Finite Group Databasebeta

← C7×#23904 #13201120

Subgroups — C7×#23904 #13201120

Order 1792.

2032 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic, complemented
2C2×C7×#535C211yesmaximal, characteristic, complemented
3C2×C7×B(4,𝔽2)C212yesmaximal, complemented
4C7×#2613C212yesmaximal, complemented
5C7×#2021C214yesmaximal, complemented
6C7×#1899C212yesmaximal, complemented
7C7×#1889C212yesmaximal, complemented
8C7×#1782C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C22×C7×#41C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic, complemented
12C7×#547—24no—
13C7×#546—24no—
14C7×#536—21no—
15C7×#535—28no—
16C7×#534—21no—
17C7×#534C2212yes—
18C2×C7×#127—21no—
19C2×C7×#124—21no—
20C7×#525—21no—
21C2×C7×#123—21no—
22C2×C7×#123C2212yes—
23C2×C7×#121—21no—
24C2×C7×#121—21no—
25C2×C7×#120C2214yes—
26C2×C7×#120C2212yes—
27C2×C7×#120C2212yescomplemented
28C7×#518C2211yescharacteristic
29C2×C7×#117C2211yescharacteristic, complemented
30C7×B(4,𝔽2)—28no—
31C2×C7×#99C2214yes—
32C2×C7×#99C2212yes—
33C2×C7×#99C2212yescomplemented
34C7×#394—24no—
35C7×#392—24no—
36C7×#381C2212yescomplemented
37C7×#379C2214yescomplemented
38C7×#342C2214yescomplemented
39#23904C711yesmaximal, characteristic, direct product factor, Sylow
40C25×C7C2312yes—
41C7×H(5,𝔽2)—24no—
42C7×H(5,𝔽2)—42no—
43C2×C7×#41—24no—
44C2×C7×#41—24no—
45C2×C7×#41—21no—
46C2×C7×#41—22no—
47C2×C7×#41—42no—
48C22×C7×Q8D411yescharacteristic, complemented
49C22×C7×D4—24no—
50C22×C7×D4—21no—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—22no—
54C22×C7×D4—24no—
55C22×C7×D4—21no—
56C22×C7×D4—21no—
57C22×C7×D4—24no—
58C22×C7×D4—42no—
59C22×C7×D4D411yescharacteristic, complemented
60C22×C7×D4C2311yescharacteristic
61C22×C7×D4C2312yes—
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7D411yescharacteristic, complemented
64C23×C4×C7D411yescharacteristic, complemented
65C23×C4×C7C2311yescharacteristic
66C7×C4⋊Q8—22no—
67C7×#127—28no—
68C7×#124—28no—
69C7×#124—22no—
70C7×#123—24no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×#123—24no—
76C7×#123—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—24no—
79C7×#121—28no—
80C7×#121—28no—
81C7×#121—22no—
82C7×#121—24no—
83C7×#121—24no—
84C7×#120—22no—
85C7×#120—24no—
86C7×#120—22no—
87C7×#120—22no—
88C7×#120—24no—
89C7×#120—24no—
90C7×#120—28no—
91C7×#120—44no—
92C7×#120D418yescomplemented
93C4×C7×D4—24no—
94C4×C7×D4—24no—
95C7×#117—21no—
96C7×#117—24no—
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4D414yescomplemented
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⋊C4—28no—
106C2×C7×C22⋊C4D414yescomplemented
107C2×C7×C22⋊C4C2314yes—
108C2×C7×C22⋊C4C2312yes—
109C2×C7×C22⋊C4C2312yes—
110C2×C42×C7—21no—
111C7×#99—24no—
112C7×#99—28no—
113C7×#99—42no—
114C7×#95—24no—
115#3485C2×C711yescharacteristic, complemented
116C2×#535C2×C711yescharacteristic, complemented
117C2×B(4,𝔽2)C2×C712yescomplemented
118#2613C2×C712yescomplemented
119#2021C2×C714yescomplemented
120#1899C2×C712yescomplemented
121#1889C2×C712yescomplemented
122#1782C2×C711yescharacteristic, complemented
123C24×C7—21no—
124C24×C7—24no—
125C24×C7—22no—
126C24×C7—22no—
127C24×C7—24no—
128C24×C7—84no—
