Finite Group Databasebeta

← C7×#24709 #13201925

Subgroups — C7×#24709 #13201925

Order 1792.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#474C211yesmaximal, characteristic, complemented
2C2×C7×#432C211yesmaximal, characteristic, complemented
3C7×#2857C211yesmaximal, characteristic, complemented
4C4×C7×#121C211yesmaximal, characteristic
5C7×#2076C212yesmaximal, complemented
6C7×#2034C212yesmaximal
7C7×#1988C211yesmaximal, characteristic, complemented
8C7×#1948C211yesmaximal, characteristic, complemented
9C7×#1944C212yesmaximal, complemented
10C7×#1936C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C2×C7×C22⋊Q8—21no—
14C2×C7×#121C2211yescharacteristic
15C2×C4×C7×D4—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×C42×C7C2211yescharacteristic
18C7×#474—24no—
19C7×#432—24no—
20C2×C7×#107C2212yes—
21C2×C7×#106C2212yes—
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×#102C2212yes—
25C2×C7×#102C2212yes—
26C2×C7×C22⋊C8C2212yes—
27C2×C7×#97C2212yes—
28C7×#391—21no—
29C7×#390C2211yescharacteristic
30C7×#389—21no—
31C7×#389C2211yescharacteristic
32C7×#389C2212yes—
33C7×#388—21no—
34C7×#386—21no—
35C7×#384C2211yescharacteristic
36C7×#380—21no—
37C7×#380C2212yes—
38C7×#380C2211yescharacteristic
39C4×C7×C4⋊C4C2211yescharacteristic
40C4×C7×C22⋊C4—21no—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24709C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×D4—21no—
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C22×C7×C8C2312yes—
54C22×C7×C8C2312yes—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—22no—
57C7×C22⋊Q8—44no—
58C7×#121—22no—
59C7×#121—22no—
60C4×C7×D4—22no—
61C4×C7×D4—22no—
62C4×C7×D4—44no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2312yes—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—22no—
80C2×C42×C7—22no—
81C2×C42×C7—21no—
82C2×C42×C7D411yescharacteristic
83C2×C42×C7D411yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C7×#107—24no—
87C7×#106—24no—
88C7×C4⋊C8—22no—
89C7×C4⋊C8D414yescomplemented
90C7×#102—24no—
91C7×#102—24no—
92C7×C22⋊C8—24no—
93C7×#97—24no—
94C7×#95—21no—
95C7×#95—21no—
96C7×#95—21no—
97C7×#95—22no—
98C7×#95—22no—
99C2×#474C2×C711yescharacteristic, complemented
100C2×#432C2×C711yescharacteristic, complemented
101#2857C2×C711yescharacteristic, complemented
102C4×#121C2×C711yescharacteristic
103#2076C2×C712yescomplemented
104#2034C2×C712yes—
105#1988C2×C711yescharacteristic, complemented
106#1948C2×C711yescharacteristic, complemented
107#1944C2×C712yescomplemented
108#1936C2×C712yescomplemented
109#1802C2×C711yescharacteristic, complemented
110C24×C7—21no—
111C24×C7C2×D411yescharacteristic
112C2×C7×Q8—42no—
113C2×C7×Q8—42no—
114C2×C7×Q8—42no—
115C2×C7×D4—24no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—42no—
122C2×C7×D4—42no—
123C2×C7×D4#4112yes—
124C22×C4×C7—22no—
125C22×C4×C7—21no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—41no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C2×C7×C8—22no—
151C2×C7×C8—22no—
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C2×C7×C8C2×D414yes—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—42no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4—44no—
168C7×C4⋊C4—42no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4#4112yes—
171C7×C22⋊C4—24no—
172C7×C22⋊C4—42no—
173C7×C22⋊C4—42no—
174C7×C22⋊C4—42no—
175C7×C22⋊C4—44no—
176C7×C22⋊C4—42no—
177C42×C7—22no—
178C42×C7—22no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—42no—
182C42×C7C2×D412yes—
183C42×C7#4112yes—
184C2×C4⋊Q8C22×C711yescharacteristic
185C2×C22⋊Q8—21no—
186C2×#121C22×C711yescharacteristic
187C2×C4×D4—21no—
188C2×C4×D4C22×C711yescharacteristic
189C22×C42C22×C711yescharacteristic
190#474—24no—
191#432—24no—
192C2×#107C22×C712yes—
193C2×#106C22×C712yes—
194C2×C4⋊C8C22×C711yescharacteristic
195C2×C4⋊C8C22×C711yescharacteristic
196C2×#102C22×C712yes—
197C2×#102C22×C712yes—
198C2×C22⋊C8C22×C712yes—
199C2×#97C22×C712yes—
200#391—21no—
201#390C22×C711yescharacteristic
202#389—21no—
203#389C22×C712yes—
204#389C22×C711yescharacteristic
205#388—21no—
206#386—21no—
207#384C22×C711yescharacteristic
208#380—21no—
209#380C22×C711yescharacteristic
210#380C22×C712yes—
211C4×C4⋊C4C22×C711yescharacteristic
212C4×C22⋊C4—21no—
213#340C22×C712yes—
214#340C22×C712yes—
215#340C22×C712yes—
216#340C22×C712yes—
217C23×C7—21no—
218C23×C7—21no—
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×C7#13712yes—
231C23×C7C22×D411yescharacteristic
232C7×Q8—84no—
233C7×D4—22no—
234C7×D4—22no—
235C7×D4—48no—
236C7×D4—41no—
237C7×D4—41no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—22no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—44no—
251C2×C4×C7—41no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—44no—
258C2×C4×C7—41no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—42no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—84no—
