Finite Group Databasebeta

← C7×#18428 #13195644

Subgroups — C7×#18428 #13195644

Order 1792.

1104 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3561C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic, complemented
3C7×#3208C211yesmaximal, characteristic, complemented
4C7×#2657C211yesmaximal, characteristic, complemented
5C7×#1600C214yesmaximal, complemented
6C7×#1596C214yesmaximal, complemented
7C7×#1506C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic, complemented
9C7×#1469C211yesmaximal, characteristic, complemented
10C7×#563—21no—
11C7×#562—21no—
12C7×#556—21no—
13C7×#551—22no—
14C7×D4×Q8—21no—
15C7×#548—21no—
16C7×#546—21no—
17C7×#543C2212yes—
18C7×#542—21no—
19C7×#542—22no—
20C7×#538C2212yes—
21C7×#536—21no—
22C7×#533C2211yescharacteristic
23C2×C7×#127—21no—
24C2×C7×#127C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#124—21no—
27C2×C7×#124C2211yescharacteristic
28C7×#496—21no—
29C7×#492—21no—
30C7×#490—21no—
31C7×#486—21no—
32C7×#460—22no—
33C7×#460—21no—
34C7×#459—21no—
35C7×#450C2214yes—
36C7×#450C2212yes—
37C7×#447C2212yes—
38C7×#432C2211yescharacteristic
39C7×#431C2211yescharacteristic
40C7×#392—22no—
41C7×#392C2214yes—
42C7×#376C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#323C2214yes—
47#18428C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4—24no—
52C22×C7×D4—21no—
53C22×C7×D4—21no—
54C22×C7×D4D411yescharacteristic
55C22×C7×D4C2312yes—
56C2×C7×SD16—24no—
57C2×C7×SD16—22no—
58C2×C7×D8—22no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8D411yescharacteristic, complemented
62C7×C4⋊Q8D411yescharacteristic, complemented
63C7×#127—21no—
64C7×#127—21no—
65C7×#127—21no—
66C7×#127—21no—
67C7×#127—44no—
68C7×#126—22no—
69C7×#126—22no—
70C7×#126—22no—
71C7×#126—21no—
72C7×#125—21no—
73C7×#125D411yescharacteristic, complemented
74C7×#124—21no—
75C7×#124—22no—
76C7×#124—22no—
77C7×#124—21no—
78C7×#124—21no—
79C7×#124—21no—
80C7×#124—21no—
81C7×#124—44no—
82C7×#124D411yescharacteristic, complemented
83C7×#123—22no—
84C7×#123—22no—
85C7×#123—22no—
86C7×#123—21no—
87C7×#123—22no—
88C7×C22⋊Q8—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—22no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—21no—
94C7×C22⋊Q8—22no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8—22no—
97C7×C22⋊Q8C2311yescharacteristic
98C7×C22⋊Q8C2312yes—
99C7×#121—21no—
100C7×#121—22no—
101C7×#121—22no—
102C7×#121C2311yescharacteristic
103C7×#120—22no—
104C4×C7×Q8—21no—
105C4×C7×Q8—21no—
106C4×C7×Q8D411yescharacteristic, complemented
107C4×C7×Q8D412yescomplemented
108C4×C7×D4—21no—
109C4×C7×D4—22no—
110C4×C7×D4—21no—
111C4×C7×D4—22no—
112C4×C7×D4D411yescharacteristic, complemented
113C7×#117—22no—
114C7×#117—21no—
115C7×#117—21no—
116C2×C7×C22⋊C4—22no—
117C2×C7×C22⋊C4—24no—
118C2×C7×C22⋊C4D412yes—
119C2×C7×C22⋊C4C2312yes—
120C2×C42×C7—21no—
121C2×C42×C7C2311yescharacteristic
122C7×C4⋊C8—21no—
123C7×C4⋊C8—21no—
124C7×#102—24no—
125C7×#102—22no—
126C7×#102—22no—
127C7×C22⋊C8C2312yes—
128C7×C22⋊C8C2312yes—
129C7×#97—21no—
130C4×C7×C8—21no—
131C7×#95—21no—
132C7×#95C2312yes—
133#3561C2×C711yescharacteristic, complemented
134#3217C2×C711yescharacteristic, complemented
135#3208C2×C711yescharacteristic, complemented
136#2657C2×C711yescharacteristic, complemented
137#1600C2×C714yescomplemented
138#1596C2×C714yescomplemented
139#1506C2×C711yescharacteristic, complemented
140#1500C2×C711yescharacteristic, complemented
141#1469C2×C711yescharacteristic, complemented
142C24×C7—22no—
143C24×C7C2×D412yes—
144C7×#41—82no—
145C2×C7×Q8—22no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—42no—
150C2×C7×Q8—82no—
151C2×C7×Q8C2×D411yescharacteristic
152C2×C7×Q8C2×D412yes—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—42no—
161C2×C7×D4—42no—
162C2×C7×D4—42no—
163C2×C7×D4—48no—
164C2×C7×D4—41no—
165C2×C7×D4—41no—
166C2×C7×D4—41no—
167C2×C7×D4—41no—
168C2×C7×D4—82no—
169C2×C7×D4—82no—
