Finite Group Databasebeta

← C7×#19020 #13196236

Subgroups — C7×#19020 #13196236

Order 1792.

960 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic, complemented
3C7×#3141C211yesmaximal, characteristic, complemented
4C2×C7×#489C211yesmaximal, characteristic, complemented
5C7×#1662C214yesmaximal, complemented
6C7×#1659C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1527C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C7×#562C2212yes—
11C7×H(3,𝔽4)C2212yes—
12C7×#559—21no—
13C7×#548—21no—
14C7×#546—22no—
15C7×#546—21no—
16C7×D42—21no—
17C7×#542—21no—
18C7×#538—22no—
19C7×#538—21no—
20C7×#535—21no—
21C7×#534—21no—
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic, complemented
24C7×#529C2211yescharacteristic
25C2×C7×#124C2211yescharacteristic
26C7×#497C2212yescomplemented
27C7×#491C2212yescomplemented
28C7×#489—22no—
29C7×#489C2214yescomplemented
30C7×#489C2214yescomplemented
31C7×#487—21no—
32C7×#486—21no—
33C7×#462—21no—
34C7×#461—21no—
35C7×#451—21no—
36C7×#448—21no—
37C7×#432C2211yescharacteristic, complemented
38C7×#417—22no—
39C2×C7×#103—21no—
40C2×C7×#102—21no—
41C4×C7×M16C2211yescharacteristic, complemented
42C2×C7×#97C2211yescharacteristic, complemented
43C7×#330C2212yescomplemented
44C7×#329C2212yescomplemented
45C7×#329C2214yescomplemented
46C7×#329C2214yescomplemented
47#19020C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C2×C7×#41D412yes—
50C22×C7×Q8—21no—
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4—22no—
53C22×C7×D4—21no—
54C22×C7×D4D411yescharacteristic
55C2×C7×Q16—22no—
56C2×C7×SD16—22no—
57C2×C7×M16—21no—
58C22×C7×C8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8D411yescharacteristic
63C7×C4⋊Q8C2312yes—
64C7×#127—21no—
65C7×#127D411yescharacteristic
66C7×#126—22no—
67C7×#126—21no—
68C7×#125C2311yescharacteristic
69C7×#124—21no—
70C7×#124—22no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#124—21no—
74C7×#124—21no—
75C7×#124D412yes—
76C7×#124C2312yes—
77C7×#124C2312yes—
78C7×#124C2311yescharacteristic
79C7×#123—22no—
80C7×#123—22no—
81C7×#123—21no—
82C7×#123—22no—
83C7×#123—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8D411yescharacteristic
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—21no—
92C7×#121—22no—
93C7×#121—22no—
94C7×#121—21no—
95C7×#121—21no—
96C7×#121D411yescharacteristic
97C7×#120—22no—
98C7×#120—22no—
99C7×#120—22no—
100C7×#120—22no—
101C7×#120—22no—
102C4×C7×Q8—21no—
103C4×C7×Q8D411yescharacteristic
104C4×C7×D4—22no—
105C4×C7×D4—22no—
106C4×C7×D4—21no—
107C4×C7×D4—21no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4D411yescharacteristic
111C7×#117—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C22⋊C4—22no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C42×C7C2311yescharacteristic
118C7×C4⋊C8—21no—
119C7×#103—24no—
120C7×#103—24no—
121C7×#103—22no—
122C7×#103—22no—
123C7×#102—24no—
124C7×#102—24no—
125C7×C22⋊C8—21no—
126C7×#97—21no—
127C7×#97C2312yes—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C4×C7×C8—21no—
131#3558C2×C711yescharacteristic
132#3218C2×C711yescharacteristic, complemented
133#3141C2×C711yescharacteristic, complemented
134C2×#489C2×C711yescharacteristic, complemented
135#1662C2×C714yescomplemented
136#1659C2×C714yescomplemented
137#1530C2×C711yescharacteristic, complemented
138#1527C2×C711yescharacteristic, complemented
139#1524C2×C711yescharacteristic, complemented
140C24×C7—22no—
141C24×C7—21no—
142C7×#41—44no—
143C7×#41—82no—
144C2×C7×Q8—21no—
145C2×C7×Q8—22no—
146C2×C7×Q8—42no—
147C2×C7×Q8—41no—
148C2×C7×Q8—82no—
149C2×C7×Q8C2×D412yes—
150C2×C7×Q8C2×D412yes—
151C2×C7×Q8C2×D411yescharacteristic
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—42no—
163C2×C7×D4—82no—
164C2×C7×D4—84no—
165C2×C7×D4C2×D412yes—
166C2×C7×D4C2×D412yes—
167C2×C7×D4C2×D411yescharacteristic
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—22no—
