Finite Group Databasebeta

← C7×#19021 #13196237

Subgroups — C7×#19021 #13196237

Order 1792.

1012 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3239C211yesmaximal, characteristic, complemented
3C2×C7×#497C211yesmaximal, characteristic, complemented
4C7×#3141C211yesmaximal, characteristic, complemented
5C7×#1661C214yesmaximal, complemented
6C7×#1659C214yesmaximal, complemented
7C7×#1530C211yesmaximal, characteristic, complemented
8C7×#1527C211yesmaximal, characteristic, complemented
9C7×#1524C211yesmaximal, characteristic, complemented
10C22×C7×Q16—21no—
11C22×C7×SD16—21no—
12C7×#562C2212yes—
13C7×H(3,𝔽4)C2212yes—
14C7×#559—21no—
15C7×#548—21no—
16C7×#546—22no—
17C7×#546—21no—
18C7×D42—21no—
19C7×#542—21no—
20C7×#538—22no—
21C7×#538—21no—
22C7×#535—21no—
23C7×#534—21no—
24C7×#532C2211yescharacteristic
25C2×C7×C4⋊Q8C2211yescharacteristic, complemented
26C7×#529C2211yescharacteristic
27C2×C7×#124C2211yescharacteristic
28C7×#501C2212yescomplemented
29C7×#497—22no—
30C7×#497C2214yescomplemented
31C7×#489C2212yescomplemented
32C7×#489C2214yescomplemented
33C7×#487—21no—
34C7×#486—21no—
35C7×#484—21no—
36C7×#481—21no—
37C7×#475—21no—
38C7×#474—21no—
39C7×#432C2211yescharacteristic, complemented
40C7×#417—22no—
41C4×C7×M16C2211yescharacteristic, complemented
42C2×C7×#97C2211yescharacteristic, complemented
43C7×#330C2212yescomplemented
44C7×#329C2212yescomplemented
45C7×#329C2214yescomplemented
46C7×#329C2214yescomplemented
47#19021C711yesmaximal, 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—24no—
56C2×C7×Q16—44no—
57C2×C7×SD16—24no—
58C2×C7×SD16—44no—
59C2×C7×M16—21no—
60C22×C7×C8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8C2312yes—
66C7×#127—21no—
67C7×#127D411yescharacteristic
68C7×#126—22no—
69C7×#126—21no—
70C7×#125C2311yescharacteristic
71C7×#124—21no—
72C7×#124—22no—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124—21no—
76C7×#124—21no—
77C7×#124D412yes—
78C7×#124C2311yescharacteristic
79C7×#124C2312yes—
80C7×#124C2312yes—
81C7×#123—22no—
82C7×#123—22no—
83C7×#123—21no—
84C7×#123—22no—
85C7×#123—22no—
86C7×C22⋊Q8—21no—
87C7×C22⋊Q8—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8D411yescharacteristic
90C7×#121—22no—
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—21no—
94C7×#121—22no—
95C7×#121—22no—
96C7×#121—21no—
97C7×#121—21no—
98C7×#121D411yescharacteristic
99C7×#120—22no—
100C7×#120—22no—
101C7×#120—22no—
102C7×#120—22no—
103C7×#120—22no—
104C4×C7×Q8—21no—
105C4×C7×Q8D411yescharacteristic
106C4×C7×D4—22no—
107C4×C7×D4—22no—
108C4×C7×D4—21no—
109C4×C7×D4—21no—
110C4×C7×D4—21no—
111C4×C7×D4—21no—
112C4×C7×D4D411yescharacteristic
113C7×#117—21no—
114C2×C7×C4⋊C4—21no—
115C2×C7×C4⋊C4—21no—
116C2×C7×C22⋊C4—22no—
117C2×C7×C22⋊C4—21no—
118C2×C7×C22⋊C4—21no—
119C2×C42×C7C2311yescharacteristic
120C7×#107—22no—
121C7×#106—22no—
122C7×C4⋊C8—21no—
123C7×#103—22no—
124C7×#103—24no—
125C7×#102—22no—
126C7×#102—24no—
127C7×C22⋊C8—21no—
128C7×#97—21no—
129C7×#97C2312yes—
130C7×#97C2312yes—
131C7×#97C2312yes—
132C4×C7×C8—21no—
133#3558C2×C711yescharacteristic
134#3239C2×C711yescharacteristic, complemented
135C2×#497C2×C711yescharacteristic, complemented
136#3141C2×C711yescharacteristic, complemented
137#1661C2×C714yescomplemented
138#1659C2×C714yescomplemented
139#1530C2×C711yescharacteristic, complemented
140#1527C2×C711yescharacteristic, complemented
141#1524C2×C711yescharacteristic, complemented
142C24×C7—22no—
143C24×C7—21no—
144C7×#41—44no—
145C7×#41—82no—
146C2×C7×Q8—21no—
147C2×C7×Q8—22no—
148C2×C7×Q8—42no—
149C2×C7×Q8—42no—
150C2×C7×Q8—42no—
151C2×C7×Q8—41no—
152C2×C7×Q8C2×D411yescharacteristic
153C2×C7×Q8C2×D412yes—
154C2×C7×Q8C2×D412yes—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—21no—
163C2×C7×D4—21no—
164C2×C7×D4—21no—
165C2×C7×D4—42no—
166C2×C7×D4—82no—
167C2×C7×D4—84no—
168C2×C7×D4C2×D411yescharacteristic
169C2×C7×D4C2×D412yes—
170C2×C7×D4C2×D412yes—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—22no—
