Finite Group Databasebeta

← C7×#24760 #13201976

Subgroups — C7×#24760 #13201976

Order 1792.

960 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
1C2×C7×#470C212yesmaximal
2C2×C7×#462C211yesmaximal, characteristic, complemented
3C2×C7×#423C211yesmaximal, characteristic
4C7×#2857C211yesmaximal, characteristic
5C4×C7×C22⋊Q8C211yesmaximal, characteristic
6C7×#2066C212yesmaximal, complemented
7C7×#2056C212yesmaximal, complemented
8C7×#1981C211yesmaximal, characteristic, complemented
9C7×#1948C211yesmaximal, characteristic, complemented
10C7×#1894C212yesmaximal
11C7×#1803C211yesmaximal, characteristic, complemented
12C22×C7×Q16C2212yes—
13C22×C7×M16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C22⋊Q8C2212yes—
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C7×#470—28no—
20C7×#462—24no—
21C2×C7×#106C2212yes—
22C7×#423—24no—
23C2×C7×C4⋊C8C2211yescharacteristic
24C2×C7×C4⋊C8C2212yes—
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#103C2212yes—
27C2×C7×#103C2212yes—
28C2×C7×#103C2212yes—
29C2×C7×#103C2212yes—
30C7×#391C2211yescharacteristic
31C7×#391C2212yes—
32C7×#389—21no—
33C7×#389—21no—
34C7×#389C2211yescharacteristic
35C7×#387—21no—
36C7×#385—21no—
37C7×#384C2211yescharacteristic
38C7×#380—21no—
39C7×#380—21no—
40C7×#380C2211yescharacteristic
41C4×C7×C4⋊C4—21no—
42C4×C7×C4⋊C4C2211yescharacteristic
43C4×C7×C22⋊C4—21no—
44C7×#340C2212yes—
45C7×#340C2212yes—
46#24760C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8C2312yes—
48C22×C7×Q8C2311yescharacteristic
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7C2311yescharacteristic
52C2×C7×Q16—24no—
53C2×C7×Q16—24no—
54C2×C7×Q16—42no—
55C2×C7×M16—24no—
56C2×C7×M16—24no—
57C2×C7×M16—42no—
58C22×C7×C8C2312yes—
59C22×C7×C8C2312yes—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—24no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8D414yes—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—22no—
83C2×C42×C7—22no—
84C2×C42×C7—22no—
85C2×C42×C7—21no—
86C2×C42×C7—21no—
87C2×C42×C7D411yescharacteristic
88C2×C42×C7D411yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C42×C7C2311yescharacteristic
91C7×#106—24no—
92C7×C4⋊C8—24no—
93C7×C4⋊C8—22no—
94C7×C4⋊C8D414yes—
95C7×#103—24no—
96C7×#103—24no—
97C7×#103—24no—
98C7×#103—42no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—22no—
105C7×#95—22no—
106C2×#470C2×C712yes—
107C2×#462C2×C711yescharacteristic, complemented
108C2×#423C2×C711yescharacteristic
109#2857C2×C711yescharacteristic
110C4×C22⋊Q8C2×C711yescharacteristic
111#2066C2×C712yescomplemented
112#2056C2×C712yescomplemented
113#1981C2×C711yescharacteristic, complemented
114#1948C2×C711yescharacteristic, complemented
115#1894C2×C712yes—
116#1803C2×C711yescharacteristic, complemented
117C24×C7C2×D411yescharacteristic
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—21no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—21no—
126C2×C7×Q8—22no—
127C2×C7×Q8—21no—
128C2×C7×Q8—22no—
129C2×C7×Q8C2×D412yes—
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—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—41no—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C7×Q16—48no—
155C7×M16—48no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8C2×D414yes—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—44no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4C2×D412yes—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—44no—
181C42×C7—22no—
182C42×C7—22no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—42no—
186C42×C7—42no—
187C42×C7C2×D412yes—
188C42×C7#4112yes—
189C22×Q16C22×C712yes—
190C22×M16C22×C712yes—
191C2×C4⋊Q8C22×C711yescharacteristic
192C2×C22⋊Q8C22×C712yes—
193C2×C22⋊Q8C22×C711yescharacteristic
194C2×C4×Q8C22×C711yescharacteristic
195C22×C42C22×C711yescharacteristic
196#470—28no—
197#462—24no—
198C2×#106C22×C712yes—
199#423—24no—
200C2×C4⋊C8C22×C712yes—
201C2×C4⋊C8C22×C711yescharacteristic
202C2×C4⋊C8C22×C711yescharacteristic
203C2×#103C22×C712yes—
204C2×#103C22×C712yes—
205C2×#103C22×C712yes—
206C2×#103C22×C712yes—
207#391C22×C711yescharacteristic
208#391C22×C712yes—
209#389—21no—
210#389—21no—
211#389C22×C711yescharacteristic
212#387—21no—
213#385—21no—
214#384C22×C711yescharacteristic
215#380—21no—
216#380—21no—
217#380C22×C711yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4C22×C711yescharacteristic
220C4×C22⋊C4—21no—
221#340C22×C712yes—
222#340C22×C712yes—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—21no—
229C23×C7#12112yes—
230C23×C7C22×D411yescharacteristic
231C7×Q8—22no—
232C7×Q8—22no—
233C7×Q8—41no—
234C7×Q8—41no—
235C7×Q8—42no—
236C7×Q8—42no—
237C7×Q8—42no—
238C7×Q8—42no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—21no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—44no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—44no—
260C2×C4×C7—41no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
