Finite Group Databasebeta

← C7×#15182 #13192398

Subgroups — C7×#15182 #13192398

Order 1792.

1990 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×C7×#391C211yesmaximal, characteristic, complemented
3C2×C7×#391C211yesmaximal, characteristic, complemented
4C2×C7×#384C211yesmaximal, characteristic, complemented
5C2×C7×#380C211yesmaximal, characteristic
6C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
7C7×#1441C214yesmaximal, complemented
8C7×#1439C214yesmaximal
9C7×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C7×Q8C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C2×C7×C22⋊Q8—24no—
13C2×C7×C22⋊Q8—24no—
14C2×C7×C22⋊Q8—24no—
15C22×C7×C4⋊C4C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2214yes—
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C7×C22⋊C4C2212yes—
23C22×C42×C7C2211yescharacteristic
24C22×C42×C7C2211yescharacteristic
25C7×#391—24no—
26C7×#391—24no—
27C7×#384—24no—
28C7×#380—24no—
29C4×C7×C22⋊C4—24no—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2214yes—
33C2×C7×#95C2214yes—
34C2×C7×#95C2214yes—
35#15182C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7C2311yescharacteristic
37C22×C7×Q8—22no—
38C22×C7×Q8—22no—
39C22×C7×Q8—22no—
40C22×C7×Q8—22no—
41C22×C7×Q8—22no—
42C22×C7×Q8—22no—
43C22×C7×Q8D414yes—
44C23×C4×C7—22no—
45C23×C4×C7—22no—
46C23×C4×C7—22no—
47C23×C4×C7—22no—
48C23×C4×C7—22no—
49C23×C4×C7—22no—
50C23×C4×C7D414yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2312yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2312yes—
58C23×C4×C7C2311yescharacteristic
59C7×C22⋊Q8—232no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—28no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—28no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—44no—
74C2×C7×C4⋊C4D414yes—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4—24no—
78C2×C7×C22⋊C4—24no—
79C2×C7×C22⋊C4—24no—
80C2×C7×C22⋊C4—24no—
81C2×C7×C22⋊C4—44no—
82C2×C7×C22⋊C4Q818yescomplemented
83C2×C42×C7—22no—
84C2×C42×C7—22no—
85C2×C42×C7—22no—
86C2×C42×C7—22no—
87C2×C42×C7—22no—
88C2×C42×C7D414yes—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—28no—
92C7×#95—28no—
93C7×#95—44no—
94C22×C22⋊Q8C2×C711yescharacteristic
95C2×#391C2×C711yescharacteristic, complemented
96C2×#391C2×C711yescharacteristic, complemented
97C2×#384C2×C711yescharacteristic, complemented
98C2×#380C2×C711yescharacteristic
99C2×C4×C22⋊C4C2×C711yescharacteristic
100#1441C2×C714yescomplemented
101#1439C2×C714yes—
102C4⋊#95C2×C711yescharacteristic, complemented
103C24×C7—21no—
104C24×C7—21no—
105C24×C7—21no—
106C24×C7—22no—
107C24×C7—21no—
108C24×C7—21no—
109C24×C7—21no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7C2×D412yes—
113C24×C7C2×Q814yes—
114C24×C7C2411yescharacteristic
115C2×C7×Q8—28no—
116C2×C7×Q8—28no—
117C2×C7×Q8—28no—
118C2×C7×Q8—44no—
119C2×C7×Q8—44no—
120C2×C7×Q8—44no—
121C2×C7×Q8—44no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
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—21no—
135C22×C4×C7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—24no—
147C22×C4×C7—22no—
148C22×C4×C7—28no—
149C22×C4×C7—28no—
150C22×C4×C7—28no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—24no—
161C22×C4×C7—44no—
162C22×C4×C7—44no—
163C22×C4×C7—44no—
164C22×C4×C7—44no—
165C22×C4×C7C2×D414yes—
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×D414yes—
168C22×C4×C7C2×D414yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×Q814yes—
174C22×C4×C7C2×Q814yes—
175C22×C4×C7#4114yes—
176C22×C4×C7#4112yes—
177C22×C4×C7#4112yes—
178C22×C4×C7#4112yes—
179C7×C4⋊C4—216no—
