Finite Group Databasebeta

← C7×#16556 #13193772

Subgroups — C7×#16556 #13193772

Order 1792.

1782 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#396C212yesmaximal, complemented
2C2×C7×#395C211yesmaximal, characteristic, complemented
3C2×C7×#393C212yesmaximal, complemented
4C2×C7×#390C211yesmaximal, characteristic
5C7×#1441C211yesmaximal, characteristic, complemented
6C7×#1439C212yesmaximal, complemented
7C7×#1439C212yesmaximal, complemented
8C7×#1438C212yesmaximal
9C7×#1436C212yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C23×C7×D4C2211yescharacteristic
12C22×C7×C4⋊C4C2212yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2212yes—
15C22×C7×C4⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2212yes—
17C22×C7×C22⋊C4C2212yes—
18C22×C7×C22⋊C4C2212yes—
19C22×C7×C22⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C7×C22⋊C4C2212yes—
22C22×C42×C7C2211yescharacteristic
23C7×#396—28no—
24C7×#395—24no—
25C7×#393—28no—
26C7×#390—24no—
27C2×C7×#95C2212yes—
28C2×C7×#95C2211yescharacteristic
29C2×C7×#95C2212yes—
30C2×C7×#95C2212yes—
31C2×C7×#95C2212yes—
32C2×C7×#95C2212yes—
33C2×C7×#95C2211yescharacteristic
34C2×C7×#95C2212yes—
35#16556C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7C2312yes—
37C22×C7×Q8—24no—
38C22×C7×Q8—22no—
39C22×C7×Q8—42no—
40C22×C7×Q8—41no—
41C22×C7×Q8—41no—
42C22×C7×D4—22no—
43C22×C7×D4—22no—
44C22×C7×D4—41no—
45C22×C7×D4—41no—
46C22×C7×D4—41no—
47C22×C7×D4—41no—
48C22×C7×D4Q814yescomplemented
49C23×C4×C7C2312yes—
50C23×C4×C7C2311yescharacteristic
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2312yes—
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2312yes—
56C23×C4×C7C2311yescharacteristic
57C2×C7×C4⋊C4—24no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—41no—
63C2×C7×C4⋊C4—41no—
64C2×C7×C4⋊C4—42no—
65C2×C7×C4⋊C4—41no—
66C2×C7×C4⋊C4—42no—
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—42no—
79C2×C7×C22⋊C4—42no—
80C2×C7×C22⋊C4—42no—
81C2×C7×C22⋊C4—42no—
82C2×C7×C22⋊C4—42no—
83C2×C7×C22⋊C4—42no—
84C2×C7×C22⋊C4—42no—
85C2×C7×C22⋊C4—42no—
86C2×C42×C7—22no—
87C2×C42×C7—41no—
88C2×C42×C7—41no—
89C7×#95—24no—
90C7×#95—24no—
91C7×#95—22no—
92C7×#95—24no—
93C7×#95—24no—
94C7×#95—42no—
95C7×#95—41no—
96C7×#95D418yescomplemented
97C2×#396C2×C712yescomplemented
98C2×#395C2×C711yescharacteristic, complemented
99C2×#393C2×C712yescomplemented
100C2×#390C2×C711yescharacteristic
101#1441C2×C711yescharacteristic, complemented
102#1439C2×C712yescomplemented
103#1439C2×C712yescomplemented
104#1438C2×C712yes—
105#1436C2×C712yescomplemented
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×C7—22no—
120C24×C7C2×Q814yes—
121C24×C7C2411yescharacteristic
122C2×C7×Q8—44no—
123C2×C7×Q8—48no—
124C2×C7×Q8—44no—
125C2×C7×Q8—48no—
126C2×C7×Q8—44no—
127C2×C7×D4—28no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—44no—
131C2×C7×D4—44no—
132C2×C7×D4—44no—
133C2×C7×D4—44no—
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
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—22no—
145C22×C4×C7—22no—
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—21no—
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—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—22no—
168C22×C4×C7—22no—
169C22×C4×C7—22no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—22no—
173C22×C4×C7—22no—
174C22×C4×C7—22no—
175C22×C4×C7—22no—
176C22×C4×C7—22no—
177C22×C4×C7C2×D414yes—
178C22×C4×C7C2×D414yes—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2×D414yes—
181C22×C4×C7C2×D412yes—
182C22×C4×C7C2×D412yes—
183C22×C4×C7C2×Q812yes—
184C22×C4×C7#4114yes—
185C22×C4×C7#4112yes—
186C7×C4⋊C4—44no—
187C7×C4⋊C4—44no—
188C7×C4⋊C4—48no—
189C7×C4⋊C4—48no—
190C7×C22⋊C4—48no—
191C7×C22⋊C4—48no—
192C7×C22⋊C4—48no—
