Finite Group Databasebeta

← C7×#24104 #13201320

Subgroups — C7×#24104 #13201320

Order 1792.

1146 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#555C211yesmaximal, characteristic
2C2×C7×#481C211yesmaximal, characteristic, complemented
3C2×C7×#428C211yesmaximal, characteristic, complemented
4C7×#2486C211yesmaximal, characteristic, complemented
5C7×#2058C212yesmaximal, complemented
6C7×#2034C212yesmaximal, complemented
7C7×#1895C211yesmaximal, characteristic, complemented
8C7×#1870C211yesmaximal, characteristic, complemented
9C7×#1862C212yesmaximal, complemented
10C7×#1854C212yesmaximal, complemented
11C7×#1825C211yesmaximal, characteristic, complemented
12C22×C7×M16C2211yescharacteristic
13C7×#555—24no—
14C7×#555—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#124—21no—
17C2×C7×#124—21no—
18C2×C7×#123—21no—
19C2×C7×#121—21no—
20C2×C7×#121C2211yescharacteristic
21C2×C7×#121C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C2×C7×#117—21no—
25C2×C7×#117C2211yescharacteristic
26C2×C7×#117C2211yescharacteristic
27C22×C7×C4⋊C4C2211yescharacteristic
28C7×#481—24no—
29C7×#428—24no—
30C2×C7×#107C2212yes—
31C2×C7×#106C2212yes—
32C2×C7×#106C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×C22⋊C8C2211yescharacteristic
37C7×#398—21no—
38C7×#397—21no—
39C7×#389C2212yes—
40C7×#381—21no—
41C7×#380C2212yes—
42C7×#380C2211yescharacteristic
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#340C2212yes—
46C7×#336C2212yes—
47#24104C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C2×C7×M16—22no—
58C2×C7×M16—24no—
59C22×C7×C8C2312yes—
60C22×C7×C8C2312yes—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—22no—
63C7×#124—28no—
64C7×#124—28no—
65C7×#123—44no—
66C7×#121—28no—
67C7×#121—22no—
68C7×#121—22no—
69C7×#121—22no—
70C7×#121D414yes—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C7×#117—28no—
76C7×#117—22no—
77C7×#117—22no—
78C7×#117—22no—
79C7×#117D414yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—22no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C4⋊C4C2311yescharacteristic
90C2×C7×C4⋊C4C2311yescharacteristic
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4D411yescharacteristic
99C2×C7×C22⋊C4D411yescharacteristic
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C2×C42×C7C2311yescharacteristic
103C7×#107—24no—
104C7×#106—24no—
105C7×#106—24no—
106C7×#102—24no—
107C7×#102—24no—
108C7×C22⋊C8—22no—
109C7×C22⋊C8—24no—
110C7×#95—22no—
111C7×#95—21no—
112C7×#95—22no—
113C2×#555C2×C711yescharacteristic
114C2×#481C2×C711yescharacteristic, complemented
115C2×#428C2×C711yescharacteristic, complemented
116#2486C2×C711yescharacteristic, complemented
117#2058C2×C712yescomplemented
118#2034C2×C712yescomplemented
119#1895C2×C711yescharacteristic, complemented
120#1870C2×C711yescharacteristic, complemented
121#1862C2×C712yescomplemented
122#1854C2×C712yescomplemented
123#1825C2×C711yescharacteristic, complemented
124C24×C7—21no—
125C24×C7C2×D411yescharacteristic
126C24×C7C2×D411yescharacteristic
127C2×C7×Q8—24no—
128C2×C7×Q8—82no—
129C2×C7×D4—22no—
130C2×C7×D4—24no—
131C2×C7×D4—24no—
132C2×C7×D4—22no—
133C2×C7×D4—21no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—41no—
137C2×C7×D4—82no—
138C2×C7×D4—82no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C7×M16—28no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8#4114yes—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—44no—
182C7×C4⋊C4C2×D412yes—
183C7×C4⋊C4C2×D412yes—
184C7×C4⋊C4C2×D412yes—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—24no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—44no—
194C42×C7—24no—
195C42×C7—24no—
196C42×C7—21no—
197C42×C7—21no—
198C22×M16C22×C711yescharacteristic
199#555—24no—
200#555—24no—
201C2×C4⋊Q8C22×C711yescharacteristic
202C2×#124—21no—
203C2×#124—21no—
204C2×#123—21no—
205C2×#121—21no—
206C2×#121C22×C711yescharacteristic
207C2×#121C22×C711yescharacteristic
208C2×C4×D4C22×C711yescharacteristic
209C2×C4×D4C22×C711yescharacteristic
210C2×#117—21no—
211C2×#117C22×C711yescharacteristic
212C2×#117C22×C711yescharacteristic
213C22×C4⋊C4C22×C711yescharacteristic
214#481—24no—
215#428—24no—
216C2×#107C22×C712yes—
217C2×#106C22×C712yes—
218C2×#106C22×C712yes—
219C2×#102C22×C712yes—
220C2×#102C22×C712yes—
221C2×C22⋊C8C22×C711yescharacteristic
222C2×C22⋊C8C22×C712yes—
223#398—21no—
224#397—21no—
225#389C22×C712yes—
226#381—21no—
227#380C22×C711yescharacteristic
228#380C22×C712yes—
229#340C22×C712yes—
230#340C22×C712yes—
231#340C22×C712yes—
232#336C22×C712yes—
233C23×C7—22no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—22no—
237C23×C7—21no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—22no—
243C23×C7—42no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—42no—
247C23×C7#12012yes—
248C23×C7C22×D411yescharacteristic
249C7×Q8—84no—
250C7×D4—24no—
251C7×D4—44no—
252C7×D4—41no—
253C7×D4—41no—
254C7×D4—84no—
