Finite Group Databasebeta

← C7×#15177 #13192393

Subgroups — C7×#15177 #13192393

Order 1792.

2770 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C212yesmaximal, complemented
2C22×C4×C7×D4C211yesmaximal, characteristic
3C2×C7×#391C211yesmaximal, characteristic, complemented
4C2×C7×#380C212yesmaximal, complemented
5C7×#1444C212yesmaximal, complemented
6C7×#1439C214yesmaximal, complemented
7C7×#1438C212yesmaximal
8C7×C4⋊#95C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2212yes—
12C2×C7×C22⋊Q8—28no—
13C2×C7×C22⋊Q8—28no—
14C2×C7×C22⋊Q8—28no—
15C2×C4×C7×D4—28no—
16C2×C4×C7×D4—24no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2214yes—
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C4⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C7×C22⋊C4C2214yes—
23C22×C7×C22⋊C4C2214yes—
24C22×C42×C7C2211yescharacteristic
25C7×#391—24no—
26C7×#380—28no—
27C2×C7×#95C2214yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32#15177C711yesmaximal, characteristic, direct product factor, Sylow
33C25×C7C2312yes—
34C22×C7×Q8—24no—
35C22×C7×Q8—24no—
36C22×C7×Q8—22no—
37C22×C7×Q8—22no—
38C22×C7×Q8D414yescomplemented
39C22×C7×D4—24no—
40C22×C7×D4—24no—
41C22×C7×D4—22no—
42C22×C7×D4—22no—
43C22×C7×D4Q814yescomplemented
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—24no—
47C23×C4×C7—24no—
48C23×C4×C7—24no—
49C23×C4×C7—24no—
50C23×C4×C7D418yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2314yes—
55C23×C4×C7C2312yes—
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C7×C22⋊Q8—264no—
59C4×C7×D4—232no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—22no—
62C2×C7×C4⋊C4—28no—
63C2×C7×C4⋊C4—28no—
64C2×C7×C4⋊C4—28no—
65C2×C7×C4⋊C4—24no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4D418yescomplemented
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—28no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4—28no—
76C2×C7×C22⋊C4—28no—
77C2×C7×C22⋊C4—28no—
78C2×C7×C22⋊C4—44no—
79C2×C42×C7—24no—
80C2×C42×C7D414yes—
81C7×#95—28no—
82C7×#95—24no—
83C7×#95—24no—
84C7×#95—24no—
85C7×#95—42no—
86C22×C22⋊Q8C2×C712yescomplemented
87C22×C4×D4C2×C711yescharacteristic
88C2×#391C2×C711yescharacteristic, complemented
89C2×#380C2×C712yescomplemented
90#1444C2×C712yescomplemented
91#1439C2×C714yescomplemented
92#1438C2×C712yes—
93C4⋊#95C2×C711yescharacteristic, complemented
94C24×C7—22no—
95C24×C7—22no—
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
105C24×C7—22no—
106C24×C7C2×D414yes—
107C24×C7C2×Q814yes—
108C24×C7#4114yes—
109C24×C7C2411yescharacteristic
110C2×C7×Q8—216no—
111C2×C7×Q8—28no—
112C2×C7×Q8—44no—
113C2×C7×Q8—48no—
114C2×C7×Q8—44no—
115C2×C7×D4—28no—
116C2×C7×D4—28no—
117C2×C7×D4—216no—
118C2×C7×D4—216no—
119C2×C7×D4—28no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—24no—
126C22×C4×C7—24no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—24no—
130C22×C4×C7—24no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—24no—
134C22×C4×C7—24no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—24no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—24no—
142C22×C4×C7—216no—
143C22×C4×C7—216no—
144C22×C4×C7—216no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—24no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—48no—
151C22×C4×C7—48no—
152C22×C4×C7—48no—
153C22×C4×C7—48no—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D418yes—
156C22×C4×C7C2×D414yes—
157C22×C4×C7C2×D414yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D414yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D414yes—
163C22×C4×C7C2×Q812yes—
164C22×C4×C7#4112yes—
165C22×C4×C7#4112yes—
166C7×C4⋊C4—28no—
167C7×C4⋊C4—232no—
168C7×C4⋊C4—216no—
169C7×C4⋊C4—44no—
170C7×C22⋊C4—216no—
171C7×C22⋊C4—232no—
172C7×C22⋊C4—416no—
173C42×C7—28no—
174C23×Q8C22×C711yescharacteristic
