Finite Group Databasebeta

← C7×#16599 #13193815

Subgroups — C7×#16599 #13193815

Order 1792.

1550 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#400C211yesmaximal, characteristic, complemented
2C2×C7×#390C211yesmaximal, characteristic
3C7×#1441C211yesmaximal, characteristic, complemented
4C7×#1439C212yesmaximal, complemented
5C7×#1439C212yesmaximal, complemented
6C7×#1439C212yesmaximal, complemented
7C7×#1439C212yesmaximal, complemented
8C7×#1438C212yesmaximal
9C7×#1436C212yesmaximal, complemented
10C23×C7×D4C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C42×C7C2211yescharacteristic
22C7×#400—24no—
23C7×#390—24no—
24C2×C7×#95C2212yes—
25C2×C7×#95C2211yescharacteristic
26C2×C7×#95C2212yes—
27C2×C7×#95C2212yes—
28C2×C7×#95C2212yes—
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2211yescharacteristic
33#16599C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7C2312yes—
35C22×C7×D4—22no—
36C22×C7×D4—22no—
37C22×C7×D4—41no—
38C22×C7×D4—41no—
39C22×C7×D4—41no—
40C22×C7×D4—41no—
41C22×C7×D4Q814yes—
42C23×C4×C7C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2312yes—
45C23×C4×C7C2311yescharacteristic
46C23×C4×C7C2311yescharacteristic
47C23×C4×C7C2312yes—
48C23×C4×C7C2312yes—
49C23×C4×C7C2311yescharacteristic
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—41no—
53C2×C7×C4⋊C4—41no—
54C2×C7×C4⋊C4—42no—
55C2×C7×C4⋊C4—42no—
56C2×C7×C4⋊C4—41no—
57C2×C7×C4⋊C4—41no—
58C2×C7×C4⋊C4—42no—
59C2×C7×C4⋊C4—41no—
60C2×C7×C4⋊C4D414yes—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—24no—
64C2×C7×C22⋊C4—24no—
65C2×C7×C22⋊C4—24no—
66C2×C7×C22⋊C4—24no—
67C2×C7×C22⋊C4—42no—
68C2×C7×C22⋊C4—42no—
69C2×C7×C22⋊C4—42no—
70C2×C7×C22⋊C4—42no—
71C2×C7×C22⋊C4—42no—
72C2×C7×C22⋊C4—42no—
73C2×C7×C22⋊C4—42no—
74C2×C7×C22⋊C4—42no—
75C2×C7×C22⋊C4—42no—
76C2×C7×C22⋊C4—42no—
77C2×C7×C22⋊C4—42no—
78C2×C7×C22⋊C4—42no—
79C2×C42×C7—22no—
80C2×C42×C7—41no—
81C2×C42×C7—41no—
82C7×#95—22no—
83C7×#95—24no—
84C7×#95—24no—
85C7×#95—24no—
86C7×#95—24no—
87C7×#95—42no—
88C7×#95—41no—
89C7×#95—42no—
90C7×#95—42no—
91C2×#400C2×C711yescharacteristic, complemented
92C2×#390C2×C711yescharacteristic
93#1441C2×C711yescharacteristic, complemented
94#1439C2×C712yescomplemented
95#1439C2×C712yescomplemented
96#1439C2×C712yescomplemented
97#1439C2×C712yescomplemented
98#1438C2×C712yes—
99#1436C2×C712yescomplemented
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—22no—
104C24×C7—22no—
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×C7C2×Q814yes—
115C24×C7C2411yescharacteristic
116C2×C7×D4—28no—
117C2×C7×D4—44no—
118C2×C7×D4—44no—
119C2×C7×D4—44no—
120C2×C7×D4—44no—
121C2×C7×D4—44no—
122C2×C7×D4—44no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—21no—
126C22×C4×C7—21no—
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—22no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
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×C7C2×D412yes—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×Q812yes—
172C22×C4×C7#4114yes—
173C22×C4×C7#4114yes—
174C22×C4×C7#4112yes—
175C7×C4⋊C4—44no—
176C7×C4⋊C4—44no—
177C7×C4⋊C4—48no—
178C7×C4⋊C4—44no—
179C7×C22⋊C4—48no—
180C7×C22⋊C4—48no—
181C7×C22⋊C4—48no—
182C7×C22⋊C4—48no—
183C7×C22⋊C4—48no—