170C2×C7×D4—84no—
171C2×C7×D4—84no—
172C2×C7×D4C2×D411yescharacteristic
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7—21no—
178C22×C4×C7C2×D411yescharacteristic
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D412yes—
181C22×C4×C7C2411yescharacteristic
182C7×SD16—82no—
183C7×SD16—84no—
184C7×D8—82no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4C2×D411yescharacteristic
201C7×C4⋊C4C2×D411yescharacteristic
202C7×C4⋊C4C2×D412yes—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—21no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—21no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—21no—
218C7×C22⋊C4—48no—
219C42×C7—24no—
220C42×C7—22no—
221C42×C7—22no—
222C42×C7—21no—
223C42×C7C2×D411yescharacteristic
224C42×C7C2×D411yescharacteristic
225C42×C7C2×D411yescharacteristic
226C42×C7C2×D411yescharacteristic
227#563—21no—
228#562—21no—
229#556—21no—
230#551—22no—
231D4×Q8—21no—
232#548—21no—
233#546—21no—
234#543C22×C712yes—
235#542—21no—
236#542—22no—
237#538C22×C712yes—
238#536—21no—
239#533C22×C711yescharacteristic
240C2×#127—21no—
241C2×#127C22×C711yescharacteristic
242#529C22×C711yescharacteristic
243C2×#124—21no—
244C2×#124C22×C711yescharacteristic
245#496—21no—
246#492—21no—
247#490—21no—
248#486—21no—
249#460—22no—
250#460—21no—
251#459—21no—
252#450C22×C712yes—
253#450C22×C714yes—
254#447C22×C712yes—
255#432C22×C711yescharacteristic
256#431C22×C711yescharacteristic
257#392—22no—
258#392C22×C714yes—
259#376C22×C711yescharacteristic
260#328C22×C712yes—
261#328C22×C714yes—
262#327C22×C712yes—
263#323C22×C714yes—
264C23×C7—22no—
265C23×C7—21no—
266C23×C7—24no—
267C23×C7—42no—
268C23×C7—42no—
269C23×C7—42no—
270C23×C7—42no—
271C23×C7—44no—
272C23×C7—42no—
273C23×C7—44no—
274C23×C7C22×D411yescharacteristic
275C7×Q8—41no—
276C7×Q8—42no—
277C7×Q8—42no—
278C7×Q8—42no—
279C7×Q8—81no—
280C7×Q8—82no—
281C7×D4—41no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—44no—
285C7×D4—42no—
286C7×D4—42no—
287C7×D4—81no—
288C7×D4—82no—
289C7×D4—82no—
290C7×D4—82no—
291C7×D4—82no—
292C7×D4—82no—
293C7×D4—816no—
294C7×D4—82no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—82no—
310C2×C4×C7—82no—
311C2×C4×C7—82no—
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7#12011yescharacteristic
315C2×C4×C7#12011yescharacteristic
316C2×C4×C7C22×D411yescharacteristic
317C2×C4×C7C22×D411yescharacteristic
318C7×C8—42no—
319C7×C8—42no—
320C2×#41—21no—
321C22×Q8—21no—
322C22×Q8C7×D411yescharacteristic
323C22×D4—24no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4C7×D411yescharacteristic
327C22×D4C23×C712yes—
328C2×SD16—24no—
329C2×SD16—22no—
330C2×D8—22no—
331C4⋊Q8—21no—
332C4⋊Q8—22no—
333C4⋊Q8C7×D411yescharacteristic, complemented
334C4⋊Q8C7×D411yescharacteristic, complemented
335#127—21no—
336#127—21no—
337#127—21no—
338#127—21no—
339#127—44no—
340#126—22no—
341#126—22no—
342#126—22no—
343#126—21no—
344#125—21no—
345#125C7×D411yescharacteristic, complemented
346#124—21no—
347#124—22no—
348#124—22no—
349#124—21no—
350#124—21no—
351#124—21no—
352#124—21no—
353#124—44no—
354#124C7×D411yescharacteristic, complemented
355#123—22no—
356#123—22no—
357#123—22no—
358#123—21no—
359#123—22no—
360C22⋊Q8—22no—
361C22⋊Q8—21no—
362C22⋊Q8—22no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8C23×C711yescharacteristic
370C22⋊Q8C23×C712yes—
371#121—21no—
372#121—22no—
373#121—22no—
374#121C23×C711yescharacteristic
375#120—22no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8C7×D411yescharacteristic, complemented
379C4×Q8C7×D412yescomplemented
380C4×D4—21no—
381C4×D4—22no—
382C4×D4—21no—
383C4×D4—22no—
384C4×D4C7×D411yescharacteristic, complemented
385#117—22no—
386#117—21no—
387#117—21no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4C7×D412yes—
391C2×C22⋊C4C23×C712yes—
392C2×C42—21no—
393C2×C42C23×C711yescharacteristic
394C4⋊C8—21no—
395C4⋊C8—21no—
396#102—22no—
397#102—22no—
