Finite Group Databasebeta

← C7×#22043 #13199259

Subgroups — C7×#22043 #13199259

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#450C211yesmaximal, characteristic
2C4×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C7×#390C211yesmaximal, characteristic
4C7×#1968C212yesmaximal, complemented
5C7×#1934C212yesmaximal, complemented
6C7×#1926C212yesmaximal, complemented
7C7×#1880C212yesmaximal, complemented
8C7×#1872C211yesmaximal, characteristic, complemented
9C7×#1853C211yesmaximal, characteristic, complemented
10C7×#1786C211yesmaximal, characteristic
11C4×C7×C22⋊C8C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×C4×C7×Q8C2211yescharacteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4—21no—
18C22×C7×C22⋊C4—21no—
19C22×C42×C7C2211yescharacteristic
20C7×#450C418yescomplemented
21C2×C7×#106C2212yescomplemented
22C2×C7×#103C2212yescomplemented
23C2×C7×#102C2212yes—
24C2×C7×#102C2212yes—
25C2×C7×#102C2212yes—
26C2×C7×C22⋊C8C2211yescharacteristic
27C2×C7×C22⋊C8C2211yescharacteristic
28C2×C4×C7×C8C2212yescomplemented
29C7×#391C2211yescharacteristic, complemented
30C7×#390—21no—
31C7×#390—21no—
32C7×#390C2212yes—
33C7×#390C2212yes—
34C7×#389C2211yescharacteristic, complemented
35C7×#387—21no—
36C7×#385—21no—
37C7×#380C2211yescharacteristic
38C4×C7×C4⋊C4—21no—
39C4×C7×C4⋊C4C2211yescharacteristic, complemented
40C4×C7×C22⋊C4—21no—
41C7×#340C2212yescomplemented
42C7×#340C2212yescomplemented
43C7×#336C2212yescomplemented
44#22043C711yesmaximal, characteristic, direct product factor, Sylow
45C25×C7—21no—
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4—22no—
49C22×C7×D4—42no—
50C22×C7×D4—42no—
51C22×C7×D4C2×C414yes—
52C22×C7×D4C2312yes—
53C23×C4×C7—21no—
54C23×C4×C7D411yescharacteristic
55C23×C4×C7D411yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C2×C7×SD16—42no—
58C2×C7×SD16—42no—
59C2×C7×SD16C2×C418yescomplemented
60C22×C7×C8C2312yes—
61C22×C7×C8C2312yes—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8C2×C414yes—
64C4×C7×Q8—42no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—42no—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—44no—
81C2×C7×C22⋊C4—44no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7—42no—
85C2×C42×C7D411yescharacteristic
86C2×C42×C7D411yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C7×#106—42no—
90C7×#103—42no—
91C7×#102—42no—
92C7×#102—42no—
93C7×#102C2×C418yescomplemented
94C7×C22⋊C8—22no—
95C7×C22⋊C8C2×C414yes—
96C4×C7×C8—42no—
97C7×#95—21no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C2×#450C2×C711yescharacteristic
103C4×C22⋊Q8C2×C711yescharacteristic, complemented
104C2×#390C2×C711yescharacteristic
105#1968C2×C712yescomplemented
106#1934C2×C712yescomplemented
107#1926C2×C712yescomplemented
108#1880C2×C712yescomplemented
109#1872C2×C711yescharacteristic, complemented
110#1853C2×C711yescharacteristic, complemented
111#1786C2×C711yescharacteristic
112C4×C22⋊C8C2×C711yescharacteristic, complemented
113C24×C7—24no—
114C24×C7—22no—
115C24×C7—42no—
116C24×C7—42no—
117C24×C7—42no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—42no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8C22×C412yes—
126C2×C7×D4—24no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—48no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—48no—
136C2×C7×D4C22×C414yes—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—41no—
147C22×C4×C7—41no—
148C22×C4×C7—41no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7SD1614yescomplemented
153C22×C4×C7C22×C412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7#4112yes—
160C22×C4×C7C2411yescharacteristic
161C7×SD16—48no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8C22×C414yes—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4C22×C412yes—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—44no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—44no—
