Finite Group Databasebeta

← C7×#23642 #13200858

Subgroups — C7×#23642 #13200858

Order 1792.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic, complemented
2C2×C7×#407C211yesmaximal, characteristic
3C7×#2280C211yesmaximal, characteristic
4C2×C7×#380C211yesmaximal, characteristic
5C7×#2077C212yesmaximal, complemented
6C7×#2077C212yesmaximal, complemented
7C7×#2076C212yesmaximal, complemented
8C7×#2075C212yesmaximal, complemented
9C7×#1736C211yesmaximal, characteristic, complemented
10C7×#1736C211yesmaximal, characteristic, complemented
11C7×#1736C211yesmaximal, characteristic, complemented
12C24×C4×C7C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×M16C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C22⋊C4—21no—
20C7×#428—24no—
21C2×C7×#107C2212yescomplemented
22C2×C7×#106C2212yescomplemented
23C7×#407—24no—
24C2×C7×C22⋊C8C2212yes—
25C2×C7×C22⋊C8C2212yes—
26C7×#381—21no—
27C7×#380—21no—
28C7×#380—21no—
29C7×#380C2211yescharacteristic
30C7×#380C2212yes—
31C7×#380C2211yescharacteristic
32C7×#380C2212yes—
33C7×#380C2211yescharacteristic
34C7×#380C2211yescharacteristic
35C7×#379—21no—
36C4×C7×C22⋊C4—21no—
37C4×C7×C22⋊C4—21no—
38C2×C7×#95—21no—
39C7×#340C2212yescomplemented
40C7×#340C2212yescomplemented
41C7×#340C2212yescomplemented
42C7×#340C2212yescomplemented
43C7×#340C2212yescomplemented
44C7×#340C2212yescomplemented
45#23642C711yesmaximal, characteristic, direct product factor, Sylow
46C25×C7D411yescharacteristic
47C23×C4×C7—22no—
48C23×C4×C7—22no—
49C23×C4×C7—22no—
50C23×C4×C7—22no—
51C23×C4×C7—22no—
52C23×C4×C7—22no—
53C23×C4×C7—21no—
54C23×C4×C7—21no—
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7D411yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C2×C7×M16—22no—
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16D414yes—
67C22×C7×C8C2312yes—
68C22×C7×C8C2312yes—
69C7×#117—22no—
70C7×#117—22no—
71C7×#117—22no—
72C7×#117—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—22no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2311yescharacteristic
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4—44no—
94C2×C42×C7—21no—
95C2×C42×C7—21no—
96C7×#107—24no—
97C7×#106—24no—
98C7×C22⋊C8—24no—
99C7×C22⋊C8—24no—
100C7×#95—22no—
101C7×#95—22no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C2×#428C2×C711yescharacteristic, complemented
107C2×#407C2×C711yescharacteristic
108#2280C2×C711yescharacteristic
109C2×#380C2×C711yescharacteristic
110#2077C2×C712yescomplemented
111#2077C2×C712yescomplemented
112#2076C2×C712yescomplemented
113#2075C2×C712yescomplemented
114#1736C2×C711yescharacteristic, complemented
115#1736C2×C711yescharacteristic, complemented
116#1736C2×C711yescharacteristic, complemented
117C24×C7—22no—
118C24×C7—22no—
119C24×C7—22no—
120C24×C7—22no—
121C24×C7—22no—
122C24×C7—41no—
123C24×C7—41no—
124C24×C7C2×D411yescharacteristic
125C24×C7C2×D411yescharacteristic
126C24×C7C2×D411yescharacteristic
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7—44no—
157C22×C4×C7—44no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D412yes—
168C22×C4×C7#4112yes—
169C22×C4×C7#4112yes—
170C22×C4×C7#4112yes—
171C22×C4×C7#4112yes—
172C22×C4×C7#4112yes—
173C22×C4×C7C2411yescharacteristic
174C7×M16—28no—
175C7×M16—44no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8C2×D414yes—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—42no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—44no—
199C7×C22⋊C4—42no—
200C7×C22⋊C4—42no—
201C7×C22⋊C4—42no—
202C7×C22⋊C4—44no—
203C7×C22⋊C4—42no—
204C42×C7—42no—
205C42×C7—42no—
