Finite Group Databasebeta

← C7×#25946 #13203162

Subgroups — C7×#25946 #13203162

Order 1792.

1176 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C2×C7×#490C211yesmaximal, characteristic, complemented
3C2×C7×#459C211yesmaximal, characteristic
4C7×#2548C211yesmaximal, characteristic, complemented
5C7×#2025C212yesmaximal, complemented
6C7×#2023C212yesmaximal, complemented
7C7×#1989C211yesmaximal, characteristic, complemented
8C7×#1985C211yesmaximal, characteristic, complemented
9C7×#1879C212yesmaximal, complemented
10C7×#1879C212yesmaximal, complemented
11C7×#1762C211yesmaximal, characteristic, complemented
12C22×C7×D8C2212yes—
13C7×#554—24no—
14C7×#554—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#127C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C2×C7×#125C2211yescharacteristic
19C2×C7×#124—21no—
20C2×C7×C22⋊Q8—21no—
21C2×C7×C22⋊Q8—21no—
22C2×C7×C22⋊Q8—21no—
23C2×C4×C7×Q8C2211yescharacteristic
24C2×C4×C7×D4C2211yescharacteristic
25C2×C4×C7×D4C2211yescharacteristic
26C2×C4×C7×D4C2211yescharacteristic
27C22×C7×C4⋊C4—21no—
28C7×#490—24no—
29C7×#459—24no—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#102C2212yes—
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#97C2211yescharacteristic
36C7×#394—21no—
37C7×#394—21no—
38C7×#390—21no—
39C7×#390C2212yes—
40C7×#390C2212yes—
41C4×C7×C4⋊C4C2211yescharacteristic
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#336C2212yes—
46C7×#336C2212yes—
47#25946C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×D4—21no—
51C22×C7×D4—21no—
52C22×C7×D4C2312yes—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7—21no—
57C2×C7×D8—24no—
58C2×C7×D8—24no—
59C2×C7×D8—24no—
60C22×C7×C8C2312yes—
61C22×C7×C8C2312yes—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×#127—22no—
65C7×#127—22no—
66C7×#125—22no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#125D414yes—
70C7×#124—44no—
71C7×C22⋊Q8—28no—
72C7×C22⋊Q8—28no—
73C7×C22⋊Q8—28no—
74C4×C7×Q8—22no—
75C4×C7×Q8—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C4×C7×D4—22no—
80C4×C7×D4—22no—
81C4×C7×D4D414yes—
82C2×C7×C4⋊C4—28no—
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⋊C4D411yescharacteristic
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C22⋊C4—21no—
96C2×C7×C22⋊C4—21no—
97C2×C7×C22⋊C4—21no—
98C2×C7×C22⋊C4—22no—
99C2×C7×C22⋊C4—22no—
100C2×C7×C22⋊C4—22no—
101C2×C7×C22⋊C4—22no—
102C2×C42×C7—21no—
103C2×C42×C7C2311yescharacteristic
104C2×C42×C7C2311yescharacteristic
105C2×C42×C7C2311yescharacteristic
106C7×C4⋊C8—22no—
107C7×#102—24no—
108C7×#102—24no—
109C7×#102—24no—
110C7×#102—24no—
111C7×#97—22no—
112C7×#95—21no—
113C2×#554C2×C711yescharacteristic, complemented
114C2×#490C2×C711yescharacteristic, complemented
115C2×#459C2×C711yescharacteristic
116#2548C2×C711yescharacteristic, complemented
117#2025C2×C712yescomplemented
118#2023C2×C712yescomplemented
119#1989C2×C711yescharacteristic, complemented
120#1985C2×C711yescharacteristic, complemented
121#1879C2×C712yescomplemented
122#1879C2×C712yescomplemented
123#1762C2×C711yescharacteristic, complemented
124C24×C7—21no—
125C24×C7—22no—
126C2×C7×Q8—24no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—41no—
130C2×C7×Q8—82no—
131C2×C7×D4—21no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—82no—
144C2×C7×D4C2×D412yes—
145C2×C7×D4#4114yes—
146C22×C4×C7—24no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—82no—
156C22×C4×C7—82no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C7×D8—48no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—24no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4C2×D412yes—
185C7×C4⋊C4C2×D412yes—
186C7×C4⋊C4C2×D412yes—
187C7×C22⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—44no—
192C7×C22⋊C4—82no—
193C7×C22⋊C4—82no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—42no—
199C42×C7C2×D412yes—
200C42×C7C2×D412yes—
201C22×D8C22×C712yes—
202#554—24no—
203#554—24no—
204C2×C4⋊Q8C22×C711yescharacteristic
205C2×#127C22×C711yescharacteristic
206C2×#125C22×C711yescharacteristic
207C2×#125C22×C711yescharacteristic
208C2×#124—21no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8—21no—
211C2×C22⋊Q8—21no—
212C2×C4×Q8C22×C711yescharacteristic
213C2×C4×D4C22×C711yescharacteristic
214C2×C4×D4C22×C711yescharacteristic
215C2×C4×D4C22×C711yescharacteristic
216C22×C4⋊C4—21no—
217#490—24no—
218#459—24no—
219C2×C4⋊C8C22×C711yescharacteristic
220C2×#102C22×C712yes—
221C2×#102C22×C712yes—
222C2×#102C22×C712yes—
223C2×#102C22×C712yes—
224C2×#97C22×C711yescharacteristic
225#394—21no—
226#394—21no—
227#390—21no—
228#390C22×C712yes—
229#390C22×C712yes—
230C4×C4⋊C4C22×C711yescharacteristic
231#340C22×C712yes—
232#340C22×C712yes—
233#340C22×C712yes—
234#336C22×C712yes—
235#336C22×C712yes—
236C23×C7—22no—
237C23×C7—24no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7C22×D411yescharacteristic
249C7×Q8—44no—
250C7×Q8—84no—
