Finite Group Databasebeta

← C7×#18427 #13195643

Subgroups — C7×#18427 #13195643

Order 1792.

996 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3558C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic, complemented
3C7×#3206C211yesmaximal, characteristic, complemented
4C7×#2640C211yesmaximal, characteristic, complemented
5C7×#1606C214yesmaximal, complemented
6C7×#1601C214yesmaximal, complemented
7C7×#1505C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic, complemented
9C7×#1469C211yesmaximal, characteristic, complemented
10C7×#562—21no—
11C7×H(3,𝔽4)—21no—
12C7×#559—21no—
13C7×#548—21no—
14C7×#546—21no—
15C7×#546—22no—
16C7×D42—21no—
17C7×#542—21no—
18C7×#538—22no—
19C7×#538C2212yes—
20C7×#535—21no—
21C7×#534C2212yes—
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic, complemented
24C7×#529C2211yescharacteristic
25C2×C7×#124C2211yescharacteristic
26C7×#497—21no—
27C7×#492—21no—
28C7×#491—21no—
29C7×#487—21no—
30C7×#462—21no—
31C7×#461—22no—
32C7×#461—21no—
33C7×#451C2212yescomplemented
34C7×#448C2212yes—
35C7×#448C2214yescomplemented
36C7×#432C2211yescharacteristic, complemented
37C7×#431C2211yescharacteristic, complemented
38C7×#397—22no—
39C7×#393C2214yescomplemented
40C7×#391—21no—
41C7×#390—21no—
42C7×#376C2211yescharacteristic, complemented
43C7×#328C2212yescomplemented
44C7×#327C2212yes—
45C7×#327C2214yescomplemented
46C7×#323C2214yescomplemented
47#18427C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C2×C7×#41—21no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×Q8C2312yes—
52C22×C7×D4—21no—
53C22×C7×D4—22no—
54C22×C7×D4D411yescharacteristic
55C2×C7×Q16—22no—
56C2×C7×SD16—24no—
57C2×C7×SD16—22no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8D411yescharacteristic
63C7×#127—21no—
64C7×#127D411yescharacteristic
65C7×#126—22no—
66C7×#126—21no—
67C7×#125D411yescharacteristic
68C7×#124—21no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#124—22no—
72C7×#124—21no—
73C7×#124—21no—
74C7×#124—21no—
75C7×#124—21no—
76C7×#124—21no—
77C7×#124D411yescharacteristic
78C7×#123—22no—
79C7×#123—21no—
80C7×#123—22no—
81C7×#123—22no—
82C7×#123—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8C2311yescharacteristic
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—21no—
91C7×#121—22no—
92C7×#121—21no—
93C7×#121—22no—
94C7×#121C2311yescharacteristic
95C7×#121C2312yes—
96C7×#120—22no—
97C7×#120—22no—
98C7×#120—22no—
99C7×#120—22no—
100C7×#120—22no—
101C4×C7×Q8—21no—
102C4×C7×Q8D411yescharacteristic
103C4×C7×D4—22no—
104C4×C7×D4—21no—
105C4×C7×D4—21no—
106C4×C7×D4—22no—
107C4×C7×D4—21no—
108C4×C7×D4D411yescharacteristic
109C4×C7×D4D412yes—
110C7×#117—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—21no—
113C2×C7×C22⋊C4—22no—
114C2×C7×C22⋊C4—24no—
115C2×C7×C22⋊C4D412yes—
116C2×C7×C22⋊C4C2312yes—
117C2×C42×C7—21no—
118C2×C42×C7C2311yescharacteristic
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×#103—24no—
122C7×#103—22no—
123C7×#103—22no—
124C7×C22⋊C8C2312yes—
125C7×C22⋊C8C2312yes—
126C7×#97—21no—
127C4×C7×C8—21no—
128C7×#95—21no—
129C7×#95—24no—
130C7×#95C2312yes—
131#3558C2×C711yescharacteristic
132#3218C2×C711yescharacteristic, complemented
133#3206C2×C711yescharacteristic, complemented
134#2640C2×C711yescharacteristic, complemented
135#1606C2×C714yescomplemented
136#1601C2×C714yescomplemented
137#1505C2×C711yescharacteristic, complemented
138#1500C2×C711yescharacteristic, complemented
139#1469C2×C711yescharacteristic, complemented
140C24×C7—22no—
141C24×C7C2×D412yes—
142C7×#41—82no—
143C7×#41—82no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—22no—
148C2×C7×Q8—22no—
149C2×C7×Q8—42no—
150C2×C7×Q8—82no—
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—21no—
163C2×C7×D4—42no—
164C2×C7×D4—82no—
165C2×C7×D4—84no—
166C2×C7×D4C2×D411yescharacteristic
167C2×C7×D4C2×D412yes—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D412yes—
