Finite Group Databasebeta

← C7×#22047 #13199263

Subgroups — C7×#22047 #13199263

Order 1792.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#448C211yesmaximal, characteristic
2C4×C7×#121C211yesmaximal, characteristic
3C2×C7×#391C211yesmaximal, characteristic, complemented
4C7×#1973C212yesmaximal, complemented
5C7×#1935C212yesmaximal, complemented
6C7×#1928C212yesmaximal, complemented
7C7×#1877C212yesmaximal, complemented
8C7×#1874C211yesmaximal, characteristic
9C7×#1852C211yesmaximal, characteristic, complemented
10C7×#1787C211yesmaximal, characteristic, complemented
11C7×#1749C211yesmaximal, characteristic, complemented
12C23×C7×Q8C2211yescharacteristic
13C22×C7×SD16C2212yes—
14C2×C7×#121C2211yescharacteristic
15C2×C4×C7×D4—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C7×#448C418yescomplemented
20C2×C7×#107C2212yes—
21C2×C7×#103C2212yes—
22C2×C7×#103C2212yes—
23C2×C7×#103C2212yes—
24C2×C7×#102C2212yes—
25C2×C7×C22⋊C8C2211yescharacteristic
26C2×C7×C22⋊C8C2211yescharacteristic
27C2×C7×#97C2212yes—
28C7×#391—21no—
29C7×#391—21no—
30C7×#391C2212yes—
31C7×#391C2212yes—
32C7×#390C2211yescharacteristic
33C7×#389C2211yescharacteristic
34C7×#388—21no—
35C7×#386—21no—
36C7×#380C2211yescharacteristic
37C4×C7×C4⋊C4C2211yescharacteristic
38C4×C7×C22⋊C4—21no—
39C2×C7×#95—21no—
40C2×C7×#95—21no—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#336C2212yes—
44#22047C711yesmaximal, characteristic, direct product factor, Sylow
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8—42no—
49C22×C7×Q8C2×C414yescomplemented
50C22×C7×Q8C2312yes—
51C22×C7×D4—21no—
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C2×C7×SD16—42no—
60C2×C7×SD16—42no—
61C2×C7×SD16C2×C418yescomplemented
62C22×C7×C8C2312yes—
63C22×C7×C8C2312yes—
64C7×#121—22no—
65C7×#121C2×C414yes—
66C4×C7×D4—42no—
67C4×C7×D4—44no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C42×C7—21no—
81C2×C42×C7—42no—
82C2×C42×C7D411yescharacteristic
83C2×C42×C7D411yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C7×#107—42no—
87C7×#103—42no—
88C7×#103—42no—
89C7×#103C2×C418yescomplemented
90C7×#102—42no—
91C7×C22⋊C8—22no—
92C7×C22⋊C8C2×C414yescomplemented
93C7×#97—42no—
94C7×#95—22no—
95C7×#95—22no—
96C7×#95—22no—
97C7×#95—22no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C2×#448C2×C711yescharacteristic
102C4×#121C2×C711yescharacteristic
103C2×#391C2×C711yescharacteristic, complemented
104#1973C2×C712yescomplemented
105#1935C2×C712yescomplemented
106#1928C2×C712yescomplemented
107#1877C2×C712yescomplemented
108#1874C2×C711yescharacteristic
109#1852C2×C711yescharacteristic, complemented
110#1787C2×C711yescharacteristic, complemented
111#1749C2×C711yescharacteristic, complemented
112C24×C7—21no—
113C24×C7C2×D411yescharacteristic
114C2×C7×Q8—24no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—48no—
121C2×C7×Q8—42no—
122C2×C7×Q8—42no—
123C2×C7×Q8—48no—
124C2×C7×Q8C22×C414yes—
125C2×C7×D4—24no—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—42no—
130C2×C7×D4—42no—
131C2×C7×D4C22×C412yes—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—24no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—41no—
144C22×C4×C7—41no—
145C22×C4×C7—41no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7C22×C412yes—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7#4112yes—
161C22×C4×C7C2411yescharacteristic