176C22×C4×C7—41no—
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C7×Q16—48no—
181C7×SD16—48no—
182C7×M16—42no—
183C2×C7×C8—24no—
184C2×C7×C8—22no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—41no—
198C7×C4⋊C4—41no—
199C7×C4⋊C4C2×D411yescharacteristic
200C7×C4⋊C4C2×D411yescharacteristic
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—21no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—21no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C42×C7—22no—
218C42×C7—21no—
219C42×C7C2×D411yescharacteristic
220C42×C7C2×D411yescharacteristic
221C42×C7C2×D411yescharacteristic
222C42×C7C2411yescharacteristic
223C22×Q16—21no—
224C22×SD16—21no—
225#562C22×C712yes—
226H(3,𝔽4)C22×C712yes—
227#559—21no—
228#548—21no—
229#546—22no—
230#546—21no—
231D42—21no—
232#542—21no—
233#538—22no—
234#538—21no—
235#535—21no—
236#534—21no—
237#532C22×C711yescharacteristic
238C2×C4⋊Q8C22×C711yescharacteristic, complemented
239#529C22×C711yescharacteristic
240C2×#124C22×C711yescharacteristic
241#501C22×C712yescomplemented
242#497—22no—
243#497C22×C714yescomplemented
244#489C22×C714yescomplemented
245#489C22×C712yescomplemented
246#487—21no—
247#486—21no—
248#484—21no—
249#481—21no—
250#475—21no—
251#474—21no—
252#432C22×C711yescharacteristic, complemented
253#417—22no—
254C4×M16C22×C711yescharacteristic, complemented
255C2×#97C22×C711yescharacteristic, complemented
256#330C22×C712yescomplemented
257#329C22×C712yescomplemented
258#329C22×C714yescomplemented
259#329C22×C714yescomplemented
260C23×C7—21no—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—42no—
264C23×C7—81no—
265C23×C7—81no—
266C23×C7—82no—
267C23×C7—82no—
268C23×C7—82no—
269C23×C7#12011yescharacteristic
270C7×Q8—42no—
271C7×Q8—41no—
272C7×Q8—42no—
273C7×Q8—42no—
274C7×Q8—42no—
275C7×Q8—42no—
276C7×Q8—82no—
277C7×D4—42no—
278C7×D4—41no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—44no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—81no—
285C7×D4—82no—
286C7×D4—82no—
287C7×D4—82no—
288C7×D4—82no—
289C7×D4—82no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—41no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—82no—
301C2×C4×C7—81no—
302C2×C4×C7—82no—
303C2×C4×C7—82no—
304C2×C4×C7—82no—
305C2×C4×C7#12011yescharacteristic
306C2×C4×C7#12011yescharacteristic
307C2×C4×C7#12011yescharacteristic
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C7×C8—24no—
312C7×C8—42no—
313C7×C8—42no—
314C2×#41—21no—
315C2×#41C7×D412yes—
316C22×Q8—21no—
317C22×Q8C7×D411yescharacteristic
318C22×D4—22no—
319C22×D4—21no—
320C22×D4C7×D411yescharacteristic
321C2×Q16—24no—
322C2×Q16—44no—
323C2×SD16—24no—
324C2×SD16—44no—
325C2×M16—21no—
326C22×C8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8C7×D411yescharacteristic
331C4⋊Q8C23×C712yes—
332#127—21no—
333#127C7×D411yescharacteristic
334#126—22no—
335#126—21no—
336#125C23×C711yescharacteristic
337#124—21no—
338#124—22no—
339#124—22no—
340#124—22no—
341#124—21no—
342#124—21no—
343#124C7×D412yes—
344#124C23×C712yes—
345#124C23×C711yescharacteristic
346#124C23×C712yes—
347#123—22no—
348#123—22no—
349#123—21no—
350#123—22no—
351#123—22no—
352C22⋊Q8—21no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8C7×D411yescharacteristic
356#121—22no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121—22no—
361#121—22no—
362#121—21no—
363#121—21no—
364#121C7×D411yescharacteristic
365#120—22no—
366#120—22no—
367#120—22no—
368#120—22no—
369#120—22no—
370C4×Q8—21no—
371C4×Q8C7×D411yescharacteristic
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4C7×D411yescharacteristic
379#117—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42C23×C711yescharacteristic
386#107—22no—
387#106—22no—
388C4⋊C8—21no—
389#103—24no—
390#103—22no—
391#102—24no—
392#102—22no—
393C22⋊C8—21no—
394#97—21no—
395#97C23×C712yes—
396#97C23×C712yes—
397#97C23×C712yes—
398C4×C8—21no—
399C22×C7—21no—
400C22×C7—21no—