182C7×C22⋊C4—44no—
183C7×C22⋊C4—44no—
184C42×C7—22no—
185C42×C7—22no—
186C42×C7—41no—
187C42×C7—41no—
188C42×C7—42no—
189C42×C7—42no—
190C23×D4C22×C711yescharacteristic
191C22×SD16C22×C712yes—
192C2×C22⋊Q8C22×C711yescharacteristic
193C2×C4×Q8C22×C711yescharacteristic, complemented
194C22×C4⋊C4C22×C711yescharacteristic
195C22×C22⋊C4—21no—
196C22×C22⋊C4—21no—
197C22×C42C22×C711yescharacteristic
198#450C4×C718yescomplemented
199C2×#106C22×C712yescomplemented
200C2×#103C22×C712yescomplemented
201C2×#102C22×C712yes—
202C2×#102C22×C712yes—
203C2×#102C22×C712yes—
204C2×C22⋊C8C22×C711yescharacteristic
205C2×C22⋊C8C22×C711yescharacteristic
206C2×C4×C8C22×C712yescomplemented
207#391C22×C711yescharacteristic, complemented
208#390—21no—
209#390—21no—
210#390C22×C712yes—
211#390C22×C712yes—
212#389C22×C711yescharacteristic, complemented
213#387—21no—
214#385—21no—
215#380C22×C711yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4C22×C711yescharacteristic, complemented
218C4×C22⋊C4—21no—
219#340C22×C712yescomplemented
220#340C22×C712yescomplemented
221#336C22×C712yescomplemented
222C23×C7—21no—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—24no—
227C23×C7—48no—
228C23×C7—48no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—48no—
236C23×C7—84no—
237C23×C7—84no—
238C23×C7—84no—
239C23×C7—84no—
240C23×C7C4×D412yes—
241C23×C7C2×SD1612yes—
242C23×C7#13512yes—
243C23×C7C22×D411yescharacteristic
244C7×Q8—24no—
245C7×Q8—42no—
246C7×D4—24no—
247C7×D4—24no—
248C7×D4—48no—
249C7×D4—42no—
250C7×D4—42no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—44no—
267C2×C4×C7—44no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—41no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—42no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—84no—
282C2×C4×C7C4×D412yes—
283C2×C4×C7C2×SD1612yes—
284C2×C4×C7C2×SD1612yes—
285C2×C4×C7AGL(1,ℤ/8)12yes—
286C2×C4×C7C23×C411yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7H(5,𝔽2)12yes—
289C2×C4×C7H(5,𝔽2)12yes—
290C7×C8—24no—
291C7×C8—42no—
292C25—21no—
293C22×Q8C23×C711yescharacteristic
294C22×D4—21no—
295C22×D4—22no—
296C22×D4—42no—
297C22×D4—42no—
298C22×D4C2×C4×C714yes—
299C22×D4C23×C712yes—
300C23×C4—21no—
301C23×C4C7×D411yescharacteristic
302C23×C4C7×D411yescharacteristic
303C23×C4C23×C711yescharacteristic
304C2×SD16—42no—
305C2×SD16—42no—
306C2×SD16C2×C4×C718yescomplemented
307C22×C8C23×C712yes—
308C22×C8C23×C712yes—
309C22⋊Q8—22no—
310C22⋊Q8C2×C4×C714yes—
311C4×Q8—42no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4C23×C712yes—
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C4⋊C4C23×C711yescharacteristic
320C2×C4⋊C4C23×C711yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—42no—
332C2×C42C7×D411yescharacteristic
333C2×C42C7×D411yescharacteristic
334C2×C42C23×C711yescharacteristic
335C2×C42C23×C711yescharacteristic
336#106—42no—
337#103—42no—
338#102—42no—
339#102—42no—
340#102C2×C4×C718yescomplemented
341C22⋊C8—22no—
342C22⋊C8C2×C4×C714yes—
343C4×C8—42no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—44no—
356C22×C7—44no—
357C22×C7—44no—
358C22×C7—816no—
359C22×C7—82no—
360C22×C7—82no—
361C22×C7—82no—
362C22×C7—82no—
363C22×C7C4×SD1614yes—
364C22×C7C2×C4×D411yescharacteristic
365C22×C7#53411yescharacteristic
366C22×C7#53511yescharacteristic
367C22×C7C22×SD1611yescharacteristic
368C22×C7C2×#13511yescharacteristic
369C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C7#57511yescharacteristic
371C4×C7—22no—
372C4×C7—22no—
373C4×C7—22no—
374C4×C7—22no—
375C4×C7—42no—
376C4×C7—41no—
377C4×C7—42no—
378C4×C7—81no—
379C4×C7—82no—
380C4×C7—81no—
381C4×C7#51812yes—
382C4×C7#53812yes—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C2×C7×D411yescharacteristic