398#102—24no—
399C22⋊C8C23×C712yes—
400C22⋊C8C23×C712yes—
401#97—21no—
402C4×C8—21no—
403#95—21no—
404#95C23×C712yes—
405C22×C7—21no—
406C22×C7—21no—
407C22×C7—41no—
408C22×C7—42no—
409C22×C7—41no—
410C22×C7—42no—
411C22×C7—42no—
412C22×C7—41no—
413C22×C7—84no—
414C22×C7—84no—
415C22×C7—84no—
416C22×C7—82no—
417C22×C7—82no—
418C22×C7—82no—
419C22×C7C2×#12011yescharacteristic
420C22×C7#57712yes—
421C4×C7—21no—
422C4×C7—21no—
423C4×C7—21no—
424C4×C7—21no—
425C4×C7—21no—
426C4×C7—44no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—42no—
431C4×C7—41no—
432C4×C7—42no—
433C4×C7#45312yes—
434C24—22no—
435C24C2×C7×D412yes—
436#41—82no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—42no—
442C2×Q8—82no—
443C2×Q8C2×C7×D411yescharacteristic
444C2×Q8C2×C7×D412yes—
445C2×D4—21no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—48no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4—41no—
460C2×D4—82no—
461C2×D4—82no—
462C2×D4—84no—
463C2×D4—84no—
464C2×D4C2×C7×D411yescharacteristic
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D412yes—
473C22×C4C24×C711yescharacteristic
474SD16—82no—
475SD16—84no—
476D8—82no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4C2×C7×D411yescharacteristic
493C4⋊C4C2×C7×D411yescharacteristic
494C4⋊C4C2×C7×D412yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—21no—
510C22⋊C4—48no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42C2×C7×D411yescharacteristic
516C42C2×C7×D411yescharacteristic
517C42C2×C7×D411yescharacteristic
518C42C2×C7×D411yescharacteristic
519C2×C7—22no—
520C2×C7—81no—
521C2×C7—82no—
522C2×C7—162no—
523C2×C7#301011yescharacteristic
524C2×C7C2×#45311yescharacteristic
525C2×C7#302711yescharacteristic
526C23—22no—
527C23—21no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23C22×C7×D411yescharacteristic
537Q8—41no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—81no—
542Q8—82no—
543D4—41no—
544D4—42no—
545D4—42no—
546D4—44no—
547D4—42no—
548D4—42no—
549D4—81no—
550D4—82no—
551D4—82no—
552D4—82no—
553D4—82no—
554D4—82no—
555D4—816no—
556D4—82no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C7×#12011yescharacteristic
577C2×C4C7×#12011yescharacteristic
578C2×C4C22×C7×D411yescharacteristic
579C2×C4C22×C7×D411yescharacteristic
580C8—42no—
581C8—42no—
582C7#1842811yescharacteristic, direct product factor, Sylow
583C22—21no—
584C22—21no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—41no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×C7×#12011yescharacteristic
598C22C7×#57712yes—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—44no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4C7×#45312yes—
612C2—22no—
613C2—82no—
614C2—81no—
615C2—162no—
616C2C7×#301011yescharacteristic
617C2C2×C7×#45311yescharacteristic
618C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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
1. C7×#3561C2semidirect
2. C7×#3217C2semidirect
3. C7×#3208C2semidirect
4. C7×#2657C2semidirect
5. C7×#1600C2semidirect
6. C7×#1596C2semidirect
7. C7×#1506C2semidirect
8. C7×#1500C2semidirect
9. C7×#1469C2semidirect
47. #18428C7direct (N × H)
61. C7×C4⋊Q8D4semidirect
62. C7×C4⋊Q8D4semidirect
73. C7×#125D4semidirect
82. C7×#124D4semidirect
106. C4×C7×Q8D4semidirect
107. C4×C7×Q8D4semidirect
112. C4×C7×D4D4semidirect
133. #3561C2×C7semidirect
134. #3217C2×C7semidirect
135. #3208C2×C7semidirect
136. #2657C2×C7semidirect
137. #1600C2×C7semidirect
138. #1596C2×C7semidirect
139. #1506C2×C7semidirect
140. #1500C2×C7semidirect
141. #1469C2×C7semidirect
333. C4⋊Q8C7×D4semidirect
334. C4⋊Q8C7×D4semidirect
345. #125C7×D4semidirect
354. #124C7×D4semidirect
378. C4×Q8C7×D4semidirect
379. C4×Q8C7×D4semidirect
384. C4×D4C7×D4semidirect
582. C7#18428direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35612
C7×#32172
C7×#32082
C7×#26572
C7×#16002
C7×#15962
C7×#15062
C7×#15002
C7×#14692
#184287