129C24×C7C2×D412yes—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D414yes—
132C24×C7C2×D411yescharacteristic
133C24×C7C2411yescharacteristic
134C7×#41—416no—
135C7×#41—48no—
136C7×#41—42no—
137C7×#41—42no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—41no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—24no—
145C2×C7×D4—22no—
146C2×C7×D4—21no—
147C2×C7×D4—24no—
148C2×C7×D4—22no—
149C2×C7×D4—24no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—22no—
155C2×C7×D4—24no—
156C2×C7×D4—24no—
157C2×C7×D4—28no—
158C2×C7×D4—28no—
159C2×C7×D4—22no—
160C2×C7×D4—41no—
161C2×C7×D4—416no—
162C2×C7×D4—42no—
163C2×C7×D4—42no—
164C2×C7×D4—48no—
165C2×C7×D4—42no—
166C2×C7×D4—42no—
167C2×C7×D4—48no—
168C2×C7×D4—42no—
169C2×C7×D4—42no—
170C2×C7×D4—84no—
171C2×C7×D4—84no—
172C2×C7×D4—88no—
173C2×C7×D4C2×D414yes—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—21no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—22no—
180C22×C4×C7—22no—
181C22×C4×C7—22no—
182C22×C4×C7—22no—
183C22×C4×C7—22no—
184C22×C4×C7—24no—
185C22×C4×C7—41no—
186C22×C4×C7—41no—
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D414yes—
190C22×C4×C7C2×D411yescharacteristic
191C22×C4×C7C2×D411yescharacteristic
192C22×C4×C7C2×D411yescharacteristic
193C22×C4×C7C2×D414yes—
194C7×C4⋊C4—24no—
195C7×C4⋊C4—24no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—24no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—28no—
201C7×C22⋊C4—24no—
202C7×C22⋊C4—24no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—28no—
208C7×C22⋊C4—24no—
209C7×C22⋊C4—24no—
210C7×C22⋊C4—24no—
211C7×C22⋊C4—48no—
212C7×C22⋊C4—88no—
213C7×C22⋊C4C2×D414yes—
214C42×C7—24no—
215C42×C7—22no—
216C2×H(5,𝔽2)C22×C711yescharacteristic
217C22×#41C22×C711yescharacteristic
218C23×D4C22×C711yescharacteristic, complemented
219#547—24no—
220#546—24no—
221#536—21no—
222#535—28no—
223#534—21no—
224#534C22×C712yes—
225C2×#127—21no—
226C2×#124—21no—
227#525—21no—
228C2×#123—21no—
229C2×#123C22×C712yes—
230C2×#121—21no—
231C2×#121—21no—
232C2×#120C22×C714yes—
233C2×#120C22×C712yes—
234C2×#120C22×C712yescomplemented
235#518C22×C711yescharacteristic
236C2×#117C22×C711yescharacteristic, complemented
237B(4,𝔽2)—28no—
238C2×#99C22×C712yes—
239C2×#99C22×C714yes—
240C2×#99C22×C712yescomplemented
241#394—24no—
242#392—24no—
243#381C22×C712yescomplemented
244#379C22×C714yescomplemented
245#342C22×C714yescomplemented
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—24no—
249C23×C7—21no—
250C23×C7—22no—
251C23×C7—21no—
252C23×C7—28no—
253C23×C7—21no—
254C23×C7—24no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7—42no—
258C23×C7—48no—
259C23×C7—48no—
260C23×C7—44no—
261C23×C7—44no—
262C23×C7—42no—
263C23×C7—44no—
264C23×C7—42no—
265C23×C7—42no—
266C23×C7—44no—
267C23×C7—84no—
268C23×C7—84no—
269C23×C7—88no—
270C23×C7—82no—
271C23×C7#12012yes—
272C23×C7C22×D411yescharacteristic
273C23×C7C22×D412yes—
274C23×C7C22×D412yes—
275C23×C7H(5,𝔽2)12yes—
276C7×Q8—44no—
277C7×Q8—42no—
278C7×D4—28no—
279C7×D4—44no—
280C7×D4—48no—
281C7×D4—48no—
282C7×D4—42no—
283C7×D4—44no—
284C7×D4—42no—
285C7×D4—42no—
286C7×D4—44no—
287C7×D4—44no—
288C7×D4—84no—
289C7×D4—88no—
290C7×D4—88no—
291C7×D4—816no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—28no—
