Finite Group Databasebeta

← C7×#16514 #13193730

Subgroups — C7×#16514 #13193730

Order 1792.

1774 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#396C212yesmaximal, complemented
2C2×C7×#394C211yesmaximal, characteristic
3C2×C7×#394C212yesmaximal
4C2×C7×#388C212yesmaximal, complemented
5C2×C7×#384C211yesmaximal, characteristic, complemented
6C7×#1441C212yesmaximal, complemented
7C7×#1441C211yesmaximal, characteristic, complemented
8C7×#1439C212yesmaximal, complemented
9C7×#1439C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C7×C4⋊C4C2212yes—
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C7×C22⋊C4C2212yes—
23C22×C42×C7C2211yescharacteristic
24C7×#396—28no—
25C7×#394—24no—
26C7×#394—28no—
27C7×#388—28no—
28C7×#384—24no—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2211yescharacteristic
34C2×C7×#95C2211yescharacteristic
35C2×C7×#95C2212yes—
36#16514C711yesmaximal, characteristic, direct product factor, Sylow
37C25×C7C2312yes—
38C22×C7×D4—24no—
39C22×C7×D4—22no—
40C22×C7×D4—42no—
41C22×C7×D4—41no—
42C22×C7×D4—41no—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2312yes—
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2311yescharacteristic
49C23×C4×C7C2311yescharacteristic
50C23×C4×C7C2312yes—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—24no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—41no—
61C2×C7×C4⋊C4—41no—
62C2×C7×C4⋊C4—42no—
63C2×C7×C4⋊C4—42no—
64C2×C7×C4⋊C4—41no—
65C2×C7×C4⋊C4—41no—
66C2×C7×C4⋊C4—41no—
67C2×C7×C4⋊C4D414yes—
68C2×C7×C22⋊C4—24no—
69C2×C7×C22⋊C4—24no—
70C2×C7×C22⋊C4—24no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—24no—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—24no—
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—42no—
82C2×C7×C22⋊C4—42no—
83C2×C7×C22⋊C4—42no—
84C2×C7×C22⋊C4—42no—
85C2×C7×C22⋊C4D418yes—
86C2×C42×C7—22no—
87C2×C42×C7—22no—
88C2×C42×C7D414yescomplemented
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—24no—
92C7×#95—22no—
93C7×#95—24no—
94C7×#95—41no—
95C7×#95—42no—
96C2×#396C2×C712yescomplemented
97C2×#394C2×C711yescharacteristic
98C2×#394C2×C712yes—
99C2×#388C2×C712yescomplemented
100C2×#384C2×C711yescharacteristic, complemented
101#1441C2×C712yescomplemented
102#1441C2×C711yescharacteristic, complemented
103#1439C2×C712yescomplemented
104#1439C2×C712yescomplemented
105C24×C7—22no—
106C24×C7—22no—
107C24×C7—22no—
108C24×C7—22no—
109C24×C7—22no—
110C24×C7—22no—
111C24×C7—22no—
112C24×C7—22no—
113C24×C7—22no—
114C24×C7—22no—
115C24×C7—22no—
116C24×C7—22no—
117C24×C7—22no—
118C24×C7—22no—
119C24×C7C2×D414yes—
120C24×C7C2411yescharacteristic
121C2×C7×D4—48no—
122C2×C7×D4—44no—
123C2×C7×D4—44no—
124C2×C7×D4—48no—
125C2×C7×D4—44no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
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—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7C2×D414yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D412yes—
173C22×C4×C7C2×D414yes—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7#4112yes—
177C22×C4×C7#4114yes—
178C7×C4⋊C4—44no—
179C7×C4⋊C4—48no—
180C7×C4⋊C4—48no—
181C7×C4⋊C4—44no—
182C7×C4⋊C4—44no—
183C7×C4⋊C4—48no—
184C7×C22⋊C4—216no—
185C7×C22⋊C4—48no—
186C7×C22⋊C4—48no—
187C7×C22⋊C4—48no—
188C7×C22⋊C4—48no—
189C7×C22⋊C4—48no—
190C42×C7—28no—
191C23×D4C22×C711yescharacteristic
192C22×C4⋊C4C22×C712yes—
