Finite Group Databasebeta

← C7×#19015 #13196231

Subgroups — C7×#19015 #13196231

Order 1792.

1012 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×#3238C211yesmaximal, characteristic, complemented
3C7×#3152C211yesmaximal, characteristic, complemented
4C2×C7×#489C211yesmaximal, characteristic, complemented
5C7×#1661C214yesmaximal, complemented
6C7×#1659C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1527C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C2×C7×#137—22no—
11C7×#562C2212yes—
12C7×H(3,𝔽4)C2212yes—
13C7×#559—21no—
14C7×#548—21no—
15C7×#546—22no—
16C7×#546—21no—
17C7×D42—21no—
18C7×#542—21no—
19C7×#538—22no—
20C7×#538—21no—
21C7×#535—21no—
22C7×#534—21no—
23C7×#532C2211yescharacteristic
24C2×C7×C4⋊Q8C2211yescharacteristic, complemented
25C7×#529C2211yescharacteristic
26C2×C7×#124C2211yescharacteristic
27C7×#501C2212yescomplemented
28C7×#497C2214yescomplemented
29C7×#494—21no—
30C7×#492—21no—
31C7×#489—22no—
32C7×#489C2212yescomplemented
33C7×#489C2214yescomplemented
34C7×#483—21no—
35C7×#480—21no—
36C7×#477—21no—
37C7×#476—21no—
38C7×#432C2211yescharacteristic, complemented
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#19015C711yesmaximal, 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
55C7×#137—48no—
56C2×C7×Q16—24no—
57C2×C7×SD16—24no—
58C2×C7×M16—21no—
59C22×C7×C8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8D411yescharacteristic
64C7×C4⋊Q8C2312yes—
65C7×#127—21no—
66C7×#127D411yescharacteristic
67C7×#126—22no—
68C7×#126—21no—
69C7×#125C2311yescharacteristic
70C7×#124—21no—
71C7×#124—22no—
72C7×#124—22no—
73C7×#124—22no—
74C7×#124—21no—
75C7×#124—21no—
76C7×#124D412yes—
77C7×#124C2312yes—
78C7×#124C2311yescharacteristic
79C7×#124C2312yes—
80C7×#123—22no—
81C7×#123—22no—
82C7×#123—21no—
83C7×#123—22no—
84C7×#123—22no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8D411yescharacteristic
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—22no—
92C7×#121—21no—
93C7×#121—22no—
94C7×#121—22no—
95C7×#121—21no—
96C7×#121—21no—
97C7×#121D411yescharacteristic
98C7×#120—22no—
99C7×#120—22no—
100C7×#120—22no—
101C7×#120—22no—
102C7×#120—22no—
103C4×C7×Q8—21no—
104C4×C7×Q8D411yescharacteristic
105C4×C7×D4—22no—
106C4×C7×D4—22no—
107C4×C7×D4—21no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—21no—
111C4×C7×D4D411yescharacteristic
112C7×#117—21no—
113C2×C7×C4⋊C4—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C22⋊C4—22no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4—21no—
118C2×C42×C7C2311yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×C4⋊C8—21no—
122C7×#103—22no—
123C7×#103—24no—
124C7×#102—22no—
125C7×#102—24no—
126C7×C22⋊C8—21no—
127C7×#97—21no—
128C7×#97C2312yes—
129C7×#97C2312yes—
130C7×#97C2312yes—
131C4×C7×C8—21no—
132#3558C2×C711yescharacteristic
133#3238C2×C711yescharacteristic, complemented
134#3152C2×C711yescharacteristic, complemented
135C2×#489C2×C711yescharacteristic, complemented
136#1661C2×C714yescomplemented
137#1659C2×C714yescomplemented
138#1530C2×C711yescharacteristic, complemented
139#1527C2×C711yescharacteristic, complemented
140#1524C2×C711yescharacteristic, complemented
141C24×C7—22no—
142C24×C7—21no—
143C7×#41—44no—
144C7×#41—82no—
145C2×C7×Q8—21no—
146C2×C7×Q8—22no—
147C2×C7×Q8—42no—
148C2×C7×Q8—44no—
149C2×C7×Q8—41no—
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—42no—
164C2×C7×D4—82no—
165C2×C7×D4—84no—
166C2×C7×D4C2×D411yescharacteristic
