Finite Group Databasebeta

← C7×#24183 #13201399

Subgroups — C7×#24183 #13201399

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3507C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C7×#3037C211yesmaximal, characteristic
4C7×#2327C211yesmaximal, characteristic, complemented
5C7×#2038C212yesmaximal, complemented
6C7×#2005C212yesmaximal, complemented
7C7×#1995C211yesmaximal, characteristic, complemented
8C7×#1963C211yesmaximal, characteristic
9C7×#1882C212yesmaximal, complemented
10C7×#1860C212yesmaximal, complemented
11C7×#1801C211yesmaximal, characteristic, complemented
12C2×C7×#137C2212yes—
13C7×#559—21no—
14C7×#557—21no—
15C7×#556—21no—
16C7×#556—22no—
17C7×#554—21no—
18C7×#552—22no—
19C7×#539—22no—
20C7×#536C2212yes—
21C7×#533C2211yescharacteristic
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529—22no—
25C7×#520C2212yes—
26C2×C4×C7×Q8C2211yescharacteristic
27C2×C4×C7×Q8C2211yescharacteristic
28C7×#479—22no—
29C7×#477—21no—
30C7×#475—21no—
31C7×#464—22no—
32C7×#462—21no—
33C7×#460—21no—
34C7×#428C2212yes—
35C7×#423C2211yescharacteristic
36C7×#423C2211yescharacteristic
37C2×C7×#104C2212yes—
38C2×C7×#104C2212yes—
39C7×#419C2212yes—
40C7×#417C2212yes—
41C2×C7×#101C2212yes—
42C7×#391C2211yescharacteristic
43C7×#391C2211yescharacteristic
44C7×#387—22no—
45C7×#387—22no—
46C7×#382C2211yescharacteristic
47C7×#376C2211yescharacteristic
48C7×#343C2212yes—
49C7×#339C2212yes—
50C7×#339C2212yes—
51#24183C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41C2312yes—
53C22×C7×Q8C2311yescharacteristic
54C22×C7×Q8C2311yescharacteristic
55C7×#137—24no—
56C7×#137—42no—
57C2×C7×Q16—22no—
58C2×C7×SD16—22no—
59C2×C7×M16C2312yes—
60C2×C7×M16C2312yes—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×C4⋊Q8D411yescharacteristic
65C7×C4⋊Q8D411yescharacteristic
66C7×C4⋊Q8D411yescharacteristic
67C7×#127D411yescharacteristic
68C7×#126—24no—
69C7×#126—24no—
70C7×#125—22no—
71C7×#125—24no—
72C7×#125D412yes—
73C7×#124—22no—
74C7×#124—22no—
75C7×#124D412yes—
76C7×#123—24no—
77C7×#123—24no—
78C7×C22⋊Q8—24no—
79C7×C22⋊Q8—21no—
80C7×C22⋊Q8—21no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×#121—21no—
84C7×#121—21no—
85C4×C7×Q8—22no—
86C4×C7×Q8—22no—
87C4×C7×Q8—21no—
88C4×C7×Q8—21no—
89C4×C7×Q8—21no—
90C4×C7×Q8—21no—
91C4×C7×Q8—21no—
92C4×C7×Q8—21no—
93C4×C7×Q8—21no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×Q8—21no—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—24no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C7×#117—24no—
103C7×#117—22no—
104C7×#117C2312yes—
105C2×C7×C4⋊C4—22no—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4D412yes—
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C42×C7—21no—
112C2×C42×C7D412yes—
113C2×C42×C7C2311yescharacteristic
114C2×C42×C7C2311yescharacteristic
115C2×C42×C7C2311yescharacteristic
116C7×#107—22no—
117C7×#106—22no—
118C7×C4⋊C8—21no—
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×C4⋊C8—21no—
122C7×#104—24no—
123C7×#104—42no—
124C7×#103—22no—
125C7×#103—22no—
126C7×#103—22no—
127C7×#102—22no—
128C7×#101—42no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—22no—
132C7×#95—22no—
133C7×#95—22no—
134#3507C2×C711yescharacteristic
135#3078C2×C711yescharacteristic, complemented
136#3037C2×C711yescharacteristic
137#2327C2×C711yescharacteristic, complemented
138#2038C2×C712yescomplemented
139#2005C2×C712yescomplemented
140#1995C2×C711yescharacteristic, complemented
141#1963C2×C711yescharacteristic
142#1882C2×C712yescomplemented
143#1860C2×C712yescomplemented
144#1801C2×C711yescharacteristic, complemented
145C7×#41—22no—
146C7×#41—22no—