193C22×C4⋊C4C22×C712yes—
194C22×C4⋊C4C22×C711yescharacteristic
195C22×C4⋊C4C22×C712yes—
196C22×C4⋊C4C22×C711yescharacteristic
197C22×C4⋊C4C22×C711yescharacteristic
198C22×C22⋊C4C22×C712yes—
199C22×C22⋊C4C22×C712yes—
200C22×C22⋊C4C22×C712yes—
201C22×C22⋊C4C22×C712yes—
202C22×C22⋊C4C22×C712yes—
203C22×C22⋊C4C22×C712yes—
204C22×C42C22×C711yescharacteristic
205#396—28no—
206#394—24no—
207#394—28no—
208#388—28no—
209#384—24no—
210C2×#95C22×C711yescharacteristic
211C2×#95C22×C712yes—
212C2×#95C22×C712yes—
213C2×#95C22×C712yes—
214C2×#95C22×C712yes—
215C2×#95C22×C712yes—
216C2×#95C22×C711yescharacteristic
217C23×C7—24no—
218C23×C7—24no—
219C23×C7—24no—
220C23×C7—24no—
221C23×C7—24no—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7—42no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—42no—
244C23×C7—42no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—42no—
249C23×C7—42no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7C22×D411yescharacteristic
253C23×C7C22×D411yescharacteristic
254C23×C7C22×D412yes—
255C23×C7C2×#4111yescharacteristic
256C23×C7C2×#4112yes—
257C23×C7H(5,𝔽2)11yescharacteristic
258C23×C7H(5,𝔽2)11yescharacteristic
259C23×C7H(5,𝔽2)12yes—
260C23×C72−1+411yescharacteristic
261C23×C72−1+411yescharacteristic
262C23×C72−1+412yes—
263C7×D4—416no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7—24no—
276C2×C4×C7—24no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—24no—
280C2×C4×C7—24no—
281C2×C4×C7—24no—
282C2×C4×C7—24no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—42no—
297C2×C4×C7—42no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—42no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—42no—
310C2×C4×C7—42no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7#12114yes—
315C2×C4×C7#12118yes—
316C25C23×C712yes—
317C22×D4—24no—
318C22×D4—22no—
319C22×D4—42no—
320C22×D4—41no—
321C22×D4—41no—
322C23×C4C23×C712yes—
323C23×C4C23×C712yes—
324C23×C4C23×C712yes—
325C23×C4C23×C711yescharacteristic
326C23×C4C23×C712yes—
327C23×C4C23×C711yescharacteristic
328C23×C4C23×C712yes—
329C23×C4C23×C711yescharacteristic
330C2×C4⋊C4—24no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—41no—
341C2×C4⋊C4—42no—
342C2×C4⋊C4—42no—
343C2×C4⋊C4—41no—
344C2×C4⋊C4—41no—
345C2×C4⋊C4—41no—
346C2×C4⋊C4C7×D414yes—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—42no—
361C2×C22⋊C4—42no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—42no—
364C2×C22⋊C4C7×D418yes—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C7×D414yescomplemented
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—41no—
374#95—42no—
375C22×C7—44no—
376C22×C7—44no—
377C22×C7—44no—
378C22×C7—44no—
379C22×C7—44no—
380C22×C7—44no—
381C22×C7—44no—
382C22×C7—82no—
383C22×C7—82no—
384C22×C7—82no—
385C22×C7—82no—
386C22×C7—82no—
387C22×C7—82no—
388C22×C7—82no—
389C22×C7—82no—
390C22×C7C2×#12111yescharacteristic
391C22×C7C2×#12112yes—
392C22×C7#53411yescharacteristic
393C22×C7#53512yes—
394C22×C7#53611yescharacteristic
395C22×C7#53612yes—
396C22×C7#53711yescharacteristic
397C22×C7#53712yes—
398C22×C7#53711yescharacteristic
399C22×C7#53911yescharacteristic
400C22×C7#53911yescharacteristic
401C22×C7#54212yes—
402C22×C7#54212yes—
403C22×C7D4212yes—
404C22×C7D4211yescharacteristic
405C22×C7#54712yes—
406C22×C7#54712yes—
407C22×C7#54812yes—
408C22×C7#55312yes—
409C22×C7#55511yescharacteristic