299C2×C4×C7—24no—
300C2×C4×C7—21no—
301C2×C4×C7—24no—
302C2×C4×C7—21no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—24no—
306C2×C4×C7—44no—
307C2×C4×C7—44no—
308C2×C4×C7—44no—
309C2×C4×C7—42no—
310C2×C4×C7—44no—
311C2×C4×C7—44no—
312C2×C4×C7—48no—
313C2×C4×C7—44no—
314C2×C4×C7—82no—
315C2×C4×C7#12012yes—
316C2×C4×C7#12012yes—
317C2×C4×C7#12012yes—
318C25C23×C712yes—
319H(5,𝔽2)—24no—
320H(5,𝔽2)—42no—
321C2×#41—24no—
322C2×#41—21no—
323C2×#41—24no—
324C2×#41—22no—
325C2×#41—42no—
326C22×Q8C7×D411yescharacteristic, complemented
327C22×D4—21no—
328C22×D4—24no—
329C22×D4—22no—
330C22×D4—22no—
331C22×D4—22no—
332C22×D4—24no—
333C22×D4—21no—
334C22×D4—21no—
335C22×D4—24no—
336C22×D4—42no—
337C22×D4C7×D411yescharacteristic, complemented
338C22×D4C23×C712yes—
339C22×D4C23×C711yescharacteristic
340C22×D4C23×C711yescharacteristic
341C23×C4C7×D411yescharacteristic, complemented
342C23×C4C7×D411yescharacteristic, complemented
343C23×C4C23×C711yescharacteristic
344C4⋊Q8—22no—
345#127—28no—
346#124—28no—
347#124—22no—
348#123—24no—
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—24no—
354#123—22no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357#121—28no—
358#121—28no—
359#121—22no—
360#121—24no—
361#121—24no—
362#120—22no—
363#120—24no—
364#120—22no—
365#120—22no—
366#120—24no—
367#120—24no—
368#120—28no—
369#120—44no—
370#120C7×D418yescomplemented
371C4×D4—24no—
372C4×D4—24no—
373#117—21no—
374#117—24no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4C7×D414yescomplemented
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—28no—
384C2×C22⋊C4C7×D414yescomplemented
385C2×C22⋊C4C23×C714yes—
386C2×C22⋊C4C23×C712yes—
387C2×C22⋊C4C23×C712yes—
388C2×C42—21no—
389#99—24no—
390#99—28no—
391#99—42no—
392#95—24no—
393C22×C7—24no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—44no—
401C22×C7—44no—
402C22×C7—48no—
403C22×C7—48no—
404C22×C7—42no—
405C22×C7—41no—
406C22×C7—42no—
407C22×C7—44no—
408C22×C7—42no—
409C22×C7—44no—
410C22×C7—81no—
411C22×C7—84no—
412C22×C7—81no—
413C22×C7—88no—
414C22×C7—84no—
415C22×C7—84no—
416C22×C7—162no—
417C22×C7C2×#12011yescharacteristic
418C22×C7C2×#12011yescharacteristic
419C22×C7#53411yescharacteristic
420C22×C7D4214yes—
421C4×C7—22no—
422C4×C7—22no—
423C4×C7—22no—
424C4×C7—44no—
425C4×C7—42no—
426C4×C7—41no—
427C4×C7—44no—
428C4×C7—84no—
429C24—21no—
430C24—24no—
431C24—22no—
432C24—22no—
433C24—24no—
434C24—84no—
435C24C2×C7×D412yes—
436C24C2×C7×D411yescharacteristic
437C24C2×C7×D414yes—
438C24C2×C7×D411yescharacteristic
439C24C24×C711yescharacteristic
440#41—416no—
441#41—48no—
442#41—42no—
443#41—42no—
444C2×Q8—21no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—28no—
464C2×D4—28no—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4—416no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—48no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—48no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—84no—
477C2×D4—84no—
478C2×D4—88no—
479C2×D4C2×C7×D414yes—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4C2×C7×D412yes—
494C22×C4C2×C7×D411yescharacteristic
495C22×C4C2×C7×D414yes—
496C22×C4C2×C7×D411yescharacteristic
