Finite Group Databasebeta

← C7×#24327 #13201543

Subgroups — C7×#24327 #13201543

Order 1792.

1146 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic
4C7×#2538C211yesmaximal, characteristic, complemented
5C7×#2059C212yesmaximal, complemented
6C7×#2025C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1984C211yesmaximal, characteristic, complemented
9C7×#1879C212yesmaximal
10C7×#1864C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×D8C2211yescharacteristic
13C7×#554—24no—
14C7×#554—24no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8—22no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2212yes—
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4—21no—
25C7×#476—24no—
26C4×C7×D8—24no—
27C2×C7×#107C2211yescharacteristic
28C2×C7×#106C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C7×#102C2211yescharacteristic
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C4×C7×C8C2211yescharacteristic
35C7×#396—22no—
36C7×#390C2212yes—
37C7×#389C2212yes—
38C7×#384C2211yescharacteristic
39C7×#380—21no—
40C7×#340C2212yes—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#336C2212yes—
44C7×#336C2212yes—
45#24327C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—22no—
52C23×C4×C7—21no—
53C2×C7×D8—22no—
54C2×C7×D8—22no—
55C2×C7×D8Q814yescomplemented
56C22×C7×C8C2311yescharacteristic
57C22×C7×C8C2312yes—
58C22×C7×C8C2311yescharacteristic
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8D414yes—
61C7×#125—24no—
62C7×#125—24no—
63C7×C22⋊Q8—28no—
64C7×C22⋊Q8—216no—
65C7×C22⋊Q8—44no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×D4—24no—
69C4×C7×D4—24no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—22no—
73C4×C7×D4D414yes—
74C2×C7×C4⋊C4—28no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4D411yescharacteristic
82C2×C7×C4⋊C4D412yes—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—22no—
92C2×C42×C7C2312yes—
93C2×C42×C7C2311yescharacteristic
94C7×#107—22no—
95C7×#106—24no—
96C7×C4⋊C8—22no—
97C7×#102—22no—
98C7×#102—24no—
99C7×#102—24no—
100C7×#102—22no—
101C4×C7×C8—22no—
102C7×#95—22no—
103C2×#554C2×C711yescharacteristic, complemented
104C2×#476C2×C711yescharacteristic, complemented
105C2×C4×D8C2×C711yescharacteristic
106#2538C2×C711yescharacteristic, complemented
107#2059C2×C712yescomplemented
108#2025C2×C712yescomplemented
109#1990C2×C711yescharacteristic, complemented
110#1984C2×C711yescharacteristic, complemented
111#1879C2×C712yes—
112#1864C2×C712yescomplemented
113#1802C2×C711yescharacteristic, complemented
114C24×C7—21no—
115C24×C7—21no—
116C2×C7×Q8—22no—
117C2×C7×Q8—24no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—81no—
121C2×C7×Q8—81no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4C2×D412yes—
131C2×C7×D4C2×Q812yes—
132C2×C7×D4C2×Q812yes—
133C2×C7×D4#4112yes—
134C22×C4×C7—21no—
135C22×C4×C7—22no—
136C22×C4×C7—28no—
137C22×C4×C7—24no—
138C22×C4×C7—21no—
139C22×C4×C7—22no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—82no—
146C22×C4×C7—82no—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C7×D8—28no—
153C2×C7×C8—21no—
154C2×C7×C8—21no—
155C2×C7×C8—21no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—22no—
161C2×C7×C8C2×Q812yes—
162C7×C4⋊C4—28no—
163C7×C4⋊C4—28no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—44no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—81no—