271C2×C4×C7#12112yes—
272C2×C4×C7#12112yes—
273C2×C4×C7AGL(1,ℤ/8)12yes—
274C2×C4×C7AGL(1,ℤ/8)12yes—
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C72−1+411yescharacteristic
280C2×C4×C72−1+411yescharacteristic
281C7×C8—24no—
282C7×C8—42no—
283C22×Q8—21no—
284C22×D4—21no—
285C22×D4C23×C711yescharacteristic
286C23×C4—21no—
287C23×C4C7×D411yescharacteristic
288C23×C4C7×D411yescharacteristic
289C23×C4C23×C711yescharacteristic
290C22×C8C23×C712yes—
291C22×C8C23×C712yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C22⋊Q8—44no—
295#121—22no—
296#121—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C23×C711yescharacteristic
306C2×C4⋊C4C23×C712yes—
307C2×C4⋊C4C23×C711yescharacteristic
308C2×C4⋊C4C23×C712yes—
309C2×C4⋊C4C23×C711yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42C7×D411yescharacteristic
320C2×C42C7×D411yescharacteristic
321C2×C42C23×C711yescharacteristic
322C2×C42C23×C711yescharacteristic
323#107—24no—
324#106—24no—
325C4⋊C8—22no—
326C4⋊C8C7×D414yescomplemented
327#102—24no—
328#102—24no—
329C22⋊C8—24no—
330#97—24no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—22no—
335#95—22no—
336C22×C7—22no—
337C22×C7—22no—
338C22×C7—22no—
339C22×C7—41no—
340C22×C7—41no—
341C22×C7—41no—
342C22×C7—41no—
343C22×C7—42no—
344C22×C7—42no—
345C22×C7—42no—
346C22×C7—81no—
347C22×C7—81no—
348C22×C7—81no—
349C22×C7—81no—
350C22×C7C2×#12111yescharacteristic
351C22×C7#52511yescharacteristic
352C22×C7#53611yescharacteristic
353C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
355C22×C7C2×#13711yescharacteristic
356C22×C7#57511yescharacteristic
357C4×C7—22no—
358C4×C7—42no—
359C4×C7—41no—
360C4×C7—41no—
361C4×C7—41no—
362C4×C7—82no—
363C4×C7—81no—
364C4×C7—81no—
365C4×C7—82no—
366C4×C7#46914yes—
367C4×C7#54711yescharacteristic
368C4×C7#54811yescharacteristic
369C4×C7#55211yescharacteristic
370C4×C7#55611yescharacteristic
371C24—21no—
372C24C2×C7×D411yescharacteristic
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4C7×#4112yes—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C24×C711yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8C2×C7×D414yes—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4C7×#4112yes—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—42no—
443C42C2×C7×D412yes—
444C42C7×#4112yes—
445C2×C7—42no—
446C2×C7—82no—
447C2×C7#241511yescharacteristic
448C2×C7C2×#46911yescharacteristic
449C2×C7#305611yescharacteristic
450C2×C7#311011yescharacteristic
451C2×C7#311511yescharacteristic
452C2×C7#323511yescharacteristic
453C2×C7#323911yescharacteristic
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23C7×#13712yes—
468C23C22×C7×D411yescharacteristic
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—48no—
473D4—41no—
474D4—41no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4C7×#12112yes—
509C2×C4C7×#12112yes—
510C2×C4C7×AGL(1,ℤ/8)12yes—
511C2×C4C7×AGL(1,ℤ/8)12yes—
512C2×C4C22×C7×D411yescharacteristic
513C2×C4C2×C7×#4111yescharacteristic
514C2×C4C2×C7×#4111yescharacteristic
515C2×C4C2×C7×#4111yescharacteristic
516C2×C4C7×2−1+411yescharacteristic
517C2×C4C7×2−1+411yescharacteristic
518C8—24no—
519C8—42no—
520C7#2470911yescharacteristic, direct product factor, Sylow
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22C2×C7×#12111yescharacteristic
536C22C7×#52511yescharacteristic
537C22C7×#53611yescharacteristic
538C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
539C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
540C22C2×C7×#13711yescharacteristic
541C22C7×#57511yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—81no—
548C4—81no—
549C4—82no—
550C4—82no—
551C4C7×#46914yes—
552C4C7×#54711yescharacteristic
553C4C7×#54811yescharacteristic
554C4C7×#55211yescharacteristic
555C4C7×#55611yescharacteristic
556C2—42no—
557C2—82no—
558C2C7×#241511yescharacteristic
559C2C2×C7×#46911yescharacteristic
560C2C7×#305611yescharacteristic
561C2C7×#311011yescharacteristic
562C2C7×#311511yescharacteristic
563C2C7×#323511yescharacteristic
564C2C7×#323911yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#474C2semidirect
2. C2×C7×#432C2semidirect
3. C7×#2857C2semidirect
5. C7×#2076C2semidirect
7. C7×#1988C2semidirect
8. C7×#1948C2semidirect
9. C7×#1944C2semidirect
10. C7×#1936C2semidirect
11. C7×#1802C2semidirect
45. #24709C7direct (N × H)
89. C7×C4⋊C8D4semidirect
99. C2×#474C2×C7semidirect
100. C2×#432C2×C7semidirect
101. #2857C2×C7semidirect
103. #2076C2×C7semidirect
105. #1988C2×C7semidirect
106. #1948C2×C7semidirect
107. #1944C2×C7semidirect
108. #1936C2×C7semidirect
109. #1802C2×C7semidirect
326. C4⋊C8C7×D4semidirect
520. C7#24709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4742
C2×C7×#4322
C7×#28572
C4×C7×#1212
C7×#20762
C7×#20342
C7×#19882
C7×#19482
C7×#19442
C7×#19362
C7×#18022
#247097