Finite Group Databasebeta

← C7×#24059 #13201275

Subgroups — C7×#24059 #13201275

Order 1792.

1348 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic
2C7×#2623C212yesmaximal, complemented
3C7×#2311C211yesmaximal, characteristic, complemented
4C7×#2044C214yesmaximal, complemented
5C7×#1899C212yesmaximal, complemented
6C7×#1867C212yesmaximal, complemented
7C7×#1790C212yesmaximal, complemented
8C7×#1782C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C22×C7×#41C2211yescharacteristic
11C7×#547—22no—
12C7×#547—22no—
13C7×#546—22no—
14C7×#546—22no—
15C7×#536—21no—
16C7×#534—21no—
17C7×#534C2212yes—
18C7×#525—21no—
19C2×C7×#123—21no—
20C2×C7×#123C2212yes—
21C7×#518C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic, complemented
23C22×C7×C4⋊C4C2211yescharacteristic, complemented
24C2×C7×#99C2212yes—
25C2×C7×#99C2214yes—
26C2×C7×#99C2212yescomplemented
27C7×#400—22no—
28C7×#399—22no—
29C7×#398—22no—
30C7×#397—22no—
31C7×#391—21no—
32C7×#390—21no—
33C7×#386C2214yes—
34C7×#386C2212yes—
35C7×#386C2212yescomplemented
36C7×#385C2214yescomplemented
37C7×#381C2212yescomplemented
38C7×#380—21no—
39C7×#380—21no—
40C7×#342C2214yescomplemented
41#24059C711yesmaximal, characteristic, direct product factor, Sylow
42C7×H(5,𝔽2)—24no—
43C7×H(5,𝔽2)—42no—
44C2×C7×#41—24no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C2×C7×#41—21no—
48C2×C7×#41—22no—
49C2×C7×#41—42no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×D4—21no—
52C22×C7×D4—22no—
53C22×C7×D4D411yescharacteristic
54C22×C7×D4C2311yescharacteristic
55C22×C7×D4C2312yes—
56C22×C7×D4C2311yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C7×C4⋊Q8—22no—
62C7×#124—22no—
63C7×#123—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×C22⋊Q8—21no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×#121—21no—
77C7×#121—21no—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121—22no—
81C7×#121—22no—
82C7×#120—22no—
83C7×#120—22no—
84C7×#120—22no—
85C4×C7×D4—22no—
86C4×C7×D4—22no—
87C4×C7×D4—22no—
88C4×C7×D4—22no—
89C7×#117—21no—
90C7×#117—42no—
91C2×C7×C4⋊C4—21no—
92C2×C7×C4⋊C4—21no—
93C2×C7×C4⋊C4—21no—
94C2×C7×C4⋊C4—21no—
95C2×C7×C4⋊C4—22no—
96C2×C7×C4⋊C4—22no—
97C2×C7×C4⋊C4—21no—
98C2×C7×C4⋊C4—24no—
99C2×C7×C4⋊C4D412yes—
100C2×C7×C4⋊C4D412yes—
101C2×C7×C22⋊C4—21no—
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—22no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4—24no—
107C2×C7×C22⋊C4D412yes—
108C2×C7×C22⋊C4D412yes—
109C2×C7×C22⋊C4C2312yes—
110C2×C7×C22⋊C4C2314yes—
111C2×C7×C22⋊C4C2312yes—
112C2×C42×C7—21no—
113C7×#99—24no—
114C7×#99—44no—
115C7×#99—42no—
116C7×#95—22no—
117C7×#95—22no—
118C7×#95—22no—
119C7×#95—22no—
120C7×#95—24no—
121C7×#95—24no—
122#3485C2×C711yescharacteristic
123#2623C2×C712yescomplemented
124#2311C2×C711yescharacteristic, complemented
125#2044C2×C714yescomplemented
126#1899C2×C712yescomplemented
127#1867C2×C712yescomplemented
128#1790C2×C712yescomplemented
129#1782C2×C711yescharacteristic, complemented
130C24×C7—21no—
131C24×C7C2×D412yes—
132C24×C7C2×D411yescharacteristic
133C24×C7C2×D411yescharacteristic
134C24×C7C2411yescharacteristic
135C7×#41—48no—
136C7×#41—48no—
137C7×#41—48no—
138C7×#41—42no—
139C7×#41—42no—
140C2×C7×Q8—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—41no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—21no—
148C2×C7×D4—22no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—24no—
154C2×C7×D4—22no—
155C2×C7×D4—24no—
156C2×C7×D4—22no—
157C2×C7×D4—22no—
158C2×C7×D4—22no—
159C2×C7×D4—41no—
160C2×C7×D4—48no—
161C2×C7×D4—42no—
162C2×C7×D4—42no—
163C2×C7×D4#4114yes—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—21no—
170C22×C4×C7—21no—
171C22×C4×C7—21no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—22no—
176C22×C4×C7—24no—
177C22×C4×C7—22no—
178C22×C4×C7—22no—
179C22×C4×C7—24no—
180C22×C4×C7—41no—
181C22×C4×C7—41no—
182C22×C4×C7—82no—