193C7×C22⋊C4—48no—
194C7×C22⋊C4—48no—
195C7×C22⋊C4—48no—
196C42×C7—44no—
197C23×Q8C22×C711yescharacteristic
198C23×D4C22×C711yescharacteristic
199C22×C4⋊C4C22×C712yes—
200C22×C4⋊C4C22×C711yescharacteristic
201C22×C4⋊C4C22×C712yes—
202C22×C4⋊C4C22×C711yescharacteristic
203C22×C22⋊C4C22×C712yes—
204C22×C22⋊C4C22×C712yes—
205C22×C22⋊C4C22×C712yes—
206C22×C22⋊C4C22×C712yes—
207C22×C22⋊C4C22×C712yes—
208C22×C22⋊C4C22×C712yes—
209C22×C42C22×C711yescharacteristic
210#396—28no—
211#395—24no—
212#393—28no—
213#390—24no—
214C2×#95C22×C712yes—
215C2×#95C22×C712yes—
216C2×#95C22×C712yes—
217C2×#95C22×C711yescharacteristic
218C2×#95C22×C712yes—
219C2×#95C22×C712yes—
220C2×#95C22×C711yescharacteristic
221C2×#95C22×C712yes—
222C23×C7—24no—
223C23×C7—24no—
224C23×C7—24no—
225C23×C7—24no—
226C23×C7—24no—
227C23×C7—24no—
228C23×C7—24no—
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×C7—42no—
253C23×C7—42no—
254C23×C7—42no—
255C23×C7—42no—
256C23×C7—42no—
257C23×C7C22×D411yescharacteristic
258C23×C7C22×D412yes—
259C23×C7C22×Q811yescharacteristic
260C23×C7C2×#4111yescharacteristic
261C23×C7C2×#4112yes—
262C23×C7H(5,𝔽2)11yescharacteristic
263C23×C7H(5,𝔽2)11yescharacteristic
264C23×C7H(5,𝔽2)11yescharacteristic
265C23×C7H(5,𝔽2)12yes—
266C23×C7H(5,𝔽2)11yescharacteristic
267C23×C72−1+412yes—
268C7×Q8—416no—
269C7×D4—416no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
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—24no—
284C2×C4×C7—24no—
285C2×C4×C7—24no—
286C2×C4×C7—24no—
287C2×C4×C7—24no—
288C2×C4×C7—24no—
289C2×C4×C7—24no—
290C2×C4×C7—24no—
291C2×C4×C7—24no—
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—41no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—41no—
306C2×C4×C7—42no—
307C2×C4×C7—42no—
308C2×C4×C7—42no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—42no—
312C2×C4×C7—42no—
313C2×C4×C7—42no—
314C2×C4×C7—42no—
315C2×C4×C7—42no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—42no—
319C2×C4×C7—42no—
320C2×C4×C7—42no—
321C2×C4×C7C22⋊Q814yes—
322C25C23×C712yes—
323C22×Q8—24no—
324C22×Q8—22no—
325C22×Q8—42no—
326C22×Q8—41no—
327C22×Q8—41no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—41no—
331C22×D4—41no—
332C22×D4—41no—
333C22×D4—41no—
334C22×D4C7×Q814yescomplemented
335C23×C4C23×C712yes—
336C23×C4C23×C711yescharacteristic
337C23×C4C23×C712yes—
338C23×C4C23×C711yescharacteristic
339C23×C4C23×C712yes—
340C23×C4C23×C712yes—
341C23×C4C23×C712yes—
342C23×C4C23×C711yescharacteristic
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4C7×D414yes—
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—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—42no—
365C2×C22⋊C4—42no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—42no—
369C2×C22⋊C4—42no—
370C2×C22⋊C4—42no—
371C2×C22⋊C4—42no—
372C2×C42—22no—
373C2×C42—41no—
374C2×C42—41no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—42no—
381#95—41no—
382#95C7×D418yescomplemented
383C22×C7—44no—
384C22×C7—44no—
385C22×C7—44no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—44no—
389C22×C7—44no—
390C22×C7—82no—
391C22×C7—82no—
392C22×C7—82no—
393C22×C7—82no—
394C22×C7—82no—
395C22×C7—82no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7C2×C22⋊Q811yescharacteristic
399C22×C7#53212yes—
400C22×C7#53411yescharacteristic
401C22×C7#53511yescharacteristic
402C22×C7#53512yes—
403C22×C7#53712yes—
404C22×C7#53811yescharacteristic
405C22×C7#53811yescharacteristic
406C22×C7#53812yes—
407C22×C7#54212yes—
408C22×C7#54212yes—
409C22×C7#54311yescharacteristic
410C22×C7#54311yescharacteristic
411C22×C7D4212yes—
412C22×C7D4211yescharacteristic
413C22×C7#54612yes—
414C22×C7#54812yes—
415C22×C7D4×Q812yes—
416C22×C7#55412yes—
417C22×C7#55511yescharacteristic
