Finite Group Databasebeta

← C7×#19014 #13196230

Subgroups — C7×#19014 #13196230

Order 1792.

1140 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C7×#3151C211yesmaximal, characteristic, complemented
4C2×C7×#496C211yesmaximal, characteristic, complemented
5C7×#1656C214yesmaximal, complemented
6C7×#1655C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1527C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C2×C7×AGL(1,ℤ/8)—22no—
11C22×C7×SD16—21no—
12C22×C7×D8—21no—
13C7×#562C2212yescomplemented
14C7×H(3,𝔽4)C2212yescomplemented
15C7×#559—21no—
16C7×#548—21no—
17C7×#546—22no—
18C7×#546—21no—
19C7×D42—21no—
20C7×#542—21no—
21C7×#538—22no—
22C7×#538—21no—
23C7×#535—21no—
24C7×#534—21no—
25C7×#532C2211yescharacteristic
26C2×C7×#127C2211yescharacteristic, complemented
27C7×#529C2211yescharacteristic
28C2×C7×#124C2211yescharacteristic
29C7×#496—22no—
30C7×#496C2212yescomplemented
31C7×#496C2214yescomplemented
32C7×#496C2214yescomplemented
33C7×#493—21no—
34C7×#492—21no—
35C7×#490C2212yescomplemented
36C7×#460—21no—
37C7×#459—21no—
38C7×#450—21no—
39C7×#447—21no—
40C7×#432C2211yescharacteristic, complemented
41C4×C7×M16C2211yescharacteristic, complemented
42C2×C7×#97C2211yescharacteristic, complemented
43C7×#330C2212yescomplemented
44C7×#329C2212yescomplemented
45C7×#329C2214yescomplemented
46C7×#329C2214yescomplemented
47#19014C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C2×C7×#41D412yes—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4—21no—
54C22×C7×D4—21no—
55C22×C7×D4—21no—
56C22×C7×D4D411yescharacteristic
57C7×AGL(1,ℤ/8)—48no—
58C2×C7×SD16—24no—
59C2×C7×SD16—24no—
60C2×C7×SD16—22no—
61C2×C7×SD16—44no—
62C2×C7×D8—24no—
63C2×C7×D8—24no—
64C2×C7×D8—22no—
65C2×C7×D8—44no—
66C2×C7×M16—21no—
67C22×C7×C8—21no—
68C7×C4⋊Q8D411yescharacteristic
69C7×#127—21no—
70C7×#127—21no—
71C7×#127—21no—
72C7×#127—21no—
73C7×#127D411yescharacteristic
74C7×#127C2312yes—
75C7×#126—22no—
76C7×#126—21no—
77C7×#125C2311yescharacteristic
78C7×#124—21no—
79C7×#124—22no—
80C7×#124—22no—
81C7×#124—22no—
82C7×#124—21no—
83C7×#124—21no—
84C7×#124D412yes—
85C7×#124C2312yes—
86C7×#124C2311yescharacteristic
87C7×#124C2312yes—
88C7×#123—22no—
89C7×#123—22no—
90C7×#123—21no—
91C7×#123—22no—
92C7×#123—22no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8D411yescharacteristic
97C7×#121—22no—
98C7×#121—22no—
99C7×#121—22no—
100C7×#121—21no—
101C7×#121—22no—
102C7×#121—22no—
103C7×#121—21no—
104C7×#121—21no—
105C7×#121D411yescharacteristic
106C7×#120—22no—
107C7×#120—22no—
108C7×#120—22no—
109C7×#120—22no—
110C7×#120—22no—
111C4×C7×Q8—21no—
112C4×C7×Q8D411yescharacteristic
113C4×C7×D4—22no—
114C4×C7×D4—22no—
115C4×C7×D4—21no—
116C4×C7×D4—21no—
117C4×C7×D4—21no—
118C4×C7×D4—21no—
119C4×C7×D4D411yescharacteristic
120C7×#117—21no—
121C2×C7×C22⋊C4—22no—
122C2×C7×C22⋊C4—21no—
123C2×C7×C22⋊C4—21no—
124C2×C42×C7C2311yescharacteristic
125C7×C4⋊C8—21no—
126C7×#102—22no—
127C7×#102—22no—
128C7×C22⋊C8—21no—
129C7×#97—21no—
130C7×#97C2312yes—
131C7×#97C2312yes—
132C7×#97C2312yes—
133C4×C7×C8—21no—
134#3558C2×C711yescharacteristic, complemented
135#3217C2×C711yescharacteristic, complemented
136#3151C2×C711yescharacteristic, complemented