183C22×C4×C7C2×D412yes—
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D412yes—
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D411yescharacteristic
188C22×C4×C7C2×D411yescharacteristic
189C22×C4×C7C2×D411yescharacteristic
190C22×C4×C7C2×D412yes—
191C22×C4×C7C2×D412yes—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—42no—
201C7×C4⋊C4—41no—
202C7×C4⋊C4—41no—
203C7×C4⋊C4—42no—
204C7×C4⋊C4—84no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—24no—
217C7×C22⋊C4—24no—
218C7×C22⋊C4—24no—
219C7×C22⋊C4—82no—
220C7×C22⋊C4—82no—
221C7×C22⋊C4—84no—
222C42×C7—22no—
223C42×C7—42no—
224C2×H(5,𝔽2)C22×C711yescharacteristic
225C22×#41C22×C711yescharacteristic
226#547—22no—
227#547—22no—
228#546—22no—
229#546—22no—
230#536—21no—
231#534—21no—
232#534C22×C712yes—
233#525—21no—
234C2×#123—21no—
235C2×#123C22×C712yes—
236#518C22×C711yescharacteristic
237C2×#117C22×C711yescharacteristic, complemented
238C22×C4⋊C4C22×C711yescharacteristic, complemented
239C2×#99C22×C712yes—
240C2×#99C22×C714yes—
241C2×#99C22×C712yescomplemented
242#400—22no—
243#399—22no—
244#398—22no—
245#397—22no—
246#391—21no—
247#390—21no—
248#386C22×C712yes—
249#386C22×C714yes—
250#386C22×C712yescomplemented
251#385C22×C714yescomplemented
252#381C22×C712yescomplemented
253#380—21no—
254#380—21no—
255#342C22×C714yescomplemented
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—22no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7—24no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—42no—
270C23×C7—42no—
271C23×C7—42no—
272C23×C7—81no—
273C23×C7—81no—
274C23×C7#12312yes—
275C23×C7C22×D411yescharacteristic
276C23×C7C22×D412yes—
277C23×C7C2×#4112yes—
278C23×C7H(5,𝔽2)12yes—
279C7×Q8—44no—
280C7×Q8—42no—
281C7×D4—28no—
282C7×D4—44no—
283C7×D4—42no—
284C7×D4—44no—
285C7×D4—42no—
286C7×D4—42no—
287C7×D4—44no—
288C7×D4—44no—
289C7×D4—82no—
290C7×D4—82no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—24no—
298C2×C4×C7—24no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—24no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—44no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—44no—
316C2×C4×C7—44no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—44no—
320C2×C4×C7—44no—
321C2×C4×C7—42no—
322C2×C4×C7—42no—
323C2×C4×C7—42no—
324C2×C4×C7—44no—
325C2×C4×C7—81no—
326C2×C4×C7—81no—
327C2×C4×C7#12011yescharacteristic
328C2×C4×C7#12011yescharacteristic
329C2×C4×C7#12012yes—
330C2×C4×C7#12312yes—
331H(5,𝔽2)—24no—
332H(5,𝔽2)—42no—
333C2×#41—24no—
334C2×#41—21no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—42no—
339C22×Q8C7×D411yescharacteristic
340C22×D4—21no—
341C22×D4—22no—
342C22×D4C7×D411yescharacteristic
343C22×D4C23×C712yes—
344C22×D4C23×C711yescharacteristic
345C22×D4C23×C711yescharacteristic
346C23×C4C7×D411yescharacteristic
347C23×C4C7×D411yescharacteristic
348C23×C4C23×C711yescharacteristic
349C23×C4C23×C712yes—
350C4⋊Q8—22no—
351#124—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355#123—22no—
356#123—22no—
357#123—22no—
358#123—22no—
359#123—22no—
360#123—22no—
361C22⋊Q8—21no—
362C22⋊Q8—21no—
363C22⋊Q8—22no—
364C22⋊Q8—22no—
365#121—21no—
366#121—21no—
367#121—22no—
368#121—22no—
369#121—22no—
370#121—22no—
371#120—22no—
372#120—22no—
373#120—22no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378#117—21no—
379#117—42no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4C7×D412yes—
389C2×C4⋊C4C7×D412yes—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4C7×D412yes—
397C2×C22⋊C4C7×D412yes—
398C2×C22⋊C4C23×C712yes—
399C2×C22⋊C4C23×C714yes—
400C2×C22⋊C4C23×C712yes—
401C2×C42—21no—
402#99—24no—
403#99—44no—
404#99—42no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—24no—
410#95—24no—
411C22×C7—22no—
412C22×C7—21no—
413C22×C7—22no—
414C22×C7—21no—
415C22×C7—22no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7—42no—
424C22×C7—41no—