251C7×D4—22no—
252C7×D4—24no—
253C7×D4—42no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—42no—
257C7×D4—84no—
258C7×D4—84no—
259C7×D4#12414yescomplemented
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—48no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
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—84no—
285C2×C4×C7#12012yes—
286C2×C4×C7#12012yes—
287C2×C4×C7AGL(1,ℤ/8)12yes—
288C2×C4×C7C22×D411yescharacteristic
289C2×C4×C7C22×D411yescharacteristic
290C2×C4×C7C2×#4112yes—
291C2×C4×C72−1+412yes—
292C7×C8—42no—
293C7×C8—42no—
294C22×Q8—21no—
295C22×Q8C7×D411yescharacteristic
296C22×D4—21no—
297C22×D4—21no—
298C22×D4C23×C712yes—
299C22×D4C23×C711yescharacteristic
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—21no—
303C2×D8—24no—
304C2×D8—24no—
305C2×D8—24no—
306C22×C8C23×C712yes—
307C22×C8C23×C712yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#127—22no—
311#127—22no—
312#125—22no—
313#125—22no—
314#125—22no—
315#125C7×D414yes—
316#124—44no—
317C22⋊Q8—28no—
318C22⋊Q8—28no—
319C22⋊Q8—28no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4C7×D414yes—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C7×D411yescharacteristic
335C2×C4⋊C4C7×D411yescharacteristic
336C2×C4⋊C4C7×D411yescharacteristic
337C2×C4⋊C4C23×C712yes—
338C2×C4⋊C4C23×C711yescharacteristic
339C2×C4⋊C4C23×C711yescharacteristic
340C2×C4⋊C4C23×C711yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C42—21no—
349C2×C42C23×C711yescharacteristic
350C2×C42C23×C711yescharacteristic
351C2×C42C23×C711yescharacteristic
352C4⋊C8—22no—
353#102—24no—
354#102—24no—
355#102—24no—
356#102—24no—
357#97—22no—
358#95—21no—
359C22×C7—24no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—82no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7—82no—
368C22×C7—82no—
369C22×C7—82no—
370C22×C7—82no—
371C22×C7C2×#12011yescharacteristic
372C22×C7#54812yes—
373C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C7#57511yescharacteristic
375C22×C7#57612yes—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7—41no—
380C4×C7—42no—
381C4×C7—41no—
382C4×C7—41no—
383C4×C7—42no—
384C4×C7—42no—
385C4×C7—82no—
386C4×C7C2×#12411yescharacteristic
387C4×C7#53611yescharacteristic
388C4×C7#54812yes—
389C24—21no—
390C24—22no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—41no—
395C2×Q8—82no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—82no—
409C2×D4C2×C7×D412yes—
410C2×D4C7×#4114yes—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C24×C711yescharacteristic
429D8—48no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4C2×C7×D412yes—
450C4⋊C4C2×C7×D412yes—
451C4⋊C4C2×C7×D412yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—42no—
464C42C2×C7×D412yes—
465C42C2×C7×D412yes—
466C2×C7—44no—
467C2×C7—162no—
468C2×C7#243611yescharacteristic
469C2×C7#300611yescharacteristic
470C2×C7#300811yescharacteristic
471C2×C7#336612yes—
472C2×C7#337012yes—
473C23—22no—
474C23—24no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23C22×C7×D411yescharacteristic
486Q8—44no—
487Q8—84no—
488D4—22no—
489D4—24no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—84no—
495D4—84no—
496D4C7×#12414yescomplemented
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—84no—
521C2×C4—84no—
522C2×C4C7×#12012yes—
523C2×C4C7×#12012yes—
524C2×C4C7×AGL(1,ℤ/8)12yes—
525C2×C4C22×C7×D411yescharacteristic
526C2×C4C22×C7×D411yescharacteristic
527C2×C4C2×C7×#4112yes—
528C2×C4C7×2−1+412yes—
529C8—42no—
530C8—42no—
531C7#2594611yescharacteristic, direct product factor, Sylow
532C22—24no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22C2×C7×#12011yescharacteristic
545C22C7×#54812yes—
546C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
547C22C7×#57511yescharacteristic
548C22C7×#57612yes—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—82no—
559C4C2×C7×#12411yescharacteristic
560C4C7×#53611yescharacteristic
561C4C7×#54812yes—
562C2—44no—
563C2—162no—
564C2C7×#243611yescharacteristic
565C2C7×#300611yescharacteristic
566C2C7×#300811yescharacteristic
567C2C7×#336612yes—
568C2C7×#337012yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#554C2semidirect
2. C2×C7×#490C2semidirect
4. C7×#2548C2semidirect
5. C7×#2025C2semidirect
6. C7×#2023C2semidirect
7. C7×#1989C2semidirect
8. C7×#1985C2semidirect
9. C7×#1879C2semidirect
10. C7×#1879C2semidirect
11. C7×#1762C2semidirect
47. #25946C7direct (N × H)
113. C2×#554C2×C7semidirect
114. C2×#490C2×C7semidirect
116. #2548C2×C7semidirect
117. #2025C2×C7semidirect
118. #2023C2×C7semidirect
119. #1989C2×C7semidirect
120. #1985C2×C7semidirect
121. #1879C2×C7semidirect
122. #1879C2×C7semidirect
123. #1762C2×C7semidirect
259. C7×D4#124semidirect
496. D4C7×#124semidirect
531. C7#25946direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5542
C2×C7×#4902
C2×C7×#4592
C7×#25482
C7×#20252
C7×#20232
C7×#19892
C7×#19852
C7×#18792
C7×#17622
#259467