206C24×C4C22×C711yescharacteristic
207C22×M16C22×C711yescharacteristic
208C22×M16C22×C711yescharacteristic
209C2×#117C22×C711yescharacteristic
210C2×#117C22×C711yescharacteristic
211C22×C4⋊C4C22×C711yescharacteristic
212C22×C4⋊C4C22×C711yescharacteristic
213C22×C22⋊C4—21no—
214#428—24no—
215C2×#107C22×C712yescomplemented
216C2×#106C22×C712yescomplemented
217#407—24no—
218C2×C22⋊C8C22×C712yes—
219C2×C22⋊C8C22×C712yes—
220#381—21no—
221#380—21no—
222#380—21no—
223#380C22×C711yescharacteristic
224#380C22×C712yes—
225#380C22×C711yescharacteristic
226#380C22×C712yes—
227#380C22×C711yescharacteristic
228#380C22×C711yescharacteristic
229#379—21no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C2×#95—21no—
233#340C22×C712yescomplemented
234#340C22×C712yescomplemented
235#340C22×C712yescomplemented
236#340C22×C712yescomplemented
237#340C22×C712yescomplemented
238#340C22×C712yescomplemented
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—28no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7—21no—
247C23×C7—21no—
248C23×C7—21no—
249C23×C7—21no—
250C23×C7—21no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—44no—
257C23×C7—44no—
258C23×C7—44no—
259C23×C7—44no—
260C23×C7—44no—
261C23×C7—44no—
262C23×C7#12112yes—
263C23×C7#12312yes—
264C23×C7#12312yes—
265C23×C7AGL(1,ℤ/8)12yes—
266C23×C7#13712yes—
267C23×C7C22×D411yescharacteristic
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—44no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—44no—
284C2×C4×C7—44no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
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—84no—
304C2×C4×C7—84no—
305C2×C4×C7#12112yes—
306C2×C4×C7#12314yes—
307C2×C4×C7#12312yes—
308C2×C4×C7#12312yes—
309C2×C4×C7#12414yes—
310C2×C4×C7#12614yes—
311C2×C4×C7#12614yes—
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C7×C8—24no—
319C7×C8—42no—
320C25C7×D411yescharacteristic
321C23×C4—22no—
322C23×C4—22no—
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—21no—
328C23×C4—21no—
329C23×C4C7×D411yescharacteristic
330C23×C4C7×D411yescharacteristic
331C23×C4C7×D411yescharacteristic
332C23×C4C23×C711yescharacteristic
333C23×C4C23×C711yescharacteristic
334C23×C4C23×C711yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16—22no—
340C2×M16C7×D414yes—
341C22×C8C23×C712yes—
342C22×C8C23×C712yes—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C23×C712yes—
354C2×C4⋊C4C23×C712yes—
355C2×C4⋊C4C23×C711yescharacteristic
356C2×C4⋊C4C23×C711yescharacteristic
357C2×C4⋊C4C23×C711yescharacteristic
358C2×C4⋊C4C23×C711yescharacteristic
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—44no—
368C2×C42—21no—
369C2×C42—21no—
370#107—24no—
371#106—24no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—22no—
384C22×C7—22no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—22no—
388C22×C7—22no—
389C22×C7—22no—
390C22×C7—22no—
391C22×C7—22no—
392C22×C7—416no—
393C22×C7—41no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7#39714yes—
402C22×C7C2×#12111yescharacteristic
403C22×C7C2×#12311yescharacteristic
404C22×C7C2×#12311yescharacteristic
405C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C7C2×#13711yescharacteristic
407C22×C7#57511yescharacteristic
408C22×C7#57511yescharacteristic
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—81no—
412C4×C7—81no—
413C4×C7—82no—
414C4×C7—81no—
415C4×C7—82no—
416C4×C7—81no—
417C4×C7#39714yes—
418C4×C7C2×#12311yescharacteristic
419C4×C7C2×#12411yescharacteristic
420C4×C7C2×#12611yescharacteristic
421C4×C7C2×#12611yescharacteristic