177C22×C4×C7C2411yescharacteristic
178C7×Q16—82no—
179C7×SD16—84no—
180C7×SD16—82no—
181C2×C7×C8—22no—
182C2×C7×C8—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—41no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D411yescharacteristic
196C7×C4⋊C4C2×D411yescharacteristic
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—21no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—21no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—84no—
214C42×C7—22no—
215C42×C7—21no—
216C42×C7—42no—
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219C42×C7C2×D411yescharacteristic
220C42×C7C2×D411yescharacteristic
221#562—21no—
222H(3,𝔽4)—21no—
223#559—21no—
224#548—21no—
225#546—21no—
226#546—22no—
227D42—21no—
228#542—21no—
229#538—22no—
230#538C22×C712yes—
231#535—21no—
232#534C22×C712yes—
233#532C22×C711yescharacteristic
234C2×C4⋊Q8C22×C711yescharacteristic, complemented
235#529C22×C711yescharacteristic
236C2×#124C22×C711yescharacteristic
237#497—21no—
238#492—21no—
239#491—21no—
240#487—21no—
241#462—21no—
242#461—22no—
243#461—21no—
244#451C22×C712yescomplemented
245#448C22×C712yes—
246#448C22×C714yescomplemented
247#432C22×C711yescharacteristic, complemented
248#431C22×C711yescharacteristic, complemented
249#397—22no—
250#393C22×C714yescomplemented
251#391—21no—
252#390—21no—
253#376C22×C711yescharacteristic, complemented
254#328C22×C712yescomplemented
255#327C22×C712yes—
256#327C22×C714yescomplemented
257#323C22×C714yescomplemented
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—22no—
261C23×C7—24no—
262C23×C7—42no—
263C23×C7—42no—
264C23×C7—82no—
265C23×C7—82no—
266C23×C7—82no—
267C23×C7C22×D411yescharacteristic
268C7×Q8—41no—
269C7×Q8—44no—
270C7×Q8—42no—
271C7×Q8—42no—
272C7×Q8—42no—
273C7×Q8—81no—
274C7×Q8—81no—
275C7×D4—42no—
276C7×D4—41no—
277C7×D4—42no—
278C7×D4—44no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—81no—
282C7×D4—81no—
283C7×D4—82no—
284C7×D4—82no—
285C7×D4—82no—
286C7×D4—82no—
287C7×D4—82no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—41no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—44no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—82no—
304C2×C4×C7—82no—
305C2×C4×C7—82no—
306C2×C4×C7#12011yescharacteristic
307C2×C4×C7#12011yescharacteristic
308C2×C4×C7#12011yescharacteristic
309C2×C4×C7#12011yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C7×C8—42no—
313C7×C8—42no—
314C2×#41—21no—
315C2×#41—21no—
316C22×Q8C7×D411yescharacteristic
317C22×Q8C23×C712yes—
318C22×D4—21no—
319C22×D4—22no—
320C22×D4C7×D411yescharacteristic
321C2×Q16—22no—
322C2×SD16—22no—
323C2×SD16—24no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8C7×D411yescharacteristic
329#127—21no—
330#127C7×D411yescharacteristic
331#126—22no—
332#126—21no—
333#125C7×D411yescharacteristic
334#124—21no—
335#124—22no—
336#124—22no—
337#124—22no—
338#124—21no—
339#124—21no—
340#124—21no—
341#124—21no—
342#124—21no—
343#124C7×D411yescharacteristic
344#123—22no—
345#123—21no—
346#123—22no—
347#123—22no—
348#123—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8C23×C711yescharacteristic
353#121—22no—
354#121—22no—
355#121—22no—
356#121—21no—
357#121—22no—
358#121—21no—
359#121—22no—
360#121C23×C711yescharacteristic
361#121C23×C712yes—
362#120—22no—
363#120—22no—
364#120—22no—
365#120—22no—
366#120—22no—
367C4×Q8—21no—
368C4×Q8C7×D411yescharacteristic
369C4×D4—22no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—22no—
373C4×D4—21no—
374C4×D4C7×D411yescharacteristic
375C4×D4C7×D412yes—
376#117—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4C7×D412yes—
382C2×C22⋊C4C23×C712yes—
383C2×C42—21no—
384C2×C42C23×C711yescharacteristic
385C4⋊C8—21no—
386C4⋊C8—21no—
387#103—22no—
388#103—24no—
389#103—22no—
390C22⋊C8C23×C712yes—
391C22⋊C8C23×C712yes—
392#97—21no—
393C4×C8—21no—
394#95—21no—
395#95—24no—
396#95C23×C712yes—
397C22×C7—21no—
398C22×C7—21no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—42no—