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—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7#12112yes—
273C2×C4×C7#12112yes—
274C2×C4×C7#12112yes—
275C2×C4×C7#13712yes—
276C2×C4×C7#13712yes—
277C2×C4×C7#13712yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C7×C8—24no—
285C7×C8—42no—
286C22×Q8C23×C712yes—
287C22×Q8C23×C711yescharacteristic
288C23×C4C7×D411yescharacteristic
289C23×C4C7×D411yescharacteristic
290C23×C4C23×C711yescharacteristic
291C2×Q16—24no—
292C2×Q16—24no—
293C2×Q16—42no—
294C2×M16—24no—
295C2×M16—24no—
296C2×M16—42no—
297C22×C8C23×C712yes—
298C22×C8C23×C712yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8C7×D414yes—
305C4×Q8—22no—
306C4×Q8—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C23×C711yescharacteristic
316C2×C4⋊C4C23×C712yes—
317C2×C4⋊C4C23×C711yescharacteristic
318C2×C4⋊C4C23×C711yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42C7×D411yescharacteristic
327C2×C42C7×D411yescharacteristic
328C2×C42C23×C711yescharacteristic
329C2×C42C23×C711yescharacteristic
330#106—24no—
331C4⋊C8—24no—
332C4⋊C8—22no—
333C4⋊C8C7×D414yes—
334#103—24no—
335#103—24no—
336#103—24no—
337#103—42no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—41no—
349C22×C7—41no—
350C22×C7—41no—
351C22×C7—41no—
352C22×C7C2×#12111yescharacteristic
353C22×C7C2×#12111yescharacteristic
354C22×C7#53711yescharacteristic
355C22×C7C2×#13711yescharacteristic
356C22×C7C2×#13711yescharacteristic
357C22×C7C2×#13711yescharacteristic
358C22×C7#57511yescharacteristic
359C4×C7—22no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—42no—
363C4×C7—42no—
364C4×C7—41no—
365C4×C7—82no—
366C4×C7—81no—
367C4×C7—82no—
368C4×C7—81no—
369C4×C7#47014yescomplemented
370C4×C7D4211yescharacteristic
371C4×C7#54711yescharacteristic
372C4×C7#54811yescharacteristic
373C4×C7#55311yescharacteristic
374C24C2×C7×D411yescharacteristic
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8C2×C7×D412yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—41no—
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×D411yescharacteristic
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D412yes—
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409C22×C4C7×#4112yes—
410C22×C4C24×C711yescharacteristic
411Q16—48no—
412M16—48no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8C2×C7×D414yes—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4C2×C7×D412yes—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C42—22no—
439C42—22no—
440C42—21no—
441C42—21no—
442C42—42no—
443C42—42no—
444C42C2×C7×D412yes—
445C42C7×#4112yes—
446C2×C7—42no—
447C2×C7#241611yescharacteristic
448C2×C7#303711yescharacteristic
449C2×C7#303911yescharacteristic
450C2×C7C2×#47011yescharacteristic
451C2×C7#305611yescharacteristic
452C2×C7#325211yescharacteristic
453C2×C7#325511yescharacteristic
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23C7×#12112yes—
461C23C22×C7×D411yescharacteristic
462Q8—22no—
463Q8—22no—
464Q8—41no—
465Q8—41no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4C7×#12112yes—
504C2×C4C7×#12112yes—
505C2×C4C7×#12112yes—
506C2×C4C7×#13712yes—
507C2×C4C7×#13712yes—
508C2×C4C7×#13712yes—
509C2×C4C22×C7×D411yescharacteristic
510C2×C4C22×C7×D411yescharacteristic
511C2×C4C2×C7×#4111yescharacteristic
512C2×C4C2×C7×#4111yescharacteristic
513C2×C4C7×H(5,𝔽2)11yescharacteristic
514C2×C4C7×2−1+411yescharacteristic
515C8—24no—
516C8—42no—
517C7#2476011yescharacteristic, direct product factor, Sylow
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C2×C7×#12111yescharacteristic
526C22C2×C7×#12111yescharacteristic
527C22C7×#53711yescharacteristic
528C22C2×C7×#13711yescharacteristic
529C22C2×C7×#13711yescharacteristic
530C22C2×C7×#13711yescharacteristic
531C22C7×#57511yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—42no—
536C4—42no—
537C4—41no—
538C4—82no—
539C4—81no—
540C4—82no—
541C4—81no—
542C4C7×#47014yescomplemented
543C4C7×D4211yescharacteristic
544C4C7×#54711yescharacteristic
545C4C7×#54811yescharacteristic
546C4C7×#55311yescharacteristic
547C2—42no—
548C2C7×#241611yescharacteristic
549C2C7×#303711yescharacteristic
550C2C7×#303911yescharacteristic
551C2C2×C7×#47011yescharacteristic
552C2C7×#305611yescharacteristic
553C2C7×#325211yescharacteristic
554C2C7×#325511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC24×C7C2×D4
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
2. C2×C7×#462C2semidirect
6. C7×#2066C2semidirect
7. C7×#2056C2semidirect
8. C7×#1981C2semidirect
9. C7×#1948C2semidirect
11. C7×#1803C2semidirect
46. #24760C7direct (N × H)
107. C2×#462C2×C7semidirect
111. #2066C2×C7semidirect
112. #2056C2×C7semidirect
113. #1981C2×C7semidirect
114. #1948C2×C7semidirect
116. #1803C2×C7semidirect
369. C4×C7#470semidirect
517. C7#24760direct (N × H)
542. C4C7×#470semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4702
C2×C7×#4622
C2×C7×#4232
C7×#28572
C4×C7×C22⋊Q82
C7×#20662
C7×#20562
C7×#19812
C7×#19482
C7×#18942
C7×#18032
#247607