167C2×C7×D4C2×D412yes—
168C2×C7×D4C2×D412yes—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—22no—
174C22×C4×C7—41no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C7×Q16—48no—
179C7×SD16—48no—
180C7×M16—24no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C2×C7×C8—42no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—41no—
196C7×C4⋊C4—41no—
197C7×C4⋊C4C2×D411yescharacteristic
198C7×C4⋊C4C2×D411yescharacteristic
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—21no—
212C7×C22⋊C4—21no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C42×C7—22no—
216C42×C7—21no—
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219C42×C7C2×D411yescharacteristic
220C42×C7C2411yescharacteristic
221C2×#137—22no—
222#562C22×C712yes—
223H(3,𝔽4)C22×C712yes—
224#559—21no—
225#548—21no—
226#546—22no—
227#546—21no—
228D42—21no—
229#542—21no—
230#538—22no—
231#538—21no—
232#535—21no—
233#534—21no—
234#532C22×C711yescharacteristic
235C2×C4⋊Q8C22×C711yescharacteristic, complemented
236#529C22×C711yescharacteristic
237C2×#124C22×C711yescharacteristic
238#501C22×C712yescomplemented
239#497C22×C714yescomplemented
240#494—21no—
241#492—21no—
242#489—22no—
243#489C22×C714yescomplemented
244#489C22×C712yescomplemented
245#483—21no—
246#480—21no—
247#477—21no—
248#476—21no—
249#432C22×C711yescharacteristic, complemented
250C2×#103—21no—
251C2×#102—21no—
252C4×M16C22×C711yescharacteristic, complemented
253C2×#97C22×C711yescharacteristic, complemented
254#330C22×C712yescomplemented
255#329C22×C712yescomplemented
256#329C22×C714yescomplemented
257#329C22×C714yescomplemented
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—22no—
261C23×C7—42no—
262C23×C7—81no—
263C23×C7—81no—
264C23×C7—82no—
265C23×C7—82no—
266C23×C7—82no—
267C23×C7#12011yescharacteristic
268C7×Q8—42no—
269C7×Q8—42no—
270C7×Q8—41no—
271C7×Q8—44no—
272C7×Q8—42no—
273C7×Q8—82no—
274C7×D4—42no—
275C7×D4—42no—
276C7×D4—41no—
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—24no—
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
318#137—48no—
319C2×Q16—24no—
320C2×SD16—24no—
321C2×M16—21no—
322C22×C8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8C7×D411yescharacteristic
327C4⋊Q8C23×C712yes—
328#127—21no—
329#127C7×D411yescharacteristic
330#126—22no—
331#126—21no—
332#125C23×C711yescharacteristic
333#124—21no—
334#124—22no—
335#124—22no—
336#124—22no—
337#124—21no—
338#124—21no—
339#124C7×D412yes—
340#124C23×C712yes—
341#124C23×C711yescharacteristic
342#124C23×C712yes—
343#123—22no—
344#123—22no—
345#123—21no—
346#123—22no—
347#123—22no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8C7×D411yescharacteristic
352#121—22no—
353#121—22no—
354#121—22no—
355#121—21no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—21no—
360#121C7×D411yescharacteristic
361#120—22no—
362#120—22no—
363#120—22no—
364#120—22no—
365#120—22no—
366C4×Q8—21no—
367C4×Q8C7×D411yescharacteristic
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C7×D411yescharacteristic
375#117—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C42C23×C711yescharacteristic
382#107—22no—
383#106—22no—
384C4⋊C8—21no—
385#103—24no—
386#103—22no—
387#102—24no—
388#102—22no—
389C22⋊C8—21no—
390#97—21no—
391#97C23×C712yes—
392#97C23×C712yes—
393#97C23×C712yes—
394C4×C8—21no—
395C22×C7—21no—
396C22×C7—21no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—41no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—162no—
