Finite Group Databasebeta

← C7×#20703 #13197919

Subgroups — C7×#20703 #13197919

Order 1792.

1374 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#520C211yesmaximal, characteristic
2C2×C7×#408C211yesmaximal, characteristic
3C7×#2311C211yesmaximal, characteristic
4C7×#1825C212yesmaximal, complemented
5C7×#1825C212yesmaximal, complemented
6C7×#1824C212yesmaximal, complemented
7C7×#1823C212yesmaximal, complemented
8C7×#1736C211yesmaximal, characteristic, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C7×#1735C211yesmaximal, characteristic, complemented
11C2×C7×#340C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C23×C7×C8C2211yescharacteristic
15C7×#520—24no—
16C7×#520C418yescomplemented
17C2×C4×C7×Q8—21no—
18C2×C4×C7×D4—22no—
19C2×C4×C7×D4—21no—
20C2×C7×#117C2211yescharacteristic
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2212yes—
23C22×C7×C4⋊C4C2211yescharacteristic
24C22×C7×C4⋊C4C2211yescharacteristic
25C22×C7×C4⋊C4C2212yes—
26C7×#408—24no—
27C2×C7×C22⋊C8C2212yes—
28C2×C7×C22⋊C8C2212yes—
29C7×#391—21no—
30C7×#390—21no—
31C7×#386—22no—
32C7×#380C2212yes—
33C7×#380C2212yes—
34C7×#340C2212yes—
35C7×#340C2212yes—
36C7×#340C2211yescharacteristic
37C7×#340C2211yescharacteristic
38C7×#340C2211yescharacteristic
39C7×#340C2211yescharacteristic
40C7×#340C2212yes—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44#20703C711yesmaximal, characteristic, direct product factor, Sylow
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—22no—
48C2×C7×#41—22no—
49C2×C7×#41C2×C414yes—
50C2×C7×#41D414yes—
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4D412yes—
53C22×C7×D4D411yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—22no—
56C23×C4×C7—21no—
57C23×C4×C7C2312yes—
58C23×C4×C7C2311yescharacteristic
59C2×C7×M16—22no—
60C2×C7×M16—22no—
61C2×C7×M16—22no—
62C22×C7×C8—22no—
63C22×C7×C8—22no—
64C22×C7×C8—22no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2312yes—
68C4×C7×Q8—28no—
69C4×C7×D4—28no—
70C4×C7×D4—216no—
71C7×#117—22no—
72C7×#117—24no—
73C7×#117—22no—
74C7×#117—22no—
75C7×#117C2×C414yes—
76C7×#117C2×C418yescomplemented
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—24no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4C2×C414yes—
85C2×C7×C4⋊C4C2×C418yescomplemented
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C4⋊C4C2311yescharacteristic
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2312yes—
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4—22no—
97C2×C42×C7—21no—
98C2×C42×C7—22no—
99C2×C42×C7—21no—
100C7×C22⋊C8—24no—
101C7×C22⋊C8—24no—
102C7×#95—22no—
103C7×#95—22no—
104C2×#520C2×C711yescharacteristic
105C2×#408C2×C711yescharacteristic
106#2311C2×C711yescharacteristic
107#1825C2×C712yescomplemented
108#1825C2×C712yescomplemented
109#1824C2×C712yescomplemented
110#1823C2×C712yescomplemented
111#1736C2×C711yescharacteristic, complemented
112#1736C2×C711yescharacteristic, complemented
113#1735C2×C711yescharacteristic, complemented
114C2×#340C2×C711yescharacteristic, complemented
115C24×C7C2×D412yes—
116C24×C7C2×D411yescharacteristic
117C7×#41—28no—
118C7×#41—28no—
119C7×#41—28no—
120C7×#41—28no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8C22⋊C414yes—
124C2×C7×D4—24no—
125C2×C7×D4—24no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4C22⋊C418yes—
129C2×C7×D4C22⋊C414yes—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—24no—
138C22×C4×C7—28no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—42no—
143C22×C4×C7—81no—
144C22×C4×C7—82no—
145C22×C4×C7—81no—
146C22×C4×C7—82no—
147C22×C4×C7—81no—
148C22×C4×C7—81no—
149C22×C4×C7C22×C414yes—
150C22×C4×C7C22×C412yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D412yes—