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—41no—
428C24—41no—
429C24C2×C7×D411yescharacteristic
430C24C2×C7×D411yescharacteristic
431C24C2×C7×D411yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C2×C7×D412yes—
473C22×C4C7×#4112yes—
474C22×C4C7×#4112yes—
475C22×C4C7×#4112yes—
476C22×C4C7×#4112yes—
477C22×C4C7×#4112yes—
478C22×C4C24×C711yescharacteristic
479M16—28no—
480M16—44no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8C2×C7×D414yes—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C42—42no—
510C42—42no—
511C2×C7—24no—
512C2×C7—42no—
513C2×C7—42no—
514C2×C7C2×#39711yescharacteristic
515C2×C7#305511yescharacteristic
516C2×C7#305611yescharacteristic
517C2×C7#309211yescharacteristic
518C2×C7#309311yescharacteristic
519C2×C7#309411yescharacteristic
520C2×C7#309511yescharacteristic
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—28no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C7×#12112yes—
545C23C7×#12312yes—
546C23C7×#12312yes—
547C23C7×AGL(1,ℤ/8)12yes—
548C23C7×#13712yes—
549C23C22×C7×D411yescharacteristic
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4C7×#12112yes—
588C2×C4C7×#12314yes—
589C2×C4C7×#12312yes—
590C2×C4C7×#12312yes—
591C2×C4C7×#12414yes—
592C2×C4C7×#12614yes—
593C2×C4C7×#12614yes—
594C2×C4C22×C7×D411yescharacteristic
595C2×C4C2×C7×#4111yescharacteristic
596C2×C4C2×C7×#4111yescharacteristic
597C2×C4C2×C7×#4111yescharacteristic
598C2×C4C2×C7×#4111yescharacteristic
599C2×C4C2×C7×#4111yescharacteristic
600C8—24no—
601C8—42no—
602C7#2364211yescharacteristic, direct product factor, Sylow
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22C7×#39714yes—
625C22C2×C7×#12111yescharacteristic
626C22C2×C7×#12311yescharacteristic
627C22C2×C7×#12311yescharacteristic
628C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
629C22C2×C7×#13711yescharacteristic
630C22C7×#57511yescharacteristic
631C22C7×#57511yescharacteristic
632C4—41no—
633C4—41no—
634C4—82no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4—82no—
640C4C7×#39714yes—
641C4C2×C7×#12311yescharacteristic
642C4C2×C7×#12411yescharacteristic
643C4C2×C7×#12611yescharacteristic
644C4C2×C7×#12611yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×C7×#39711yescharacteristic
649C2C7×#305511yescharacteristic
650C2C7×#305611yescharacteristic
651C2C7×#309211yescharacteristic
652C2C7×#309311yescharacteristic
653C2C7×#309411yescharacteristic
654C2C7×#309511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC25×C7D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#428C2semidirect
5. C7×#2077C2semidirect
6. C7×#2077C2semidirect
7. C7×#2076C2semidirect
8. C7×#2075C2semidirect
9. C7×#1736C2semidirect
10. C7×#1736C2semidirect
11. C7×#1736C2semidirect
21. C2×C7×#107C22semidirect
22. C2×C7×#106C22semidirect
39. C7×#340C22semidirect
40. C7×#340C22semidirect
41. C7×#340C22semidirect
42. C7×#340C22semidirect
43. C7×#340C22semidirect
44. C7×#340C22semidirect
45. #23642C7direct (N × H)
106. C2×#428C2×C7semidirect
110. #2077C2×C7semidirect
111. #2077C2×C7semidirect
112. #2076C2×C7semidirect
113. #2075C2×C7semidirect
114. #1736C2×C7semidirect
115. #1736C2×C7semidirect
116. #1736C2×C7semidirect
215. C2×#107C22×C7semidirect
216. C2×#106C22×C7semidirect
233. #340C22×C7semidirect
234. #340C22×C7semidirect
235. #340C22×C7semidirect
236. #340C22×C7semidirect
237. #340C22×C7semidirect
238. #340C22×C7semidirect
602. C7#23642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×C7×#4072
C7×#22802
C2×C7×#3802
C7×#20772
C7×#20762
C7×#20752
C7×#17362
#236427