
34C7×#468—21no—
35C7×#467—22no—
36C7×#452—21no—
37C7×#451—21no—
38C7×#451C2212yes—
39C7×#451C2211yescharacteristic
40C7×#448C2211yescharacteristic
41C7×#407C2211yescharacteristic
42C2×C7×C22⋊C8C2211yescharacteristic
43C7×#400—21no—
44C7×#400C2212yes—
45C7×#397—22no—
46C7×#395—21no—
47C7×#395C2212yes—
48C7×#381C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18455C711yesmaximal, characteristic, direct product factor, Sylow
56C2×C7×#41D411yescharacteristic
57C22×C7×Q8—21no—
58C22×C7×Q8D411yescharacteristic
59C22×C7×Q8D411yescharacteristic
60C22×C7×Q8C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C2×C7×Q16—22no—
63C2×C7×Q16—21no—
64C2×C7×SD16—21no—
65C2×C7×M16—21no—
66C22×C7×C8—21no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—22no—
69C7×C4⋊Q8—22no—
70C7×C4⋊Q8—22no—
71C7×#126—22no—
72C7×#126—22no—
73C7×#126—22no—
74C7×#126—22no—
75C7×#125—21no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—22no—
80C7×#124—22no—
81C7×#123—22no—
82C7×#123—22no—
83C7×#123—22no—
84C7×#123—21no—
85C7×#123—22no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—21no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—22no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—21no—
92C7×C22⋊Q8—21no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—21no—
96C7×C22⋊Q8—21no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—21no—
99C7×C22⋊Q8—21no—
100C7×C22⋊Q8—21no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8D412yes—
103C7×C22⋊Q8D411yescharacteristic
104C7×C22⋊Q8C2312yes—
105C7×C22⋊Q8C2311yescharacteristic
106C7×#121D411yescharacteristic
107C7×#121C2311yescharacteristic
108C7×#120—21no—
109C4×C7×Q8—21no—
110C4×C7×Q8—21no—
111C4×C7×D4—22no—
112C4×C7×D4—21no—
113C4×C7×D4—21no—
114C4×C7×D4—22no—
115C7×#117—21no—
116C7×#117—22no—
117C7×#117—22no—
118C7×#117D411yescharacteristic
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4—21no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4D411yescharacteristic
124C2×C7×C4⋊C4C2311yescharacteristic
125C2×C7×C4⋊C4C2312yes—
126C2×C7×C22⋊C4—21no—
127C2×C7×C22⋊C4—21no—
128C2×C7×C22⋊C4—21no—
129C2×C7×C22⋊C4D412yes—
130C7×#107—22no—
131C7×#107—21no—
132C7×#106—21no—
133C7×#103—22no—
134C7×#103—21no—
135C7×#103—22no—
136C7×#103—21no—
137C7×#103—21no—
138C7×#102—21no—
139C7×C22⋊C8—21no—
140C7×C22⋊C8—21no—
141C7×C22⋊C8C2311yescharacteristic
142C7×C22⋊C8C2312yes—
143C7×C22⋊C8C2311yescharacteristic
144C7×#95—21no—
145C7×#95—22no—
146C7×#95—22no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3538C2×C711yescharacteristic
150#3196C2×C711yescharacteristic
151#3189C2×C711yescharacteristic, complemented
152#2817C2×C711yescharacteristic, complemented
153#1617C2×C712yescomplemented
154#1615C2×C712yescomplemented
155#1605C2×C712yescomplemented
156#1603C2×C712yescomplemented
157#1516C2×C711yescharacteristic
158#1511C2×C711yescharacteristic, complemented
159#1472C2×C711yescharacteristic, complemented
160C24×C7C2×D411yescharacteristic
161C7×#41—81no—
162C2×C7×Q8—22no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8—21no—
168C2×C7×Q8—81no—
169C2×C7×Q8—81no—
170C2×C7×Q8—82no—
171C2×C7×Q8—81no—
172C2×C7×Q8C2×D412yes—
173C2×C7×Q8C2×D411yescharacteristic
174C2×C7×D4—21no—
175C2×C7×D4—21no—
176C2×C7×D4—21no—
177C2×C7×D4C2×D411yescharacteristic
178C22×C4×C7—21no—
179C22×C4×C7—22no—
180C22×C4×C7—21no—
181C22×C4×C7—21no—
182C22×C4×C7—22no—
183C22×C4×C7—21no—
184C22×C4×C7—21no—
185C22×C4×C7—81no—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2411yescharacteristic
194C7×Q16—82no—
195C7×Q16—81no—
196C7×SD16—81no—
197C7×M16—81no—
198C2×C7×C8—21no—
199C2×C7×C8—22no—
200C2×C7×C8—21no—
201C2×C7×C8—81no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—22no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—22no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4—21no—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—22no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—21no—