180C7×C4⋊C4—28no—
181C7×C4⋊C4—416no—
182C7×C4⋊C4—44no—
183C7×C4⋊C4—44no—
184C7×C22⋊C4—216no—
185C7×C22⋊C4—216no—
186C7×C22⋊C4—48no—
187C42×C7—28no—
188C42×C7—44no—
189C23×Q8C22×C711yescharacteristic
190C24×C4C22×C711yescharacteristic
191C2×C22⋊Q8—24no—
192C2×C22⋊Q8—24no—
193C2×C22⋊Q8—24no—
194C22×C4⋊C4C22×C712yes—
195C22×C4⋊C4C22×C711yescharacteristic
196C22×C4⋊C4C22×C711yescharacteristic
197C22×C4⋊C4C22×C714yes—
198C22×C4⋊C4C22×C711yescharacteristic
199C22×C22⋊C4C22×C712yes—
200C22×C22⋊C4C22×C712yes—
201C22×C22⋊C4C22×C712yes—
202C22×C42C22×C711yescharacteristic
203C22×C42C22×C711yescharacteristic
204#391—24no—
205#391—24no—
206#384—24no—
207#380—24no—
208C4×C22⋊C4—24no—
209C2×#95C22×C714yes—
210C2×#95C22×C714yes—
211C2×#95C22×C712yes—
212C2×#95C22×C712yes—
213C2×#95C22×C714yes—
214C23×C7—22no—
215C23×C7—22no—
216C23×C7—24no—
217C23×C7—22no—
218C23×C7—22no—
219C23×C7—24no—
220C23×C7—22no—
221C23×C7—22no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7C4⋊Q814yes—
237C23×C7C22×D411yescharacteristic
238C23×C7C22×D411yescharacteristic
239C23×C7C22×D411yescharacteristic
240C23×C7C22×D411yescharacteristic
241C23×C7C22×Q812yes—
242C23×C7C2×#4111yescharacteristic
243C23×C7C2×#4111yescharacteristic
244C23×C7C2×#4111yescharacteristic
245C23×C7C2×#4112yes—
246C23×C7H(5,𝔽2)12yes—
247C23×C72−1+412yes—
248C7×Q8—416no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—28no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—41no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—44no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—416no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—44no—
289C2×C4×C7—44no—
290C2×C4×C7—44no—
291C2×C4×C7—44no—
292C2×C4×C7#12014yes—
293C2×C4×C7#12114yes—
294C2×C4×C7#12114yes—
295C2×C4×C7#12114yes—
296C2×C4×C7C22⋊Q818yescomplemented
297C2×C4×C7#12314yes—
298C2×C4×C7#12314yes—
299C2×C4×C7#12414yes—
300C25C23×C711yescharacteristic
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—22no—
305C22×Q8—22no—
306C22×Q8—22no—
307C22×Q8C7×D414yes—
308C23×C4—22no—
309C23×C4—22no—
310C23×C4—22no—
311C23×C4—22no—
312C23×C4—22no—
313C23×C4—22no—
314C23×C4C7×D414yes—
315C23×C4C23×C712yes—
316C23×C4C23×C712yes—
317C23×C4C23×C712yes—
318C23×C4C23×C712yes—
319C23×C4C23×C711yescharacteristic
320C23×C4C23×C712yes—
321C23×C4C23×C712yes—
322C23×C4C23×C711yescharacteristic
323C22⋊Q8—232no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4C7×D414yes—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4C7×Q818yescomplemented
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42C7×D414yes—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—44no—
358C22×C7—24no—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—42no—
362C22×C7—42no—
363C22×C7—44no—
364C22×C7—44no—
365C22×C7—42no—
366C22×C7—44no—
367C22×C7—42no—
368C22×C7—44no—
369C22×C7C2×#12011yescharacteristic
370C22×C7C2×#12111yescharacteristic
371C22×C7C2×#12111yescharacteristic
372C22×C7C2×#12111yescharacteristic
373C22×C7C2×C22⋊Q812yes—
374C22×C7C2×#12311yescharacteristic
375C22×C7C2×#12311yescharacteristic
376C22×C7C2×#12411yescharacteristic
377C22×C7C2×C4⋊Q811yescharacteristic
378C22×C7#53211yescharacteristic
379C22×C7#53312yes—
380C22×C7#53511yescharacteristic
381C22×C7#53611yescharacteristic
382C22×C7#53712yes—
383C22×C7#54212yes—
384C22×C7#54612yes—
385C22×C7D4×Q812yes—
386C22×C7D4×Q812yes—
387C22×C7#55012yes—
388C22×C7#55212yes—
389C22×C7#55412yes—
390C22×C7#55412yes—
391C22×C7#55612yes—
392C4×C7—28no—
393C4×C7—42no—
394C4×C7—44no—
395C4×C7—44no—
396C4×C7—44no—
397C4×C7—82no—
398C4×C7—82no—