162C7×SD16—48no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8C22×C414yes—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4C22×C412yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C42×C7—22no—
183C42×C7—22no—
184C42×C7—41no—
185C42×C7—41no—
186C42×C7—42no—
187C23×Q8C22×C711yescharacteristic
188C22×SD16C22×C712yes—
189C2×#121C22×C711yescharacteristic
190C2×C4×D4—21no—
191C2×C4×D4C22×C711yescharacteristic
192C22×C4⋊C4C22×C711yescharacteristic
193C22×C42C22×C711yescharacteristic
194#448C4×C718yescomplemented
195C2×#107C22×C712yes—
196C2×#103C22×C712yes—
197C2×#103C22×C712yes—
198C2×#103C22×C712yes—
199C2×#102C22×C712yes—
200C2×C22⋊C8C22×C711yescharacteristic
201C2×C22⋊C8C22×C711yescharacteristic
202C2×#97C22×C712yes—
203#391—21no—
204#391—21no—
205#391C22×C712yes—
206#391C22×C712yes—
207#390C22×C711yescharacteristic
208#389C22×C711yescharacteristic
209#388—21no—
210#386—21no—
211#380C22×C711yescharacteristic
212C4×C4⋊C4C22×C711yescharacteristic
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—21no—
216#340C22×C712yes—
217#340C22×C712yes—
218#336C22×C712yes—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—21no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7C4×D412yes—
231C23×C7AGL(1,ℤ/8)12yes—
232C23×C7#13712yes—
233C23×C7C22×D411yescharacteristic
234C7×Q8—24no—
235C7×Q8—24no—
236C7×Q8—48no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×D4—24no—
240C7×D4—48no—
241C7×D4—42no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—22no—
249C2×C4×C7—24no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—44no—
260C2×C4×C7—44no—
261C2×C4×C7—48no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—84no—
276C2×C4×C7—84no—
277C2×C4×C7C4×D412yes—
278C2×C4×C7#13712yes—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7H(5,𝔽2)12yes—
282C2×C4×C7H(5,𝔽2)12yes—
283C7×C8—24no—
284C7×C8—42no—
285C22×Q8—21no—
286C22×Q8—22no—
287C22×Q8—42no—
288C22×Q8—42no—
289C22×Q8C2×C4×C714yescomplemented
290C22×Q8C23×C712yes—
291C22×D4—21no—
292C22×D4C23×C711yescharacteristic
293C23×C4—21no—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4C7×D411yescharacteristic
297C23×C4C7×D411yescharacteristic
298C23×C4C23×C711yescharacteristic
299C2×SD16—42no—
300C2×SD16—42no—
301C2×SD16C2×C4×C718yescomplemented
302C22×C8C23×C712yes—
303C22×C8C23×C712yes—
304#121—22no—
305#121C2×C4×C714yes—
306C4×D4—42no—
307C4×D4—44no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4C23×C712yes—
312C2×C4⋊C4C23×C711yescharacteristic
313C2×C4⋊C4C23×C711yescharacteristic
314C2×C4⋊C4C23×C711yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C42—21no—
321C2×C42—42no—
322C2×C42C7×D411yescharacteristic
323C2×C42C7×D411yescharacteristic
324C2×C42C23×C711yescharacteristic
325C2×C42C23×C711yescharacteristic
326#107—42no—
327#103—42no—
328#103—42no—
329#103C2×C4×C718yescomplemented
330#102—42no—
331C22⋊C8—22no—
332C22⋊C8C2×C4×C714yescomplemented
333#97—42no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—21no—
340#95—21no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7—81no—
354C22×C7#44014yes—
355C22×C7C2×C4×D411yescharacteristic
356C22×C7#53411yescharacteristic
357C22×C7#53511yescharacteristic
358C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C7C2×#13711yescharacteristic
360C22×C7C2×#13711yescharacteristic
361C22×C7#57511yescharacteristic
362C4×C7—22no—
363C4×C7—22no—
364C4×C7—44no—
365C4×C7—41no—