137C2×#496C2×C711yescharacteristic, complemented
138#1656C2×C714yescomplemented
139#1655C2×C714yescomplemented
140#1530C2×C711yescharacteristic, complemented
141#1527C2×C711yescharacteristic, complemented
142#1524C2×C711yescharacteristic, complemented
143C24×C7—22no—
144C24×C7—21no—
145C24×C7—41no—
146C7×#41—44no—
147C7×#41—82no—
148C2×C7×Q8—21no—
149C2×C7×Q8—42no—
150C2×C7×Q8C2×D411yescharacteristic
151C2×C7×Q8C2×D412yes—
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—22no—
154C2×C7×D4—21no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4—22no—
164C2×C7×D4—22no—
165C2×C7×D4—22no—
166C2×C7×D4—42no—
167C2×C7×D4—42no—
168C2×C7×D4—42no—
169C2×C7×D4—44no—
170C2×C7×D4—41no—
171C2×C7×D4—41no—
172C2×C7×D4—41no—
173C2×C7×D4—82no—
174C2×C7×D4—82no—
175C2×C7×D4—84no—
176C2×C7×D4C2×D411yescharacteristic
177C2×C7×D4C2×D412yes—
178C2×C7×D4C2×D412yes—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—21no—
182C22×C4×C7—21no—
183C22×C4×C7—22no—
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D411yescharacteristic
187C7×SD16—48no—
188C7×SD16—48no—
189C7×SD16—82no—
190C7×D8—48no—
191C7×D8—48no—
192C7×D8—82no—
193C7×M16—24no—
194C2×C7×C8—24no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C2×C7×C8—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4C2×D411yescharacteristic
207C7×C4⋊C4C2×D411yescharacteristic
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—21no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—21no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—21no—
221C7×C22⋊C4—21no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C42×C7—22no—
225C42×C7—21no—
226C42×C7C2×D411yescharacteristic
227C42×C7C2×D411yescharacteristic
228C42×C7C2×D411yescharacteristic
229C42×C7C2411yescharacteristic
230C2×AGL(1,ℤ/8)—22no—
231C22×SD16—21no—
232C22×D8—21no—
233#562C22×C712yescomplemented
234H(3,𝔽4)C22×C712yescomplemented
235#559—21no—
236#548—21no—
237#546—22no—
238#546—21no—
239D42—21no—
240#542—21no—
241#538—22no—
242#538—21no—
243#535—21no—
244#534—21no—
245#532C22×C711yescharacteristic
246C2×#127C22×C711yescharacteristic, complemented
247#529C22×C711yescharacteristic
248C2×#124C22×C711yescharacteristic
249#496—22no—
250#496C22×C714yescomplemented
251#496C22×C714yescomplemented
252#496C22×C712yescomplemented
253#493—21no—
254#492—21no—
255#490C22×C712yescomplemented
256#460—21no—
257#459—21no—
258#450—21no—
259#447—21no—
260#432C22×C711yescharacteristic, complemented
261C4×M16C22×C711yescharacteristic, complemented
262C2×#97C22×C711yescharacteristic, complemented
263#330C22×C712yescomplemented
264#329C22×C712yescomplemented
265#329C22×C714yescomplemented
266#329C22×C714yescomplemented
267C23×C7—21no—
268C23×C7—22no—
269C23×C7—22no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7—81no—
273C23×C7—81no—
274C23×C7—82no—
275C23×C7—82no—
276C23×C7—82no—
277C23×C7—82no—
278C23×C7—82no—
279C23×C7—82no—
280C23×C7#12011yescharacteristic
281C7×Q8—42no—
282C7×Q8—42no—
283C7×Q8—41no—
284C7×Q8—42no—
285C7×D4—42no—
286C7×D4—42no—
287C7×D4—41no—
288C7×D4—42no—
289C7×D4—44no—
290C7×D4—42no—
291C7×D4—42no—
292C7×D4—42no—
293C7×D4—42no—
294C7×D4—44no—
295C7×D4—42no—
296C7×D4—42no—
297C7×D4—81no—