147C7×#41—22no—
148C7×#41—22no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8C2×D412yes—
163C2×C7×Q8#4112yes—
164C2×C7×D4—22no—
165C2×C7×D4—21no—
166C2×C7×D4—21no—
167C2×C7×D4C2×D412yes—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—24no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D412yes—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2411yescharacteristic
179C7×Q16—44no—
180C7×SD16—44no—
181C7×M16—22no—
182C7×M16—22no—
183C7×M16—22no—
184C7×M16—22no—
185C2×C7×C8—22no—
186C2×C7×C8—22no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—28no—
189C7×C4⋊C4—24no—
190C7×C4⋊C4—24no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—41no—
204C7×C4⋊C4—41no—
205C7×C4⋊C4C2×D412yes—
206C7×C4⋊C4C2×D412yes—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C42×C7—24no—
214C42×C7—22no—
215C42×C7—22no—
216C42×C7—22no—
217C42×C7—21no—
218C42×C7—21no—
219C42×C7—21no—
220C42×C7—41no—
221C42×C7—41no—
222C42×C7C2×D411yescharacteristic
223C42×C7C2×D411yescharacteristic
224C42×C7C2×D411yescharacteristic
225C42×C7C2×D411yescharacteristic
226C42×C7#4112yes—
227C2×#137C22×C712yes—
228#559—21no—
229#557—21no—
230#556—21no—
231#556—22no—
232#554—21no—
233#552—22no—
234#539—22no—
235#536C22×C712yes—
236#533C22×C711yescharacteristic
237#532C22×C711yescharacteristic
238C2×C4⋊Q8C22×C711yescharacteristic
239#529—22no—
240#520C22×C712yes—
241C2×C4×Q8C22×C711yescharacteristic
242C2×C4×Q8C22×C711yescharacteristic
243#479—22no—
244#477—21no—
245#475—21no—
246#464—22no—
247#462—21no—
248#460—21no—
249#428C22×C712yes—
250#423C22×C711yescharacteristic
251#423C22×C711yescharacteristic
252C2×#104C22×C712yes—
253C2×#104C22×C712yes—
254#419C22×C712yes—
255#417C22×C712yes—
256C2×#101C22×C712yes—
257#391C22×C711yescharacteristic
258#391C22×C711yescharacteristic
259#387—22no—
260#387—22no—
261#382C22×C711yescharacteristic
262#376C22×C711yescharacteristic
263#343C22×C712yes—
264#339C22×C712yes—
265#339C22×C712yes—
266C23×C7—22no—
267C23×C7C22×D411yescharacteristic
268C7×Q8—22no—
269C7×Q8—22no—
270C7×Q8—22no—
271C7×Q8—22no—
272C7×Q8—42no—
273C7×Q8—41no—
274C7×Q8—41no—
275C7×Q8—41no—
276C7×Q8—41no—
277C7×Q8—41no—
278C7×Q8—41no—
279C7×Q8—81no—
280C7×D4—22no—
281C7×D4—22no—
282C7×D4—42no—
283C7×D4—42no—
284C7×D4—42no—
285C7×D4—81no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—84no—
310C2×C4×C7#12011yescharacteristic
311C2×C4×C7#12011yescharacteristic
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7H(5,𝔽2)11yescharacteristic
319C2×C4×C72−1+411yescharacteristic
320C7×C8—42no—
321C7×C8—42no—
322C2×#41C23×C712yes—
323C22×Q8C23×C711yescharacteristic
324C22×Q8C23×C711yescharacteristic
325#137—24no—
326#137—42no—
327C2×Q16—22no—
328C2×SD16—22no—
329C2×M16C23×C712yes—
330C2×M16C23×C712yes—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334C4⋊Q8C7×D411yescharacteristic
335C4⋊Q8C7×D411yescharacteristic
336C4⋊Q8C7×D411yescharacteristic
337#127C7×D411yescharacteristic
338#126—24no—
339#126—24no—
340#125—22no—
341#125—24no—
342#125C7×D412yes—
343#124—22no—
344#124—22no—
345#124C7×D412yes—
346#123—24no—
347#123—24no—
348C22⋊Q8—24no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353#121—21no—
354#121—21no—
355C4×Q8—22no—
356C4×Q8—22no—
357C4×Q8—21no—
358C4×Q8—21no—
359C4×Q8—21no—
360C4×Q8—21no—
361C4×Q8—21no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—24no—
370C4×D4—21no—
371C4×D4—21no—
372#117—24no—
373#117—22no—
374#117C23×C712yes—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C7×D412yes—
379C2×C4⋊C4C23×C711yescharacteristic
380C2×C4⋊C4C23×C711yescharacteristic
381C2×C42—21no—
382C2×C42C7×D412yes—