184C7×C22⋊C4—48no—
185C42×C7—44no—
186C23×D4C22×C711yescharacteristic
187C22×C4⋊C4C22×C711yescharacteristic
188C22×C4⋊C4C22×C712yes—
189C22×C4⋊C4C22×C711yescharacteristic
190C22×C4⋊C4C22×C711yescharacteristic
191C22×C22⋊C4C22×C712yes—
192C22×C22⋊C4C22×C712yes—
193C22×C22⋊C4C22×C712yes—
194C22×C22⋊C4C22×C712yes—
195C22×C22⋊C4C22×C712yes—
196C22×C22⋊C4C22×C712yes—
197C22×C42C22×C711yescharacteristic
198#400—24no—
199#390—24no—
200C2×#95C22×C712yes—
201C2×#95C22×C711yescharacteristic
202C2×#95C22×C712yes—
203C2×#95C22×C712yes—
204C2×#95C22×C712yes—
205C2×#95C22×C712yes—
206C2×#95C22×C712yes—
207C2×#95C22×C712yes—
208C2×#95C22×C711yescharacteristic
209C23×C7—24no—
210C23×C7—24no—
211C23×C7—24no—
212C23×C7—24no—
213C23×C7—24no—
214C23×C7—24no—
215C23×C7—24no—
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×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×C7C22×D411yescharacteristic
245C23×C7C22×Q811yescharacteristic
246C23×C7C2×#4111yescharacteristic
247C23×C7C2×#4112yes—
248C23×C7C2×#4112yes—
249C23×C7H(5,𝔽2)11yescharacteristic
250C23×C7H(5,𝔽2)11yescharacteristic
251C23×C7H(5,𝔽2)11yescharacteristic
252C23×C7H(5,𝔽2)11yescharacteristic
253C23×C7H(5,𝔽2)12yes—
254C23×C7H(5,𝔽2)12yes—
255C7×D4—416no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—42no—
278C2×C4×C7—42no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
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×C7C22⋊Q814yes—
308C25C23×C712yes—
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—41no—
312C22×D4—41no—
313C22×D4—41no—
314C22×D4—41no—
315C22×D4C7×Q814yes—
316C23×C4C23×C712yes—
317C23×C4C23×C712yes—
318C23×C4C23×C712yes—
319C23×C4C23×C711yescharacteristic
320C23×C4C23×C711yescharacteristic
321C23×C4C23×C712yes—
322C23×C4C23×C712yes—
323C23×C4C23×C711yescharacteristic
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—41no—
327C2×C4⋊C4—41no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—41no—
331C2×C4⋊C4—41no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4C7×D414yes—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—42no—
343C2×C22⋊C4—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4—42no—
347C2×C22⋊C4—42no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C42—22no—
354C2×C42—41no—
355C2×C42—41no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—42no—
362#95—41no—
363#95—42no—
364#95—42no—
365C22×C7—44no—
366C22×C7—44no—
367C22×C7—44no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—44no—
371C22×C7—44no—
372C22×C7—82no—
373C22×C7—82no—
374C22×C7—82no—
375C22×C7—82no—
376C22×C7—82no—
377C22×C7—82no—
378C22×C7—82no—
379C22×C7—82no—
380C22×C7C2×C22⋊Q811yescharacteristic
381C22×C7#52912yes—
382C22×C7#53411yescharacteristic
383C22×C7#53511yescharacteristic
384C22×C7#53811yescharacteristic
385C22×C7#53811yescharacteristic
386C22×C7#53812yes—
387C22×C7#53812yes—
388C22×C7#53812yes—
389C22×C7#54012yes—
390C22×C7#54012yes—
391C22×C7#54311yescharacteristic
392C22×C7#54311yescharacteristic
393C22×C7#54612yes—
394C22×C7#55012yes—
395C22×C7#55312yes—
396C22×C7#55412yes—
397C22×C7#55412yes—
398C22×C7#55511yescharacteristic
399C22×C7#55911yescharacteristic
400C22×C7#56011yescharacteristic
401C22×C7#56012yes—
402C22×C7H(3,𝔽4)11yescharacteristic
