Finite Group Databasebeta

← C7×#18335 #13195551

Subgroups — C7×#18335 #13195551

Order 1792.

1194 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic, complemented
2C7×#3217C212yesmaximal, complemented
3C7×#2541C211yesmaximal, characteristic, complemented
4C7×#1600C214yesmaximal, complemented
5C7×#1596C212yesmaximal, complemented
6C7×#1595C212yesmaximal, complemented
7C7×#1509C212yesmaximal, complemented
8C7×#1466C211yesmaximal, characteristic, complemented
9C7×#543C2211yescharacteristic
10C7×#539—22no—
11C7×#539—21no—
12C7×#538—22no—
13C7×#538C2212yes—
14C7×#534—21no—
15C7×#534C2211yescharacteristic
16C2×C7×#127—21no—
17C2×C7×#127C2211yescharacteristic, complemented
18C7×#529—22no—
19C7×#529C2212yes—
20C7×#525—21no—
21C7×#525—22no—
22C2×C7×#121—21no—
23C2×C7×#121—21no—
24C2×C4×C7×D4—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#496—22no—
27C7×#490—22no—
28C7×#460—22no—
29C7×#459—22no—
30C7×#450C2214yescomplemented
31C7×#447C2214yes—
32C7×#432C2212yescomplemented
33C7×#392C2212yes—
34C7×#392C2212yescomplemented
35C7×#390—21no—
36C4×C7×C4⋊C4C2211yescharacteristic, complemented
37C7×#328C2214yescomplemented
38C7×#327C2214yes—
39C7×#323C2212yes—
40C7×#323C2212yescomplemented
41#18335C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×D4—21no—
44C22×C7×D4—22no—
45C22×C7×D4—22no—
46C22×C7×D4—21no—
47C22×C7×D4—21no—
48C22×C7×D4D411yescharacteristic
49C22×C7×D4C2312yes—
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C2×C7×SD16—24no—
54C2×C7×D8—24no—
55C7×#127—22no—
56C7×#127—22no—
57C7×#127—44no—
58C7×#126—22no—
59C7×#126—22no—
60C7×#126—22no—
61C7×#125—22no—
62C7×#125D412yescomplemented
63C7×#124—22no—
64C7×#124D412yescomplemented
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×#123—22no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8C2312yes—
77C7×#121—22no—
78C7×#121—21no—
79C7×#121—22no—
80C7×#121—22no—
81C7×#121—21no—
82C7×#121—44no—
83C7×#121—44no—
84C7×#121C2312yes—
85C7×#120—22no—
86C7×#120—22no—
87C7×#120—22no—
88C4×C7×Q8D412yescomplemented
89C4×C7×D4—22no—
90C4×C7×D4—22no—
91C4×C7×D4—22no—
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4D412yescomplemented
97C7×#117—21no—
98C7×#117—22no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4D411yescharacteristic
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4C2311yescharacteristic
107C2×C7×C22⋊C4C2311yescharacteristic
108C2×C42×C7—21no—
109C2×C42×C7—21no—
110C2×C42×C7C2311yescharacteristic
111C7×C4⋊C8—22no—
112C7×#102—24no—
113C7×#102—24no—
114C7×C22⋊C8C2314yes—
115C7×#97—22no—
116C7×#95C2312yes—
117#3459C2×C711yescharacteristic, complemented
118#3217C2×C712yescomplemented
119#2541C2×C711yescharacteristic, complemented
120#1600C2×C714yescomplemented
121#1596C2×C712yescomplemented
122#1595C2×C712yescomplemented
123#1509C2×C712yescomplemented
124#1466C2×C711yescharacteristic, complemented
125C24×C7—21no—
126C24×C7—22no—
127C24×C7C2×D411yescharacteristic
128C24×C7C2×D411yescharacteristic
129C7×#41—82no—
130C2×C7×Q8—21no—
131C2×C7×Q8C2×D412yes—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—21no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—24no—
140C2×C7×D4—41no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—44no—
144C2×C7×D4—44no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4—42no—
148C2×C7×D4—42no—
149C2×C7×D4—44no—
150C2×C7×D4—44no—
151C2×C7×D4—42no—
152C2×C7×D4—41no—
153C2×C7×D4—41no—
154C2×C7×D4—82no—
155C2×C7×D4—82no—
156C2×C7×D4—82no—
157C2×C7×D4C2×D412yes—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—42no—
169C22×C4×C7—42no—