176C7×C4⋊C4C2×D414yes—
177C7×C4⋊C4C2×D412yes—
178C7×C22⋊C4—28no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—44no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—82no—
183C42×C7—22no—
184C42×C7—22no—
185C42×C7C2×D412yes—
186C42×C7C2×D412yes—
187C22×D8C22×C711yescharacteristic
188#554—24no—
189#554—24no—
190C2×C4⋊Q8C22×C711yescharacteristic
191C2×#125C22×C712yes—
192C2×C22⋊Q8—21no—
193C2×C22⋊Q8—21no—
194C2×C22⋊Q8—22no—
195C2×C4×Q8C22×C711yescharacteristic
196C2×C4×D4C22×C712yes—
197C2×C4×D4C22×C711yescharacteristic
198C2×C4×D4C22×C711yescharacteristic
199C22×C4⋊C4—21no—
200#476—24no—
201C4×D8—24no—
202C2×#107C22×C711yescharacteristic
203C2×#106C22×C712yes—
204C2×C4⋊C8C22×C711yescharacteristic
205C2×#102C22×C712yes—
206C2×#102C22×C711yescharacteristic
207C2×#102C22×C712yes—
208C2×#102C22×C711yescharacteristic
209C2×C4×C8C22×C711yescharacteristic
210#396—22no—
211#390C22×C712yes—
212#389C22×C712yes—
213#384C22×C711yescharacteristic
214#380—21no—
215#340C22×C712yes—
216#340C22×C712yes—
217#340C22×C712yes—
218#336C22×C712yes—
219#336C22×C712yes—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7C22×D411yescharacteristic
235C7×Q8—44no—
236C7×Q8—84no—
237C7×D4—22no—
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—22no—
241C7×D4—22no—
242C7×D4—41no—
243C7×D4—41no—
244C7×D4C22⋊Q814yescomplemented
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—48no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—88no—
267C2×C4×C7—84no—
268C2×C4×C7#12012yes—
269C2×C4×C7#12012yes—
270C2×C4×C7#13512yes—
271C2×C4×C7AGL(1,ℤ/8)12yes—
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×Q811yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C72−1+411yescharacteristic
277C2×C4×C72−1+411yescharacteristic
278C7×C8—22no—
279C7×C8—41no—
280C7×C8—42no—
281C22×Q8—21no—
282C22×Q8C7×D411yescharacteristic
283C22×D4C23×C711yescharacteristic
284C22×D4C23×C711yescharacteristic
285C23×C4—21no—
286C23×C4—22no—
287C23×C4—21no—
288C2×D8—22no—
289C2×D8—22no—
290C2×D8C7×Q814yescomplemented
291C22×C8C23×C711yescharacteristic
292C22×C8C23×C712yes—
293C22×C8C23×C711yescharacteristic
294C4⋊Q8—22no—
295C4⋊Q8C7×D414yes—
296#125—24no—
297#125—24no—
298C22⋊Q8—28no—
299C22⋊Q8—216no—
300C22⋊Q8—44no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4C7×D414yes—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C7×D411yescharacteristic
317C2×C4⋊C4C7×D412yes—
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C4⋊C4C23×C712yes—
320C2×C4⋊C4C23×C711yescharacteristic
321C2×C4⋊C4C23×C712yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C42C23×C712yes—
328C2×C42C23×C711yescharacteristic
329#107—22no—
330#106—24no—
331C4⋊C8—22no—
332#102—22no—
333#102—24no—
334#102—22no—
335#102—24no—
336C4×C8—22no—
337#95—22no—
338C22×C7—24no—
339C22×C7—42no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7C2×#12011yescharacteristic
353C22×C7#54811yescharacteristic
354C22×C7D4×Q811yescharacteristic
355C22×C7C2×#13511yescharacteristic
356C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
357C22×C7#57611yescharacteristic
358C22×C7#57711yescharacteristic
359C4×C7—22no—
360C4×C7—42no—
361C4×C7—42no—
362C4×C7—42no—
363C4×C7—44no—
364C4×C7—82no—
365C4×C7—81no—
366C4×C7#44914yes—
367C4×C7C2×C22⋊Q811yescharacteristic
368C4×C7#53611yescharacteristic
369C4×C7#54811yescharacteristic