173C22×C4×C7—41no—
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C7×Q16—82no—
178C7×SD16—82no—
179C7×M16—42no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C2×C7×C8—42no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—41no—
195C7×C4⋊C4—41no—
196C7×C4⋊C4C2×D411yescharacteristic
197C7×C4⋊C4C2×D411yescharacteristic
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—21no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—21no—
211C7×C22⋊C4—21no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C42×C7—22no—
215C42×C7—21no—
216C42×C7C2×D411yescharacteristic
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219C42×C7C2411yescharacteristic
220#562C22×C712yes—
221H(3,𝔽4)C22×C712yes—
222#559—21no—
223#548—21no—
224#546—22no—
225#546—21no—
226D42—21no—
227#542—21no—
228#538—22no—
229#538—21no—
230#535—21no—
231#534—21no—
232#532C22×C711yescharacteristic
233C2×C4⋊Q8C22×C711yescharacteristic, complemented
234#529C22×C711yescharacteristic
235C2×#124C22×C711yescharacteristic
236#497C22×C712yescomplemented
237#491C22×C712yescomplemented
238#489—22no—
239#489C22×C714yescomplemented
240#489C22×C714yescomplemented
241#487—21no—
242#486—21no—
243#462—21no—
244#461—21no—
245#451—21no—
246#448—21no—
247#432C22×C711yescharacteristic, complemented
248#417—22no—
249C2×#103—21no—
250C2×#102—21no—
251C4×M16C22×C711yescharacteristic, complemented
252C2×#97C22×C711yescharacteristic, complemented
253#330C22×C712yescomplemented
254#329C22×C712yescomplemented
255#329C22×C714yescomplemented
256#329C22×C714yescomplemented
257C23×C7—21no—
258C23×C7—22no—
259C23×C7—22no—
260C23×C7—42no—
261C23×C7—81no—
262C23×C7—81no—
263C23×C7—82no—
264C23×C7—82no—
265C23×C7—82no—
266C23×C7#12011yescharacteristic
267C7×Q8—41no—
268C7×Q8—42no—
269C7×Q8—42no—
270C7×Q8—42no—
271C7×Q8—42no—
272C7×Q8—42no—
273C7×Q8—82no—
274C7×D4—41no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4—44no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—81no—
282C7×D4—82no—
283C7×D4—82no—
284C7×D4—82no—
285C7×D4—82no—
286C7×D4—82no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—41no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—82no—
298C2×C4×C7—81no—
299C2×C4×C7—82no—
300C2×C4×C7—82no—
301C2×C4×C7—82no—
302C2×C4×C7#12011yescharacteristic
303C2×C4×C7#12011yescharacteristic
304C2×C4×C7#12011yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C22×D411yescharacteristic
308C7×C8—42no—
309C7×C8—42no—
310C7×C8—42no—
311C2×#41—21no—
312C2×#41C7×D412yes—
313C22×Q8—21no—
314C22×Q8C7×D411yescharacteristic
315C22×D4—22no—
316C22×D4—21no—
317C22×D4C7×D411yescharacteristic
318C2×Q16—22no—
319C2×SD16—22no—
320C2×M16—21no—
321C22×C8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8C7×D411yescharacteristic
326C4⋊Q8C23×C712yes—
327#127—21no—
328#127C7×D411yescharacteristic
329#126—22no—
330#126—21no—
331#125C23×C711yescharacteristic
332#124—21no—
333#124—22no—
334#124—22no—
335#124—22no—
336#124—21no—
337#124—21no—
338#124C7×D412yes—
339#124C23×C712yes—
340#124C23×C711yescharacteristic
341#124C23×C712yes—
342#123—22no—
343#123—22no—
344#123—21no—
345#123—22no—
346#123—22no—
347C22⋊Q8—21no—
348C22⋊Q8—22no—
349C22⋊Q8—22no—
350C22⋊Q8C7×D411yescharacteristic
351#121—22no—
352#121—22no—
353#121—22no—
354#121—21no—
355#121—22no—
356#121—22no—
357#121—21no—
358#121—21no—
359#121C7×D411yescharacteristic
360#120—22no—
361#120—22no—
362#120—22no—
363#120—22no—
364#120—22no—
365C4×Q8—21no—
366C4×Q8C7×D411yescharacteristic
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4C7×D411yescharacteristic
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42C23×C711yescharacteristic
381C4⋊C8—21no—
382#103—24no—
383#103—24no—
384#103—22no—
385#103—22no—
386#102—24no—
387#102—24no—
388C22⋊C8—21no—
389#97—21no—
390#97C23×C712yes—
391#97C23×C712yes—
392#97C23×C712yes—
393C4×C8—21no—