399C4×C7#39418yes—
400C24—21no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—22no—
408C24—22no—
409C24C2×C7×D412yes—
410C24C2×C7×Q814yes—
411C24C24×C711yescharacteristic
412C2×Q8—28no—
413C2×Q8—28no—
414C2×Q8—28no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4C2×C7×D414yes—
463C22×C4C2×C7×D412yes—
464C22×C4C2×C7×D414yes—
465C22×C4C2×C7×D414yes—
466C22×C4C2×C7×D412yes—
467C22×C4C2×C7×D412yes—
468C22×C4C2×C7×D412yes—
469C22×C4C2×C7×D412yes—
470C22×C4C2×C7×Q814yes—
471C22×C4C2×C7×Q814yes—
472C22×C4C7×#4114yes—
473C22×C4C7×#4112yes—
474C22×C4C7×#4112yes—
475C22×C4C7×#4112yes—
476C4⋊C4—216no—
477C4⋊C4—28no—
478C4⋊C4—416no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—48no—
484C42—28no—
485C42—44no—
486C2×C7—44no—
487C2×C7C2×#39411yescharacteristic
488C2×C7#241712yes—
489C2×C7#242312yes—
490C2×C7#243111yescharacteristic
491C2×C7#243411yescharacteristic
492C2×C7#249212yes—
493C2×C7#249911yescharacteristic
494C2×C7#250211yescharacteristic
495C2×C7#253812yes—
496C2×C7#262011yescharacteristic
497C2×C7#285711yescharacteristic
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C7×C4⋊Q814yes—
521C23C22×C7×D411yescharacteristic
522C23C22×C7×D411yescharacteristic
523C23C22×C7×D411yescharacteristic
524C23C22×C7×D411yescharacteristic
525C23C22×C7×Q812yes—
526C23C2×C7×#4111yescharacteristic
527C23C2×C7×#4111yescharacteristic
528C23C2×C7×#4111yescharacteristic
529C23C2×C7×#4112yes—
530C23C7×H(5,𝔽2)12yes—
531C23C7×2−1+412yes—
532Q8—416no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4C7×#12014yes—
577C2×C4C7×#12114yes—
578C2×C4C7×#12114yes—
579C2×C4C7×#12114yes—
580C2×C4C7×C22⋊Q818yescomplemented
581C2×C4C7×#12314yes—
582C2×C4C7×#12314yes—
583C2×C4C7×#12414yes—
584C7#1518211yescharacteristic, direct product factor, Sylow
585C22—24no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—44no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22C2×C7×#12011yescharacteristic
597C22C2×C7×#12111yescharacteristic
598C22C2×C7×#12111yescharacteristic
599C22C2×C7×#12111yescharacteristic
600C22C2×C7×C22⋊Q812yes—
601C22C2×C7×#12311yescharacteristic
602C22C2×C7×#12311yescharacteristic
603C22C2×C7×#12411yescharacteristic
604C22C2×C7×C4⋊Q811yescharacteristic
605C22C7×#53211yescharacteristic
606C22C7×#53312yes—
607C22C7×#53511yescharacteristic
608C22C7×#53611yescharacteristic
609C22C7×#53712yes—
610C22C7×#54212yes—
611C22C7×#54612yes—
612C22C7×D4×Q812yes—
613C22C7×D4×Q812yes—
614C22C7×#55012yes—
615C22C7×#55212yes—
616C22C7×#55412yes—
617C22C7×#55412yes—
618C22C7×#55612yes—
619C4—28no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—82no—
625C4—82no—
626C4C7×#39418yes—
627C2—44no—
628C2C2×C7×#39411yescharacteristic
629C2C7×#241712yes—
630C2C7×#242312yes—
631C2C7×#243111yescharacteristic
632C2C7×#243411yescharacteristic
633C2C7×#249212yes—
634C2C7×#249911yescharacteristic
635C2C7×#250211yescharacteristic
636C2C7×#253812yes—
637C2C7×#262011yescharacteristic
638C2C7×#285711yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#391C2semidirect
3. C2×C7×#391C2semidirect
4. C2×C7×#384C2semidirect
7. C7×#1441C2semidirect
9. C7×C4⋊#95C2semidirect
35. #15182C7direct (N × H)
82. C2×C7×C22⋊C4Q8semidirect
95. C2×#391C2×C7semidirect
96. C2×#391C2×C7semidirect
97. C2×#384C2×C7semidirect
100. #1441C2×C7semidirect
102. C4⋊#95C2×C7semidirect
296. C2×C4×C7C22⋊Q8semidirect
346. C2×C22⋊C4C7×Q8semidirect
580. C2×C4C7×C22⋊Q8semidirect
584. C7#15182direct (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
C2×C7×#3912
C2×C7×#3842
C2×C7×#3802
C2×C4×C7×C22⋊C42
C7×#14412
C7×#14392
C7×C4⋊#952
#151827