366C4×C7—42no—
367C4×C7—41no—
368C4×C7—41no—
369C4×C7—81no—
370C4×C7—81no—
371C4×C7—82no—
372C4×C7#51812yes—
373C4×C7#53812yes—
374C24—21no—
375C24C2×C7×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8C22×C4×C714yes—
387C2×D4—24no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C22×C4×C712yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C22×C4×C712yes—
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C2×C7×D411yescharacteristic
420C22×C4C2×C7×D411yescharacteristic
421C22×C4C2×C7×D411yescharacteristic
422C22×C4C7×#4112yes—
423C22×C4C24×C711yescharacteristic
424SD16—48no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8C22×C4×C714yes—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4C22×C4×C712yes—
439C22⋊C4—24no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C42—22no—
445C42—22no—
446C42—41no—
447C42—41no—
448C42—42no—
449C2×C7—24no—
450C2×C7—82no—
451C2×C7#232111yescharacteristic
452C2×C7C2×#44011yescharacteristic
453C2×C7C4×#13711yescharacteristic
454C2×C7#319311yescharacteristic
455C2×C7#319611yescharacteristic
456C2×C7#321311yescharacteristic
457C2×C7#321811yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C4×C7×D412yes—
470C23C7×AGL(1,ℤ/8)12yes—
471C23C7×#13712yes—
472C23C22×C7×D411yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—48no—
476Q8—42no—
477Q8—42no—
478D4—24no—
479D4—48no—
480D4—42no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4C4×C7×D412yes—
517C2×C4C7×#13712yes—
518C2×C4C23×C4×C711yescharacteristic
519C2×C4C2×C7×#4111yescharacteristic
520C2×C4C7×H(5,𝔽2)12yes—
521C2×C4C7×H(5,𝔽2)12yes—
522C8—24no—
523C8—42no—
524C7#2204711yescharacteristic, direct product factor, Sylow
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22C7×#44014yes—
539C22C2×C4×C7×D411yescharacteristic
540C22C7×#53411yescharacteristic
541C22C7×#53511yescharacteristic
542C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
543C22C2×C7×#13711yescharacteristic
544C22C2×C7×#13711yescharacteristic
545C22C7×#57511yescharacteristic
546C4—22no—
547C4—22no—
548C4—41no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—44no—
553C4—81no—
554C4—82no—
555C4—81no—
556C4C7×#51812yes—
557C4C7×#53812yes—
558C2—24no—
559C2—82no—
560C2C7×#232111yescharacteristic
561C2C2×C7×#44011yescharacteristic
562C2C4×C7×#13711yescharacteristic
563C2C7×#319311yescharacteristic
564C2C7×#319611yescharacteristic
565C2C7×#321311yescharacteristic
566C2C7×#321811yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C7×#391C2semidirect
4. C7×#1973C2semidirect
5. C7×#1935C2semidirect
6. C7×#1928C2semidirect
7. C7×#1877C2semidirect
9. C7×#1852C2semidirect
10. C7×#1787C2semidirect
11. C7×#1749C2semidirect
19. C7×#448C4semidirect
44. #22047C7direct (N × H)
49. C22×C7×Q8C2×C4semidirect
61. C2×C7×SD16C2×C4semidirect
89. C7×#103C2×C4semidirect
92. C7×C22⋊C8C2×C4semidirect
103. C2×#391C2×C7semidirect
104. #1973C2×C7semidirect
105. #1935C2×C7semidirect
106. #1928C2×C7semidirect
107. #1877C2×C7semidirect
109. #1852C2×C7semidirect
110. #1787C2×C7semidirect
111. #1749C2×C7semidirect
194. #448C4×C7semidirect
289. C22×Q8C2×C4×C7semidirect
301. C2×SD16C2×C4×C7semidirect
329. #103C2×C4×C7semidirect
332. C22⋊C8C2×C4×C7semidirect
524. C7#22047direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4482
C4×C7×#1212
C2×C7×#3912
C7×#19732
C7×#19352
C7×#19282
C7×#18772
C7×#18742
C7×#18522
C7×#17872
C7×#17492
#220477