219C7×C4⋊C4—21no—
220C7×C4⋊C4—21no—
221C7×C4⋊C4—41no—
222C7×C4⋊C4—41no—
223C7×C4⋊C4—82no—
224C7×C4⋊C4C2×D411yescharacteristic
225C7×C4⋊C4C2×D412yes—
226C7×C4⋊C4C2×D411yescharacteristic
227C7×C22⋊C4—22no—
228C7×C22⋊C4—22no—
229C7×C22⋊C4—22no—
230C7×C22⋊C4—22no—
231C7×C22⋊C4—22no—
232C7×C22⋊C4—22no—
233C7×C22⋊C4—21no—
234C7×C22⋊C4—21no—
235C7×C22⋊C4—22no—
236C7×C22⋊C4—21no—
237C7×C22⋊C4—21no—
238C7×C22⋊C4—21no—
239C7×C22⋊C4—41no—
240C7×C22⋊C4—41no—
241C7×C22⋊C4—81no—
242C42×C7—22no—
243C42×C7—22no—
244C42×C7—22no—
245C42×C7—21no—
246#563—22no—
247#552—21no—
248#552—21no—
249#551—22no—
250D4×Q8—21no—
251D4×Q8—21no—
252#544—21no—
253#544C22×C712yes—
254#541—21no—
255#539—21no—
256#539C22×C712yes—
257#536—21no—
258#525C22×C711yescharacteristic
259C2×C22⋊Q8C22×C711yescharacteristic
260C2×C22⋊Q8C22×C711yescharacteristic
261C2×C22⋊Q8C22×C711yescharacteristic
262#485—21no—
263#484—21no—
264#484C22×C712yes—
265#484C22×C711yescharacteristic
266#481C22×C711yescharacteristic
267#470—21no—
268#468—21no—
269#467—22no—
270#452—21no—
271#451—21no—
272#451C22×C712yes—
273#451C22×C711yescharacteristic
274#448C22×C711yescharacteristic
275#407C22×C711yescharacteristic
276C2×C22⋊C8C22×C711yescharacteristic
277#400—21no—
278#400C22×C712yes—
279#397—22no—
280#395—21no—
281#395C22×C712yes—
282#381C22×C711yescharacteristic
283#328C22×C712yes—
284#328C22×C712yes—
285#328C22×C712yes—
286#327C22×C712yes—
287#324C22×C712yes—
288#324C22×C712yes—
289C23×C7—22no—
290C23×C7—21no—
291C23×C7—41no—
292C23×C7—41no—
293C23×C7#12011yescharacteristic
294C23×C7#12011yescharacteristic
295C23×C7C22×D411yescharacteristic
296C7×Q8—42no—
297C7×Q8—41no—
298C7×Q8—42no—
299C7×Q8—42no—
300C7×Q8—41no—
301C7×Q8—41no—
302C7×Q8—42no—
303C7×Q8—81no—
304C7×Q8—82no—
305C7×D4—41no—
306C7×D4—42no—
307C7×D4—42no—
308C7×D4—81no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—22no—
312C2×C4×C7—22no—
313C2×C4×C7—21no—
314C2×C4×C7—22no—
315C2×C4×C7—22no—
316C2×C4×C7—24no—
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—42no—
324C2×C4×C7—42no—
325C2×C4×C7—42no—
326C2×C4×C7—42no—
327C2×C4×C7—42no—
328C2×C4×C7—42no—
329C2×C4×C7—42no—
330C2×C4×C7—41no—
331C2×C4×C7—41no—
332C2×C4×C7—42no—
333C2×C4×C7—81no—
334C2×C4×C7—82no—
335C2×C4×C7#12011yescharacteristic
336C2×C4×C7#12011yescharacteristic
337C2×C4×C7C22×D411yescharacteristic
338C2×C4×C7C22×D411yescharacteristic
339C7×C8—42no—
340C7×C8—41no—
341C7×C8—41no—
342C2×#41C7×D411yescharacteristic
343C22×Q8—21no—
344C22×Q8C7×D411yescharacteristic
345C22×Q8C7×D411yescharacteristic
346C22×Q8C23×C711yescharacteristic
347C23×C4C23×C711yescharacteristic
348C2×Q16—22no—
349C2×Q16—21no—
350C2×SD16—21no—
351C2×M16—21no—
352C22×C8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—22no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357#126—22no—
358#126—22no—
359#126—22no—
360#126—22no—
361#125—21no—
362#125—22no—
363#125—22no—
364#125—22no—
365#125—22no—
366#124—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—21no—
371#123—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C22⋊Q8—21no—
388C22⋊Q8C7×D412yes—
389C22⋊Q8C7×D411yescharacteristic
390C22⋊Q8C23×C712yes—
391C22⋊Q8C23×C711yescharacteristic
392#121C7×D411yescharacteristic
393#121C23×C711yescharacteristic
394#120—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×D4—22no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4—22no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117C7×D411yescharacteristic
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4C7×D411yescharacteristic
409C2×C4⋊C4C7×D411yescharacteristic
410C2×C4⋊C4C23×C711yescharacteristic
411C2×C4⋊C4C23×C712yes—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4C7×D412yes—
