Finite Group Databasebeta

← C7×#20846 #13198062

Subgroups — C7×#20846 #13198062

Order 1792.

1630 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#117C211yesmaximal, characteristic
2C2×C7×#428C212yesmaximal, complemented
3C2×C7×#407C211yesmaximal, characteristic
4C2×C7×#380C211yesmaximal, characteristic
5C7×#1854C214yesmaximal, complemented
6C7×#1853C212yesmaximal, complemented
7C7×#1852C212yesmaximal, complemented
8C7×#1736C212yesmaximal, complemented
9C24×C4×C7C2211yescharacteristic
10C22×C7×M16C2212yes—
11C2×C7×#117—21no—
12C2×C7×#117—21no—
13C2×C7×#117—28no—
14C2×C7×#117C2212yes—
15C2×C7×#117C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4—21no—
21C22×C7×C22⋊C4—21no—
22C22×C42×C7—21no—
23C7×#428—28no—
24C2×C7×#107C2214yescomplemented
25C2×C7×#106C2214yescomplemented
26C7×#407C418yescomplemented
27C2×C7×C22⋊C8C2212yes—
28C2×C7×C22⋊C8C2212yes—
29C7×#380—21no—
30C7×#380—21no—
31C7×#380C2212yes—
32C7×#380C2212yes—
33C2×C7×#95—21no—
34C7×#340C2214yescomplemented
35C7×#340C2214yescomplemented
36#20846C711yesmaximal, characteristic, direct product factor, Sylow
37C25×C7D411yescharacteristic
38C23×C4×C7—24no—
39C23×C4×C7—24no—
40C23×C4×C7—21no—
41C23×C4×C7—21no—
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7C2×C414yes—
45C23×C4×C7D411yescharacteristic
46C23×C4×C7D411yescharacteristic
47C23×C4×C7D411yescharacteristic
48C23×C4×C7Q814yes—
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C2×C7×M16—24no—
53C2×C7×M16—24no—
54C2×C7×M16C2×C418yescomplemented
55C22×C7×C8C2314yes—
56C7×#117—28no—
57C7×#117—28no—
58C7×#117—24no—
59C7×#117—24no—
60C7×#117—24no—
61C7×#117D418yes—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—24no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—44no—
80C2×C42×C7—28no—
81C2×C42×C7—22no—
82C2×C42×C7—22no—
83C7×#107—28no—
84C7×#106—28no—
85C7×C22⋊C8C2×C418yescomplemented
86C7×C22⋊C8C2×C418yescomplemented
87C7×#95—22no—
88C7×#95—22no—
89C22×#117C2×C711yescharacteristic
90C2×#428C2×C712yescomplemented
91C2×#407C2×C711yescharacteristic
92C2×#380C2×C711yescharacteristic
93#1854C2×C714yescomplemented
94#1853C2×C712yescomplemented
95#1852C2×C712yescomplemented
96#1736C2×C712yescomplemented
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—41no—
102C24×C7—41no—
103C24×C7C4⋊C414yes—
104C24×C7C2×D411yescharacteristic
105C24×C7C2×D411yescharacteristic
106C24×C7C2×D411yescharacteristic
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—216no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—216no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—24no—
120C22×C4×C7—24no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—42no—
126C22×C4×C7—42no—
127C22×C4×C7—42no—
128C22×C4×C7—42no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7—42no—
132C22×C4×C7—42no—
133C22×C4×C7—42no—
134C22×C4×C7—42no—
135C22×C4×C7C4⋊C414yes—
136C22×C4×C7C4⋊C414yes—
137C22×C4×C7C4⋊C414yes—
138C22×C4×C7C22×C412yes—
139C22×C4×C7C22×C412yes—
140C22×C4×C7C22×C412yes—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D414yes—
145C22×C4×C7C2×Q812yes—
146C22×C4×C7C2×Q812yes—
147C22×C4×C7C2×Q812yes—
148C22×C4×C7#4114yes—
149C22×C4×C7C2411yescharacteristic
150C7×M16—216no—
151C2×C7×C8—24no—
152C2×C7×C8—24no—
153C2×C7×C8C22×C418yes—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—42no—
161C7×C4⋊C4—42no—
162C7×C4⋊C4C2×D414yes—
163C7×C4⋊C4C2×D414yes—
164C7×C22⋊C4—28no—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—44no—
167C7×C22⋊C4—44no—
168C42×C7—28no—
169C42×C7—28no—
170C24×C4C22×C711yescharacteristic
171C22×M16C22×C712yes—
172C2×#117—21no—
173C2×#117—21no—
174C2×#117—28no—
175C2×#117C22×C712yes—
