Finite Group Databasebeta

← C7×#24761 #13201977

Subgroups — C7×#24761 #13201977

Order 1792.

1100 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
1C2×C7×#468C212yesmaximal, complemented
2C2×C7×#460C211yesmaximal, characteristic, complemented
3C2×C7×#423C211yesmaximal, characteristic, complemented
4C7×#2856C211yesmaximal, characteristic
5C4×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
6C7×#2067C212yesmaximal, complemented
7C7×#2055C212yesmaximal, complemented
8C7×#1982C211yesmaximal, characteristic, complemented
9C7×#1947C211yesmaximal, characteristic, complemented
10C7×#1895C212yesmaximal, complemented
11C7×#1803C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C22×C7×M16C2212yes—
14C2×C7×#127C2211yescharacteristic
15C2×C7×C22⋊Q8C2211yescharacteristic
16C2×C7×#121—21no—
17C2×C7×#121—21no—
18C2×C7×#121C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C22×C42×C7C2211yescharacteristic
21C7×#468—28no—
22C7×#460—24no—
23C2×C7×#107C2212yes—
24C7×#423—24no—
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×C4⋊C8C2212yes—
27C2×C7×C4⋊C8C2211yescharacteristic
28C2×C7×#103C2212yes—
29C2×C7×#102C2212yes—
30C2×C7×#102C2212yes—
31C2×C7×#102C2212yes—
32C7×#391C2211yescharacteristic
33C7×#390—21no—
34C7×#390—21no—
35C7×#390C2212yes—
36C7×#389C2211yescharacteristic
37C7×#387—21no—
38C7×#385—21no—
39C7×#384C2211yescharacteristic
40C7×#380C2211yescharacteristic
41C4×C7×C4⋊C4—21no—
42C4×C7×C4⋊C4C2211yescharacteristic
43C4×C7×C22⋊C4—21no—
44C7×#340C2212yes—
45C7×#340C2212yes—
46#24761C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—22no—
49C22×C7×D4—21no—
50C22×C7×D4—21no—
51C22×C7×D4C2312yes—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C2×C7×SD16—24no—
56C2×C7×SD16—24no—
57C2×C7×SD16—42no—
58C2×C7×M16—24no—
59C2×C7×M16—24no—
60C2×C7×M16—42no—
61C22×C7×C8C2312yes—
62C22×C7×C8C2312yes—
63C7×#127—22no—
64C7×#127—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8D414yescomplemented
67C7×#121—24no—
68C7×#121—24no—
69C7×#121—44no—
70C7×#121—44no—
71C4×C7×Q8—22no—
72C4×C7×Q8—22no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7—21no—
90C2×C42×C7—21no—
91C2×C42×C7D411yescharacteristic
92C2×C42×C7D411yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C2×C42×C7C2311yescharacteristic
95C7×#107—24no—
96C7×C4⋊C8—24no—
97C7×C4⋊C8—22no—
98C7×C4⋊C8D414yescomplemented
99C7×#103—24no—
100C7×#102—24no—
101C7×#102—24no—
102C7×#102—42no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C2×#468C2×C712yescomplemented
109C2×#460C2×C711yescharacteristic, complemented
110C2×#423C2×C711yescharacteristic, complemented
111#2856C2×C711yescharacteristic
112C4×C22⋊Q8C2×C711yescharacteristic, complemented
113#2067C2×C712yescomplemented
114#2055C2×C712yescomplemented
115#1982C2×C711yescharacteristic, complemented
116#1947C2×C711yescharacteristic, complemented
117#1895C2×C712yescomplemented
118#1803C2×C711yescharacteristic, complemented
119C24×C7—22no—
120C24×C7C2×D411yescharacteristic
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8C2×D412yes—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—44no—
136C2×C7×D4—44no—
137C2×C7×D4—44no—
138C2×C7×D4—42no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4—42no—
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—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—41no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C7×SD16—48no—
166C7×M16—48no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C2×C7×C8C2×D414yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4C2×D412yes—
186C7×C22⋊C4—24no—