155C22×C4×C7#4112yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4112yes—
158C22×C4×C7#4112yes—
159C22×C4×C7C2411yescharacteristic
160C7×M16—44no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—22no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—44no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—28no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4C22×C414yes—
184C7×C4⋊C4C22×C412yes—
185C7×C4⋊C4C22×C412yes—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—82no—
190C7×C22⋊C4—82no—
191C42×C7—24no—
192C42×C7—28no—
193C42×C7—42no—
194C22×#41C22×C711yescharacteristic
195C22×M16C22×C711yescharacteristic
196C23×C8C22×C711yescharacteristic
197#520—24no—
198#520C4×C718yescomplemented
199C2×C4×Q8—21no—
200C2×C4×D4—22no—
201C2×C4×D4—21no—
202C2×#117C22×C711yescharacteristic
203C2×#117C22×C711yescharacteristic
204C2×#117C22×C712yes—
205C22×C4⋊C4C22×C711yescharacteristic
206C22×C4⋊C4C22×C711yescharacteristic
207C22×C4⋊C4C22×C712yes—
208#408—24no—
209C2×C22⋊C8C22×C712yes—
210C2×C22⋊C8C22×C712yes—
211#391—21no—
212#390—21no—
213#386—22no—
214#380C22×C712yes—
215#380C22×C712yes—
216#340C22×C711yescharacteristic
217#340C22×C712yes—
218#340C22×C712yes—
219#340C22×C712yes—
220#340C22×C711yescharacteristic
221#340C22×C712yes—
222#340C22×C712yes—
223#340C22×C711yescharacteristic
224#340C22×C711yescharacteristic
225#340C22×C712yes—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7C2×C22⋊C414yes—
237C23×C7C2×C22⋊C412yes—
238C23×C7#12312yes—
239C23×C7#12312yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—28no—
242C7×D4—216no—
243C7×D4—28no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—42no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—84no—
269C2×C4×C7—88no—
270C2×C4×C7—84no—
271C2×C4×C7C2×C22⋊C414yes—
272C2×C4×C7C2×C22⋊C412yes—
273C2×C4×C7#11714yes—
274C2×C4×C7#11714yes—
275C2×C4×C7#12314yes—
276C2×C4×C7#12314yes—
277C2×C4×C7#12312yes—
278C2×C4×C7#12312yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C7×C8—41no—
286C7×C8—41no—
287C7×C8—42no—
288C2×#41—22no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41C2×C4×C714yes—
293C2×#41C7×D414yes—
294C22×Q8C7×D411yescharacteristic
295C22×D4C7×D412yes—
296C22×D4C7×D411yescharacteristic
297C23×C4—21no—
298C23×C4—22no—
299C23×C4—21no—
300C23×C4C23×C712yes—
301C23×C4C23×C711yescharacteristic
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8—22no—
308C22×C8C23×C711yescharacteristic
309C22×C8C23×C711yescharacteristic
310C22×C8C23×C712yes—
311C4×Q8—28no—
312C4×D4—28no—
313C4×D4—216no—
314#117—22no—
315#117—24no—
316#117—22no—
317#117—22no—
318#117C2×C4×C714yes—
319#117C2×C4×C718yescomplemented
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4C2×C4×C714yes—
328C2×C4⋊C4C2×C4×C718yescomplemented
329C2×C4⋊C4C23×C712yes—
330C2×C4⋊C4C23×C711yescharacteristic
331C2×C4⋊C4C23×C711yescharacteristic
332C2×C4⋊C4C23×C712yes—
333C2×C4⋊C4C23×C711yescharacteristic
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—21no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345#95—22no—
346#95—22no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—24no—
354C22×C7—24no—
355C22×C7—24no—
356C22×C7—42no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7#38114yes—
361C22×C7C22×C22⋊C411yescharacteristic
362C22×C7C2×#12311yescharacteristic
363C22×C7C2×#12311yescharacteristic
364C22×C7#57611yescharacteristic
365C22×C7#57711yescharacteristic
366C22×C7#57711yescharacteristic
367C22×C7#57811yescharacteristic
368C4×C7—24no—
369C4×C7—42no—
370C4×C7—44no—
371C4×C7—42no—
372C4×C7—82no—
373C4×C7—81no—
374C4×C7—81no—
375C4×C7#38114yes—
376C4×C7C2×#11711yescharacteristic
377C4×C7C2×#11711yescharacteristic
378C4×C7C2×#12311yescharacteristic
379C4×C7C2×#12311yescharacteristic
380C24C2×C7×D412yes—