255C7×D4—84no—
256C2×C4×C7—21no—
257C2×C4×C7—22no—
258C2×C4×C7—21no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—48no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
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—84no—
284C2×C4×C7#12012yes—
285C2×C4×C7AGL(1,ℤ/8)12yes—
286C2×C4×C7C22×D411yescharacteristic
287C2×C4×C7C22×D411yescharacteristic
288C2×C4×C7C2×#4112yes—
289C2×C4×C72−1+412yes—
290C7×C8—24no—
291C7×C8—42no—
292C22×Q8—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4C7×D411yescharacteristic
296C22×D4C23×C711yescharacteristic
297C23×C4—21no—
298C23×C4—21no—
299C23×C4C23×C711yescharacteristic
300C23×C4C23×C711yescharacteristic
301C2×M16—22no—
302C2×M16—24no—
303C22×C8C23×C712yes—
304C22×C8C23×C712yes—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307#124—28no—
308#124—28no—
309#123—44no—
310#121—28no—
311#121—22no—
312#121—22no—
313#121—22no—
314#121C7×D414yes—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319#117—28no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117C7×D414yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4C7×D411yescharacteristic
330C2×C4⋊C4C23×C712yes—
331C2×C4⋊C4C23×C712yes—
332C2×C4⋊C4C23×C711yescharacteristic
333C2×C4⋊C4C23×C711yescharacteristic
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C7×D411yescharacteristic
343C2×C22⋊C4C7×D411yescharacteristic
344C2×C42—21no—
345C2×C42—21no—
346C2×C42C23×C711yescharacteristic
347#107—24no—
348#106—24no—
349#106—24no—
350#102—24no—
351#102—24no—
352C22⋊C8—22no—
353C22⋊C8—24no—
354#95—22no—
355#95—21no—
356#95—22no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—22no—
360C22×C7—42no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—41no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7—42no—
370C22×C7—82no—
371C22×C7—82no—
372C22×C7C2×#12011yescharacteristic
373C22×C7#54712yes—
374C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C7#57511yescharacteristic
376C22×C7#57712yes—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—41no—
382C4×C7—42no—
383C4×C7—42no—
384C4×C7—82no—
385C4×C7—82no—
386C4×C7#52511yescharacteristic
387C4×C7#53611yescharacteristic
388C4×C7#54712yes—
389C24—21no—
390C24C2×C7×D411yescharacteristic
391C24C2×C7×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4C2×C7×D412yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D412yes—
426C22×C4C2×C7×D412yes—
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C24×C711yescharacteristic
429M16—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C7×#4114yes—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4C2×C7×D412yes—
448C4⋊C4C2×C7×D412yes—
449C4⋊C4C2×C7×D412yes—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—22no—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C42—24no—
460C42—24no—
461C42—21no—
462C42—21no—
463C2×C7—42no—
464C2×C7—81no—
465C2×C7—82no—
466C2×C7#240611yescharacteristic
467C2×C7#300611yescharacteristic
468C2×C7#300911yescharacteristic
469C2×C7#334812yes—
470C2×C7#335412yes—
471C23—22no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23C7×#12012yes—
486C23C22×C7×D411yescharacteristic
487Q8—84no—
488D4—24no—
489D4—44no—
490D4—41no—
491D4—41no—
492D4—84no—
493D4—84no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4C7×#12012yes—
523C2×C4C7×AGL(1,ℤ/8)12yes—
524C2×C4C22×C7×D411yescharacteristic
525C2×C4C22×C7×D411yescharacteristic
526C2×C4C2×C7×#4112yes—
527C2×C4C7×2−1+412yes—
528C8—24no—
529C8—42no—
530C7#2410411yescharacteristic, direct product factor, Sylow
531C22—22no—
532C22—22no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22C2×C7×#12011yescharacteristic
547C22C7×#54712yes—
548C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
549C22C7×#57511yescharacteristic
550C22C7×#57712yes—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4C7×#52511yescharacteristic
561C4C7×#53611yescharacteristic
562C4C7×#54712yes—
563C2—42no—
564C2—81no—
565C2—82no—
566C2C7×#240611yescharacteristic
567C2C7×#300611yescharacteristic
568C2C7×#300911yescharacteristic
569C2C7×#334812yes—
570C2C7×#335412yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#555C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#481C2semidirect
3. C2×C7×#428C2semidirect
4. C7×#2486C2semidirect
5. C7×#2058C2semidirect
6. C7×#2034C2semidirect
7. C7×#1895C2semidirect
8. C7×#1870C2semidirect
9. C7×#1862C2semidirect
10. C7×#1854C2semidirect
11. C7×#1825C2semidirect
47. #24104C7direct (N × H)
114. C2×#481C2×C7semidirect
115. C2×#428C2×C7semidirect
116. #2486C2×C7semidirect
117. #2058C2×C7semidirect
118. #2034C2×C7semidirect
119. #1895C2×C7semidirect
120. #1870C2×C7semidirect
121. #1862C2×C7semidirect
122. #1854C2×C7semidirect
123. #1825C2×C7semidirect
530. C7#24104direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5552
C2×C7×#4812
C2×C7×#4282
C7×#24862
C7×#20582
C7×#20342
C7×#18952
C7×#18702
C7×#18622
C7×#18542
C7×#18252
#241047