Finite Group Databasebeta

← C7×#18320 #13195536

Subgroups — C7×#18320 #13195536

Order 1792.

1008 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#539C211yesmaximal, characteristic, complemented
2C7×#3193C211yesmaximal, characteristic, complemented
3C7×#3192C211yesmaximal, characteristic
4C7×#2591C211yesmaximal, characteristic, complemented
5C7×#1615C214yesmaximal, complemented
6C7×#1597C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1507C211yesmaximal, characteristic, complemented
9C7×#1463C211yesmaximal, characteristic, complemented
10C7×#539—22no—
11C7×#539—22no—
12C7×#539—22no—
13C7×#539C2214yes—
14C2×C7×#126—21no—
15C2×C7×#125—21no—
16C2×C7×#123—21no—
17C2×C7×#123—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2211yescharacteristic
20C2×C7×#121C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C22×C7×C4⋊C4C2211yescharacteristic
24C7×#484C2212yescomplemented
25C7×#481—21no—
26C7×#481C2212yes—
27C7×#480—21no—
28C7×#469—22no—
29C7×#468—22no—
30C7×#450C2212yescomplemented
31C7×#448—21no—
32C7×#447—21no—
33C7×#447C2212yes—
34C7×#407C2211yescharacteristic
35C7×#407C2211yescharacteristic
36C7×#400C2212yescomplemented
37C7×#394C2212yescomplemented
38C7×#388—22no—
39C7×#386—21no—
40C7×#386—21no—
41C4×C7×C22⋊C4C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yes—
44C7×#324C2214yescomplemented
45#18320C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×D4—21no—
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C2×C7×SD16—22no—
56C2×C7×D8—22no—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C7×#126—24no—
60C7×#126—24no—
61C7×#125—24no—
62C7×#125—24no—
63C7×#123—24no—
64C7×#123—24no—
65C7×#123—24no—
66C7×#123—24no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8D412yes—
72C7×C22⋊Q8C2312yes—
73C7×#121—21no—
74C7×#121—21no—
75C7×#121—21no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121—21no—
79C7×#121D412yes—
80C7×#121C2312yes—
81C4×C7×D4—24no—
82C4×C7×D4—24no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—21no—
90C2×C7×C4⋊C4—21no—
91C2×C7×C4⋊C4D412yes—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C4⋊C4D411yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C4⋊C4C2312yes—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—21no—
99C2×C7×C22⋊C4—21no—
100C2×C7×C22⋊C4—21no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4—22no—
105C2×C42×C7—21no—
106C2×C42×C7—21no—
107C7×#107—22no—
108C7×#106—22no—
109C7×#103—22no—
110C7×#102—22no—
111C7×#102—22no—
112C7×#102—22no—
113C7×C22⋊C8—21no—
114C7×C22⋊C8—21no—
115C7×C22⋊C8C2312yes—
116C7×C22⋊C8C2312yes—
117C7×#95—22no—
118C7×#95—22no—
119C7×#95C2312yes—
120C2×#539C2×C711yescharacteristic, complemented
121#3193C2×C711yescharacteristic, complemented
122#3192C2×C711yescharacteristic
123#2591C2×C711yescharacteristic, complemented
124#1615C2×C714yescomplemented
125#1597C2×C714yescomplemented
126#1508C2×C711yescharacteristic, complemented
127#1507C2×C711yescharacteristic, complemented
128#1463C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7—21no—
131C24×C7C2×D411yescharacteristic
132C2×C7×Q8—21no—
133C2×C7×Q8—81no—
134C2×C7×Q8C2×D412yes—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—44no—
141C2×C7×D4—44no—
142C2×C7×D4—41no—
143C2×C7×D4—41no—
144C2×C7×D4—81no—
145C2×C7×D4—81no—
146C2×C7×D4C2×D412yes—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—24no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—42no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7—82no—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2411yescharacteristic
169C7×SD16—82no—
170C7×D8—82no—
171C7×M16—81no—
172C7×M16—81no—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4C2×D412yes—
189C7×C4⋊C4C2×D412yes—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—24no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—44no—
198C7×C22⋊C4—42no—
199C7×C22⋊C4—41no—
200C7×C22⋊C4—41no—
201C7×C22⋊C4—82no—
202C42×C7—24no—
203C42×C7—81no—
204#539—22no—
205#539—22no—
206#539—22no—
207#539C22×C714yes—
208C2×#126—21no—
209C2×#125—21no—
210C2×#123—21no—
211C2×#123—21no—
212C2×C22⋊Q8—21no—
213C2×C22⋊Q8C22×C711yescharacteristic
214C2×#121C22×C711yescharacteristic
215C2×#121C22×C711yescharacteristic
216C2×C4×D4—21no—
217C22×C4⋊C4C22×C711yescharacteristic
218#484C22×C712yescomplemented
219#481—21no—
220#481C22×C712yes—
221#480—21no—
222#469—22no—
223#468—22no—
224#450C22×C712yescomplemented
225#448—21no—
226#447—21no—
227#447C22×C712yes—
228#407C22×C711yescharacteristic
229#407C22×C711yescharacteristic
230#400C22×C712yescomplemented
231#394C22×C712yescomplemented
232#388—22no—
233#386—21no—
234#386—21no—
235C4×C22⋊C4C22×C711yescharacteristic
236#328C22×C714yescomplemented
237#327C22×C714yes—
238#324C22×C714yescomplemented
239C23×C7—22no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—41no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—41no—
249C23×C7—42no—
250C23×C7#12011yescharacteristic
251C23×C7#12011yescharacteristic
