Finite Group Databasebeta

← C7×#20652 #13197868

Subgroups — C7×#20652 #13197868

Order 1792.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#520C211yesmaximal, characteristic
2C2×C7×#424C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic
4C2×C7×#416C211yesmaximal, characteristic
5C7×#2317C211yesmaximal, characteristic, complemented
6C7×#1805C212yesmaximal, complemented
7C7×#1804C212yesmaximal, complemented
8C7×#1803C212yesmaximal, complemented
9C7×#1802C212yesmaximal, complemented
10C7×#1736C211yesmaximal, characteristic, complemented
11C2×C7×#340C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C7×#520—24no—
16C7×#520C418yescomplemented
17C2×C4×C7×Q8C2212yes—
18C2×C4×C7×D4—22no—
19C2×C4×C7×D4C2212yes—
20C2×C7×#117—21no—
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic
23C22×C7×C4⋊C4—21no—
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C7×#424—24no—
27C2×C7×C4⋊C8C2212yes—
28C2×C7×C4⋊C8C2212yes—
29C7×#417—24no—
30C7×#416—24no—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2212yes—
33C2×C7×#103C2211yescharacteristic
34C2×C7×#102C2211yescharacteristic
35C2×C7×#102C2212yes—
36C2×C7×#102C2211yescharacteristic
37C7×#389—21no—
38C7×#389—21no—
39C7×#385—22no—
40C7×#384C2212yes—
41C7×#382C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46#20652C711yesmaximal, characteristic, direct product factor, Sylow
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41C2×C414yes—
51C2×C7×#41D414yes—
52C2×C7×#41Q814yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×D4—21no—
55C22×C7×D4C2311yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7—22no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8—22no—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2312yes—
70C4×C7×Q8—24no—
71C4×C7×Q8C2×C418yescomplemented
72C4×C7×D4—216no—
73C4×C7×D4—24no—
74C4×C7×D4C2×C418yescomplemented
75C7×#117—28no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117C2×C414yes—
79C7×#117D414yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—28no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4C2×C414yes—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4D411yescharacteristic
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4D412yes—
99C2×C42×C7—22no—
100C2×C42×C7—21no—
101C2×C42×C7C2312yes—
102C7×C4⋊C8—24no—
103C7×C4⋊C8—24no—
104C7×#103—22no—
105C7×#103—22no—
106C7×#103—24no—
107C7×#102—22no—
108C7×#102—22no—
109C7×#102—24no—
110C7×#95—22no—
111C7×#95—22no—
112C2×#520C2×C711yescharacteristic
113C2×#424C2×C711yescharacteristic, complemented
114C2×#417C2×C711yescharacteristic
115C2×#416C2×C711yescharacteristic
116#2317C2×C711yescharacteristic, complemented
117#1805C2×C712yescomplemented
118#1804C2×C712yescomplemented
119#1803C2×C712yescomplemented
120#1802C2×C712yescomplemented
121#1736C2×C711yescharacteristic, complemented
122C2×#340C2×C711yescharacteristic, complemented
123C24×C7—21no—
124C24×C7C2×D411yescharacteristic
125C7×#41—28no—
126C7×#41—28no—
127C7×#41—28no—
128C7×#41C4⋊C4116yescomplemented
129C2×C7×Q8—21no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8C22×C412yes—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8C2×Q812yes—
135C2×C7×D4—24no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C22×C412yes—
142C2×C7×D4C2×D412yes—
143C2×C7×D4C2×Q812yes—
144C22×C4×C7—21no—
145C22×C4×C7—24no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—24no—
150C22×C4×C7—28no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—42no—
155C22×C4×C7—82no—
156C22×C4×C7—82no—
157C22×C4×C7—81no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7—81no—
161C22×C4×C7C22×C412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×Q812yes—
168C22×C4×C7#4112yes—
169C22×C4×C7#4112yes—
170C22×C4×C7C2411yescharacteristic
171C7×M16—44no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—22no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—22no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—22no—