394C22×C7—21no—
395C22×C7—21no—
396C22×C7—41no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—162no—
406C22×C7—164no—
407C22×C7#45512yes—
408C22×C7C2×#12011yescharacteristic
409C4×C7—21no—
410C4×C7—21no—
411C4×C7—21no—
412C4×C7—21no—
413C4×C7—21no—
414C4×C7—41no—
415C4×C7—41no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—41no—
419C4×C7—82no—
420C4×C7#45512yes—
421C24—22no—
422C24—21no—
423#41—44no—
424#41—82no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—41no—
429C2×Q8—82no—
430C2×Q8C2×C7×D412yes—
431C2×Q8C2×C7×D412yes—
432C2×Q8C2×C7×D411yescharacteristic
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—82no—
445C2×D4—84no—
446C2×D4C2×C7×D412yes—
447C2×D4C2×C7×D412yes—
448C2×D4C2×C7×D411yescharacteristic
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4C2×C7×D411yescharacteristic
456C22×C4C2×C7×D411yescharacteristic
457C22×C4C2×C7×D411yescharacteristic
458Q16—82no—
459SD16—82no—
460M16—42no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—42no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4C2×C7×D411yescharacteristic
478C4⋊C4C2×C7×D411yescharacteristic
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—22no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C42—22no—
496C42—21no—
497C42C2×C7×D411yescharacteristic
498C42C2×C7×D411yescharacteristic
499C42C2×C7×D411yescharacteristic
500C42C24×C711yescharacteristic
501C2×C7—22no—
502C2×C7—82no—
503C2×C7—81no—
504C2×C7—82no—
505C2×C7C2×#45511yescharacteristic
506C2×C7C2×#45511yescharacteristic
507C2×C7#302111yescharacteristic
508C23—21no—
509C23—22no—
510C23—22no—
511C23—42no—
512C23—81no—
513C23—81no—
514C23—82no—
515C23—82no—
516C23—82no—
517C23C7×#12011yescharacteristic
518Q8—41no—
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—82no—
525D4—41no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—81no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4C7×#12011yescharacteristic
554C2×C4C7×#12011yescharacteristic
555C2×C4C7×#12011yescharacteristic
556C2×C4C22×C7×D411yescharacteristic
557C2×C4C22×C7×D411yescharacteristic
558C2×C4C22×C7×D411yescharacteristic
559C8—42no—
560C8—42no—
561C8—42no—
562C7#1902011yescharacteristic, direct product factor, Sylow
563C22—21no—
564C22—21no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22C7×#45512yes—
577C22C2×C7×#12011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4C7×#45512yes—
590C2—22no—
591C2—81no—
592C2—82no—
593C2—82no—
594C2C2×C7×#45511yescharacteristic
595C2C2×C7×#45511yescharacteristic
596C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3558C2
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
2. C7×#3218C2semidirect
3. C7×#3141C2semidirect
4. C2×C7×#489C2semidirect
5. C7×#1662C2semidirect
6. C7×#1659C2semidirect
7. C7×#1530C2semidirect
8. C7×#1527C2semidirect
9. C7×#1524C2semidirect
23. C2×C7×C4⋊Q8C22semidirect
26. C7×#497C22semidirect
27. C7×#491C22semidirect
29. C7×#489C22semidirect
30. C7×#489C22semidirect
37. C7×#432C22semidirect
41. C4×C7×M16C22semidirect
42. C2×C7×#97C22semidirect
43. C7×#330C22semidirect
44. C7×#329C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19020C7direct (N × H)
132. #3218C2×C7semidirect
133. #3141C2×C7semidirect
134. C2×#489C2×C7semidirect
135. #1662C2×C7semidirect
136. #1659C2×C7semidirect
137. #1530C2×C7semidirect
138. #1527C2×C7semidirect
139. #1524C2×C7semidirect
233. C2×C4⋊Q8C22×C7semidirect
236. #497C22×C7semidirect
237. #491C22×C7semidirect
239. #489C22×C7semidirect
240. #489C22×C7semidirect
247. #432C22×C7semidirect
251. C4×M16C22×C7semidirect
252. C2×#97C22×C7semidirect
253. #330C22×C7semidirect
254. #329C22×C7semidirect
255. #329C22×C7semidirect
256. #329C22×C7semidirect
562. C7#19020direct (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×#32182
C7×#31412
C2×C7×#4892
C7×#16622
C7×#16592
C7×#15302
C7×#15272
C7×#15242
#190207