401C22×C7—41no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—41no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—42no—
410C22×C7—162no—
411C22×C7—164no—
412C22×C7#45512yes—
413C22×C7C2×#12011yescharacteristic
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—21no—
417C4×C7—21no—
418C4×C7—21no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—41no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—82no—
425C4×C7#45312yes—
426C24—22no—
427C24—21no—
428#41—44no—
429#41—82no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—41no—
436C2×Q8C2×C7×D411yescharacteristic
437C2×Q8C2×C7×D412yes—
438C2×Q8C2×C7×D412yes—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—82no—
451C2×D4—84no—
452C2×D4C2×C7×D411yescharacteristic
453C2×D4C2×C7×D412yes—
454C2×D4C2×C7×D412yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4C2×C7×D411yescharacteristic
462C22×C4C2×C7×D411yescharacteristic
463C22×C4C2×C7×D411yescharacteristic
464Q16—48no—
465SD16—48no—
466M16—42no—
467C2×C8—24no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4C2×C7×D411yescharacteristic
484C4⋊C4C2×C7×D411yescharacteristic
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—21no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C42—22no—
502C42—21no—
503C42C2×C7×D411yescharacteristic
504C42C2×C7×D411yescharacteristic
505C42C2×C7×D411yescharacteristic
506C42C24×C711yescharacteristic
507C2×C7—22no—
508C2×C7—82no—
509C2×C7—81no—
510C2×C7—82no—
511C2×C7C2×#45311yescharacteristic
512C2×C7C2×#45511yescharacteristic
513C2×C7#302211yescharacteristic
514C23—21no—
515C23—22no—
516C23—22no—
517C23—42no—
518C23—81no—
519C23—81no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23C7×#12011yescharacteristic
524Q8—42no—
525Q8—41no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—82no—
531D4—42no—
532D4—41no—
533D4—42no—
534D4—42no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—81no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543D4—82no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4C7×#12011yescharacteristic
560C2×C4C7×#12011yescharacteristic
561C2×C4C7×#12011yescharacteristic
562C2×C4C22×C7×D411yescharacteristic
563C2×C4C22×C7×D411yescharacteristic
564C2×C4C22×C7×D411yescharacteristic
565C8—24no—
566C8—42no—
567C8—42no—
568C7#1902111yescharacteristic, direct product factor, Sylow
569C22—21no—
570C22—21no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—162no—
581C22—164no—
582C22C7×#45512yes—
583C22C2×C7×#12011yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—42no—
591C4—41no—
592C4—42no—
593C4—41no—
594C4—82no—
595C4C7×#45312yes—
596C2—22no—
597C2—81no—
598C2—82no—
599C2—82no—
600C2C2×C7×#45311yescharacteristic
601C2C2×C7×#45511yescharacteristic
602C2C7×#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×#3239C2semidirect
3. C2×C7×#497C2semidirect
4. C7×#3141C2semidirect
5. C7×#1661C2semidirect
6. C7×#1659C2semidirect
7. C7×#1530C2semidirect
8. C7×#1527C2semidirect
9. C7×#1524C2semidirect
25. C2×C7×C4⋊Q8C22semidirect
28. C7×#501C22semidirect
30. C7×#497C22semidirect
31. C7×#489C22semidirect
32. C7×#489C22semidirect
39. C7×#432C22semidirect
41. C4×C7×M16C22semidirect
42. C2×C7×#97C22semidirect
43. C7×#330C22semidirect
44. C7×#329C22semidirect
45. C7×#329C22semidirect
46. C7×#329C22semidirect
47. #19021C7direct (N × H)
134. #3239C2×C7semidirect
135. C2×#497C2×C7semidirect
136. #3141C2×C7semidirect
137. #1661C2×C7semidirect
138. #1659C2×C7semidirect
139. #1530C2×C7semidirect
140. #1527C2×C7semidirect
141. #1524C2×C7semidirect
238. C2×C4⋊Q8C22×C7semidirect
241. #501C22×C7semidirect
243. #497C22×C7semidirect
244. #489C22×C7semidirect
245. #489C22×C7semidirect
252. #432C22×C7semidirect
254. C4×M16C22×C7semidirect
255. C2×#97C22×C7semidirect
256. #330C22×C7semidirect
257. #329C22×C7semidirect
258. #329C22×C7semidirect
259. #329C22×C7semidirect
568. C7#19021direct (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×#32392
C2×C7×#4972
C7×#31412
C7×#16612
C7×#16592
C7×#15302
C7×#15272
C7×#15242
#190217