181C2×C7×C8—44no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—28no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—82no—
193C7×C4⋊C4—82no—
194C7×C4⋊C4C22⋊C414yes—
195C7×C4⋊C4C22⋊C414yes—
196C7×C4⋊C4C22×C414yes—
197C7×C22⋊C4—28no—
198C7×C22⋊C4—42no—
199C7×C22⋊C4C22⋊C418yes—
200C42×C7—28no—
201C42×C7—42no—
202C42×C7C22×C414yes—
203C42×C7C2×D414yes—
204C22×#41C22×C711yescharacteristic
205C22×M16C22×C711yescharacteristic
206C23×C8C22×C711yescharacteristic
207#520—24no—
208#520C4×C718yescomplemented
209C2×C4×Q8C22×C712yes—
210C2×C4×D4—22no—
211C2×C4×D4C22×C712yes—
212C2×#117—21no—
213C2×#117C22×C711yescharacteristic
214C2×#117C22×C711yescharacteristic
215C22×C4⋊C4—21no—
216C22×C4⋊C4C22×C711yescharacteristic
217C22×C4⋊C4C22×C711yescharacteristic
218#424—24no—
219C2×C4⋊C8C22×C712yes—
220C2×C4⋊C8C22×C712yes—
221#417—24no—
222#416—24no—
223C2×#103C22×C711yescharacteristic
224C2×#103C22×C712yes—
225C2×#103C22×C711yescharacteristic
226C2×#102C22×C711yescharacteristic
227C2×#102C22×C712yes—
228C2×#102C22×C711yescharacteristic
229#389—21no—
230#389—21no—
231#385—22no—
232#384C22×C712yes—
233#382C22×C712yes—
234#340C22×C712yes—
235#340C22×C712yes—
236#340C22×C712yes—
237#340C22×C712yes—
238C23×C7—21no—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7C2×C22⋊C412yes—
250C23×C7#12112yes—
251C23×C7C22⋊Q812yes—
252C23×C7C22×D411yescharacteristic
253C7×Q8—22no—
254C7×Q8—22no—
255C7×Q8—22no—
256C7×Q8C2×C4⋊C414yes—
257C7×D4—216no—
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—22no—
261C7×D4C2×C4⋊C414yes—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—216no—
270C2×C4×C7—24no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—41no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—88no—
283C2×C4×C7—84no—
284C2×C4×C7—84no—
285C2×C4×C7C2×C22⋊C414yes—
286C2×C4×C7C2×C22⋊C414yes—
287C2×C4×C7C2×C22⋊C412yes—
288C2×C4×C7C2×C4⋊C414yes—
289C2×C4×C7#11714yes—
290C2×C4×C7#12114yes—
291C2×C4×C7#12112yes—
292C2×C4×C7C22⋊Q814yes—
293C2×C4×C7C22⋊Q812yes—
294C2×C4×C7C23×C411yescharacteristic
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C7×C8—41no—
301C7×C8—41no—
302C7×C8—42no—
303C2×#41—22no—
304C2×#41—22no—
305C2×#41—22no—
306C2×#41C2×C4×C714yes—
307C2×#41C7×D414yes—
308C2×#41C7×Q814yes—
309C22×Q8C23×C711yescharacteristic
310C22×D4—21no—
311C22×D4C23×C711yescharacteristic
312C23×C4—21no—
313C23×C4—21no—
314C23×C4—22no—
315C23×C4—21no—
316C23×C4C23×C711yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8C23×C711yescharacteristic
324C22×C8C23×C711yescharacteristic
325C22×C8C23×C712yes—
326C4×Q8—24no—
327C4×Q8C2×C4×C718yescomplemented
328C4×D4—216no—
329C4×D4—24no—
330C4×D4C2×C4×C718yescomplemented
331#117—28no—
332#117—22no—
333#117—22no—
334#117C2×C4×C714yes—
335#117C7×D414yes—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2×C4×C714yes—
346C2×C4⋊C4C7×D411yescharacteristic
347C2×C4⋊C4C7×D411yescharacteristic
348C2×C4⋊C4C23×C712yes—
349C2×C4⋊C4C23×C711yescharacteristic
350C2×C4⋊C4C23×C711yescharacteristic
351C2×C4⋊C4C23×C712yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C7×D412yes—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42C23×C712yes—
358C4⋊C8—24no—
359C4⋊C8—24no—
360#103—22no—
361#103—22no—
362#103—24no—
363#102—22no—
364#102—22no—
365#102—24no—
366#95—22no—
367#95—22no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—22no—
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7#38014yes—
382C22×C7C22×C22⋊C411yescharacteristic
383C22×C7C2×#12111yescharacteristic
384C22×C7C2×C22⋊Q811yescharacteristic
385C22×C7#57611yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57711yescharacteristic
388C22×C7#57811yescharacteristic
389C4×C7—24no—
390C4×C7—24no—
391C4×C7—24no—
392C4×C7—44no—
393C4×C7—81no—
394C4×C7—82no—
395C4×C7—81no—
396C4×C7#38014yes—
397C4×C7C22×C4⋊C411yescharacteristic
398C4×C7C2×#11711yescharacteristic
399C4×C7C2×#12111yescharacteristic