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22×C4×C712yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—48no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C22×C4×C714yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C7×SD1614yescomplemented
423C22×C4C22×C4×C712yes—
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C7×#4112yes—
430C22×C4C24×C711yescharacteristic
431SD16—48no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8C22×C4×C714yes—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4C22×C4×C712yes—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C2×C7—24no—
461C2×C7—84no—
462C2×C7#232111yescharacteristic
463C2×C7C2×C4×SD1611yescharacteristic
464C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
465C2×C7#318711yescharacteristic
466C2×C7#318811yescharacteristic
467C2×C7#320811yescharacteristic
468C2×C7#321111yescharacteristic
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—48no—
475C23—48no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—48no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23C4×C7×D412yes—
488C23C2×C7×SD1612yes—
489C23C7×#13512yes—
490C23C22×C7×D411yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—24no—
494D4—24no—
495D4—48no—
496D4—42no—
497D4—42no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4C4×C7×D412yes—
530C2×C4C2×C7×SD1612yes—
531C2×C4C2×C7×SD1612yes—
532C2×C4C7×AGL(1,ℤ/8)12yes—
533C2×C4C23×C4×C711yescharacteristic
534C2×C4C2×C7×#4111yescharacteristic
535C2×C4C7×H(5,𝔽2)12yes—
536C2×C4C7×H(5,𝔽2)12yes—
537C8—24no—
538C8—42no—
539C7#2204311yescharacteristic, direct product factor, Sylow
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—816no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22C4×C7×SD1614yes—
555C22C2×C4×C7×D411yescharacteristic
556C22C7×#53411yescharacteristic
557C22C7×#53511yescharacteristic
558C22C22×C7×SD1611yescharacteristic
559C22C2×C7×#13511yescharacteristic
560C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
561C22C7×#57511yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—82no—
571C4—81no—
572C4C7×#51812yes—
573C4C7×#53812yes—
574C2—24no—
575C2—84no—
576C2C7×#232111yescharacteristic
577C2C2×C4×C7×SD1611yescharacteristic
578C2C4×C7×AGL(1,ℤ/8)11yescharacteristic
579C2C7×#318711yescharacteristic
580C2C7×#318811yescharacteristic
581C2C7×#320811yescharacteristic
582C2C7×#321111yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C7×C22⋊Q8C2semidirect
4. C7×#1968C2semidirect
5. C7×#1934C2semidirect
6. C7×#1926C2semidirect
7. C7×#1880C2semidirect
8. C7×#1872C2semidirect
9. C7×#1853C2semidirect
11. C4×C7×C22⋊C8C2semidirect
15. C2×C4×C7×Q8C22semidirect
20. C7×#450C4semidirect
21. C2×C7×#106C22semidirect
22. C2×C7×#103C22semidirect
28. C2×C4×C7×C8C22semidirect
29. C7×#391C22semidirect
34. C7×#389C22semidirect
39. C4×C7×C4⋊C4C22semidirect
41. C7×#340C22semidirect
42. C7×#340C22semidirect
43. C7×#336C22semidirect
44. #22043C7direct (N × H)
59. C2×C7×SD16C2×C4semidirect
93. C7×#102C2×C4semidirect
103. C4×C22⋊Q8C2×C7semidirect
105. #1968C2×C7semidirect
106. #1934C2×C7semidirect
107. #1926C2×C7semidirect
108. #1880C2×C7semidirect
109. #1872C2×C7semidirect
110. #1853C2×C7semidirect
112. C4×C22⋊C8C2×C7semidirect
152. C22×C4×C7SD16semidirect
193. C2×C4×Q8C22×C7semidirect
198. #450C4×C7semidirect
199. C2×#106C22×C7semidirect
200. C2×#103C22×C7semidirect
206. C2×C4×C8C22×C7semidirect
207. #391C22×C7semidirect
212. #389C22×C7semidirect
217. C4×C4⋊C4C22×C7semidirect
219. #340C22×C7semidirect
220. #340C22×C7semidirect
221. #336C22×C7semidirect
306. C2×SD16C2×C4×C7semidirect
340. #102C2×C4×C7semidirect
422. C22×C4C7×SD16semidirect
539. C7#22043direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4502
C4×C7×C22⋊Q82
C2×C7×#3902
C7×#19682
C7×#19342
C7×#19262
C7×#18802
C7×#18722
C7×#18532
C7×#17862
C4×C7×C22⋊C82
#220437