176C2×#117C22×C712yes—
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C711yescharacteristic
179C22×C4⋊C4C22×C711yescharacteristic
180C22×C4⋊C4C22×C711yescharacteristic
181C22×C22⋊C4—21no—
182C22×C22⋊C4—21no—
183C22×C42—21no—
184#428—28no—
185C2×#107C22×C714yescomplemented
186C2×#106C22×C714yescomplemented
187#407C4×C718yescomplemented
188C2×C22⋊C8C22×C712yes—
189C2×C22⋊C8C22×C712yes—
190#380—21no—
191#380—21no—
192#380C22×C712yes—
193#380C22×C712yes—
194C2×#95—21no—
195#340C22×C714yescomplemented
196#340C22×C714yescomplemented
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—21no—
206C23×C7—21no—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—21no—
211C23×C7—28no—
212C23×C7—28no—
213C23×C7—44no—
214C23×C7—44no—
215C23×C7—44no—
216C23×C7—44no—
217C23×C7—44no—
218C23×C7C2×C4⋊C412yes—
219C23×C7C2×C4⋊C412yes—
220C23×C7C2×C4⋊C412yes—
221C23×C7#12012yes—
222C23×C7#12312yes—
223C23×C7AGL(1,ℤ/8)12yes—
224C23×C7#13712yes—
225C23×C7C22×D411yescharacteristic
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—28no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—28no—
236C2×C4×C7—28no—
237C2×C4×C7—24no—
238C2×C4×C7—24no—
239C2×C4×C7—48no—
240C2×C4×C7—48no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—84no—
252C2×C4×C7—84no—
253C2×C4×C7C2×C4⋊C412yes—
254C2×C4×C7C2×C4⋊C412yes—
255C2×C4×C7C2×C4⋊C412yes—
256C2×C4×C7C4×D418yes—
257C2×C4×C7#12012yes—
258C2×C4×C7C22⋊Q818yes—
259C2×C4×C7#12312yes—
260C2×C4×C7C23×C411yescharacteristic
261C2×C4×C7C22×D412yes—
262C2×C4×C7C22×Q811yescharacteristic
263C2×C4×C7C2×#4112yes—
264C7×C8—28no—
265C25C7×D411yescharacteristic
266C23×C4—24no—
267C23×C4—24no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×C4×C714yes—
273C23×C4C7×D411yescharacteristic
274C23×C4C7×D411yescharacteristic
275C23×C4C7×D411yescharacteristic
276C23×C4C7×Q814yes—
277C23×C4C23×C711yescharacteristic
278C23×C4C23×C711yescharacteristic
279C23×C4C23×C711yescharacteristic
280C2×M16—24no—
281C2×M16—24no—
282C2×M16C2×C4×C718yescomplemented
283C22×C8C23×C714yes—
284#117—28no—
285#117—28no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117C7×D418yes—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4C23×C712yes—
299C2×C4⋊C4C23×C712yes—
300C2×C4⋊C4C23×C712yes—
301C2×C4⋊C4C23×C712yes—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—44no—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—22no—
311#107—28no—
312#106—28no—
313C22⋊C8C2×C4×C718yescomplemented
314C22⋊C8C2×C4×C718yescomplemented
315#95—22no—
316#95—22no—
317C22×C7—22no—
318C22×C7—22no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—22no—
326C22×C7—22no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—22no—
330C22×C7—22no—
331C22×C7—22no—
332C22×C7—416no—
333C22×C7—41no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7—41no—
337C22×C7#38514yes—
338C22×C7#42814yescomplemented
339C22×C7C22×C4⋊C411yescharacteristic
340C22×C7C2×#12011yescharacteristic
341C22×C7C2×#12311yescharacteristic
342C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
343C22×C7C2×#13711yescharacteristic
344C22×C7#57511yescharacteristic
345C22×C7#57511yescharacteristic
346C4×C7—22no—
347C4×C7—22no—
348C4×C7—44no—
349C4×C7—44no—
350C4×C7—82no—
351C4×C7—82no—
352C4×C7#38514yes—
353C4×C7C2×C4×D412yes—
354C4×C7C2×C22⋊Q812yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—41no—
360C24—41no—
361C24C7×C4⋊C414yes—
362C24C2×C7×D411yescharacteristic
363C24C2×C7×D411yescharacteristic
364C24C2×C7×D411yescharacteristic
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—216no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4C7×C4⋊C414yes—
394C22×C4C7×C4⋊C414yes—