418C22×C7#56011yescharacteristic
419C22×C7#56212yes—
420C22×C7#56311yescharacteristic
421C4×C7—24no—
422C4×C7—42no—
423C4×C7—42no—
424C4×C7—44no—
425C4×C7—44no—
426C4×C7—44no—
427C4×C7—44no—
428C4×C7—82no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24C2×C7×Q814yes—
444C24C24×C711yescharacteristic
445C2×Q8—44no—
446C2×Q8—48no—
447C2×Q8—44no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×D4—28no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
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—21no—
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×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4C2×C7×D414yes—
501C22×C4C2×C7×D414yes—
502C22×C4C2×C7×D412yes—
503C22×C4C2×C7×D414yes—
504C22×C4C2×C7×D412yes—
505C22×C4C2×C7×D412yes—
506C22×C4C2×C7×Q812yes—
507C22×C4C7×#4114yes—
508C22×C4C7×#4112yes—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C42—44no—
520C2×C7—84no—
521C2×C7#245212yes—
522C2×C7#255511yescharacteristic
523C2×C7#262611yescharacteristic
524C2×C7#266212yes—
525C2×C7#266911yescharacteristic
526C2×C7#267612yes—
527C2×C7#268311yescharacteristic
528C2×C7#269212yes—
529C2×C7#273111yescharacteristic
530C2×C7#278411yescharacteristic
531C2×C7#283011yescharacteristic
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
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—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23C22×C7×D411yescharacteristic
568C23C22×C7×D412yes—
569C23C22×C7×Q811yescharacteristic
570C23C2×C7×#4111yescharacteristic
571C23C2×C7×#4112yes—
572C23C7×H(5,𝔽2)11yescharacteristic
573C23C7×H(5,𝔽2)11yescharacteristic
574C23C7×H(5,𝔽2)11yescharacteristic
575C23C7×H(5,𝔽2)12yes—
576C23C7×H(5,𝔽2)11yescharacteristic
577C23C7×2−1+412yes—
578Q8—416no—
579D4—416no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
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—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4C7×C22⋊Q814yes—
632C7#1655611yescharacteristic, direct product factor, Sylow
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22C2×C7×C22⋊Q811yescharacteristic
649C22C7×#53212yes—
650C22C7×#53411yescharacteristic
651C22C7×#53512yes—
652C22C7×#53511yescharacteristic
653C22C7×#53712yes—
654C22C7×#53811yescharacteristic
655C22C7×#53812yes—
656C22C7×#53811yescharacteristic
657C22C7×#54212yes—
658C22C7×#54212yes—
659C22C7×#54311yescharacteristic
660C22C7×#54311yescharacteristic
661C22C7×D4212yes—
662C22C7×D4211yescharacteristic
663C22C7×#54612yes—
664C22C7×#54812yes—
665C22C7×D4×Q812yes—
666C22C7×#55412yes—
667C22C7×#55511yescharacteristic
668C22C7×#56011yescharacteristic
669C22C7×#56212yes—
670C22C7×#56311yescharacteristic
671C4—24no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—82no—
679C2—84no—
680C2C7×#245212yes—
681C2C7×#255511yescharacteristic
682C2C7×#262611yescharacteristic
683C2C7×#266212yes—
684C2C7×#266911yescharacteristic
685C2C7×#267612yes—
686C2C7×#268311yescharacteristic
687C2C7×#269212yes—
688C2C7×#273111yescharacteristic
689C2C7×#278411yescharacteristic
690C2C7×#283011yescharacteristic
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
2. C2×C7×#395C2semidirect
3. C2×C7×#393C2semidirect
5. C7×#1441C2semidirect
6. C7×#1439C2semidirect
7. C7×#1439C2semidirect
9. C7×#1436C2semidirect
35. #16556C7direct (N × H)
48. C22×C7×D4Q8semidirect
96. C7×#95D4semidirect
97. C2×#396C2×C7semidirect
98. C2×#395C2×C7semidirect
99. C2×#393C2×C7semidirect
101. #1441C2×C7semidirect
102. #1439C2×C7semidirect
103. #1439C2×C7semidirect
105. #1436C2×C7semidirect
334. C22×D4C7×Q8semidirect
382. #95C7×D4semidirect
632. C7#16556direct (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×#3952
C2×C7×#3932
C2×C7×#3902
C7×#14412
C7×#14392
C7×#14382
C7×#14362
#165567