Finite Group Databasebeta

← C7×#18289 #13195505

Subgroups — C7×#18289 #13195505

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#539C211yesmaximal, characteristic
2C7×#3189C211yesmaximal, characteristic, complemented
3C7×#3187C211yesmaximal, characteristic
4C7×#2627C211yesmaximal, characteristic, complemented
5C7×#1615C214yesmaximal, complemented
6C7×#1603C214yesmaximal, complemented
7C7×#1520C211yesmaximal, characteristic, complemented
8C7×#1519C211yesmaximal, characteristic
9C2×C7×#324C211yesmaximal, 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×C22⋊Q8C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×D4—21no—
23C22×C7×C4⋊C4C2211yescharacteristic
24C7×#484—21no—
25C7×#484C2212yescomplemented
26C7×#483—21no—
27C7×#481C2212yes—
28C7×#467—22no—
29C7×#465—22no—
30C7×#451—21no—
31C7×#451C2212yescomplemented
32C7×#450—21no—
33C7×#448C2212yes—
34C2×C7×C22⋊C8C2211yescharacteristic
35C2×C7×C22⋊C8C2211yescharacteristic
36C7×#400C2212yescomplemented
37C7×#398—22no—
38C7×#395C2212yescomplemented
39C7×#385—21no—
40C7×#381—21no—
41C7×#380C2211yescharacteristic
42C7×#328C2214yescomplemented
43C7×#327C2214yes—
44C7×#324C2214yescomplemented
45#18289C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C2×C7×Q16—22no—
55C2×C7×SD16—22no—
56C22×C7×C8—21no—
57C22×C7×C8—21no—
58C7×#126—24no—
59C7×#126—24no—
60C7×#125—24no—
61C7×#125—24no—
62C7×#123—24no—
63C7×#123—24no—
64C7×#123—24no—
65C7×#123—24no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—21no—
72C7×C22⋊Q8—21no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8D412yes—
75C7×C22⋊Q8C2312yes—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121D412yes—
79C7×#121C2312yes—
80C4×C7×D4—24no—
81C4×C7×D4—24no—
82C2×C7×C4⋊C4—21no—
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⋊C4—21no—
92C2×C7×C4⋊C4—22no—
93C2×C7×C4⋊C4D412yes—
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4D411yescharacteristic
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2312yes—
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—21no—
104C2×C7×C22⋊C4—21no—
105C2×C42×C7—21no—
106C7×#107—22no—
107C7×#106—22no—
108C7×#103—22no—
109C7×#103—22no—
110C7×#103—22no—
111C7×#102—22no—
112C7×C22⋊C8—21no—
113C7×C22⋊C8—21no—
114C7×C22⋊C8C2312yes—
115C7×C22⋊C8C2312yes—
116C7×#95—22no—
117C7×#95—22no—
118C7×#95—22no—
119C7×#95C2312yes—
120C2×#539C2×C711yescharacteristic
121#3189C2×C711yescharacteristic, complemented
122#3187C2×C711yescharacteristic
123#2627C2×C711yescharacteristic, complemented
124#1615C2×C714yescomplemented
125#1603C2×C714yescomplemented
126#1520C2×C711yescharacteristic, complemented
127#1519C2×C711yescharacteristic
128C2×#324C2×C711yescharacteristic, complemented
129C24×C7—21no—
130C24×C7C2×D411yescharacteristic
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—81no—
135C2×C7×Q8—81no—
136C2×C7×Q8C2×D412yes—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—21no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—81no—
143C2×C7×D4C2×D412yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—24no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—21no—
157C22×C4×C7—81no—
158C22×C4×C7—81no—
159C22×C4×C7—82no—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2411yescharacteristic
168C7×Q16—82no—
169C7×SD16—82no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C2×C7×C8—81no—
173C2×C7×C8—81no—
174C7×C4⋊C4—24no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4—41no—
189C7×C4⋊C4—81no—
190C7×C4⋊C4—82no—
191C7×C4⋊C4C2×D412yes—
192C7×C4⋊C4C2×D412yes—
193C7×C22⋊C4—24no—
194C7×C22⋊C4—24no—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—41no—
201C7×C22⋊C4—41no—
202C7×C22⋊C4—42no—
203C42×C7—24no—
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×C22⋊Q8C22×C711yescharacteristic
215C2×#121C22×C711yescharacteristic
216C2×C4×D4—21no—
217C22×C4⋊C4C22×C711yescharacteristic
218#484—21no—
219#484C22×C712yescomplemented
220#483—21no—
221#481C22×C712yes—
222#467—22no—
223#465—22no—
224#451—21no—
225#451C22×C712yescomplemented
226#450—21no—
227#448C22×C712yes—
228C2×C22⋊C8C22×C711yescharacteristic
229C2×C22⋊C8C22×C711yescharacteristic
230#400C22×C712yescomplemented
231#398—22no—
232#395C22×C712yescomplemented
233#385—21no—
234#381—21no—
235#380C22×C711yescharacteristic
236#328C22×C714yescomplemented
237#327C22×C714yes—
238#324C22×C714yescomplemented
239C23×C7—22no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7#12011yescharacteristic
246C23×C7#12011yescharacteristic
247C23×C7#13512yes—
248C23×C7#13712yes—
249C23×C7C22×D411yescharacteristic
250C7×Q8—42no—
251C7×Q8—42no—
252C7×Q8—42no—
253C7×Q8—81no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—42no—
257C7×D4—44no—
258C7×D4—81no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