175C23×D4C22×C711yescharacteristic
176C24×C4C22×C712yes—
177C2×C22⋊Q8—28no—
178C2×C22⋊Q8—28no—
179C2×C22⋊Q8—28no—
180C2×C4×D4—28no—
181C2×C4×D4—24no—
182C22×C4⋊C4C22×C711yescharacteristic
183C22×C4⋊C4C22×C714yes—
184C22×C4⋊C4C22×C711yescharacteristic
185C22×C4⋊C4C22×C712yes—
186C22×C22⋊C4C22×C712yes—
187C22×C22⋊C4C22×C714yes—
188C22×C22⋊C4C22×C714yes—
189C22×C42C22×C711yescharacteristic
190#391—24no—
191#380—28no—
192C2×#95C22×C712yes—
193C2×#95C22×C712yes—
194C2×#95C22×C712yes—
195C2×#95C22×C714yes—
196C2×#95C22×C712yes—
197C23×C7—24no—
198C23×C7—24no—
199C23×C7—24no—
200C23×C7—24no—
201C23×C7—24no—
202C23×C7—24no—
203C23×C7—24no—
204C23×C7—24no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—24no—
208C23×C7—24no—
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—42no—
216C23×C7—42no—
217C23×C7—42no—
218C23×C7—42no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7—42no—
222C23×C7—42no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7C22⋊Q818yescomplemented
232C23×C7C22×D412yes—
233C23×C7C22×D411yescharacteristic
234C23×C7C22×D411yescharacteristic
235C23×C7C22×D412yes—
236C23×C7C22×Q811yescharacteristic
237C23×C7C2×#4111yescharacteristic
238C23×C7C2×#4112yes—
239C23×C7C2×#4111yescharacteristic
240C23×C7H(5,𝔽2)11yescharacteristic
241C23×C72−1+412yes—
242C23×C72−1+411yescharacteristic
243C7×Q8—416no—
244C7×D4—232no—
245C2×C4×C7—22no—
246C2×C4×C7—24no—
247C2×C4×C7—22no—
248C2×C4×C7—24no—
249C2×C4×C7—22no—
250C2×C4×C7—28no—
251C2×C4×C7—24no—
252C2×C4×C7—28no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—28no—
259C2×C4×C7—28no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—44no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—44no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7—432no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—44no—
278C2×C4×C7—44no—
279C2×C4×C7#12014yes—
280C2×C4×C7#12018yes—
281C2×C4×C7#12118yes—
282C2×C4×C7#12114yes—
283C2×C4×C7C22⋊Q814yescomplemented
284C2×C4×C7#12714yes—
285C25C23×C712yes—
286C22×Q8—24no—
287C22×Q8—24no—
288C22×Q8—22no—
289C22×Q8—22no—
290C22×Q8C7×D414yescomplemented
291C22×D4—24no—
292C22×D4—24no—
293C22×D4—22no—
294C22×D4—22no—
295C22×D4C7×Q814yescomplemented
296C23×C4—24no—
297C23×C4—24no—
298C23×C4—24no—
299C23×C4—24no—
300C23×C4—24no—
301C23×C4—24no—
302C23×C4C7×D418yes—
303C23×C4C23×C712yes—
304C23×C4C23×C712yes—
305C23×C4C23×C714yes—
306C23×C4C23×C711yescharacteristic
307C23×C4C23×C712yes—
308C23×C4C23×C711yescharacteristic
309C23×C4C23×C711yescharacteristic
310C22⋊Q8—264no—
311C4×D4—232no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4C7×D418yescomplemented
322C2×C22⋊C4—24no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—44no—
331C2×C42—24no—
332C2×C42C7×D414yes—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—42no—
338C22×C7—28no—
339C22×C7—28no—
340C22×C7—28no—
341C22×C7—44no—
342C22×C7—44no—
343C22×C7—44no—
344C22×C7—44no—
345C22×C7—44no—
346C22×C7—44no—
347C22×C7—44no—
348C22×C7—44no—
349C22×C7—44no—
350C22×C7—44no—
351C22×C7—44no—
352C22×C7—44no—
353C22×C7C2×#12011yescharacteristic
354C22×C7C2×#12012yes—
355C22×C7C2×#12112yes—
356C22×C7C2×#12111yescharacteristic
357C22×C7C2×C22⋊Q812yes—
358C22×C7C2×C22⋊Q811yescharacteristic
359C22×C7#52512yes—
360C22×C7#52511yescharacteristic
361C22×C7C2×#12711yescharacteristic
362C22×C7#53612yes—
363C22×C7#53611yescharacteristic
364C22×C7#53712yes—
365C22×C7#53711yescharacteristic
366C22×C7#54612yes—
367C22×C7#54712yes—
368C22×C7#54812yes—
369C22×C7#54812yes—
370C22×C7D4×Q811yescharacteristic
371C22×C7D4×Q812yes—
372C22×C7#55011yescharacteristic
373C22×C7#55212yes—
374C22×C7#55411yescharacteristic
375C22×C7#55611yescharacteristic
376C4×C7—24no—
377C4×C7—42no—
378C4×C7—44no—
379C4×C7—44no—
380C4×C7—48no—
381C4×C7—82no—
382C4×C7#39218yes—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24C2×C7×D414yes—