381C24C2×C7×D411yescharacteristic
382#41—28no—
383#41—28no—
384#41—28no—
385#41—28no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8C7×C22⋊C414yes—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4C7×C22⋊C418yes—
394C2×D4C7×C22⋊C414yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—82no—
410C22×C4—81no—
411C22×C4—82no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4C22×C4×C714yes—
415C22×C4C22×C4×C712yes—
416C22×C4C2×C7×D412yes—
417C22×C4C2×C7×D414yes—
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C2×C7×D412yes—
420C22×C4C7×#4112yes—
421C22×C4C7×#4112yes—
422C22×C4C7×#4112yes—
423C22×C4C7×#4112yes—
424C22×C4C24×C711yescharacteristic
425M16—44no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—44no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4C22×C4×C714yes—
449C4⋊C4C22×C4×C712yes—
450C4⋊C4C22×C4×C712yes—
451C22⋊C4—24no—
452C22⋊C4—28no—
453C22⋊C4—42no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C42—24no—
457C42—28no—
458C42—42no—
459C2×C7—24no—
460C2×C7—44no—
461C2×C7C2×#38111yescharacteristic
462C2×C7#289711yescharacteristic
463C2×C7#289811yescharacteristic
464C2×C7#309611yescharacteristic
465C2×C7#309711yescharacteristic
466C2×C7#309811yescharacteristic
467C2×C7#309911yescharacteristic
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23C2×C7×C22⋊C414yes—
479C23C2×C7×C22⋊C412yes—
480C23C7×#12312yes—
481C23C7×#12312yes—
482C23C22×C7×D411yescharacteristic
483Q8—28no—
484D4—216no—
485D4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—88no—
512C2×C4—84no—
513C2×C4C2×C7×C22⋊C414yes—
514C2×C4C2×C7×C22⋊C412yes—
515C2×C4C7×#11714yes—
516C2×C4C7×#11714yes—
517C2×C4C7×#12312yes—
518C2×C4C7×#12312yes—
519C2×C4C7×#12314yes—
520C2×C4C7×#12314yes—
521C2×C4C23×C4×C711yescharacteristic
522C2×C4C22×C7×D411yescharacteristic
523C2×C4C2×C7×#4111yescharacteristic
524C2×C4C2×C7×#4111yescharacteristic
525C2×C4C2×C7×#4111yescharacteristic
526C2×C4C2×C7×#4111yescharacteristic
527C8—41no—
528C8—41no—
529C8—42no—
530C7#2070311yescharacteristic, direct product factor, Sylow
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22C7×#38114yes—
545C22C22×C7×C22⋊C411yescharacteristic
546C22C2×C7×#12311yescharacteristic
547C22C2×C7×#12311yescharacteristic
548C22C7×#57611yescharacteristic
549C22C7×#57711yescharacteristic
550C22C7×#57711yescharacteristic
551C22C7×#57811yescharacteristic
552C4—24no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—81no—
559C4C7×#38114yes—
560C4C2×C7×#11711yescharacteristic
561C4C2×C7×#11711yescharacteristic
562C4C2×C7×#12311yescharacteristic
563C4C2×C7×#12311yescharacteristic
564C2—24no—
565C2—44no—
566C2C2×C7×#38111yescharacteristic
567C2C7×#289711yescharacteristic
568C2C7×#289811yescharacteristic
569C2C7×#309611yescharacteristic
570C2C7×#309711yescharacteristic
571C2C7×#309811yescharacteristic
572C2C7×#309911yescharacteristic
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 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1825C2semidirect
5. C7×#1825C2semidirect
6. C7×#1824C2semidirect
7. C7×#1823C2semidirect
8. C7×#1736C2semidirect
9. C7×#1736C2semidirect
10. C7×#1735C2semidirect
11. C2×C7×#340C2semidirect
16. C7×#520C4semidirect
44. #20703C7direct (N × H)
76. C7×#117C2×C4semidirect
85. C2×C7×C4⋊C4C2×C4semidirect
107. #1825C2×C7semidirect
108. #1825C2×C7semidirect
109. #1824C2×C7semidirect
110. #1823C2×C7semidirect
111. #1736C2×C7semidirect
112. #1736C2×C7semidirect
113. #1735C2×C7semidirect
114. C2×#340C2×C7semidirect
198. #520C4×C7semidirect
319. #117C2×C4×C7semidirect
328. C2×C4⋊C4C2×C4×C7semidirect
530. C7#20703direct (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×#4082
C7×#23112
C7×#18252
C7×#18242
C7×#18232
C7×#17362
C7×#17352
C2×C7×#3402
#207037