410C22×C7#56011yescharacteristic
411C22×C7#56312yes—
412C22×C7#56311yescharacteristic
413C4×C7—24no—
414C4×C7—28no—
415C4×C7—42no—
416C4×C7—42no—
417C4×C7—44no—
418C4×C7—82no—
419C4×C7—82no—
420C4×C7—82no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24C2×C7×D414yes—
436C24C24×C711yescharacteristic
437C2×D4—48no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—48no—
441C2×D4—44no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4C2×C7×D414yes—
486C22×C4C2×C7×D412yes—
487C22×C4C2×C7×D412yes—
488C22×C4C2×C7×D412yes—
489C22×C4C2×C7×D414yes—
490C22×C4C2×C7×D412yes—
491C22×C4C2×C7×D414yes—
492C22×C4C7×#4112yes—
493C22×C4C7×#4114yes—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C22⋊C4—216no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C42—28no—
507C2×C7—84no—
508C2×C7#241612yes—
509C2×C7#253711yescharacteristic
510C2×C7#261811yescharacteristic
511C2×C7#262012yes—
512C2×C7#266911yescharacteristic
513C2×C7#267412yes—
514C2×C7#268011yescharacteristic
515C2×C7#271812yes—
516C2×C7#273111yescharacteristic
517C2×C7#279511yescharacteristic
518C2×C7#282311yescharacteristic
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C22×C7×D411yescharacteristic
555C23C22×C7×D411yescharacteristic
556C23C22×C7×D412yes—
557C23C2×C7×#4111yescharacteristic
558C23C2×C7×#4112yes—
559C23C7×H(5,𝔽2)11yescharacteristic
560C23C7×H(5,𝔽2)11yescharacteristic
561C23C7×H(5,𝔽2)12yes—
562C23C7×2−1+411yescharacteristic
563C23C7×2−1+411yescharacteristic
564C23C7×2−1+412yes—
565D4—416no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4C7×#12114yes—
617C2×C4C7×#12118yes—
618C7#1651411yescharacteristic, direct product factor, Sylow
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×C7×#12111yescharacteristic
635C22C2×C7×#12112yes—
636C22C7×#53411yescharacteristic
637C22C7×#53512yes—
638C22C7×#53611yescharacteristic
639C22C7×#53612yes—
640C22C7×#53712yes—
641C22C7×#53711yescharacteristic
642C22C7×#53711yescharacteristic
643C22C7×#53911yescharacteristic
644C22C7×#53911yescharacteristic
645C22C7×#54212yes—
646C22C7×#54212yes—
647C22C7×D4212yes—
648C22C7×D4211yescharacteristic
649C22C7×#54712yes—
650C22C7×#54712yes—
651C22C7×#54812yes—
652C22C7×#55312yes—
653C22C7×#55511yescharacteristic
654C22C7×#56011yescharacteristic
655C22C7×#56312yes—
656C22C7×#56311yescharacteristic
657C4—28no—
658C4—24no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—84no—
666C2C7×#241612yes—
667C2C7×#253711yescharacteristic
668C2C7×#261811yescharacteristic
669C2C7×#262012yes—
670C2C7×#266911yescharacteristic
671C2C7×#267412yes—
672C2C7×#268011yescharacteristic
673C2C7×#271812yes—
674C2C7×#273111yescharacteristic
675C2C7×#279511yescharacteristic
676C2C7×#282311yescharacteristic
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. C2×C7×#396C2semidirect
4. C2×C7×#388C2semidirect
5. C2×C7×#384C2semidirect
6. C7×#1441C2semidirect
7. C7×#1441C2semidirect
8. C7×#1439C2semidirect
9. C7×#1439C2semidirect
36. #16514C7direct (N × H)
88. C2×C42×C7D4semidirect
96. C2×#396C2×C7semidirect
99. C2×#388C2×C7semidirect
100. C2×#384C2×C7semidirect
101. #1441C2×C7semidirect
102. #1441C2×C7semidirect
103. #1439C2×C7semidirect
104. #1439C2×C7semidirect
367. C2×C42C7×D4semidirect
618. C7#16514direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#3962
C2×C7×#3942
C2×C7×#3882
C2×C7×#3842
C7×#14412
C7×#14392
#165147