400C4×C7C2×C22⋊Q811yescharacteristic
401C24—21no—
402C24C2×C7×D411yescharacteristic
403#41—28no—
404#41—28no—
405#41—28no—
406#41C7×C4⋊C4116yescomplemented
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8C22×C4×C712yes—
411C2×Q8C2×C7×D412yes—
412C2×Q8C2×C7×Q812yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4C22×C4×C712yes—
420C2×D4C2×C7×D412yes—
421C2×D4C2×C7×Q812yes—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4C22×C4×C712yes—
440C22×C4C2×C7×D412yes—
441C22×C4C2×C7×D412yes—
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D412yes—
444C22×C4C2×C7×D412yes—
445C22×C4C2×C7×Q812yes—
446C22×C4C7×#4112yes—
447C22×C4C7×#4112yes—
448C22×C4C24×C711yescharacteristic
449M16—44no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—44no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4C7×C22⋊C414yes—
473C4⋊C4C7×C22⋊C414yes—
474C4⋊C4C22×C4×C714yes—
475C22⋊C4—28no—
476C22⋊C4—42no—
477C22⋊C4C7×C22⋊C418yes—
478C42—28no—
479C42—42no—
480C42C22×C4×C714yes—
481C42C2×C7×D414yes—
482C2×C7—24no—
483C2×C7—44no—
484C2×C7C2×#38011yescharacteristic
485C2×C7#289711yescharacteristic
486C2×C7#289811yescharacteristic
487C2×C7#304511yescharacteristic
488C2×C7#304611yescharacteristic
489C2×C7#308311yescharacteristic
490C2×C7#308411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C2×C7×C22⋊C412yes—
503C23C7×#12112yes—
504C23C7×C22⋊Q812yes—
505C23C22×C7×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8C2×C7×C4⋊C414yes—
510D4—216no—
511D4—22no—
512D4—22no—
513D4—22no—
514D4C2×C7×C4⋊C414yes—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—216no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—88no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4C2×C7×C22⋊C414yes—
539C2×C4C2×C7×C22⋊C414yes—
540C2×C4C2×C7×C22⋊C412yes—
541C2×C4C2×C7×C4⋊C414yes—
542C2×C4C7×#11714yes—
543C2×C4C7×#12112yes—
544C2×C4C7×#12114yes—
545C2×C4C7×C22⋊Q812yes—
546C2×C4C7×C22⋊Q814yes—
547C2×C4C23×C4×C711yescharacteristic
548C2×C4C22×C7×D411yescharacteristic
549C2×C4C22×C7×D411yescharacteristic
550C2×C4C22×C7×Q811yescharacteristic
551C2×C4C2×C7×#4111yescharacteristic
552C2×C4C2×C7×#4111yescharacteristic
553C8—41no—
554C8—41no—
555C8—42no—
556C7#2065211yescharacteristic, direct product factor, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22C7×#38014yes—
571C22C22×C7×C22⋊C411yescharacteristic
572C22C2×C7×#12111yescharacteristic
573C22C2×C7×C22⋊Q811yescharacteristic
574C22C7×#57611yescharacteristic
575C22C7×#57711yescharacteristic
576C22C7×#57711yescharacteristic
577C22C7×#57811yescharacteristic
578C4—24no—
579C4—24no—
580C4—24no—
581C4—44no—
582C4—82no—
583C4—81no—
584C4—81no—
585C4C7×#38014yes—
586C4C22×C7×C4⋊C411yescharacteristic
587C4C2×C7×#11711yescharacteristic
588C4C2×C7×#12111yescharacteristic
589C4C2×C7×C22⋊Q811yescharacteristic
590C2—24no—
591C2—44no—
592C2C2×C7×#38011yescharacteristic
593C2C7×#289711yescharacteristic
594C2C7×#289811yescharacteristic
595C2C7×#304511yescharacteristic
596C2C7×#304611yescharacteristic
597C2C7×#308311yescharacteristic
598C2C7×#308411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#424C2semidirect
5. C7×#2317C2semidirect
6. C7×#1805C2semidirect
7. C7×#1804C2semidirect
8. C7×#1803C2semidirect
9. C7×#1802C2semidirect
10. C7×#1736C2semidirect
11. C2×C7×#340C2semidirect
16. C7×#520C4semidirect
46. #20652C7direct (N × H)
71. C4×C7×Q8C2×C4semidirect
74. C4×C7×D4C2×C4semidirect
113. C2×#424C2×C7semidirect
116. #2317C2×C7semidirect
117. #1805C2×C7semidirect
118. #1804C2×C7semidirect
119. #1803C2×C7semidirect
120. #1802C2×C7semidirect
121. #1736C2×C7semidirect
122. C2×#340C2×C7semidirect
128. C7×#41C4⋊C4semidirect
208. #520C4×C7semidirect
327. C4×Q8C2×C4×C7semidirect
330. C4×D4C2×C4×C7semidirect
406. #41C7×C4⋊C4semidirect
556. C7#20652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5202
C2×C7×#4242
C2×C7×#4172
C2×C7×#4162
C7×#23172
C7×#18052
C7×#18042
C7×#18032
C7×#18022
C7×#17362
C2×C7×#3402
#206527