416#107—22no—
417#107—21no—
418#106—21no—
419#103—22no—
420#103—21no—
421#103—21no—
422#103—22no—
423#103—21no—
424#102—21no—
425C22⋊C8—21no—
426C22⋊C8—21no—
427C22⋊C8C23×C712yes—
428C22⋊C8C23×C711yescharacteristic
429C22⋊C8C23×C711yescharacteristic
430#95—21no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C23×C712yes—
435C22×C7—21no—
436C22×C7—21no—
437C22×C7—21no—
438C22×C7—21no—
439C22×C7—21no—
440C22×C7—21no—
441C22×C7—21no—
442C22×C7—21no—
443C22×C7—41no—
444C22×C7—41no—
445C22×C7—41no—
446C22×C7—82no—
447C22×C7C2×#12011yescharacteristic
448C22×C7#57812yes—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—21no—
452C4×C7—21no—
453C4×C7—42no—
454C4×C7—41no—
455C4×C7—42no—
456C4×C7—41no—
457C4×C7—41no—
458C4×C7—42no—
459C4×C7—42no—
460C4×C7—82no—
461C4×C7—81no—
462C4×C7—81no—
463C24C2×C7×D411yescharacteristic
464#41—81no—
465C2×Q8—22no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—81no—
472C2×Q8—81no—
473C2×Q8—82no—
474C2×Q8—81no—
475C2×Q8C2×C7×D412yes—
476C2×Q8C2×C7×D411yescharacteristic
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4C2×C7×D411yescharacteristic
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—81no—
489C22×C4C2×C7×D412yes—
490C22×C4C2×C7×D412yes—
491C22×C4C2×C7×D411yescharacteristic
492C22×C4C2×C7×D411yescharacteristic
493C22×C4C2×C7×D411yescharacteristic
494C22×C4C2×C7×D411yescharacteristic
495C22×C4C2×C7×D411yescharacteristic
496C22×C4C24×C711yescharacteristic
497Q16—82no—
498Q16—81no—
499SD16—81no—
500M16—81no—
501C2×C8—21no—
502C2×C8—22no—
503C2×C8—21no—
504C2×C8—81no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—82no—
527C4⋊C4C2×C7×D411yescharacteristic
528C4⋊C4C2×C7×D412yes—
529C4⋊C4C2×C7×D411yescharacteristic
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—21no—
538C22⋊C4—22no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—81no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—21no—
549C2×C7—22no—
550C2×C7—41no—
551C2×C7—41no—
552C2×C7—81no—
553C2×C7#301111yescharacteristic
554C2×C7#302011yescharacteristic
555C2×C7#302611yescharacteristic
556C23—22no—
557C23—21no—
558C23—41no—
559C23—41no—
560C23C7×#12011yescharacteristic
561C23C7×#12011yescharacteristic
562C23C22×C7×D411yescharacteristic
563Q8—42no—
564Q8—41no—
565Q8—42no—
566Q8—42no—
567Q8—41no—
568Q8—41no—
569Q8—42no—
570Q8—81no—
571Q8—82no—
572D4—41no—
573D4—42no—
574D4—42no—
575D4—81no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—81no—
601C2×C4—82no—
602C2×C4C7×#12011yescharacteristic
603C2×C4C7×#12011yescharacteristic
604C2×C4C22×C7×D411yescharacteristic
605C2×C4C22×C7×D411yescharacteristic
606C8—42no—
607C8—41no—
608C8—41no—
609C7#1845511yescharacteristic, direct product factor, Sylow
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—82no—
622C22C2×C7×#12011yescharacteristic
623C22C7×#57812yes—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—42no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C2—22no—
639C2—41no—
640C2—41no—
641C2—81no—
642C2C7×#301111yescharacteristic
643C2C7×#302011yescharacteristic
644C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#525C22
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
3. C7×#3189C2semidirect
4. C7×#2817C2semidirect
5. C7×#1617C2semidirect
6. C7×#1615C2semidirect
7. C7×#1605C2semidirect
8. C7×#1603C2semidirect
10. C7×#1511C2semidirect
11. C7×#1472C2semidirect
55. #18455C7direct (N × H)
151. #3189C2×C7semidirect
152. #2817C2×C7semidirect
153. #1617C2×C7semidirect
154. #1615C2×C7semidirect
155. #1605C2×C7semidirect
156. #1603C2×C7semidirect
158. #1511C2×C7semidirect
159. #1472C2×C7semidirect
609. C7#18455direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35382
C7×#31962
C7×#31892
C7×#28172
C7×#16172
C7×#16152
C7×#16052
C7×#16032
C7×#15162
C7×#15112
C7×#14722
#184557