Finite Group Databasebeta

← C7×#24117 #13201333

Subgroups — C7×#24117 #13201333

Order 1792.

1238 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic, complemented
4C7×#2537C211yesmaximal, characteristic, complemented
5C7×#2034C212yesmaximal, complemented
6C7×#1999C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1954C211yesmaximal, characteristic, complemented
9C7×#1875C212yesmaximal, complemented
10C7×#1854C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×D8C2211yescharacteristic
13C7×#555—24no—
14C7×#555—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#124—22no—
17C2×C7×#121—21no—
18C2×C7×#121—21no—
19C2×C7×#121C2212yes—
20C2×C7×#121C2212yes—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117—21no—
25C2×C7×#117C2212yes—
26C7×#476—24no—
27C4×C7×D8—24no—
28C2×C7×#107C2211yescharacteristic
29C2×C7×#106C2212yes—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#102C2212yes—
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2211yescharacteristic
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2212yes—
37C2×C4×C7×C8C2211yescharacteristic
38C7×#394—22no—
39C7×#384C2211yescharacteristic
40C7×#380—21no—
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24117C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×D4—22no—
48C22×C7×D4—22no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7—21no—
54C23×C4×C7C2312yes—
55C2×C7×D8—22no—
56C2×C7×D8—22no—
57C2×C7×D8—22no—
58C22×C7×C8C2311yescharacteristic
59C22×C7×C8C2312yes—
60C22×C7×C8C2311yescharacteristic
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8D414yescomplemented
63C7×#124—216no—
64C7×#121—28no—
65C7×#121—24no—
66C7×#121—24no—
67C7×#121—24no—
68C7×#121—42no—
69C7×#121—44no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4D414yescomplemented
76C7×#117—28no—
77C7×#117—24no—
78C7×#117—24no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4D411yescharacteristic
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4D412yes—
94C2×C42×C7—22no—
95C2×C42×C7C2311yescharacteristic
96C7×#107—22no—
97C7×#106—24no—
98C7×C4⋊C8—22no—
99C7×#102—22no—
100C7×#102—24no—
101C7×#102—22no—
102C7×#102—42no—
103C7×C22⋊C8—24no—
104C7×C22⋊C8—42no—
105C4×C7×C8—22no—
106C7×#95—22no—
107C2×#555C2×C711yescharacteristic, complemented
108C2×#476C2×C711yescharacteristic, complemented
109C2×C4×D8C2×C711yescharacteristic, complemented
110#2537C2×C711yescharacteristic, complemented
111#2034C2×C712yescomplemented
112#1999C2×C712yescomplemented
113#1990C2×C711yescharacteristic, complemented
114#1954C2×C711yescharacteristic, complemented
115#1875C2×C712yescomplemented
116#1854C2×C712yescomplemented
117#1802C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7—21no—
120C24×C7C2×D412yes—
121C2×C7×Q8—24no—
122C2×C7×Q8—81no—
123C2×C7×Q8—81no—
124C2×C7×D4—28no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—44no—
139C2×C7×D4—44no—
140C2×C7×D4—82no—
141C2×C7×D4—82no—
142C2×C7×D4—82no—
143C2×C7×D4C2×D412yes—
144C2×C7×D4#4112yes—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C7×D8—28no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—21no—
172C2×C7×C8—22no—
173C2×C7×C8—21no—
174C2×C7×C8#4112yes—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—81no—
185C7×C4⋊C4C2×D414yes—
186C7×C4⋊C4C2×D412yes—
187C7×C22⋊C4—28no—
188C7×C22⋊C4—28no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—44no—
192C7×C22⋊C4—44no—
193C7×C22⋊C4—42no—
194C42×C7—28no—