252C23×C7C22×D411yescharacteristic
253C7×Q8—42no—
254C7×Q8—81no—
255C7×D4—42no—
256C7×D4—42no—
257C7×D4—42no—
258C7×D4—44no—
259C7×D4—44no—
260C7×D4—81no—
261C7×D4—82no—
262C7×D4—82no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—81no—
289C2×C4×C7—84no—
290C2×C4×C7#12011yescharacteristic
291C2×C4×C7#12011yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C7×C8—42no—
295C7×C8—42no—
296C22×Q8C7×D411yescharacteristic
297C22×D4—21no—
298C22×D4—21no—
299C22×D4C7×D411yescharacteristic
300C22×D4C23×C711yescharacteristic
301C23×C4—21no—
302C23×C4C7×D411yescharacteristic
303C23×C4C7×D411yescharacteristic
304C23×C4C23×C711yescharacteristic
305C2×SD16—22no—
306C2×D8—22no—
307C2×M16—21no—
308C2×M16—21no—
309#126—24no—
310#126—24no—
311#125—24no—
312#125—24no—
313#123—24no—
314#123—24no—
315#123—24no—
316#123—24no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8C7×D412yes—
322C22⋊Q8C23×C712yes—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—21no—
327#121—21no—
328#121—21no—
329#121C7×D412yes—
330#121C23×C712yes—
331C4×D4—24no—
332C4×D4—24no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C7×D412yes—
342C2×C4⋊C4C7×D411yescharacteristic
343C2×C4⋊C4C7×D411yescharacteristic
344C2×C4⋊C4C23×C711yescharacteristic
345C2×C4⋊C4C23×C712yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C42—21no—
356C2×C42—21no—
357#107—22no—
358#106—22no—
359#103—22no—
360#102—22no—
361#102—22no—
362#102—22no—
363C22⋊C8—21no—
364C22⋊C8—21no—
365C22⋊C8C23×C712yes—
366C22⋊C8C23×C712yes—
367#95—22no—
368#95—22no—
369#95C23×C712yes—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7—21no—
374C22×C7—21no—
375C22×C7—21no—
376C22×C7—21no—
377C22×C7—44no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7—84no—
382C22×C7—84no—
383C22×C7—81no—
384C22×C7—81no—
385C22×C7—81no—
386C22×C7#45412yes—
387C22×C7C2×#12011yescharacteristic
388C22×C7#57512yes—
389C4×C7—21no—
390C4×C7—21no—
391C4×C7—21no—
392C4×C7—21no—
393C4×C7—42no—
394C4×C7—42no—
395C4×C7—42no—
396C4×C7—42no—
397C4×C7—42no—
398C4×C7—82no—
399C4×C7—81no—
400C24—21no—
401C24—21no—
402C24C2×C7×D411yescharacteristic
403C2×Q8—21no—
404C2×Q8—81no—
405C2×Q8C2×C7×D412yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4C2×C7×D412yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4C2×C7×D412yes—
433C22×C4C2×C7×D412yes—
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C2×C7×D411yescharacteristic
436C22×C4C2×C7×D411yescharacteristic
437C22×C4C2×C7×D411yescharacteristic
438C22×C4C2×C7×D411yescharacteristic
439C22×C4C24×C711yescharacteristic
440SD16—82no—
441D8—82no—
442M16—81no—
443M16—81no—
444C2×C8—22no—
445C2×C8—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4C2×C7×D412yes—
460C4⋊C4C2×C7×D412yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C42—24no—
474C42—81no—
475C2×C7—22no—
476C2×C7—22no—
477C2×C7—41no—
478C2×C7—82no—
479C2×C7—161no—
480C2×C7#300511yescharacteristic
481C2×C7C2×#45411yescharacteristic
482C2×C7#302611yescharacteristic
483C23—22no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23C7×#12011yescharacteristic
495C23C7×#12011yescharacteristic
496C23C22×C7×D411yescharacteristic
497Q8—42no—
498Q8—81no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—81no—
505D4—82no—
506D4—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—84no—
534C2×C4C7×#12011yescharacteristic
535C2×C4C7×#12011yescharacteristic
536C2×C4C22×C7×D411yescharacteristic
537C2×C4C22×C7×D411yescharacteristic
538C8—42no—
539C8—42no—
540C7#1832011yescharacteristic, direct product factor, Sylow
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—84no—
553C22—84no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C7×#45412yes—
558C22C2×C7×#12011yescharacteristic
559C22C7×#57512yes—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—81no—
571C2—22no—
572C2—22no—
573C2—41no—
574C2—82no—
575C2—161no—
576C2C7×#300511yescharacteristic
577C2C2×C7×#45411yescharacteristic
578C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3192C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C7×#539C2semidirect
2. C7×#3193C2semidirect
4. C7×#2591C2semidirect
5. C7×#1615C2semidirect
6. C7×#1597C2semidirect
7. C7×#1508C2semidirect
8. C7×#1507C2semidirect
9. C7×#1463C2semidirect
24. C7×#484C22semidirect
30. C7×#450C22semidirect
36. C7×#400C22semidirect
37. C7×#394C22semidirect
42. C7×#328C22semidirect
44. C7×#324C22semidirect
45. #18320C7direct (N × H)
120. C2×#539C2×C7semidirect
121. #3193C2×C7semidirect
123. #2591C2×C7semidirect
124. #1615C2×C7semidirect
125. #1597C2×C7semidirect
126. #1508C2×C7semidirect
127. #1507C2×C7semidirect
128. #1463C2×C7semidirect
218. #484C22×C7semidirect
224. #450C22×C7semidirect
230. #400C22×C7semidirect
231. #394C22×C7semidirect
236. #328C22×C7semidirect
238. #324C22×C7semidirect
540. C7#18320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5392
C7×#31932
C7×#31922
C7×#25912
C7×#16152
C7×#15972
C7×#15082
C7×#15072
C7×#14632
#183207