187C7×C22⋊C4—44no—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—44no—
190C7×C22⋊C4—44no—
191C42×C7—22no—
192C42×C7—22no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—42no—
196C42×C7—42no—
197C42×C7C2×D412yes—
198C42×C7#4112yes—
199C22×SD16C22×C712yes—
200C22×M16C22×C712yes—
201C2×#127C22×C711yescharacteristic
202C2×C22⋊Q8C22×C711yescharacteristic
203C2×#121—21no—
204C2×#121—21no—
205C2×#121C22×C712yes—
206C2×C4×Q8C22×C711yescharacteristic
207C22×C42C22×C711yescharacteristic
208#468—28no—
209#460—24no—
210C2×#107C22×C712yes—
211#423—24no—
212C2×C4⋊C8C22×C712yes—
213C2×C4⋊C8C22×C711yescharacteristic
214C2×C4⋊C8C22×C711yescharacteristic
215C2×#103C22×C712yes—
216C2×#102C22×C712yes—
217C2×#102C22×C712yes—
218C2×#102C22×C712yes—
219#391C22×C711yescharacteristic
220#390—21no—
221#390—21no—
222#390C22×C712yes—
223#389C22×C711yescharacteristic
224#387—21no—
225#385—21no—
226#384C22×C711yescharacteristic
227#380C22×C711yescharacteristic
228C4×C4⋊C4—21no—
229C4×C4⋊C4C22×C711yescharacteristic
230C4×C22⋊C4—21no—
231#340C22×C712yes—
232#340C22×C712yes—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—21no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7#12112yes—
247C23×C7C22×D411yescharacteristic
248C7×Q8—22no—
249C7×Q8—22no—
250C7×Q8—41no—
251C7×Q8—41no—
252C7×D4—42no—
253C7×D4—42no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4—84no—
257C7×D4—88no—
258C7×D4—84no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
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—44no—
280C2×C4×C7—41no—
281C2×C4×C7—42no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7#12112yes—
290C2×C4×C7#12112yes—
291C2×C4×C7#12112yes—
292C2×C4×C7AGL(1,ℤ/8)12yes—
293C2×C4×C7AGL(1,ℤ/8)12yes—
294C2×C4×C7#13712yes—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C7×C8—24no—
302C7×C8—42no—
303C22×Q8C23×C711yescharacteristic
304C22×D4—22no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4C23×C712yes—
308C23×C4C7×D411yescharacteristic
309C23×C4C7×D411yescharacteristic
310C23×C4C23×C711yescharacteristic
311C2×SD16—24no—
312C2×SD16—24no—
313C2×SD16—42no—
314C2×M16—24no—
315C2×M16—24no—
316C2×M16—42no—
317C22×C8C23×C712yes—
318C22×C8C23×C712yes—
319#127—22no—
320#127—22no—
321C22⋊Q8—22no—
322C22⋊Q8C7×D414yescomplemented
323#121—24no—
324#121—24no—
325#121—44no—
326#121—44no—
327C4×Q8—22no—
328C4×Q8—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4C23×C711yescharacteristic
335C2×C4⋊C4C23×C712yes—
336C2×C4⋊C4C23×C711yescharacteristic
337C2×C4⋊C4C23×C711yescharacteristic
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C7×D411yescharacteristic
348C2×C42C7×D411yescharacteristic
349C2×C42C23×C711yescharacteristic
350C2×C42C23×C711yescharacteristic
351#107—24no—
352C4⋊C8—24no—
353C4⋊C8—22no—
354C4⋊C8C7×D414yescomplemented
355#103—24no—
356#102—24no—
357#102—24no—
358#102—42no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364C22×C7—22no—
365C22×C7—22no—
366C22×C7—22no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7—41no—
370C22×C7—41no—
371C22×C7—82no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7C2×#12111yescharacteristic
379C22×C7C2×#12111yescharacteristic
380C22×C7#53711yescharacteristic
381C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
382C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
383C22×C7C2×#13711yescharacteristic
384C22×C7#57511yescharacteristic
385C4×C7—22no—
386C4×C7—22no—
387C4×C7—22no—
388C4×C7—42no—
389C4×C7—42no—
390C4×C7—41no—
391C4×C7—81no—
392C4×C7—82no—
393C4×C7—81no—
394C4×C7#46814yescomplemented
395C4×C7D4211yescharacteristic
396C4×C7#54711yescharacteristic