298C7×D4—82no—
299C7×D4—82no—
300C7×D4—82no—
301C7×D4—82no—
302C7×D4—82no—
303C7×D4—82no—
304C7×D4—82no—
305C7×D4—82no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—21no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—41no—
314C2×C4×C7—42no—
315C2×C4×C7—82no—
316C2×C4×C7—81no—
317C2×C4×C7—82no—
318C2×C4×C7—82no—
319C2×C4×C7#12011yescharacteristic
320C2×C4×C7#12011yescharacteristic
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7C22×D411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C7×C8—24no—
326C7×C8—24no—
327C7×C8—42no—
328C2×#41—21no—
329C2×#41C7×D412yes—
330C22×Q8C7×D411yescharacteristic
331C22×D4—22no—
332C22×D4—21no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—21no—
336C22×D4C7×D411yescharacteristic
337AGL(1,ℤ/8)—48no—
338C2×SD16—24no—
339C2×SD16—24no—
340C2×SD16—22no—
341C2×SD16—44no—
342C2×D8—24no—
343C2×D8—24no—
344C2×D8—22no—
345C2×D8—44no—
346C2×M16—21no—
347C22×C8—21no—
348C4⋊Q8C7×D411yescharacteristic
349#127—21no—
350#127—21no—
351#127—21no—
352#127—21no—
353#127C7×D411yescharacteristic
354#127C23×C712yes—
355#126—22no—
356#126—21no—
357#125C23×C711yescharacteristic
358#124—21no—
359#124—22no—
360#124—22no—
361#124—22no—
362#124—21no—
363#124—21no—
364#124C7×D412yes—
365#124C23×C712yes—
366#124C23×C711yescharacteristic
367#124C23×C712yes—
368#123—22no—
369#123—22no—
370#123—21no—
371#123—22no—
372#123—22no—
373C22⋊Q8—21no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8C7×D411yescharacteristic
377#121—22no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121—22no—
382#121—22no—
383#121—21no—
384#121—21no—
385#121C7×D411yescharacteristic
386#120—22no—
387#120—22no—
388#120—22no—
389#120—22no—
390#120—22no—
391C4×Q8—21no—
392C4×Q8C7×D411yescharacteristic
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C7×D411yescharacteristic
400#117—21no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C42C23×C711yescharacteristic
405C4⋊C8—21no—
406#102—22no—
407#102—22no—
408C22⋊C8—21no—
409#97—21no—
410#97C23×C712yes—
411#97C23×C712yes—
412#97C23×C712yes—
413C4×C8—21no—
414C22×C7—21no—
415C22×C7—21no—
416C22×C7—41no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—82no—
426C22×C7—82no—
427C22×C7—82no—
428C22×C7—162no—
429C22×C7—164no—
430C22×C7—164no—
431C22×C7#45312yes—
432C22×C7C2×#12011yescharacteristic
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—21no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—42no—
441C4×C7—42no—
442C4×C7—41no—
443C4×C7#45312yes—
444C24—22no—
445C24—21no—
446C24—41no—
447#41—44no—
448#41—82no—
449C2×Q8—21no—
450C2×Q8—42no—
451C2×Q8C2×C7×D411yescharacteristic
452C2×Q8C2×C7×D412yes—
453C2×Q8C2×C7×D412yes—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—82no—
475C2×D4—82no—
476C2×D4—84no—
477C2×D4C2×C7×D411yescharacteristic
478C2×D4C2×C7×D412yes—
479C2×D4C2×C7×D412yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4C2×C7×D411yescharacteristic
486C22×C4C2×C7×D411yescharacteristic
487C22×C4C2×C7×D411yescharacteristic
488SD16—48no—
489SD16—48no—
490SD16—82no—
491D8—48no—
492D8—48no—
493D8—82no—
494M16—24no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4C2×C7×D411yescharacteristic
508C4⋊C4C2×C7×D411yescharacteristic