396C24C2×C7×Q814yes—
397C24C7×#4114yes—
398C24C24×C711yescharacteristic
399C2×Q8—216no—
400C2×Q8—28no—
401C2×Q8—44no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—216no—
407C2×D4—216no—
408C2×D4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4C2×C7×D412yes—
444C22×C4C2×C7×D418yes—
445C22×C4C2×C7×D414yes—
446C22×C4C2×C7×D414yes—
447C22×C4C2×C7×D412yes—
448C22×C4C2×C7×D414yes—
449C22×C4C2×C7×D412yes—
450C22×C4C2×C7×D412yes—
451C22×C4C2×C7×D414yes—
452C22×C4C2×C7×Q812yes—
453C22×C4C7×#4112yes—
454C22×C4C7×#4112yes—
455C4⋊C4—28no—
456C4⋊C4—232no—
457C4⋊C4—216no—
458C4⋊C4—44no—
459C22⋊C4—216no—
460C22⋊C4—232no—
461C22⋊C4—416no—
462C42—28no—
463C2×C7—48no—
464C2×C7C2×#39211yescharacteristic
465C2×C7#240712yes—
466C2×C7#240912yes—
467C2×C7#241512yes—
468C2×C7#241712yes—
469C2×C7#253611yescharacteristic
470C2×C7#253811yescharacteristic
471C2×C7#254011yescharacteristic
472C2×C7#254211yescharacteristic
473C2×C7#261611yescharacteristic
474C2×C7#284811yescharacteristic
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C7×C22⋊Q818yescomplemented
510C23C22×C7×D412yes—
511C23C22×C7×D411yescharacteristic
512C23C22×C7×D411yescharacteristic
513C23C22×C7×D412yes—
514C23C22×C7×Q811yescharacteristic
515C23C2×C7×#4111yescharacteristic
516C23C2×C7×#4112yes—
517C23C2×C7×#4111yescharacteristic
518C23C7×H(5,𝔽2)11yescharacteristic
519C23C7×2−1+412yes—
520C23C7×2−1+411yescharacteristic
521Q8—416no—
522D4—232no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—432no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4C7×#12014yes—
558C2×C4C7×#12018yes—
559C2×C4C7×#12118yes—
560C2×C4C7×#12114yes—
561C2×C4C7×C22⋊Q814yescomplemented
562C2×C4C7×#12714yes—
563C7#1517711yescharacteristic, direct product factor, Sylow
564C22—28no—
565C22—28no—
566C22—28no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22C2×C7×#12012yes—
580C22C2×C7×#12011yescharacteristic
581C22C2×C7×#12112yes—
582C22C2×C7×#12111yescharacteristic
583C22C2×C7×C22⋊Q812yes—
584C22C2×C7×C22⋊Q811yescharacteristic
585C22C7×#52512yes—
586C22C7×#52511yescharacteristic
587C22C2×C7×#12711yescharacteristic
588C22C7×#53612yes—
589C22C7×#53611yescharacteristic
590C22C7×#53712yes—
591C22C7×#53711yescharacteristic
592C22C7×#54612yes—
593C22C7×#54712yes—
594C22C7×#54812yes—
595C22C7×#54812yes—
596C22C7×D4×Q811yescharacteristic
597C22C7×D4×Q812yes—
598C22C7×#55011yescharacteristic
599C22C7×#55212yes—
600C22C7×#55411yescharacteristic
601C22C7×#55611yescharacteristic
602C4—24no—
603C4—48no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—82no—
608C4C7×#39218yes—
609C2—48no—
610C2C2×C7×#39211yescharacteristic
611C2C7×#240712yes—
612C2C7×#240912yes—
613C2C7×#241512yes—
614C2C7×#241712yes—
615C2C7×#253611yescharacteristic
616C2C7×#253811yescharacteristic
617C2C7×#254011yescharacteristic
618C2C7×#254211yescharacteristic
619C2C7×#261611yescharacteristic
620C2C7×#284811yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC23×C7×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊Q8C2semidirect
3. C2×C7×#391C2semidirect
4. C2×C7×#380C2semidirect
5. C7×#1444C2semidirect
6. C7×#1439C2semidirect
8. C7×C4⋊#95C2semidirect
32. #15177C7direct (N × H)
38. C22×C7×Q8D4semidirect
43. C22×C7×D4Q8semidirect
69. C2×C7×C4⋊C4D4semidirect
86. C22×C22⋊Q8C2×C7semidirect
88. C2×#391C2×C7semidirect
89. C2×#380C2×C7semidirect
90. #1444C2×C7semidirect
91. #1439C2×C7semidirect
93. C4⋊#95C2×C7semidirect
231. C23×C7C22⋊Q8semidirect
283. C2×C4×C7C22⋊Q8semidirect
290. C22×Q8C7×D4semidirect
295. C22×D4C7×Q8semidirect
321. C2×C4⋊C4C7×D4semidirect
509. C23C7×C22⋊Q8semidirect
561. C2×C4C7×C22⋊Q8semidirect
563. C7#15177direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C22×C4×C7×D42
C2×C7×#3912
C2×C7×#3802
C7×#14442
C7×#14392
C7×#14382
C7×C4⋊#952
#151777