383C2×C42C23×C711yescharacteristic
384C2×C42C23×C711yescharacteristic
385C2×C42C23×C711yescharacteristic
386#107—22no—
387#106—22no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392#104—24no—
393#104—42no—
394#103—22no—
395#103—22no—
396#103—22no—
397#102—22no—
398#101—42no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404C22×C7—21no—
405C22×C7—21no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7C2×#12011yescharacteristic
410C22×C7#52511yescharacteristic
411C22×C7#53711yescharacteristic
412C4×C7—21no—
413C4×C7—21no—
414C4×C7—21no—
415C4×C7—21no—
416C4×C7—42no—
417C4×C7—42no—
418C4×C7—42no—
419C4×C7—42no—
420C4×C7—42no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—81no—
424C4×C7—81no—
425C4×C7#54611yescharacteristic
426C4×C7#54711yescharacteristic
427C4×C7#54811yescharacteristic
428C4×C7#55011yescharacteristic
429#41—22no—
430#41—22no—
431#41—22no—
432#41—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C2×C7×D412yes—
447C2×Q8C7×#4112yes—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4C2×C7×D412yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4C2×C7×D412yes—
459C22×C4C2×C7×D412yes—
460C22×C4C2×C7×D411yescharacteristic
461C22×C4C2×C7×D411yescharacteristic
462C22×C4C24×C711yescharacteristic
463Q16—44no—
464SD16—44no—
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×C7×D412yes—
490C4⋊C4C2×C7×D412yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—41no—
505C42—41no—
506C42C2×C7×D411yescharacteristic
507C42C2×C7×D411yescharacteristic
508C42C2×C7×D411yescharacteristic
509C42C2×C7×D411yescharacteristic
510C42C7×#4112yes—
511C2×C7—21no—
512C2×C7—21no—
513C2×C7—82no—
514C2×C7#240711yescharacteristic
515C2×C7#302011yescharacteristic
516C2×C7#302211yescharacteristic
517C23—22no—
518C23C22×C7×D411yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—41no—
525Q8—41no—
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—81no—
531D4—22no—
532D4—22no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—81no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4C7×#12011yescharacteristic
562C2×C4C7×#12011yescharacteristic
563C2×C4C7×#12011yescharacteristic
564C2×C4C7×#12011yescharacteristic
565C2×C4C22×C7×D411yescharacteristic
566C2×C4C22×C7×D411yescharacteristic
567C2×C4C2×C7×#4111yescharacteristic
568C2×C4C2×C7×#4111yescharacteristic
569C2×C4C7×H(5,𝔽2)11yescharacteristic
570C2×C4C7×2−1+411yescharacteristic
571C8—42no—
572C8—42no—
573C7#2418311yescharacteristic, direct product factor, Sylow
574C22—21no—
575C22—21no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C7×#12011yescharacteristic
580C22C7×#52511yescharacteristic
581C22C7×#53711yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—81no—
595C4C7×#54611yescharacteristic
596C4C7×#54711yescharacteristic
597C4C7×#54811yescharacteristic
598C4C7×#55011yescharacteristic
599C2—21no—
600C2—21no—
601C2—82no—
602C2C7×#240711yescharacteristic
603C2C7×#302011yescharacteristic
604C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3078C2semidirect
4. C7×#2327C2semidirect
5. C7×#2038C2semidirect
6. C7×#2005C2semidirect
7. C7×#1995C2semidirect
9. C7×#1882C2semidirect
10. C7×#1860C2semidirect
11. C7×#1801C2semidirect
51. #24183C7direct (N × H)
135. #3078C2×C7semidirect
137. #2327C2×C7semidirect
138. #2038C2×C7semidirect
139. #2005C2×C7semidirect
140. #1995C2×C7semidirect
142. #1882C2×C7semidirect
143. #1860C2×C7semidirect
144. #1801C2×C7semidirect
573. C7#24183direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35072
C7×#30782
C7×#30372
C7×#23272
C7×#20382
C7×#20052
C7×#19952
C7×#19632
C7×#18822
C7×#18602
C7×#18012
#241837