370C4×C7D4×Q811yescharacteristic
371C24—21no—
372C24—21no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4C2×C7×D412yes—
388C2×D4C2×C7×Q812yes—
389C2×D4C2×C7×Q812yes—
390C2×D4C7×#4112yes—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4C2×C7×D412yes—
405C22×C4C2×C7×D412yes—
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C24×C711yescharacteristic
409D8—28no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8C2×C7×Q812yes—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—44no—
431C4⋊C4—42no—
432C4⋊C4—81no—
433C4⋊C4C2×C7×D414yes—
434C4⋊C4C2×C7×D412yes—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—82no—
440C42—22no—
441C42—22no—
442C42C2×C7×D412yes—
443C42C2×C7×D412yes—
444C2×C7—44no—
445C2×C7—82no—
446C2×C7#240911yescharacteristic
447C2×C7C2×#44911yescharacteristic
448C2×C7#301211yescharacteristic
449C2×C7#336011yescharacteristic
450C2×C7#336611yescharacteristic
451C2×C7#338011yescharacteristic
452C2×C7#338411yescharacteristic
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23C22×C7×D411yescharacteristic
468Q8—44no—
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—41no—
476D4—41no—
477D4C7×C22⋊Q814yescomplemented
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—88no—
500C2×C4—84no—
501C2×C4C7×#12012yes—
502C2×C4C7×#12012yes—
503C2×C4C7×#13512yes—
504C2×C4C7×AGL(1,ℤ/8)12yes—
505C2×C4C22×C7×D411yescharacteristic
506C2×C4C22×C7×D411yescharacteristic
507C2×C4C22×C7×Q811yescharacteristic
508C2×C4C2×C7×#4111yescharacteristic
509C2×C4C7×2−1+411yescharacteristic
510C2×C4C7×2−1+411yescharacteristic
511C8—22no—
512C8—41no—
513C8—42no—
514C7#2432711yescharacteristic, direct product factor, Sylow
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22C2×C7×#12011yescharacteristic
530C22C7×#54811yescharacteristic
531C22C7×D4×Q811yescharacteristic
532C22C2×C7×#13511yescharacteristic
533C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
534C22C7×#57611yescharacteristic
535C22C7×#57711yescharacteristic
536C4—22no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—44no—
541C4—81no—
542C4—82no—
543C4C7×#44914yes—
544C4C2×C7×C22⋊Q811yescharacteristic
545C4C7×#53611yescharacteristic
546C4C7×#54811yescharacteristic
547C4C7×D4×Q811yescharacteristic
548C2—44no—
549C2—82no—
550C2C7×#240911yescharacteristic
551C2C2×C7×#44911yescharacteristic
552C2C7×#301211yescharacteristic
553C2C7×#336011yescharacteristic
554C2C7×#336611yescharacteristic
555C2C7×#338011yescharacteristic
556C2C7×#338411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D8C22
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×#476C2semidirect
4. C7×#2538C2semidirect
5. C7×#2059C2semidirect
6. C7×#2025C2semidirect
7. C7×#1990C2semidirect
8. C7×#1984C2semidirect
10. C7×#1864C2semidirect
11. C7×#1802C2semidirect
45. #24327C7direct (N × H)
55. C2×C7×D8Q8semidirect
103. C2×#554C2×C7semidirect
104. C2×#476C2×C7semidirect
106. #2538C2×C7semidirect
107. #2059C2×C7semidirect
108. #2025C2×C7semidirect
109. #1990C2×C7semidirect
110. #1984C2×C7semidirect
112. #1864C2×C7semidirect
113. #1802C2×C7semidirect
244. C7×D4C22⋊Q8semidirect
290. C2×D8C7×Q8semidirect
477. D4C7×C22⋊Q8semidirect
514. C7#24327direct (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×#4762
C2×C4×C7×D82
C7×#25382
C7×#20592
C7×#20252
C7×#19902
C7×#19842
C7×#18792
C7×#18642
C7×#18022
#243277