Finite Group Databasebeta

← C7×#23905 #13201121

Subgroups — C7×#23905 #13201121

Order 1792.

1744 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic
2C2×C7×#536C211yesmaximal, characteristic, complemented
3C2×C7×#458C212yesmaximal, complemented
4C7×#2614C212yesmaximal, complemented
5C7×#2022C214yesmaximal, complemented
6C7×#1899C212yesmaximal, complemented
7C7×#1890C212yesmaximal, complemented
8C7×#1782C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic, complemented
12C7×#547—24no—
13C7×#546—24no—
14C7×#536—21no—
15C7×#536—28no—
16C7×#534—21no—
17C7×#534C2212yes—
18C2×C7×C4⋊Q8—21no—
19C2×C7×#124—21no—
20C7×#525—21no—
21C2×C7×#123—21no—
22C2×C7×#123C2214yes—
23C2×C7×#123C2212yes—
24C2×C7×#123C2212yes—
25C2×C7×#123C2212yescomplemented
26C2×C7×C22⋊Q8—21no—
27C2×C7×C22⋊Q8—21no—
28C7×#518C2211yescharacteristic
29C2×C7×#117C2211yescharacteristic, complemented
30C7×#458—28no—
31C2×C7×#99C2214yes—
32C2×C7×#99C2212yes—
33C2×C7×#99C2212yescomplemented
34C7×#395—24no—
35C7×#393—24no—
36C7×#381C2214yescomplemented
37C7×#381C2212yescomplemented
38C7×#342C2214yescomplemented
39#23905C711yesmaximal, characteristic, direct product factor, Sylow
40C7×H(5,𝔽2)—24no—
41C7×H(5,𝔽2)—42no—
42C2×C7×#41—24no—
43C2×C7×#41—24no—
44C2×C7×#41—21no—
45C2×C7×#41—22no—
46C2×C7×#41—42no—
47C22×C7×Q8—21no—
48C22×C7×Q8—21no—
49C22×C7×Q8—24no—
50C22×C7×Q8—24no—
51C22×C7×Q8—42no—
52C22×C7×Q8D411yescharacteristic
53C22×C7×D4—21no—
54C22×C7×D4—22no—
55C22×C7×D4D411yescharacteristic
56C22×C7×D4C2311yescharacteristic
57C22×C7×D4C2312yes—
58C22×C7×D4C2311yescharacteristic
59C23×C4×C7D411yescharacteristic
60C23×C4×C7D411yescharacteristic
61C23×C4×C7C2312yes—
62C23×C4×C7C2311yescharacteristic
63C7×C4⋊Q8—28no—
64C7×C4⋊Q8—22no—
65C7×#124—28no—
66C7×#124—22no—
67C7×#123—24no—
68C7×#123—22no—
69C7×#123—24no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—24no—
74C7×#123—24no—
75C7×#123—22no—
76C7×#123—24no—
77C7×#123—28no—
78C7×#123—44no—
79C7×#123D418yescomplemented
80C7×C22⋊Q8—28no—
81C7×C22⋊Q8—28no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—24no—
84C7×#121—22no—
85C7×#121—24no—
86C7×#121—24no—
87C7×#120—22no—
88C7×#120—22no—
89C7×#120—22no—
90C4×C7×D4—24no—
91C4×C7×D4—24no—
92C7×#117—21no—
93C7×#117—24no—
94C2×C7×C4⋊C4—22no—
95C2×C7×C4⋊C4—21no—
96C2×C7×C4⋊C4—22no—
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4—24no—
99C2×C7×C4⋊C4D414yes—
100C2×C7×C22⋊C4—22no—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—22no—
103C2×C7×C22⋊C4—22no—
104C2×C7×C22⋊C4—24no—
105C2×C7×C22⋊C4D414yes—
106C2×C7×C22⋊C4C2314yes—
107C2×C7×C22⋊C4C2312yes—
108C2×C7×C22⋊C4C2312yes—
109C2×C42×C7—21no—
110C7×#99—24no—
111C7×#99—28no—
112C7×#99—42no—
113C7×#95—28no—
114C7×#95—24no—
115#3485C2×C711yescharacteristic
116C2×#536C2×C711yescharacteristic, complemented
117C2×#458C2×C712yescomplemented
118#2614C2×C712yescomplemented
119#2022C2×C714yescomplemented
120#1899C2×C712yescomplemented
121#1890C2×C712yescomplemented
122#1782C2×C711yescharacteristic, complemented
123C24×C7—21no—
124C24×C7C2×D412yes—
125C24×C7C2×D411yescharacteristic
126C24×C7C2×D411yescharacteristic
127C24×C7C2411yescharacteristic
128C7×#41—416no—
129C7×#41—48no—
130C7×#41—42no—
131C7×#41—42no—
132C2×C7×Q8—24no—
133C2×C7×Q8—21no—
134C2×C7×Q8—24no—
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—41no—
138C2×C7×Q8—416no—
139C2×C7×Q8—42no—
140C2×C7×Q8—42no—
141C2×C7×Q8—42no—
142C2×C7×Q8—42no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—21no—
147C2×C7×D4—22no—
148C2×C7×D4—24no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—24no—
155C2×C7×D4—24no—
156C2×C7×D4—22no—
157C2×C7×D4—41no—
158C2×C7×D4—48no—
159C2×C7×D4—42no—
160C2×C7×D4—42no—