195C42×C7C2×D412yes—
196C42×C7C2×D412yes—
197C22×D8C22×C711yescharacteristic
198#555—24no—
199#555—24no—
200C2×C4⋊Q8C22×C711yescharacteristic
201C2×#124—22no—
202C2×#121—21no—
203C2×#121—21no—
204C2×#121C22×C712yes—
205C2×#121C22×C712yes—
206C2×C4×D4C22×C711yescharacteristic
207C2×C4×D4C22×C711yescharacteristic
208C2×C4×D4C22×C711yescharacteristic
209C2×#117—21no—
210C2×#117C22×C712yes—
211#476—24no—
212C4×D8—24no—
213C2×#107C22×C711yescharacteristic
214C2×#106C22×C712yes—
215C2×C4⋊C8C22×C711yescharacteristic
216C2×#102C22×C711yescharacteristic
217C2×#102C22×C712yes—
218C2×#102C22×C711yescharacteristic
219C2×#102C22×C712yes—
220C2×C22⋊C8C22×C712yes—
221C2×C22⋊C8C22×C712yes—
222C2×C4×C8C22×C711yescharacteristic
223#394—22no—
224#384C22×C711yescharacteristic
225#380—21no—
226#380C22×C712yes—
227#340C22×C712yes—
228#340C22×C712yes—
229#340C22×C712yes—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—22no—
236C23×C7—22no—
237C23×C7—22no—
238C23×C7—22no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7—41no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7C22×D411yescharacteristic
252C7×Q8—84no—
253C7×D4—22no—
254C7×D4—22no—
255C7×D4—22no—
256C7×D4—22no—
257C7×D4—41no—
258C7×D4—41no—
259C7×D4—41no—
260C7×D4—41no—
261C7×D4—44no—
262C7×D4—88no—
263C7×D4—88no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—24no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—48no—
280C2×C4×C7—41no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—84no—
287C2×C4×C7#12012yes—
288C2×C4×C7#12012yes—
289C2×C4×C7#13512yes—
290C2×C4×C7AGL(1,ℤ/8)12yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C7×C8—22no—
298C7×C8—41no—
299C7×C8—42no—
300C22×Q8—21no—
301C22×D4—22no—
302C22×D4—22no—
303C22×D4C7×D411yescharacteristic
304C22×D4C23×C711yescharacteristic
305C22×D4C23×C711yescharacteristic
306C23×C4—21no—
307C23×C4—21no—
308C23×C4C23×C712yes—
309C2×D8—22no—
310C2×D8—22no—
311C2×D8—22no—
312C22×C8C23×C711yescharacteristic
313C22×C8C23×C712yes—
314C22×C8C23×C711yescharacteristic
315C4⋊Q8—22no—
316C4⋊Q8C7×D414yescomplemented
317#124—216no—
318#121—28no—
319#121—24no—
320#121—24no—
321#121—24no—
322#121—42no—
323#121—44no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4C7×D414yescomplemented
330#117—28no—
331#117—24no—
332#117—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C7×D411yescharacteristic
337C2×C4⋊C4C23×C711yescharacteristic
338C2×C4⋊C4C23×C712yes—
339C2×C4⋊C4C23×C711yescharacteristic
340C2×C4⋊C4C23×C712yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C7×D412yes—
348C2×C42—22no—
349C2×C42C23×C711yescharacteristic
350#107—22no—
351#106—24no—
352C4⋊C8—22no—
353#102—22no—
354#102—22no—
355#102—24no—
356#102—42no—
357C22⋊C8—24no—
358C22⋊C8—42no—
359C4×C8—22no—
360#95—22no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—42no—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—81no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7—81no—
382C22×C7C2×#12011yescharacteristic
383C22×C7#54611yescharacteristic
384C22×C7#54711yescharacteristic
385C22×C7C2×#13511yescharacteristic
386C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C7#57611yescharacteristic
388C22×C7#57711yescharacteristic
389C4×C7—22no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—44no—
393C4×C7—81no—
394C4×C7—82no—
395C4×C7#44914yes—
396C4×C7#52511yescharacteristic
397C4×C7#53711yescharacteristic
398C4×C7#54811yescharacteristic
