Finite Group Databasebeta

← C7×#23950 #13201166

Subgroups — C7×#23950 #13201166

Order 1792.

1242 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3493C211yesmaximal, characteristic, complemented
2C7×#3005C212yesmaximal, complemented
3C7×#2297C211yesmaximal, characteristic, complemented
4C7×#2028C211yesmaximal, characteristic, complemented
5C7×#2027C211yesmaximal, characteristic, complemented
6C7×#2007C211yesmaximal, characteristic, complemented
7C7×#2006C211yesmaximal, characteristic, complemented
8C7×#1896C212yesmaximal, complemented
9C7×#1883C212yesmaximal, complemented
10C7×#1882C212yesmaximal, complemented
11C7×#1822C211yesmaximal, characteristic, complemented
12C22×C7×#41C2211yescharacteristic
13C2×C7×#137C2212yes—
14C2×C7×AGL(1,ℤ/8)C2212yes—
15C7×#556—22no—
16C7×#555—22no—
17C7×#553—22no—
18C7×#552—22no—
19C7×#544C2211yescharacteristic
20C7×#538—22no—
21C7×#537C2211yescharacteristic
22C7×#529—22no—
23C7×#525C2212yes—
24C7×#520C2212yes—
25C2×C7×#117C2211yescharacteristic
26C22×C42×C7C2211yescharacteristic
27C7×#452—22no—
28C7×#452—22no—
29C7×#449—22no—
30C7×#449—22no—
31C2×C7×#104C2212yes—
32C2×C7×#104C2212yes—
33C7×#419C2212yes—
34C7×#418C2212yes—
35C7×#407C2212yes—
36C7×#391C2211yescharacteristic
37C7×#391C2211yescharacteristic
38C7×#390C2211yescharacteristic
39C7×#390C2211yescharacteristic
40C7×#386—22no—
41C4×C7×C22⋊C4—22no—
42C7×#343C2211yescharacteristic
43C7×#343C2211yescharacteristic
44C7×#343C2212yes—
45C7×#339C2212yes—
46C7×#339C2212yes—
47#23950C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41—42no—
53C2×C7×#41C2312yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×D4—21no—
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7—21no—
58C23×C4×C7—21no—
59C23×C4×C7C2311yescharacteristic
60C7×#137—24no—
61C7×#137—42no—
62C7×AGL(1,ℤ/8)—24no—
63C7×AGL(1,ℤ/8)—42no—
64C2×C7×Q16—22no—
65C2×C7×SD16—22no—
66C2×C7×SD16—22no—
67C2×C7×D8—22no—
68C2×C7×M16C2312yes—
69C2×C7×M16C2312yes—
70C7×C4⋊Q8—22no—
71C7×#126—24no—
72C7×#126—24no—
73C7×#125—22no—
74C7×#125—22no—
75C7×#124—22no—
76C7×#124—24no—
77C7×#123—24no—
78C7×#123—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8D412yes—
82C7×C22⋊Q8D412yes—
83C7×#121—24no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121D412yes—
87C7×#121D412yes—
88C7×#120—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—22no—
91C4×C7×D4—22no—
92C4×C7×D4—22no—
93C4×C7×D4—24no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C7×#117—24no—
97C7×#117—22no—
98C7×#117—22no—
99C7×#117—22no—
100C7×#117C2312yes—
101C2×C7×C4⋊C4—22no—
102C2×C7×C4⋊C4C2311yescharacteristic
103C2×C7×C4⋊C4C2311yescharacteristic
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4D412yes—
107C2×C42×C7—21no—
108C2×C42×C7—42no—
109C2×C42×C7D412yes—
110C2×C42×C7C2311yescharacteristic
111C2×C42×C7C2311yescharacteristic
112C7×#104—42no—
113C7×#104—42no—
114C7×#103—22no—
115C7×#103—22no—
116C7×#102—22no—
117C7×#102—22no—
118C7×C22⋊C8—22no—
119C7×C22⋊C8—22no—
120C7×#95—22no—
121C7×#95—22no—
122#3493C2×C711yescharacteristic, complemented
123#3005C2×C712yescomplemented
124#2297C2×C711yescharacteristic, complemented
125#2028C2×C711yescharacteristic, complemented
126#2027C2×C711yescharacteristic, complemented
127#2007C2×C711yescharacteristic, complemented
128#2006C2×C711yescharacteristic, complemented
129#1896C2×C712yescomplemented
130#1883C2×C712yescomplemented
131#1882C2×C712yescomplemented
132#1822C2×C711yescharacteristic, complemented
133C24×C7—21no—
134C24×C7C2×D411yescharacteristic
135C7×#41—22no—
136C7×#41—22no—
137C7×#41—22no—
138C7×#41—22no—
139C7×#41—44no—
140C7×#41—44no—
141C7×#41—44no—
142C7×#41—44no—
143C2×C7×Q8—22no—
144C2×C7×Q8—21no—
145C2×C7×Q8—22no—
146C2×C7×Q8—22no—