403C4×C7—24no—
404C4×C7—44no—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—44no—
408C4×C7—82no—
409C4×C7—82no—
410C4×C7—82no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24C2×C7×Q814yes—
426C24C24×C711yescharacteristic
427C2×D4—28no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
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×C4C2×C7×D412yes—
480C22×C4C2×C7×D412yes—
481C22×C4C2×C7×D412yes—
482C22×C4C2×C7×Q812yes—
483C22×C4C7×#4114yes—
484C22×C4C7×#4114yes—
485C22×C4C7×#4112yes—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—44no—
497C2×C7—84no—
498C2×C7#245412yes—
499C2×C7#259311yescharacteristic
500C2×C7#262611yescharacteristic
501C2×C7#264812yes—
502C2×C7#270911yescharacteristic
503C2×C7#271412yes—
504C2×C7#273311yescharacteristic
505C2×C7#278012yes—
506C2×C7#278411yescharacteristic
507C2×C7#280411yescharacteristic
508C2×C7#282811yescharacteristic
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
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—
544C23C22×C7×D411yescharacteristic
545C23C22×C7×Q811yescharacteristic
546C23C2×C7×#4111yescharacteristic
547C23C2×C7×#4112yes—
548C23C2×C7×#4112yes—
549C23C7×H(5,𝔽2)11yescharacteristic
550C23C7×H(5,𝔽2)11yescharacteristic
551C23C7×H(5,𝔽2)11yescharacteristic
552C23C7×H(5,𝔽2)11yescharacteristic
553C23C7×H(5,𝔽2)12yes—
554C23C7×H(5,𝔽2)12yes—
555D4—416no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
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×C4C7×C22⋊Q814yes—
608C7#1659911yescharacteristic, direct product factor, Sylow
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C2×C7×C22⋊Q811yescharacteristic
625C22C7×#52912yes—
626C22C7×#53411yescharacteristic
627C22C7×#53511yescharacteristic
628C22C7×#53811yescharacteristic
629C22C7×#53812yes—
630C22C7×#53812yes—
631C22C7×#53812yes—
632C22C7×#53811yescharacteristic
633C22C7×#54012yes—
634C22C7×#54012yes—
635C22C7×#54311yescharacteristic
636C22C7×#54311yescharacteristic
637C22C7×#54612yes—
638C22C7×#55012yes—
639C22C7×#55312yes—
640C22C7×#55412yes—
641C22C7×#55412yes—
642C22C7×#55511yescharacteristic
643C22C7×#55911yescharacteristic
644C22C7×#56011yescharacteristic
645C22C7×#56012yes—
646C22C7×H(3,𝔽4)11yescharacteristic
647C4—24no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—44no—
652C4—82no—
653C4—82no—
654C4—82no—
655C2—84no—
656C2C7×#245412yes—
657C2C7×#259311yescharacteristic
658C2C7×#262611yescharacteristic
659C2C7×#264812yes—
660C2C7×#270911yescharacteristic
661C2C7×#271412yes—
662C2C7×#273311yescharacteristic
663C2C7×#278012yes—
664C2C7×#278411yescharacteristic
665C2C7×#280411yescharacteristic
666C2C7×#282811yescharacteristic
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×#400C2semidirect
3. C7×#1441C2semidirect
4. C7×#1439C2semidirect
5. C7×#1439C2semidirect
6. C7×#1439C2semidirect
7. C7×#1439C2semidirect
9. C7×#1436C2semidirect
33. #16599C7direct (N × H)
91. C2×#400C2×C7semidirect
93. #1441C2×C7semidirect
94. #1439C2×C7semidirect
95. #1439C2×C7semidirect
96. #1439C2×C7semidirect
97. #1439C2×C7semidirect
99. #1436C2×C7semidirect
608. C7#16599direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4002
C2×C7×#3902
C7×#14412
C7×#14392
C7×#14382
C7×#14362
#165997