425C22×C7—42no—
426C22×C7—41no—
427C22×C7—42no—
428C22×C7—81no—
429C22×C7—81no—
430C22×C7—84no—
431C22×C7—162no—
432C22×C7C2×#12011yescharacteristic
433C22×C7C2×#12311yescharacteristic
434C22×C7#53411yescharacteristic
435C22×C7#54614yes—
436C4×C7—21no—
437C4×C7—21no—
438C4×C7—22no—
439C4×C7—22no—
440C4×C7—42no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—42no—
444C4×C7—82no—
445C4×C7—84no—
446C4×C7—82no—
447C24—21no—
448C24C2×C7×D412yes—
449C24C2×C7×D411yescharacteristic
450C24C2×C7×D411yescharacteristic
451C24C24×C711yescharacteristic
452#41—48no—
453#41—48no—
454#41—48no—
455#41—42no—
456#41—42no—
457C2×Q8—21no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—41no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—24no—
471C2×D4—22no—
472C2×D4—24no—
473C2×D4—22no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—41no—
477C2×D4—48no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4C7×#4114yes—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—82no—
500C22×C4C2×C7×D412yes—
501C22×C4C2×C7×D411yescharacteristic
502C22×C4C2×C7×D412yes—
503C22×C4C2×C7×D412yes—
504C22×C4C2×C7×D411yescharacteristic
505C22×C4C2×C7×D411yescharacteristic
506C22×C4C2×C7×D411yescharacteristic
507C22×C4C2×C7×D412yes—
508C22×C4C2×C7×D412yes—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—42no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—42no—
521C4⋊C4—84no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—84no—
539C42—22no—
540C42—42no—
541C2×C7—21no—
542C2×C7—24no—
543C2×C7—21no—
544C2×C7—42no—
545C2×C7—81no—
546C2×C7—82no—
547C2×C7#240511yescharacteristic
548C2×C7#302711yescharacteristic
549C2×C7#302811yescharacteristic
550C23—21no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—24no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—81no—
567C23—81no—
568C23C7×#12312yes—
569C23C22×C7×D411yescharacteristic
570C23C22×C7×D412yes—
571C23C2×C7×#4112yes—
572C23C7×H(5,𝔽2)12yes—
573Q8—44no—
574Q8—42no—
575D4—28no—
576D4—44no—
577D4—42no—
578D4—44no—
579D4—42no—
580D4—42no—
581D4—44no—
582D4—44no—
583D4—82no—
584D4—82no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4C7×#12011yescharacteristic
622C2×C4C7×#12011yescharacteristic
623C2×C4C7×#12012yes—
624C2×C4C7×#12312yes—
625C7#2405911yescharacteristic, direct product factor, Sylow
626C22—22no—
627C22—21no—
628C22—22no—
629C22—21no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—42no—
639C22—41no—
640C22—42no—
641C22—41no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—84no—
646C22—162no—
647C22C2×C7×#12011yescharacteristic
648C22C2×C7×#12311yescharacteristic
649C22C7×#53411yescharacteristic
650C22C7×#54614yes—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—82no—
660C4—84no—
661C4—82no—
662C2—21no—
663C2—24no—
664C2—21no—
665C2—42no—
666C2—81no—
667C2—82no—
668C2C7×#240511yescharacteristic
669C2C7×#302711yescharacteristic
670C2C7×#302811yescharacteristic
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. C7×#2623C2semidirect
3. C7×#2311C2semidirect
4. C7×#2044C2semidirect
5. C7×#1899C2semidirect
6. C7×#1867C2semidirect
7. C7×#1790C2semidirect
8. C7×#1782C2semidirect
22. C2×C7×#117C22semidirect
23. C22×C7×C4⋊C4C22semidirect
26. C2×C7×#99C22semidirect
35. C7×#386C22semidirect
36. C7×#385C22semidirect
37. C7×#381C22semidirect
40. C7×#342C22semidirect
41. #24059C7direct (N × H)
123. #2623C2×C7semidirect
124. #2311C2×C7semidirect
125. #2044C2×C7semidirect
126. #1899C2×C7semidirect
127. #1867C2×C7semidirect
128. #1790C2×C7semidirect
129. #1782C2×C7semidirect
237. C2×#117C22×C7semidirect
238. C22×C4⋊C4C22×C7semidirect
241. C2×#99C22×C7semidirect
250. #386C22×C7semidirect
251. #385C22×C7semidirect
252. #381C22×C7semidirect
255. #342C22×C7semidirect
625. C7#24059direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C7×#26232
C7×#23112
C7×#20442
C7×#18992
C7×#18672
C7×#17902
C7×#17822
#240597