147C2×C7×Q8C2×D412yes—
148C2×C7×Q8#4112yes—
149C2×C7×D4—24no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—21no—
156C2×C7×D4—22no—
157C2×C7×D4—42no—
158C2×C7×D4—42no—
159C2×C7×D4C2×D412yes—
160C2×C7×D4#4112yes—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—22no—
164C22×C4×C7—24no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7—24no—
169C22×C4×C7—22no—
170C22×C4×C7—42no—
171C22×C4×C7—42no—
172C22×C4×C7—42no—
173C22×C4×C7—42no—
174C22×C4×C7—41no—
175C22×C4×C7—41no—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2411yescharacteristic
182C7×Q16—44no—
183C7×SD16—44no—
184C7×SD16—44no—
185C7×D8—44no—
186C7×M16—22no—
187C7×M16—22no—
188C7×M16—22no—
189C7×M16—22no—
190C2×C7×C8—22no—
191C2×C7×C8—22no—
192C7×C4⋊C4—24no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4C2×D412yes—
201C7×C4⋊C4C2×D412yes—
202C7×C22⋊C4—24no—
203C7×C22⋊C4—24no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—44no—
211C7×C22⋊C4—44no—
212C42×C7—24no—
213C42×C7—22no—
214C42×C7—22no—
215C42×C7—22no—
216C42×C7—22no—
217C42×C7—22no—
218C42×C7—42no—
219C22×#41C22×C711yescharacteristic
220C2×#137C22×C712yes—
221C2×AGL(1,ℤ/8)C22×C712yes—
222#556—22no—
223#555—22no—
224#553—22no—
225#552—22no—
226#544C22×C711yescharacteristic
227#538—22no—
228#537C22×C711yescharacteristic
229#529—22no—
230#525C22×C712yes—
231#520C22×C712yes—
232C2×#117C22×C711yescharacteristic
233C22×C42C22×C711yescharacteristic
234#452—22no—
235#452—22no—
236#449—22no—
237#449—22no—
238C2×#104C22×C712yes—
239C2×#104C22×C712yes—
240#419C22×C712yes—
241#418C22×C712yes—
242#407C22×C712yes—
243#391C22×C711yescharacteristic
244#391C22×C711yescharacteristic
245#390C22×C711yescharacteristic
246#390C22×C711yescharacteristic
247#386—22no—
248C4×C22⋊C4—22no—
249#343C22×C712yes—
250#343C22×C711yescharacteristic
251#343C22×C711yescharacteristic
252#339C22×C712yes—
253#339C22×C712yes—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—42no—
261C23×C7—42no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7#12012yes—
265C23×C7C22×D411yescharacteristic
266C7×Q8—22no—
267C7×Q8—24no—
268C7×Q8—22no—
269C7×Q8—42no—
270C7×Q8—82no—
271C7×D4—22no—
272C7×D4—24no—
273C7×D4—22no—
274C7×D4—48no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4—82no—
279C7×D4—82no—
280C7×D4—82no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—44no—
296C2×C4×C7—44no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—48no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—84no—
309C2×C4×C7#12011yescharacteristic
310C2×C4×C7#12011yescharacteristic
311C2×C4×C7C22×D412yes—
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7H(5,𝔽2)12yes—
315C7×C8—42no—
316C7×C8—42no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—22no—
320C2×#41—22no—
321C2×#41—42no—
322C2×#41C23×C712yes—
323C22×Q8C23×C711yescharacteristic
324C22×D4—21no—
325C22×D4C23×C711yescharacteristic
326C23×C4—21no—
327C23×C4—21no—
328C23×C4C23×C711yescharacteristic
329#137—24no—
330#137—42no—
331AGL(1,ℤ/8)—24no—
332AGL(1,ℤ/8)—42no—
333C2×Q16—22no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×D8—22no—
337C2×M16C23×C712yes—
338C2×M16C23×C712yes—
339C4⋊Q8—22no—
340#126—24no—
341#126—24no—
342#125—22no—
343#125—22no—
344#124—22no—
345#124—24no—
346#123—24no—
347#123—22no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8C7×D412yes—
351C22⋊Q8C7×D412yes—
352#121—24no—
353#121—22no—
354#121—22no—
355#121C7×D412yes—
356#121C7×D412yes—
357#120—22no—
358C4×Q8—22no—
359C4×Q8—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—24no—
363C4×D4—22no—
364C4×D4—22no—
365#117—24no—
366#117—22no—
367#117—22no—
368#117—22no—
369#117C23×C712yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4C23×C711yescharacteristic
372C2×C4⋊C4C23×C711yescharacteristic