399C4×C7#55011yescharacteristic
400C24—21no—
401C24—21no—
402C24C2×C7×D412yes—
403C2×Q8—24no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×D4—28no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×C7×D412yes—
426C2×D4C7×#4112yes—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4C2×C7×D412yes—
444C22×C4C2×C7×D411yescharacteristic
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C24×C711yescharacteristic
447D8—28no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—22no—
455C2×C8—21no—
456C2×C8C7×#4112yes—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4C2×C7×D414yes—
468C4⋊C4C2×C7×D412yes—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—42no—
476C42—28no—
477C42C2×C7×D412yes—
478C42C2×C7×D412yes—
479C2×C7—82no—
480C2×C7—82no—
481C2×C7—82no—
482C2×C7#240711yescharacteristic
483C2×C7C2×#44911yescharacteristic
484C2×C7#301211yescharacteristic
485C2×C7#330311yescharacteristic
486C2×C7#332511yescharacteristic
487C2×C7#334111yescharacteristic
488C2×C7#335411yescharacteristic
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C7×D411yescharacteristic
511Q8—84no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4—41no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—44no—
521D4—88no—
522D4—88no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4C7×#12012yes—
547C2×C4C7×#12012yes—
548C2×C4C7×#13512yes—
549C2×C4C7×AGL(1,ℤ/8)12yes—
550C2×C4C22×C7×D411yescharacteristic
551C2×C4C22×C7×D411yescharacteristic
552C2×C4C2×C7×#4111yescharacteristic
553C2×C4C2×C7×#4111yescharacteristic
554C2×C4C7×H(5,𝔽2)11yescharacteristic
555C2×C4C7×2−1+411yescharacteristic
556C8—22no—
557C8—41no—
558C8—42no—
559C7#2411711yescharacteristic, direct product factor, Sylow
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×C7×#12011yescharacteristic
582C22C7×#54611yescharacteristic
583C22C7×#54711yescharacteristic
584C22C2×C7×#13511yescharacteristic
585C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
586C22C7×#57611yescharacteristic
587C22C7×#57711yescharacteristic
588C4—22no—
589C4—42no—
590C4—42no—
591C4—44no—
592C4—81no—
593C4—82no—
594C4C7×#44914yes—
595C4C7×#52511yescharacteristic
596C4C7×#53711yescharacteristic
597C4C7×#54811yescharacteristic
598C4C7×#55011yescharacteristic
599C2—82no—
600C2—82no—
601C2—82no—
602C2C7×#240711yescharacteristic
603C2C2×C7×#44911yescharacteristic
604C2C7×#301211yescharacteristic
605C2C7×#330311yescharacteristic
606C2C7×#332511yescharacteristic
607C2C7×#334111yescharacteristic
608C2C7×#335411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#555C2semidirect
2. C2×C7×#476C2semidirect
3. C2×C4×C7×D8C2semidirect
4. C7×#2537C2semidirect
5. C7×#2034C2semidirect
6. C7×#1999C2semidirect
7. C7×#1990C2semidirect
8. C7×#1954C2semidirect
9. C7×#1875C2semidirect
10. C7×#1854C2semidirect
11. C7×#1802C2semidirect
45. #24117C7direct (N × H)
62. C7×C4⋊Q8D4semidirect
75. C4×C7×D4D4semidirect
107. C2×#555C2×C7semidirect
108. C2×#476C2×C7semidirect
109. C2×C4×D8C2×C7semidirect
110. #2537C2×C7semidirect
111. #2034C2×C7semidirect
112. #1999C2×C7semidirect
113. #1990C2×C7semidirect
114. #1954C2×C7semidirect
115. #1875C2×C7semidirect
116. #1854C2×C7semidirect
117. #1802C2×C7semidirect
316. C4⋊Q8C7×D4semidirect
329. C4×D4C7×D4semidirect
559. C7#24117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C2×C7×#4762
C2×C4×C7×D82
C7×#25372
C7×#20342
C7×#19992
C7×#19902
C7×#19542
C7×#18752
C7×#18542
C7×#18022
#241177