407C22×C7—164no—
408C22×C7#45312yes—
409C22×C7C2×#12011yescharacteristic
410C4×C7—21no—
411C4×C7—21no—
412C4×C7—21no—
413C4×C7—21no—
414C4×C7—21no—
415C4×C7—41no—
416C4×C7—42no—
417C4×C7—41no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—82no—
421C4×C7#45512yes—
422C24—22no—
423C24—21no—
424#41—44no—
425#41—82no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—44no—
430C2×Q8—41no—
431C2×Q8C2×C7×D411yescharacteristic
432C2×Q8C2×C7×D412yes—
433C2×Q8C2×C7×D412yes—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—82no—
446C2×D4—84no—
447C2×D4C2×C7×D411yescharacteristic
448C2×D4C2×C7×D412yes—
449C2×D4C2×C7×D412yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4C2×C7×D411yescharacteristic
457C22×C4C2×C7×D411yescharacteristic
458C22×C4C2×C7×D411yescharacteristic
459Q16—48no—
460SD16—48no—
461M16—24no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—42no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4C2×C7×D411yescharacteristic
479C4⋊C4C2×C7×D411yescharacteristic
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C42—22no—
497C42—21no—
498C42C2×C7×D411yescharacteristic
499C42C2×C7×D411yescharacteristic
500C42C2×C7×D411yescharacteristic
501C42C24×C711yescharacteristic
502C2×C7—22no—
503C2×C7—82no—
504C2×C7—82no—
505C2×C7—81no—
506C2×C7C2×#45311yescharacteristic
507C2×C7C2×#45511yescharacteristic
508C2×C7#302211yescharacteristic
509C23—22no—
510C23—21no—
511C23—22no—
512C23—42no—
513C23—81no—
514C23—81no—
515C23—82no—
516C23—82no—
517C23—82no—
518C23C7×#12011yescharacteristic
519Q8—42no—
520Q8—42no—
521Q8—41no—
522Q8—44no—
523Q8—42no—
524Q8—82no—
525D4—42no—
526D4—42no—
527D4—41no—
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—24no—
560C8—42no—
561C8—42no—
562C7#1901511yescharacteristic, direct product factor, Sylow
563C22—21no—
564C22—21no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—162no—
575C22—164no—
576C22C7×#45312yes—
577C22C2×C7×#12011yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—82no—
589C4C7×#45512yes—
590C2—22no—
591C2—82no—
592C2—81no—
593C2—82no—
594C2C2×C7×#45311yescharacteristic
595C2C2×C7×#45511yescharacteristic
596C2C7×#302211yescharacteristic
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×#3238C2semidirect
3. C7×#3152C2semidirect
4. C2×C7×#489C2semidirect
5. C7×#1661C2semidirect
6. C7×#1659C2semidirect
7. C7×#1530C2semidirect
8. C7×#1527C2semidirect
9. C7×#1524C2semidirect
24. C2×C7×C4⋊Q8C22semidirect
27. C7×#501C22semidirect
28. C7×#497C22semidirect
32. C7×#489C22semidirect
33. C7×#489C22semidirect
38. C7×#432C22semidirect
41. C4×C7×M16C22semidirect
42. C2×C7×#97C22semidirect
43. C7×#330C22semidirect
44. C7×#329C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19015C7direct (N × H)
133. #3238C2×C7semidirect
134. #3152C2×C7semidirect
135. C2×#489C2×C7semidirect
136. #1661C2×C7semidirect
137. #1659C2×C7semidirect
138. #1530C2×C7semidirect
139. #1527C2×C7semidirect
140. #1524C2×C7semidirect
235. C2×C4⋊Q8C22×C7semidirect
238. #501C22×C7semidirect
239. #497C22×C7semidirect
243. #489C22×C7semidirect
244. #489C22×C7semidirect
249. #432C22×C7semidirect
252. C4×M16C22×C7semidirect
253. C2×#97C22×C7semidirect
254. #330C22×C7semidirect
255. #329C22×C7semidirect
256. #329C22×C7semidirect
257. #329C22×C7semidirect
562. C7#19015direct (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×#32382
C7×#31522
C2×C7×#4892
C7×#16612
C7×#16592
C7×#15302
C7×#15272
C7×#15242
#190157