161C2×C7×D4C2×D414yes—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—24no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—24no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—24no—
177C22×C4×C7—41no—
178C22×C4×C7—41no—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D414yes—
182C22×C4×C7C2×D414yes—
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D414yes—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—24no—
189C7×C4⋊C4—28no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—24no—
192C7×C4⋊C4—28no—
193C7×C4⋊C4—24no—
194C7×C4⋊C4—48no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—24no—
197C7×C22⋊C4—28no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—28no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—24no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4—24no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—48no—
208C7×C22⋊C4C2×D414yes—
209C42×C7—24no—
210C42×C7—22no—
211C2×H(5,𝔽2)C22×C711yescharacteristic
212C22×#41C22×C711yescharacteristic
213C23×Q8C22×C711yescharacteristic, complemented
214#547—24no—
215#546—24no—
216#536—21no—
217#536—28no—
218#534—21no—
219#534C22×C712yes—
220C2×C4⋊Q8—21no—
221C2×#124—21no—
222#525—21no—
223C2×#123—21no—
224C2×#123C22×C714yes—
225C2×#123C22×C712yes—
226C2×#123C22×C712yes—
227C2×#123C22×C712yescomplemented
228C2×C22⋊Q8—21no—
229C2×C22⋊Q8—21no—
230#518C22×C711yescharacteristic
231C2×#117C22×C711yescharacteristic, complemented
232#458—28no—
233C2×#99C22×C712yes—
234C2×#99C22×C714yes—
235C2×#99C22×C712yescomplemented
236#395—24no—
237#393—24no—
238#381C22×C714yescomplemented
239#381C22×C712yescomplemented
240#342C22×C714yescomplemented
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—22no—
245C23×C7—21no—
246C23×C7—21no—
247C23×C7—24no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—44no—
253C23×C7—82no—
254C23×C7#12012yes—
255C23×C7C22×D411yescharacteristic
256C23×C7C22×D412yes—
257C23×C7C22×D412yes—
258C23×C7H(5,𝔽2)12yes—
259C7×Q8—44no—
260C7×Q8—48no—
261C7×Q8—48no—
262C7×Q8—42no—
263C7×D4—28no—
264C7×D4—44no—
265C7×D4—42no—
266C7×D4—44no—
267C7×D4—42no—
268C7×D4—42no—
269C7×D4—44no—
270C7×D4—44no—
271C7×D4—84no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—28no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—21no—
282C2×C4×C7—24no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—28no—
287C2×C4×C7—24no—
288C2×C4×C7—44no—
289C2×C4×C7—44no—
290C2×C4×C7—44no—
291C2×C4×C7—44no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—44no—
296C2×C4×C7—44no—
297C2×C4×C7—48no—
298C2×C4×C7—44no—
299C2×C4×C7—82no—
300C2×C4×C7#12012yes—
301C2×C4×C7#12012yes—
302C2×C4×C7#12012yes—
303H(5,𝔽2)—24no—
304H(5,𝔽2)—42no—
305C2×#41—24no—
306C2×#41—21no—
307C2×#41—24no—
308C2×#41—22no—
309C2×#41—42no—
310C22×Q8—21no—
311C22×Q8—21no—
312C22×Q8—24no—
313C22×Q8—24no—
314C22×Q8—42no—
315C22×Q8C7×D411yescharacteristic
316C22×D4—21no—
317C22×D4—22no—
318C22×D4C7×D411yescharacteristic
319C22×D4C23×C712yes—
320C22×D4C23×C711yescharacteristic
321C22×D4C23×C711yescharacteristic
322C23×C4C7×D411yescharacteristic
323C23×C4C7×D411yescharacteristic
324C23×C4C23×C712yes—
325C23×C4C23×C711yescharacteristic
326C4⋊Q8—28no—
327C4⋊Q8—22no—
328#124—28no—
329#124—22no—
330#123—24no—
331#123—22no—
332#123—24no—
333#123—22no—
334#123—22no—
335#123—22no—
336#123—24no—
337#123—24no—
338#123—22no—
339#123—24no—
340#123—28no—
341#123—44no—
342#123C7×D418yescomplemented
343C22⋊Q8—28no—
344C22⋊Q8—28no—
345C22⋊Q8—22no—
346C22⋊Q8—24no—
347#121—22no—
348#121—24no—
349#121—24no—
350#120—22no—
351#120—22no—
352#120—22no—
353C4×D4—24no—
354C4×D4—24no—
355#117—21no—
356#117—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4C7×D414yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4C7×D414yes—
369C2×C22⋊C4C23×C714yes—
370C2×C22⋊C4C23×C712yes—