395C22×C4C7×C4⋊C414yes—
396C22×C4C22×C4×C712yes—
397C22×C4C22×C4×C712yes—
398C22×C4C22×C4×C712yes—
399C22×C4C2×C7×D411yescharacteristic
400C22×C4C2×C7×D411yescharacteristic
401C22×C4C2×C7×D411yescharacteristic
402C22×C4C2×C7×D414yes—
403C22×C4C2×C7×Q812yes—
404C22×C4C2×C7×Q812yes—
405C22×C4C2×C7×Q812yes—
406C22×C4C7×#4114yes—
407C22×C4C24×C711yescharacteristic
408M16—216no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8C22×C4×C718yes—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4C2×C7×D414yes—
421C4⋊C4C2×C7×D414yes—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C42—28no—
427C42—28no—
428C2×C7—24no—
429C2×C7—24no—
430C2×C7—42no—
431C2×C7C2×#38511yescharacteristic
432C2×C7C2×#42811yescharacteristic
433C2×C7#291111yescharacteristic
434C2×C7#300611yescharacteristic
435C2×C7#300711yescharacteristic
436C2×C7#309411yescharacteristic
437C2×C7#309511yescharacteristic
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—28no—
448C23—21no—
449C23—28no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23C2×C7×C4⋊C412yes—
460C23C2×C7×C4⋊C412yes—
461C23C2×C7×C4⋊C412yes—
462C23C7×#12012yes—
463C23C7×#12312yes—
464C23C7×AGL(1,ℤ/8)12yes—
465C23C7×#13712yes—
466C23C22×C7×D411yescharacteristic
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—28no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4C2×C7×C4⋊C412yes—
495C2×C4C2×C7×C4⋊C412yes—
496C2×C4C2×C7×C4⋊C412yes—
497C2×C4C4×C7×D418yes—
498C2×C4C7×#12012yes—
499C2×C4C7×C22⋊Q818yes—
500C2×C4C7×#12312yes—
501C2×C4C23×C4×C711yescharacteristic
502C2×C4C22×C7×D412yes—
503C2×C4C22×C7×Q811yescharacteristic
504C2×C4C2×C7×#4112yes—
505C8—28no—
506C7#2084611yescharacteristic, direct product factor, Sylow
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—416no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22C7×#38514yes—
528C22C7×#42814yescomplemented
529C22C22×C7×C4⋊C411yescharacteristic
530C22C2×C7×#12011yescharacteristic
531C22C2×C7×#12311yescharacteristic
532C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
533C22C2×C7×#13711yescharacteristic
534C22C7×#57511yescharacteristic
535C22C7×#57511yescharacteristic
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4C7×#38514yes—
543C4C2×C4×C7×D412yes—
544C4C2×C7×C22⋊Q812yes—
545C2—24no—
546C2—24no—
547C2—42no—
548C2C2×C7×#38511yescharacteristic
549C2C2×C7×#42811yescharacteristic
550C2C7×#291111yescharacteristic
551C2C7×#300611yescharacteristic
552C2C7×#300711yescharacteristic
553C2C7×#309411yescharacteristic
554C2C7×#309511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutatorC2×C4C23×C4×C7
Frattini, cosocleC22×C4C24×C7
prime radicalC25×C7D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#428C2semidirect
5. C7×#1854C2semidirect
6. C7×#1853C2semidirect
7. C7×#1852C2semidirect
8. C7×#1736C2semidirect
24. C2×C7×#107C22semidirect
25. C2×C7×#106C22semidirect
26. C7×#407C4semidirect
34. C7×#340C22semidirect
35. C7×#340C22semidirect
36. #20846C7direct (N × H)
54. C2×C7×M16C2×C4semidirect
85. C7×C22⋊C8C2×C4semidirect
86. C7×C22⋊C8C2×C4semidirect
90. C2×#428C2×C7semidirect
93. #1854C2×C7semidirect
94. #1853C2×C7semidirect
95. #1852C2×C7semidirect
96. #1736C2×C7semidirect
185. C2×#107C22×C7semidirect
186. C2×#106C22×C7semidirect
187. #407C4×C7semidirect
195. #340C22×C7semidirect
196. #340C22×C7semidirect
282. C2×M16C2×C4×C7semidirect
313. C22⋊C8C2×C4×C7semidirect
314. C22⋊C8C2×C4×C7semidirect
338. C22×C7#428semidirect
506. C7#20846direct (N × H)
528. C22C7×#428semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1172
C2×C7×#4282
C2×C7×#4072
C2×C7×#3802
C7×#18542
C7×#18532
C7×#18522
C7×#17362
#208467