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—21no—
515C22⋊C4—22no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—21no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C42—22no—
526C42—21no—
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C2×C7×D411yescharacteristic
530C42C24×C711yescharacteristic
531C2×C7—22no—
532C2×C7—82no—
533C2×C7—81no—
534C2×C7—82no—
535C2×C7—162no—
536C2×C7C2×#45311yescharacteristic
537C2×C7C2×#45311yescharacteristic
538C2×C7#302111yescharacteristic
539C23—21no—
540C23—22no—
541C23—22no—
542C23—42no—
543C23—42no—
544C23—81no—
545C23—81no—
546C23—82no—
547C23—82no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23C7×#12011yescharacteristic
553Q8—42no—
554Q8—42no—
555Q8—41no—
556Q8—42no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—42no—
561D4—44no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—42no—
568D4—42no—
569D4—81no—
570D4—82no—
571D4—82no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—82no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—82no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4C7×#12011yescharacteristic
592C2×C4C7×#12011yescharacteristic
593C2×C4C7×#12011yescharacteristic
594C2×C4C22×C7×D411yescharacteristic
595C2×C4C22×C7×D411yescharacteristic
596C2×C4C22×C7×D411yescharacteristic
597C8—24no—
598C8—24no—
599C8—42no—
600C7#1901411yescharacteristic, direct product factor, Sylow
601C22—21no—
602C22—21no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—162no—
616C22—164no—
617C22—164no—
618C22C7×#45312yes—
619C22C2×C7×#12011yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4C7×#45312yes—
631C2—22no—
632C2—81no—
633C2—82no—
634C2—82no—
635C2—162no—
636C2C2×C7×#45311yescharacteristic
637C2C2×C7×#45311yescharacteristic
638C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3558C2semidirect
2. C7×#3217C2semidirect
3. C7×#3151C2semidirect
4. C2×C7×#496C2semidirect
5. C7×#1656C2semidirect
6. C7×#1655C2semidirect
7. C7×#1530C2semidirect
8. C7×#1527C2semidirect
9. C7×#1524C2semidirect
13. C7×#562C22semidirect
14. C7×H(3,𝔽4)C22semidirect
26. C2×C7×#127C22semidirect
30. C7×#496C22semidirect
31. C7×#496C22semidirect
32. C7×#496C22semidirect
35. C7×#490C22semidirect
40. C7×#432C22semidirect
41. C4×C7×M16C22semidirect
42. C2×C7×#97C22semidirect
43. C7×#330C22semidirect
44. C7×#329C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19014C7direct (N × H)
134. #3558C2×C7semidirect
135. #3217C2×C7semidirect
136. #3151C2×C7semidirect
137. C2×#496C2×C7semidirect
138. #1656C2×C7semidirect
139. #1655C2×C7semidirect
140. #1530C2×C7semidirect
141. #1527C2×C7semidirect
142. #1524C2×C7semidirect
233. #562C22×C7semidirect
234. H(3,𝔽4)C22×C7semidirect
246. C2×#127C22×C7semidirect
250. #496C22×C7semidirect
251. #496C22×C7semidirect
252. #496C22×C7semidirect
255. #490C22×C7semidirect
260. #432C22×C7semidirect
261. C4×M16C22×C7semidirect
262. C2×#97C22×C7semidirect
263. #330C22×C7semidirect
264. #329C22×C7semidirect
265. #329C22×C7semidirect
266. #329C22×C7semidirect
600. C7#19014direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35582
C7×#32172
C7×#31512
C2×C7×#4962
C7×#16562
C7×#16552
C7×#15302
C7×#15272
C7×#15242
#190147