Finite Group Databasebeta

← C7×#23977 #13201193

Subgroups — C7×#23977 #13201193

Order 1792.

1286 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×C7×#448C211yesmaximal, characteristic, complemented
3C2×C7×#419C211yesmaximal, characteristic, complemented
4C7×#2409C211yesmaximal, characteristic, complemented
5C7×#2024C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1895C211yesmaximal, characteristic, complemented
8C7×#1877C212yesmaximal, complemented
9C7×#1873C212yesmaximal, complemented
10C7×#1870C211yesmaximal, characteristic, complemented
11C7×#1825C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C22×C7×M16C2211yescharacteristic
15C7×#555—24no—
16C7×#555—24no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#124—21no—
19C2×C7×#124—21no—
20C2×C7×C22⋊Q8C2212yes—
21C2×C7×#121—21no—
22C2×C7×#121C2211yescharacteristic
23C2×C7×#121C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C2×C4×C7×D4C2211yescharacteristic
26C2×C7×#117—21no—
27C2×C7×#117C2211yescharacteristic
28C2×C7×#117C2211yescharacteristic
29C7×#448—24no—
30C7×#419—24no—
31C2×C7×#103C2212yes—
32C2×C7×#103C2212yes—
33C2×C7×#103C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2211yescharacteristic
36C2×C7×C22⋊C8C2212yes—
37C7×#395—21no—
38C7×#393—21no—
39C7×#391C2212yes—
40C7×#386—21no—
41C7×#385—21no—
42C7×#380C2211yescharacteristic
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#23977C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8—24no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8—42no—
54C22×C7×Q8C2312yes—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4D411yescharacteristic
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7—21no—
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2311yescharacteristic
63C2×C7×SD16—24no—
64C2×C7×SD16—24no—
65C2×C7×SD16—42no—
66C2×C7×M16—22no—
67C2×C7×M16—42no—
68C22×C7×C8C2312yes—
69C22×C7×C8C2312yes—
70C7×C4⋊Q8—22no—
71C7×C4⋊Q8—22no—
72C7×#124—28no—
73C7×#124—28no—
74C7×C22⋊Q8—24no—
75C7×C22⋊Q8—42no—
76C7×#121—28no—
77C7×#121—22no—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121D414yes—
81C4×C7×D4—22no—
82C4×C7×D4—22no—
83C4×C7×D4—22no—
84C4×C7×D4—22no—
85C7×#117—28no—
86C7×#117—22no—
87C7×#117—22no—
88C7×#117—22no—
89C7×#117D414yes—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4D411yescharacteristic
105C2×C7×C22⋊C4D411yescharacteristic
106C2×C42×C7—21no—
107C2×C42×C7—21no—
108C2×C42×C7C2311yescharacteristic
109C7×#103—24no—
110C7×#103—24no—
111C7×#103—24no—
112C7×#102—42no—
113C7×C22⋊C8—22no—
114C7×C22⋊C8—42no—
115C7×#95—21no—
116C7×#95—22no—
117C7×#95—22no—
118C2×#555C2×C711yescharacteristic
119C2×#448C2×C711yescharacteristic, complemented
120C2×#419C2×C711yescharacteristic, complemented
121#2409C2×C711yescharacteristic, complemented
122#2024C2×C712yescomplemented
123#2000C2×C712yescomplemented
124#1895C2×C711yescharacteristic, complemented
125#1877C2×C712yescomplemented
126#1873C2×C712yescomplemented
127#1870C2×C711yescharacteristic, complemented
128#1825C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C2×C7×Q8—24no—
133C2×C7×Q8—22no—
134C2×C7×Q8—28no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—44no—
140C2×C7×Q8—48no—
141C2×C7×Q8—44no—
142C2×C7×Q8—42no—
143C2×C7×Q8—82no—
144C2×C7×Q8#4114yes—
145C2×C7×D4—22no—
146C2×C7×D4—24no—
147C2×C7×D4—24no—
148C2×C7×D4—22no—
149C2×C7×D4—21no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—41no—
153C2×C7×D4—82no—
154C2×C7×D4—82no—
155C2×C7×D4C2×D412yes—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—24no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—44no—