397C4×C7#54811yescharacteristic
398C4×C7#55311yescharacteristic
399C24—22no—
400C24C2×C7×D411yescharacteristic
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8C2×C7×D412yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—41no—
437C22×C4C2×C7×D411yescharacteristic
438C22×C4C2×C7×D411yescharacteristic
439C22×C4C2×C7×D411yescharacteristic
440C22×C4C2×C7×D412yes—
441C22×C4C2×C7×D411yescharacteristic
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C7×#4112yes—
444C22×C4C24×C711yescharacteristic
445SD16—48no—
446M16—48no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8C2×C7×D414yes—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4C2×C7×D412yes—
466C22⋊C4—24no—
467C22⋊C4—44no—
468C22⋊C4—42no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C42—22no—
472C42—22no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—42no—
477C42C2×C7×D412yes—
478C42C7×#4112yes—
479C2×C7—42no—
480C2×C7—162no—
481C2×C7#241611yescharacteristic
482C2×C7#303711yescharacteristic
483C2×C7#303811yescharacteristic
484C2×C7C2×#46811yescharacteristic
485C2×C7#305511yescharacteristic
486C2×C7#325111yescharacteristic
487C2×C7#325411yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C7×#12112yes—
502C23C22×C7×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—84no—
512D4—88no—
513D4—84no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4C7×#12112yes—
545C2×C4C7×#12112yes—
546C2×C4C7×#12112yes—
547C2×C4C7×AGL(1,ℤ/8)12yes—
548C2×C4C7×AGL(1,ℤ/8)12yes—
549C2×C4C7×#13712yes—
550C2×C4C22×C7×D411yescharacteristic
551C2×C4C22×C7×D411yescharacteristic
552C2×C4C2×C7×#4111yescharacteristic
553C2×C4C2×C7×#4111yescharacteristic
554C2×C4C7×H(5,𝔽2)11yescharacteristic
555C2×C4C7×2−1+411yescharacteristic
556C8—24no—
557C8—42no—
558C7#2476111yescharacteristic, direct product factor, Sylow
559C22—22no—
560C22—22no—
561C22—22no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22C2×C7×#12111yescharacteristic
574C22C2×C7×#12111yescharacteristic
575C22C7×#53711yescharacteristic
576C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
577C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
578C22C2×C7×#13711yescharacteristic
579C22C7×#57511yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—82no—
588C4—81no—
589C4C7×#46814yescomplemented
590C4C7×D4211yescharacteristic
591C4C7×#54711yescharacteristic
592C4C7×#54811yescharacteristic
593C4C7×#55311yescharacteristic
594C2—42no—
595C2—162no—
596C2C7×#241611yescharacteristic
597C2C7×#303711yescharacteristic
598C2C7×#303811yescharacteristic
599C2C2×C7×#46811yescharacteristic
600C2C7×#305511yescharacteristic
601C2C7×#325111yescharacteristic
602C2C7×#325411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#468C2semidirect
2. C2×C7×#460C2semidirect
3. C2×C7×#423C2semidirect
5. C4×C7×C22⋊Q8C2semidirect
6. C7×#2067C2semidirect
7. C7×#2055C2semidirect
8. C7×#1982C2semidirect
9. C7×#1947C2semidirect
10. C7×#1895C2semidirect
11. C7×#1803C2semidirect
46. #24761C7direct (N × H)
66. C7×C22⋊Q8D4semidirect
98. C7×C4⋊C8D4semidirect
108. C2×#468C2×C7semidirect
109. C2×#460C2×C7semidirect
110. C2×#423C2×C7semidirect
112. C4×C22⋊Q8C2×C7semidirect
113. #2067C2×C7semidirect
114. #2055C2×C7semidirect
115. #1982C2×C7semidirect
116. #1947C2×C7semidirect
117. #1895C2×C7semidirect
118. #1803C2×C7semidirect
322. C22⋊Q8C7×D4semidirect
354. C4⋊C8C7×D4semidirect
394. C4×C7#468semidirect
558. C7#24761direct (N × H)
589. C4C7×#468semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4682
C2×C7×#4602
C2×C7×#4232
C7×#28562
C4×C7×C22⋊Q82
C7×#20672
C7×#20552
C7×#19822
C7×#19472
C7×#18952
C7×#18032
#247617