497C22×C4C2×C7×D411yescharacteristic
498C22×C4C2×C7×D411yescharacteristic
499C22×C4C2×C7×D414yes—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—28no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—48no—
518C22⋊C4—88no—
519C22⋊C4C2×C7×D414yes—
520C42—24no—
521C42—22no—
522C2×C7—21no—
523C2×C7—24no—
524C2×C7—21no—
525C2×C7—42no—
526C2×C7—84no—
527C2×C7—81no—
528C2×C7—82no—
529C2×C7#240311yescharacteristic
530C2×C7#302712yes—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—21no—
535C23—24no—
536C23—21no—
537C23—28no—
538C23—21no—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—84no—
553C23—84no—
554C23—88no—
555C23—82no—
556C23C7×#12012yes—
557C23C22×C7×D411yescharacteristic
558C23C22×C7×D412yes—
559C23C22×C7×D412yes—
560C23C7×H(5,𝔽2)12yes—
561Q8—44no—
562Q8—42no—
563D4—28no—
564D4—44no—
565D4—48no—
566D4—48no—
567D4—42no—
568D4—44no—
569D4—42no—
570D4—42no—
571D4—44no—
572D4—44no—
573D4—84no—
574D4—88no—
575D4—88no—
576D4—816no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—82no—
600C2×C4C7×#12012yes—
601C2×C4C7×#12012yes—
602C2×C4C7×#12012yes—
603C7#2390411yescharacteristic, direct product factor, Sylow
604C22—24no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—44no—
619C22—42no—
620C22—44no—
621C22—81no—
622C22—84no—
623C22—81no—
624C22—88no—
625C22—84no—
626C22—84no—
627C22—162no—
628C22C2×C7×#12011yescharacteristic
629C22C2×C7×#12011yescharacteristic
630C22C7×#53411yescharacteristic
631C22C7×D4214yes—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—44no—
636C4—42no—
637C4—41no—
638C4—44no—
639C4—84no—
640C2—21no—
641C2—24no—
642C2—21no—
643C2—42no—
644C2—84no—
645C2—81no—
646C2—82no—
647C2C7×#240311yescharacteristic
648C2C7×#302712yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
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
1. C7×#3485C2semidirect
2. C2×C7×#535C2semidirect
3. C2×C7×B(4,𝔽2)C2semidirect
4. C7×#2613C2semidirect
5. C7×#2021C2semidirect
6. C7×#1899C2semidirect
7. C7×#1889C2semidirect
8. C7×#1782C2semidirect
11. C23×C7×D4C22semidirect
27. C2×C7×#120C22semidirect
29. C2×C7×#117C22semidirect
33. C2×C7×#99C22semidirect
36. C7×#381C22semidirect
37. C7×#379C22semidirect
38. C7×#342C22semidirect
39. #23904C7direct (N × H)
48. C22×C7×Q8D4semidirect
59. C22×C7×D4D4semidirect
63. C23×C4×C7D4semidirect
64. C23×C4×C7D4semidirect
92. C7×#120D4semidirect
99. C2×C7×C4⋊C4D4semidirect
106. C2×C7×C22⋊C4D4semidirect
115. #3485C2×C7semidirect
116. C2×#535C2×C7semidirect
117. C2×B(4,𝔽2)C2×C7semidirect
118. #2613C2×C7semidirect
119. #2021C2×C7semidirect
120. #1899C2×C7semidirect
121. #1889C2×C7semidirect
122. #1782C2×C7semidirect
218. C23×D4C22×C7semidirect
234. C2×#120C22×C7semidirect
236. C2×#117C22×C7semidirect
240. C2×#99C22×C7semidirect
243. #381C22×C7semidirect
244. #379C22×C7semidirect
245. #342C22×C7semidirect
326. C22×Q8C7×D4semidirect
337. C22×D4C7×D4semidirect
341. C23×C4C7×D4semidirect
342. C23×C4C7×D4semidirect
370. #120C7×D4semidirect
377. C2×C4⋊C4C7×D4semidirect
384. C2×C22⋊C4C7×D4semidirect
603. C7#23904direct (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×#5352
C2×C7×B(4,𝔽2)2
C7×#26132
C7×#20212
C7×#18992
C7×#18892
C7×#17822
#239047