169C22×C4×C7—42no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2411yescharacteristic
179C7×SD16—48no—
180C7×M16—44no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—44no—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D412yes—
197C7×C4⋊C4C2×D412yes—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—24no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—44no—
206C7×C22⋊C4—42no—
207C42×C7—24no—
208C42×C7—24no—
209C42×C7—21no—
210C42×C7—21no—
211C23×Q8C22×C711yescharacteristic
212C22×SD16C22×C712yes—
213C22×M16C22×C711yescharacteristic
214#555—24no—
215#555—24no—
216C2×C4⋊Q8C22×C711yescharacteristic
217C2×#124—21no—
218C2×#124—21no—
219C2×C22⋊Q8C22×C712yes—
220C2×#121—21no—
221C2×#121C22×C711yescharacteristic
222C2×#121C22×C711yescharacteristic
223C2×C4×D4C22×C711yescharacteristic
224C2×C4×D4C22×C711yescharacteristic
225C2×#117—21no—
226C2×#117C22×C711yescharacteristic
227C2×#117C22×C711yescharacteristic
228#448—24no—
229#419—24no—
230C2×#103C22×C712yes—
231C2×#103C22×C712yes—
232C2×#103C22×C712yes—
233C2×#102C22×C712yes—
234C2×C22⋊C8C22×C711yescharacteristic
235C2×C22⋊C8C22×C712yes—
236#395—21no—
237#393—21no—
238#391C22×C712yes—
239#386—21no—
240#385—21no—
241#380C22×C711yescharacteristic
242#340C22×C712yes—
243#340C22×C712yes—
244#340C22×C712yes—
245#336C22×C712yes—
246C23×C7—22no—
247C23×C7—21no—
248C23×C7—21no—
249C23×C7—22no—
250C23×C7—21no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7—42no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7—42no—
260C23×C7#12012yes—
261C23×C7C22×D411yescharacteristic
262C7×Q8—24no—
263C7×Q8—24no—
264C7×Q8—48no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—84no—
268C7×D4—24no—
269C7×D4—44no—
270C7×D4—41no—
271C7×D4—41no—
272C7×D4—84no—
273C7×D4—84no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—24no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—48no—
289C2×C4×C7—48no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7#12012yes—
303C2×C4×C7#13712yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C2×#4112yes—
307C2×C4×C7H(5,𝔽2)12yes—
308C7×C8—42no—
309C7×C8—42no—
310C22×Q8—21no—
311C22×Q8—21no—
312C22×Q8—24no—
313C22×Q8—22no—
314C22×Q8—22no—
315C22×Q8—42no—
316C22×Q8C23×C712yes—
317C22×D4—21no—
318C22×D4—21no—
319C22×D4C7×D411yescharacteristic
320C22×D4C23×C711yescharacteristic
321C23×C4—21no—
322C23×C4—21no—
323C23×C4C23×C711yescharacteristic
324C23×C4C23×C711yescharacteristic
325C2×SD16—24no—
326C2×SD16—24no—
327C2×SD16—42no—
328C2×M16—22no—
329C2×M16—42no—
330C22×C8C23×C712yes—
331C22×C8C23×C712yes—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334#124—28no—
335#124—28no—
336C22⋊Q8—24no—
337C22⋊Q8—42no—
338#121—28no—
339#121—22no—
340#121—22no—
341#121—22no—
342#121C7×D414yes—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4—22no—
347#117—28no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117C7×D414yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4C7×D411yescharacteristic
355C2×C4⋊C4C23×C712yes—
356C2×C4⋊C4C23×C711yescharacteristic
357C2×C4⋊C4C23×C711yescharacteristic
358C2×C4⋊C4C23×C711yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4C7×D411yescharacteristic
367C2×C22⋊C4C7×D411yescharacteristic
368C2×C42—21no—
369C2×C42—21no—
370C2×C42C23×C711yescharacteristic
371#103—24no—
372#103—24no—
373#103—24no—
374#102—42no—
375C22⋊C8—22no—
376C22⋊C8—42no—
377#95—21no—
378#95—22no—
379#95—22no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—41no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7—82no—
394C22×C7—82no—
395C22×C7C2×#12011yescharacteristic