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4C7×D412yes—
376C2×C42—21no—
377C2×C42—42no—
378C2×C42C7×D412yes—
379C2×C42C23×C711yescharacteristic
380C2×C42C23×C711yescharacteristic
381#104—42no—
382#104—42no—
383#103—22no—
384#103—22no—
385#102—22no—
386#102—22no—
387C22⋊C8—22no—
388C22⋊C8—22no—
389#95—22no—
390#95—22no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—44no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—82no—
404C22×C7—81no—
405C22×C7—81no—
406C22×C7C2×#12011yescharacteristic
407C22×C7#52511yescharacteristic
408C22×C7#53511yescharacteristic
409C4×C7—21no—
410C4×C7—21no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7—42no—
414C4×C7—42no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7#54612yes—
420C4×C7#54612yes—
421C24—21no—
422C24C2×C7×D411yescharacteristic
423#41—22no—
424#41—22no—
425#41—22no—
426#41—22no—
427#41—44no—
428#41—44no—
429#41—44no—
430#41—44no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8C2×C7×D412yes—
436C2×Q8C7×#4112yes—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4C2×C7×D412yes—
448C2×D4C7×#4112yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4C2×C7×D411yescharacteristic
465C22×C4C2×C7×D412yes—
466C22×C4C2×C7×D412yes—
467C22×C4C2×C7×D412yes—
468C22×C4C2×C7×D412yes—
469C22×C4C24×C711yescharacteristic
470Q16—44no—
471SD16—44no—
472SD16—44no—
473D8—44no—
474M16—22no—
475M16—22no—
476M16—22no—
477M16—22no—
478C2×C8—22no—
479C2×C8—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4C2×C7×D412yes—
489C4⋊C4C2×C7×D412yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—42no—
507C2×C7—21no—
508C2×C7—21no—
509C2×C7—42no—
510C2×C7—82no—
511C2×C7—82no—
512C2×C7#240411yescharacteristic
513C2×C7#301811yescharacteristic
514C2×C7#301811yescharacteristic
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23C7×#12012yes—
526C23C22×C7×D411yescharacteristic
527Q8—22no—
528Q8—24no—
529Q8—22no—
530Q8—42no—
531Q8—82no—
532D4—22no—
533D4—24no—
534D4—22no—
535D4—48no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—82no—
541D4—82no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—84no—
570C2×C4C7×#12011yescharacteristic
571C2×C4C7×#12011yescharacteristic
572C2×C4C22×C7×D412yes—
573C2×C4C2×C7×#4111yescharacteristic
574C2×C4C2×C7×#4111yescharacteristic
575C2×C4C7×H(5,𝔽2)12yes—
576C8—42no—
577C8—42no—
578C7#2395011yescharacteristic, direct product factor, Sylow
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22C2×C7×#12011yescharacteristic
595C22C7×#52511yescharacteristic
596C22C7×#53511yescharacteristic
597C4—21no—
598C4—21no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4C7×#54612yes—
608C4C7×#54612yes—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—82no—
613C2—82no—
614C2C7×#240411yescharacteristic
615C2C7×#301811yescharacteristic
616C2C7×#301811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
LCSG ▷ C22×C4 ▷ 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×#3493C2semidirect
2. C7×#3005C2semidirect
3. C7×#2297C2semidirect
4. C7×#2028C2semidirect
5. C7×#2027C2semidirect
6. C7×#2007C2semidirect
7. C7×#2006C2semidirect
8. C7×#1896C2semidirect
9. C7×#1883C2semidirect
10. C7×#1882C2semidirect
11. C7×#1822C2semidirect
47. #23950C7direct (N × H)
122. #3493C2×C7semidirect
123. #3005C2×C7semidirect
124. #2297C2×C7semidirect
125. #2028C2×C7semidirect
126. #2027C2×C7semidirect
127. #2007C2×C7semidirect
128. #2006C2×C7semidirect
129. #1896C2×C7semidirect
130. #1883C2×C7semidirect
131. #1882C2×C7semidirect
132. #1822C2×C7semidirect
578. C7#23950direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34932
C7×#30052
C7×#22972
C7×#20282
C7×#20272
C7×#20072
C7×#20062
C7×#18962
C7×#18832
C7×#18822
C7×#18222
#239507