Finite Group Databasebeta

← C7×#24350 #13201566

Subgroups — C7×#24350 #13201566

Order 1792.

1462 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×#3501C211yesmaximal, characteristic, complemented
2C7×#2848C212yesmaximal, complemented
3C4×C7×#120C211yesmaximal, characteristic
4C7×#2062C212yesmaximal, complemented
5C7×#2029C212yesmaximal, complemented
6C7×#1996C212yesmaximal, complemented
7C7×#1889C212yesmaximal
8C7×#1867C212yesmaximal, complemented
9C7×#1799C211yesmaximal, characteristic, complemented
10C24×C4×C7C2211yescharacteristic
11C7×#554—22no—
12C7×#554—24no—
13C7×D4×Q8—22no—
14C7×#544—21no—
15C7×#543—21no—
16C7×#543C2212yes—
17C7×#533—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8—22no—
20C2×C7×C22⋊Q8C2212yes—
21C2×C7×#120C2211yescharacteristic
22C7×#518C2212yes—
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C2×C7×#99C2212yes—
26C7×#386—21no—
27C7×#386C2211yescharacteristic
28C7×#385—21no—
29C7×#385—24no—
30C7×#385C2211yescharacteristic
31C7×#385C2214yes—
32C7×#380—22no—
33C7×#380C2212yes—
34C7×#379C2211yescharacteristic
35C7×#379C2214yes—
36C4×C7×C22⋊C4—21no—
37C4×C7×C22⋊C4C2211yescharacteristic
38C7×#342C2214yes—
39C7×#342C2214yes—
40C7×#342C2212yes—
41#24350C711yesmaximal, characteristic, direct product factor, Sylow
42C25×C7C2311yescharacteristic
43C22×C7×Q8—21no—
44C22×C7×Q8D412yescomplemented
45C22×C7×D4—21no—
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7—22no—
49C23×C4×C7—22no—
50C23×C4×C7—21no—
51C23×C4×C7—44no—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—24no—
58C7×#125—22no—
59C7×#125—24no—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8—22no—
62C7×C22⋊Q8—24no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—22no—
69C7×C22⋊Q8—22no—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—24no—
72C7×C22⋊Q8—48no—
73C7×#120—22no—
74C7×#120Q814yescomplemented
75C4×C7×Q8—22no—
76C4×C7×Q8—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C4×C7×D4—24no—
80C4×C7×D4—24no—
81C4×C7×D4—44no—
82C7×#117—22no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4—24no—
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4D412yescomplemented
90C2×C7×C4⋊C4D414yescomplemented
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—24no—
97C2×C7×C22⋊C4—24no—
98C2×C7×C22⋊C4D411yescharacteristic
99C2×C7×C22⋊C4C2311yescharacteristic
100C2×C7×C22⋊C4C2314yes—
101C2×C7×C22⋊C4C2311yescharacteristic
102C2×C7×C22⋊C4C2314yes—
103C2×C7×C22⋊C4C2311yescharacteristic
104C2×C42×C7—21no—
105C2×C42×C7D411yescharacteristic
106C7×#99—24no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—24no—
111C7×#95—22no—
112#3501C2×C711yescharacteristic, complemented
113#2848C2×C712yescomplemented
114C4×#120C2×C711yescharacteristic
115#2062C2×C712yescomplemented
116#2029C2×C712yescomplemented
117#1996C2×C712yescomplemented
118#1889C2×C712yes—
119#1867C2×C712yescomplemented
120#1799C2×C711yescharacteristic, complemented
121C24×C7—22no—
122C24×C7—21no—
123C24×C7—21no—
124C24×C7—22no—
125C24×C7—42no—
126C24×C7—42no—
127C24×C7C2×D411yescharacteristic
128C24×C7C2×Q812yescomplemented
129C24×C7C2411yescharacteristic
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8—44no—
135C2×C7×Q8—82no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—24no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4C2×Q812yes—
142C2×C7×D4#4112yes—
143C22×C4×C7—24no—
144C22×C4×C7—28no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—24no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—44no—
158C22×C4×C7—42no—
159C22×C4×C7—44no—
160C22×C4×C7—42no—
161C22×C4×C7—44no—
162C22×C4×C7—42no—
163C22×C4×C7—41no—
164C22×C4×C7—42no—
165C22×C4×C7—41no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D414yes—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D414yes—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—24no—
186C7×C4⋊C4—42no—
187C7×C4⋊C4—44no—