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—41no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—81no—
287C2×C4×C7—84no—
288C2×C4×C7#12011yescharacteristic
289C2×C4×C7#12011yescharacteristic
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×D411yescharacteristic
292C7×C8—42no—
293C7×C8—42no—
294C22×Q8C7×D411yescharacteristic
295C22×Q8C23×C711yescharacteristic
296C22×D4—21no—
297C22×D4C7×D411yescharacteristic
298C23×C4—21no—
299C23×C4C7×D411yescharacteristic
300C23×C4C7×D411yescharacteristic
301C23×C4C23×C711yescharacteristic
302C2×Q16—22no—
303C2×SD16—22no—
304C22×C8—21no—
305C22×C8—21no—
306#126—24no—
307#126—24no—
308#125—24no—
309#125—24no—
310#123—24no—
311#123—24no—
312#123—24no—
313#123—24no—
314C22⋊Q8—24no—
315C22⋊Q8—24no—
316C22⋊Q8—21no—
317C22⋊Q8—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322C22⋊Q8C7×D412yes—
323C22⋊Q8C23×C712yes—
324#121—21no—
325#121—21no—
326#121C7×D412yes—
327#121C23×C712yes—
328C4×D4—24no—
329C4×D4—24no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
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—22no—
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×C42—21no—
354#107—22no—
355#106—22no—
356#103—22no—
357#103—22no—
358#103—22no—
359#102—22no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8C23×C712yes—
363C22⋊C8C23×C712yes—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C23×C712yes—
368C22×C7—21no—
369C22×C7—21no—
370C22×C7—21no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7—21no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—84no—
380C22×C7#45212yes—
381C22×C7#45212yes—
382C22×C7C2×#12011yescharacteristic
383C22×C7C2×#13511yescharacteristic
384C22×C7C2×#13711yescharacteristic
385C4×C7—21no—
386C4×C7—21no—
387C4×C7—21no—
388C4×C7—21no—
389C4×C7—42no—
390C4×C7—42no—
391C4×C7—42no—
392C4×C7—42no—
393C4×C7—42no—
394C4×C7—82no—
395C4×C7—81no—
396C4×C7—81no—
397C24—21no—
398C24C2×C7×D411yescharacteristic
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8C2×C7×D412yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—81no—
411C2×D4C2×C7×D412yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—82no—
428C22×C4C2×C7×D412yes—
429C22×C4C2×C7×D412yes—
430C22×C4C2×C7×D411yescharacteristic
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C24×C711yescharacteristic
436Q16—82no—
437SD16—82no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—81no—
441C2×C8—81no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—81no—
458C4⋊C4—82no—
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—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C42—24no—
472C2×C7—21no—
473C2×C7—22no—
474C2×C7—21no—
475C2×C7—22no—
476C2×C7—82no—
477C2×C7C2×#45211yescharacteristic
478C2×C7#302011yescharacteristic
479C2×C7#302611yescharacteristic
480C23—22no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C7×#12011yescharacteristic
487C23C7×#12011yescharacteristic
488C23C7×#13512yes—
489C23C7×#13712yes—
490C23C22×C7×D411yescharacteristic
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—81no—
495D4—42no—
496D4—42no—
497D4—42no—
498D4—44no—
499D4—81no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4C7×#12011yescharacteristic
530C2×C4C7×#12011yescharacteristic
531C2×C4C22×C7×D411yescharacteristic
532C2×C4C22×C7×D411yescharacteristic
533C8—42no—
534C8—42no—
535C7#1828911yescharacteristic, direct product factor, Sylow
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—84no—
548C22C7×#45212yes—
549C22C7×#45212yes—
550C22C2×C7×#12011yescharacteristic
551C22C2×C7×#13511yescharacteristic
552C22C2×C7×#13711yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—81no—
564C4—81no—
565C2—21no—
566C2—22no—
567C2—21no—
568C2—22no—
569C2—82no—
570C2C2×C7×#45211yescharacteristic
571C2C7×#302011yescharacteristic
572C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
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
2. C7×#3189C2semidirect
4. C7×#2627C2semidirect
5. C7×#1615C2semidirect
6. C7×#1603C2semidirect
7. C7×#1520C2semidirect
9. C2×C7×#324C2semidirect
25. C7×#484C22semidirect
31. C7×#451C22semidirect
36. C7×#400C22semidirect
38. C7×#395C22semidirect
42. C7×#328C22semidirect
44. C7×#324C22semidirect
45. #18289C7direct (N × H)
121. #3189C2×C7semidirect
123. #2627C2×C7semidirect
124. #1615C2×C7semidirect
125. #1603C2×C7semidirect
126. #1520C2×C7semidirect
128. C2×#324C2×C7semidirect
219. #484C22×C7semidirect
225. #451C22×C7semidirect
230. #400C22×C7semidirect
232. #395C22×C7semidirect
236. #328C22×C7semidirect
238. #324C22×C7semidirect
535. C7#18289direct (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×#31892
C7×#31872
C7×#26272
C7×#16152
C7×#16032
C7×#15202
C7×#15192
C2×C7×#3242
#182897