404C22×C7—42no—
405C22×C7—42no—
406C22×C7—41no—
407C22×C7—42no—
408C22×C7—84no—
409C22×C7—164no—
410C22×C7C2×#12011yescharacteristic
411C22×C7#57712yes—
412C4×C7—21no—
413C4×C7—21no—
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—21no—
417C4×C7—41no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—41no—
422C4×C7—82no—
423C4×C7—82no—
424C4×C7#45512yes—
425C24—22no—
426C24C2×C7×D412yes—
427#41—82no—
428#41—82no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—42no—
435C2×Q8—82no—
436C2×Q8C2×C7×D411yescharacteristic
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—42no—
449C2×D4—82no—
450C2×D4—84no—
451C2×D4C2×C7×D411yescharacteristic
452C2×D4C2×C7×D412yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4C2×C7×D411yescharacteristic
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D412yes—
462C22×C4C24×C711yescharacteristic
463Q16—82no—
464SD16—84no—
465SD16—82no—
466C2×C8—22no—
467C2×C8—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4C2×C7×D412yes—
480C4⋊C4C2×C7×D411yescharacteristic
481C4⋊C4C2×C7×D411yescharacteristic
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—21no—
497C22⋊C4—22no—
498C22⋊C4—84no—
499C42—22no—
500C42—21no—
501C42—42no—
502C42C2×C7×D411yescharacteristic
503C42C2×C7×D411yescharacteristic
504C42C2×C7×D411yescharacteristic
505C42C2×C7×D411yescharacteristic
506C2×C7—22no—
507C2×C7—82no—
508C2×C7—82no—
509C2×C7—81no—
510C2×C7#300911yescharacteristic
511C2×C7C2×#45511yescharacteristic
512C2×C7#302711yescharacteristic
513C23—22no—
514C23—21no—
515C23—22no—
516C23—24no—
517C23—42no—
518C23—42no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23C22×C7×D411yescharacteristic
523Q8—41no—
524Q8—44no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—81no—
529Q8—81no—
530D4—42no—
531D4—41no—
532D4—42no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—81no—
537D4—81no—
538D4—82no—
539D4—82no—
540D4—82no—
541D4—82no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4C7×#12011yescharacteristic
562C2×C4C7×#12011yescharacteristic
563C2×C4C7×#12011yescharacteristic
564C2×C4C7×#12011yescharacteristic
565C2×C4C22×C7×D411yescharacteristic
566C2×C4C22×C7×D411yescharacteristic
567C8—42no—
568C8—42no—
569C7#1842711yescharacteristic, direct product factor, Sylow
570C22—21no—
571C22—21no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—41no—
580C22—42no—
581C22—84no—
582C22—164no—
583C22C2×C7×#12011yescharacteristic
584C22C7×#57712yes—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—82no—
596C4—82no—
597C4C7×#45512yes—
598C2—22no—
599C2—82no—
600C2—81no—
601C2—82no—
602C2C7×#300911yescharacteristic
603C2C2×C7×#45511yescharacteristic
604C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3558C2
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
2. C7×#3218C2semidirect
3. C7×#3206C2semidirect
4. C7×#2640C2semidirect
5. C7×#1606C2semidirect
6. C7×#1601C2semidirect
7. C7×#1505C2semidirect
8. C7×#1500C2semidirect
9. C7×#1469C2semidirect
23. C2×C7×C4⋊Q8C22semidirect
33. C7×#451C22semidirect
35. C7×#448C22semidirect
36. C7×#432C22semidirect
37. C7×#431C22semidirect
39. C7×#393C22semidirect
42. C7×#376C22semidirect
43. C7×#328C22semidirect
45. C7×#327C22semidirect
46. C7×#323C22semidirect
47. #18427C7direct (N × H)
132. #3218C2×C7semidirect
133. #3206C2×C7semidirect
134. #2640C2×C7semidirect
135. #1606C2×C7semidirect
136. #1601C2×C7semidirect
137. #1505C2×C7semidirect
138. #1500C2×C7semidirect
139. #1469C2×C7semidirect
234. C2×C4⋊Q8C22×C7semidirect
244. #451C22×C7semidirect
246. #448C22×C7semidirect
247. #432C22×C7semidirect
248. #431C22×C7semidirect
250. #393C22×C7semidirect
253. #376C22×C7semidirect
254. #328C22×C7semidirect
256. #327C22×C7semidirect
257. #323C22×C7semidirect
569. C7#18427direct (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×#32062
C7×#26402
C7×#16062
C7×#16012
C7×#15052
C7×#15002
C7×#14692
#184277