170C22×C4×C7—82no—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2411yescharacteristic
176C7×SD16—84no—
177C7×D8—84no—
178C2×C7×C8—24no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—24no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4C2×D412yes—
190C7×C4⋊C4C2×D412yes—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—44no—
204C7×C22⋊C4—44no—
205C42×C7—22no—
206C42×C7—22no—
207C42×C7—42no—
208C42×C7C2×D412yes—
209C42×C7C2×D412yes—
210#543C22×C711yescharacteristic
211#539—22no—
212#539—21no—
213#538—22no—
214#538C22×C712yes—
215#534—21no—
216#534C22×C711yescharacteristic
217C2×#127—21no—
218C2×#127C22×C711yescharacteristic, complemented
219#529—22no—
220#529C22×C712yes—
221#525—21no—
222#525—22no—
223C2×#121—21no—
224C2×#121—21no—
225C2×C4×D4—21no—
226C2×C4×D4C22×C711yescharacteristic
227#496—22no—
228#490—22no—
229#460—22no—
230#459—22no—
231#450C22×C714yescomplemented
232#447C22×C714yes—
233#432C22×C712yescomplemented
234#392C22×C712yes—
235#392C22×C712yescomplemented
236#390—21no—
237C4×C4⋊C4C22×C711yescharacteristic, complemented
238#328C22×C714yescomplemented
239#327C22×C714yes—
240#323C22×C712yes—
241#323C22×C712yescomplemented
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—22no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7—42no—
259C23×C7—42no—
260C23×C7—81no—
261C23×C7—81no—
262C23×C7AGL(1,ℤ/8)12yes—
263C23×C7C22×D411yescharacteristic
264C7×Q8—42no—
265C7×Q8—81no—
266C7×D4—44no—
267C7×D4—42no—
268C7×D4—44no—
269C7×D4—44no—
270C7×D4—44no—
271C7×D4—42no—
272C7×D4—42no—
273C7×D4—84no—
274C7×D4—82no—
275C7×D4—82no—
276C7×D4—81no—
277C7×D4—82no—
278C7×D4—88no—
279C7×D4—88no—
280C7×D4—82no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—44no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—81no—
300C2×C4×C7—81no—
301C2×C4×C7—81no—
302C2×C4×C7—82no—
303C2×C4×C7—81no—
304C2×C4×C7#12012yes—
305C2×C4×C7#12011yescharacteristic
306C2×C4×C7#12011yescharacteristic
307C2×C4×C7C22×D412yes—
308C7×C8—44no—
309C2×#41—21no—
310C22×D4—21no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—21no—
314C22×D4—21no—
315C22×D4C7×D411yescharacteristic
316C22×D4C23×C712yes—
317C23×C4—21no—
318C23×C4C7×D411yescharacteristic
319C23×C4C7×D411yescharacteristic
320C2×SD16—24no—
321C2×D8—24no—
322#127—22no—
323#127—22no—
324#127—44no—
325#126—22no—
326#126—22no—
327#126—22no—
328#125—22no—
329#125C7×D412yescomplemented
330#124—22no—
331#124C7×D412yescomplemented
332#123—22no—
333#123—22no—
334#123—22no—
335#123—22no—
336#123—22no—
337#123—22no—
338C22⋊Q8—21no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8C23×C712yes—
344#121—22no—
345#121—21no—
346#121—22no—
347#121—22no—
348#121—21no—
349#121—44no—
350#121—44no—
351#121C23×C712yes—
352#120—22no—
353#120—22no—
354#120—22no—
355C4×Q8C7×D412yescomplemented
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4C7×D412yescomplemented
364#117—21no—
365#117—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4C7×D411yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C23×C711yescharacteristic
374C2×C22⋊C4C23×C711yescharacteristic
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C23×C711yescharacteristic
378C4⋊C8—22no—
379#102—24no—
380#102—24no—
381C22⋊C8C23×C714yes—
382#97—22no—
383#95C23×C712yes—
384C22×C7—21no—
385C22×C7—21no—
386C22×C7—22no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7—41no—
392C22×C7—42no—
393C22×C7—41no—
394C22×C7—42no—
395C22×C7—42no—
396C22×C7—41no—
397C22×C7—81no—
398C22×C7—81no—
399C22×C7—82no—
400C22×C7—82no—
401C22×C7—82no—
402C22×C7—82no—