396C22×C7#54612yes—
397C22×C7C2×#13711yescharacteristic
398C22×C7#57511yescharacteristic
399C22×C7#57712yes—
400C4×C7—22no—
401C4×C7—22no—
402C4×C7—41no—
403C4×C7—44no—
404C4×C7—42no—
405C4×C7—41no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—82no—
409C4×C7#52511yescharacteristic
410C4×C7#53511yescharacteristic
411C4×C7#54612yes—
412C24—21no—
413C24C2×C7×D411yescharacteristic
414C24C2×C7×D411yescharacteristic
415C2×Q8—24no—
416C2×Q8—22no—
417C2×Q8—28no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—44no—
423C2×Q8—48no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—82no—
427C2×Q8C7×#4114yes—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4C2×C7×D412yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4C2×C7×D411yescharacteristic
456C22×C4C2×C7×D411yescharacteristic
457C22×C4C2×C7×D411yescharacteristic
458C22×C4C2×C7×D412yes—
459C22×C4C2×C7×D412yes—
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C24×C711yescharacteristic
462SD16—48no—
463M16—44no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—44no—
478C4⋊C4C2×C7×D412yes—
479C4⋊C4C2×C7×D412yes—
480C4⋊C4C2×C7×D412yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—22no—
486C22⋊C4—24no—
487C22⋊C4—22no—
488C22⋊C4—44no—
489C22⋊C4—42no—
490C42—24no—
491C42—24no—
492C42—21no—
493C42—21no—
494C2×C7—42no—
495C2×C7—81no—
496C2×C7—82no—
497C2×C7#240411yescharacteristic
498C2×C7#300711yescharacteristic
499C2×C7#300911yescharacteristic
500C2×C7#332112yes—
501C2×C7#332912yes—
502C23—22no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—22no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23C7×#12012yes—
517C23C22×C7×D411yescharacteristic
518Q8—24no—
519Q8—24no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—84no—
524D4—24no—
525D4—44no—
526D4—41no—
527D4—41no—
528D4—84no—
529D4—84no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4C7×#12012yes—
559C2×C4C7×#13712yes—
560C2×C4C22×C7×D411yescharacteristic
561C2×C4C22×C7×D411yescharacteristic
562C2×C4C2×C7×#4112yes—
563C2×C4C7×H(5,𝔽2)12yes—
564C8—42no—
565C8—42no—
566C7#2397711yescharacteristic, direct product factor, Sylow
567C22—22no—
568C22—22no—
569C22—22no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—82no—
581C22—82no—
582C22C2×C7×#12011yescharacteristic
583C22C7×#54612yes—
584C22C2×C7×#13711yescharacteristic
585C22C7×#57511yescharacteristic
586C22C7×#57712yes—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—44no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—82no—
596C4C7×#52511yescharacteristic
597C4C7×#53511yescharacteristic
598C4C7×#54612yes—
599C2—42no—
600C2—81no—
601C2—82no—
602C2C7×#240411yescharacteristic
603C2C7×#300711yescharacteristic
604C2C7×#300911yescharacteristic
605C2C7×#332112yes—
606C2C7×#332912yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#555C2
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
2. C2×C7×#448C2semidirect
3. C2×C7×#419C2semidirect
4. C7×#2409C2semidirect
5. C7×#2024C2semidirect
6. C7×#2000C2semidirect
7. C7×#1895C2semidirect
8. C7×#1877C2semidirect
9. C7×#1873C2semidirect
10. C7×#1870C2semidirect
11. C7×#1825C2semidirect
47. #23977C7direct (N × H)
119. C2×#448C2×C7semidirect
120. C2×#419C2×C7semidirect
121. #2409C2×C7semidirect
122. #2024C2×C7semidirect
123. #2000C2×C7semidirect
124. #1895C2×C7semidirect
125. #1877C2×C7semidirect
126. #1873C2×C7semidirect
127. #1870C2×C7semidirect
128. #1825C2×C7semidirect
566. C7#23977direct (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×#4482
C2×C7×#4192
C7×#24092
C7×#20242
C7×#20002
C7×#18952
C7×#18772
C7×#18732
C7×#18702
C7×#18252
#239777