188C7×C4⋊C4—48no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—21no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—42no—
200C7×C22⋊C4—48no—
201C7×C22⋊C4—42no—
202C7×C22⋊C4—42no—
203C7×C22⋊C4—48no—
204C7×C22⋊C4C2×Q812yescomplemented
205C42×C7—24no—
206C42×C7—21no—
207C42×C7—21no—
208C42×C7—42no—
209C24×C4C22×C711yescharacteristic
210#554—22no—
211#554—24no—
212D4×Q8—22no—
213#544—21no—
214#543—21no—
215#543C22×C712yes—
216#533—21no—
217C2×C22⋊Q8—21no—
218C2×C22⋊Q8—22no—
219C2×C22⋊Q8C22×C712yes—
220C2×#120C22×C711yescharacteristic
221#518C22×C712yes—
222C2×C4×D4—21no—
223C2×C4×D4C22×C711yescharacteristic
224C2×#99C22×C712yes—
225#386—21no—
226#386C22×C711yescharacteristic
227#385—21no—
228#385—24no—
229#385C22×C711yescharacteristic
230#385C22×C714yes—
231#380—22no—
232#380C22×C712yes—
233#379C22×C714yes—
234#379C22×C711yescharacteristic
235C4×C22⋊C4—21no—
236C4×C22⋊C4C22×C711yescharacteristic
237#342C22×C712yes—
238#342C22×C714yes—
239#342C22×C714yes—
240C23×C7—24no—
241C23×C7—24no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—21no—
245C23×C7—22no—
246C23×C7—24no—
247C23×C7—21no—
248C23×C7—41no—
249C23×C7—44no—
250C23×C7—41no—
251C23×C7—42no—
252C23×C7—41no—
253C23×C7—42no—
254C23×C7—41no—
255C23×C7—44no—
256C23×C7—48no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7—42no—
262C23×C7C22⋊Q812yescomplemented
263C23×C7C22×D411yescharacteristic
264C23×C7C22×D412yes—
265C23×C7C22×Q811yescharacteristic
266C23×C7C2×#4111yescharacteristic
267C23×C72−1+412yes—
268C7×Q8—44no—
269C7×Q8—82no—
270C7×Q8—82no—
271C7×Q8—82no—
272C7×D4—24no—
273C7×D4—24no—
274C7×D4—48no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—21no—
282C2×C4×C7—24no—
283C2×C4×C7—24no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—42no—
287C2×C4×C7—41no—
288C2×C4×C7—44no—
289C2×C4×C7—48no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—44no—
296C2×C4×C7—42no—
297C2×C4×C7—41no—
298C2×C4×C7—44no—
299C2×C4×C7—44no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—44no—
303C2×C4×C7—44no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—44no—
307C2×C4×C7—81no—
308C2×C4×C7—82no—
309C2×C4×C7—82no—
310C2×C4×C7#12011yescharacteristic
311C2×C4×C7#12012yes—
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7C22⋊Q812yescomplemented
314C25C23×C711yescharacteristic
315C22×Q8—21no—
316C22×Q8C7×D412yescomplemented
317C22×D4—21no—
318C22×D4C23×C711yescharacteristic
319C23×C4—21no—
320C23×C4—22no—
321C23×C4—22no—
322C23×C4—21no—
323C23×C4—44no—
324C23×C4C7×D411yescharacteristic
325C23×C4C23×C711yescharacteristic
326C23×C4C23×C711yescharacteristic
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—24no—
330#125—22no—
331#125—24no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—24no—
335C22⋊Q8—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C22⋊Q8—48no—
345#120—22no—
346#120C7×Q814yescomplemented
347C4×Q8—22no—
348C4×Q8—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—24no—
352C4×D4—24no—
353C4×D4—44no—
354#117—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4C7×D411yescharacteristic
361C2×C4⋊C4C7×D412yescomplemented
362C2×C4⋊C4C7×D414yescomplemented
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4C7×D411yescharacteristic
371C2×C22⋊C4C23×C711yescharacteristic
372C2×C22⋊C4C23×C714yes—
373C2×C22⋊C4C23×C711yescharacteristic
374C2×C22⋊C4C23×C714yes—
375C2×C22⋊C4C23×C711yescharacteristic
376C2×C42—21no—
377C2×C42C7×D411yescharacteristic
378#99—24no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—22no—
384C22×C7—22no—
385C22×C7—21no—
386C22×C7—22no—
387C22×C7—21no—
388C22×C7—24no—
389C22×C7—28no—
390C22×C7—22no—
391C22×C7—41no—
392C22×C7—41no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—44no—
397C22×C7—44no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—44no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—81no—
407C22×C7C2×#12011yescharacteristic
408C22×C7C2×C22⋊Q811yescharacteristic