403C22×C7—84no—
404C22×C7—82no—
405C22×C7—82no—
406C22×C7—82no—
407C22×C7C2×#12011yescharacteristic
408C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C7#57511yescharacteristic
410C4×C7—22no—
411C4×C7—21no—
412C4×C7—22no—
413C4×C7—44no—
414C4×C7—42no—
415C4×C7—41no—
416C4×C7—41no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7#45312yes—
420C24—21no—
421C24—22no—
422C24C2×C7×D411yescharacteristic
423C24C2×C7×D411yescharacteristic
424#41—82no—
425C2×Q8—21no—
426C2×Q8C2×C7×D412yes—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—41no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C2×D4C2×C7×D412yes—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—82no—
466C22×C4C2×C7×D411yescharacteristic
467C22×C4C2×C7×D411yescharacteristic
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C24×C711yescharacteristic
471SD16—84no—
472D8—84no—
473C2×C8—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4C2×C7×D412yes—
485C4⋊C4C2×C7×D412yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C42—22no—
501C42—22no—
502C42—42no—
503C42C2×C7×D412yes—
504C42C2×C7×D412yes—
505C2×C7—21no—
506C2×C7—21no—
507C2×C7—42no—
508C2×C7—81no—
509C2×C7—82no—
510C2×C7—162no—
511C2×C7#300611yescharacteristic
512C2×C7C2×#45311yescharacteristic
513C2×C7#302711yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—81no—
533C23—81no—
534C23C7×AGL(1,ℤ/8)12yes—
535C23C22×C7×D411yescharacteristic
536Q8—42no—
537Q8—81no—
538D4—44no—
539D4—42no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—84no—
546D4—82no—
547D4—82no—
548D4—81no—
549D4—82no—
550D4—88no—
551D4—88no—
552D4—82no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4C7×#12012yes—
577C2×C4C7×#12011yescharacteristic
578C2×C4C7×#12011yescharacteristic
579C2×C4C22×C7×D412yes—
580C8—44no—
581C7#1833511yescharacteristic, direct product factor, Sylow
582C22—21no—
583C22—21no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×C7×#12011yescharacteristic
606C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
607C22C7×#57511yescharacteristic
608C4—22no—
609C4—21no—
610C4—22no—
611C4—44no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4C7×#45312yes—
618C2—21no—
619C2—21no—
620C2—42no—
621C2—81no—
622C2—82no—
623C2—162no—
624C2C7×#300611yescharacteristic
625C2C2×C7×#45311yescharacteristic
626C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ 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×#3459C2semidirect
2. C7×#3217C2semidirect
3. C7×#2541C2semidirect
4. C7×#1600C2semidirect
5. C7×#1596C2semidirect
6. C7×#1595C2semidirect
7. C7×#1509C2semidirect
8. C7×#1466C2semidirect
17. C2×C7×#127C22semidirect
30. C7×#450C22semidirect
32. C7×#432C22semidirect
34. C7×#392C22semidirect
36. C4×C7×C4⋊C4C22semidirect
37. C7×#328C22semidirect
40. C7×#323C22semidirect
41. #18335C7direct (N × H)
62. C7×#125D4semidirect
64. C7×#124D4semidirect
88. C4×C7×Q8D4semidirect
96. C4×C7×D4D4semidirect
117. #3459C2×C7semidirect
118. #3217C2×C7semidirect
119. #2541C2×C7semidirect
120. #1600C2×C7semidirect
121. #1596C2×C7semidirect
122. #1595C2×C7semidirect
123. #1509C2×C7semidirect
124. #1466C2×C7semidirect
218. C2×#127C22×C7semidirect
231. #450C22×C7semidirect
233. #432C22×C7semidirect
235. #392C22×C7semidirect
237. C4×C4⋊C4C22×C7semidirect
238. #328C22×C7semidirect
241. #323C22×C7semidirect
329. #125C7×D4semidirect
331. #124C7×D4semidirect
355. C4×Q8C7×D4semidirect
363. C4×D4C7×D4semidirect
581. C7#18335direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#32172
C7×#25412
C7×#16002
C7×#15962
C7×#15952
C7×#15092
C7×#14662
#183357