371C2×C22⋊C4C23×C712yes—
372C2×C42—21no—
373#99—24no—
374#99—28no—
375#99—42no—
376#95—28no—
377#95—24no—
378C22×C7—24no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—41no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—44no—
390C22×C7—81no—
391C22×C7—84no—
392C22×C7—81no—
393C22×C7—162no—
394C22×C7C2×#12011yescharacteristic
395C22×C7C2×#12011yescharacteristic
396C22×C7#53411yescharacteristic
397C22×C7D4214yes—
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—44no—
402C4×C7—42no—
403C4×C7—41no—
404C4×C7—44no—
405C4×C7—44no—
406C4×C7—84no—
407C24—21no—
408C24C2×C7×D412yes—
409C24C2×C7×D411yescharacteristic
410C24C2×C7×D411yescharacteristic
411C24C24×C711yescharacteristic
412#41—416no—
413#41—48no—
414#41—42no—
415#41—42no—
416C2×Q8—24no—
417C2×Q8—21no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8—416no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—22no—
441C2×D4—41no—
442C2×D4—48no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4C2×C7×D414yes—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4C2×C7×D412yes—
464C22×C4C2×C7×D411yescharacteristic
465C22×C4C2×C7×D414yes—
466C22×C4C2×C7×D414yes—
467C22×C4C2×C7×D411yescharacteristic
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D414yes—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—48no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—28no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—48no—
492C22⋊C4C2×C7×D414yes—
493C42—24no—
494C42—22no—
495C2×C7—21no—
496C2×C7—24no—
497C2×C7—21no—
498C2×C7—42no—
499C2×C7—81no—
500C2×C7—82no—
501C2×C7#240311yescharacteristic
502C2×C7#302812yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—24no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—82no—
516C23C7×#12012yes—
517C23C22×C7×D411yescharacteristic
518C23C22×C7×D412yes—
519C23C22×C7×D412yes—
520C23C7×H(5,𝔽2)12yes—
521Q8—44no—
522Q8—48no—
523Q8—48no—
524Q8—42no—
525D4—28no—
526D4—44no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—42no—
531D4—44no—
532D4—44no—
533D4—84no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—82no—
562C2×C4C7×#12012yes—
563C2×C4C7×#12012yes—
564C2×C4C7×#12012yes—
565C7#2390511yescharacteristic, direct product factor, Sylow
566C22—24no—
567C22—22no—
568C22—22no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—81no—
579C22—84no—
580C22—81no—
581C22—162no—
582C22C2×C7×#12011yescharacteristic
583C22C2×C7×#12011yescharacteristic
584C22C7×#53411yescharacteristic
585C22C7×D4214yes—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—44no—
590C4—42no—
591C4—41no—
592C4—44no—
593C4—44no—
594C4—84no—
595C2—21no—
596C2—24no—
597C2—21no—
598C2—42no—
599C2—81no—
600C2—82no—
601C2C7×#240311yescharacteristic
602C2C7×#302812yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#536C2semidirect
3. C2×C7×#458C2semidirect
4. C7×#2614C2semidirect
5. C7×#2022C2semidirect
6. C7×#1899C2semidirect
7. C7×#1890C2semidirect
8. C7×#1782C2semidirect
11. C23×C7×Q8C22semidirect
25. C2×C7×#123C22semidirect
29. C2×C7×#117C22semidirect
33. C2×C7×#99C22semidirect
36. C7×#381C22semidirect
37. C7×#381C22semidirect
38. C7×#342C22semidirect
39. #23905C7direct (N × H)
79. C7×#123D4semidirect
116. C2×#536C2×C7semidirect
117. C2×#458C2×C7semidirect
118. #2614C2×C7semidirect
119. #2022C2×C7semidirect
120. #1899C2×C7semidirect
121. #1890C2×C7semidirect
122. #1782C2×C7semidirect
213. C23×Q8C22×C7semidirect
227. C2×#123C22×C7semidirect
231. C2×#117C22×C7semidirect
235. C2×#99C22×C7semidirect
238. #381C22×C7semidirect
239. #381C22×C7semidirect
240. #342C22×C7semidirect
342. #123C7×D4semidirect
565. C7#23905direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C2×C7×#5362
C2×C7×#4582
C7×#26142
C7×#20222
C7×#18992
C7×#18902
C7×#17822
#239057