409C22×C7#53611yescharacteristic
410C22×C7#54812yes—
411C22×C7D4×Q812yes—
412C4×C7—22no—
413C4×C7—21no—
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7—41no—
418C4×C7—44no—
419C4×C7—44no—
420C4×C7—41no—
421C4×C7—84no—
422C4×C7—81no—
423C4×C7B(4,𝔽2)12yes—
424C24—22no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—42no—
429C24—42no—
430C24C2×C7×D411yescharacteristic
431C24C2×C7×Q812yescomplemented
432C24C24×C711yescharacteristic
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—44no—
438C2×Q8—82no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×C7×Q812yes—
445C2×D4C7×#4112yes—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—41no—
467C22×C4—42no—
468C22×C4—41no—
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D412yes—
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D411yescharacteristic
473C22×C4C2×C7×D414yes—
474C22×C4C2×C7×D411yescharacteristic
475C22×C4C2×C7×D414yes—
476C22×C4C2×C7×D411yescharacteristic
477C22×C4C2×C7×D412yes—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—24no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—22no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—42no—
507C22⋊C4C2×C7×Q812yescomplemented
508C42—24no—
509C42—21no—
510C42—21no—
511C42—42no—
512C2×C7—21no—
513C2×C7—22no—
514C2×C7—21no—
515C2×C7—22no—
516C2×C7—44no—
517C2×C7—42no—
518C2×C7#240911yescharacteristic
519C2×C7C2×B(4,𝔽2)11yescharacteristic
520C2×C7#302511yescharacteristic
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—22no—
527C23—24no—
528C23—21no—
529C23—41no—
530C23—44no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—44no—
537C23—48no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23C7×C22⋊Q812yescomplemented
544C23C22×C7×D411yescharacteristic
545C23C22×C7×D412yes—
546C23C22×C7×Q811yescharacteristic
547C23C2×C7×#4111yescharacteristic
548C23C7×2−1+412yes—
549Q8—44no—
550Q8—82no—
551Q8—82no—
552Q8—82no—
553D4—24no—
554D4—24no—
555D4—48no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4C7×#12011yescharacteristic
592C2×C4C7×#12012yes—
593C2×C4C7×#12011yescharacteristic
594C2×C4C7×C22⋊Q812yescomplemented
595C7#2435011yescharacteristic, direct product factor, Sylow
596C22—22no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—24no—
601C22—28no—
602C22—22no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—81no—
619C22C2×C7×#12011yescharacteristic
620C22C2×C7×C22⋊Q811yescharacteristic
621C22C7×#53611yescharacteristic
622C22C7×#54812yes—
623C22C7×D4×Q812yes—
624C4—22no—
625C4—21no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—44no—
631C4—44no—
632C4—41no—
633C4—84no—
634C4—81no—
635C4C7×B(4,𝔽2)12yes—
636C2—21no—
637C2—22no—
638C2—21no—
639C2—22no—
640C2—44no—
641C2—42no—
642C2C7×#240911yescharacteristic
643C2C2×C7×B(4,𝔽2)11yescharacteristic
644C2C7×#302511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC2×C7×#120C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3501C2semidirect
2. C7×#2848C2semidirect
4. C7×#2062C2semidirect
5. C7×#2029C2semidirect
6. C7×#1996C2semidirect
8. C7×#1867C2semidirect
9. C7×#1799C2semidirect
41. #24350C7direct (N × H)
44. C22×C7×Q8D4semidirect
74. C7×#120Q8semidirect
89. C2×C7×C4⋊C4D4semidirect
90. C2×C7×C4⋊C4D4semidirect
112. #3501C2×C7semidirect
113. #2848C2×C7semidirect
115. #2062C2×C7semidirect
116. #2029C2×C7semidirect
117. #1996C2×C7semidirect
119. #1867C2×C7semidirect
120. #1799C2×C7semidirect
128. C24×C7C2×Q8semidirect
204. C7×C22⋊C4C2×Q8semidirect
262. C23×C7C22⋊Q8semidirect
313. C2×C4×C7C22⋊Q8semidirect
316. C22×Q8C7×D4semidirect
346. #120C7×Q8semidirect
361. C2×C4⋊C4C7×D4semidirect
362. C2×C4⋊C4C7×D4semidirect
431. C24C2×C7×Q8semidirect
507. C22⋊C4C2×C7×Q8semidirect
543. C23C7×C22⋊Q8semidirect
594. C2×C4C7×C22⋊Q8semidirect
595. C7#24350direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35012
C7×#28482
C4×C7×#1202
C7×#20622
C7×#20292
C7×#19962
C7×#18892
C7×#18672
C7×#17992
#243507