Finite Group Databasebeta

← C7×#24759 #13201975

Subgroups — C7×#24759 #13201975

Order 1792.

1044 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#468C212yesmaximal
2C2×C7×#461C211yesmaximal, characteristic, complemented
3C2×C7×#423C211yesmaximal, characteristic, complemented
4C7×#2857C211yesmaximal, characteristic, complemented
5C4×C7×#121C211yesmaximal, characteristic
6C7×#2068C212yesmaximal, complemented
7C7×#2055C212yesmaximal, complemented
8C7×#1981C211yesmaximal, characteristic, complemented
9C7×#1947C211yesmaximal, characteristic, complemented
10C7×#1894C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C22×C7×M16C2212yes—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×C22⋊Q8C2212yes—
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C7×#468—28no—
21C7×#461—24no—
22C2×C7×#107C2212yes—
23C7×#423—24no—
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×C4⋊C8C2212yes—
26C2×C7×C4⋊C8C2211yescharacteristic
27C2×C7×#103C2212yes—
28C2×C7×#103C2212yes—
29C2×C7×#103C2212yes—
30C2×C7×#102C2212yes—
31C7×#391C2212yes—
32C7×#390C2211yescharacteristic
33C7×#389—21no—
34C7×#389—21no—
35C7×#389C2211yescharacteristic
36C7×#388—21no—
37C7×#386—21no—
38C7×#384C2211yescharacteristic
39C7×#380—21no—
40C7×#380—21no—
41C7×#380C2211yescharacteristic
42C4×C7×C4⋊C4C2211yescharacteristic
43C4×C7×C22⋊C4—21no—
44C7×#340C2212yes—
45C7×#340C2212yes—
46#24759C711yesmaximal, characteristic, direct product factor, Sylow
47C22×C7×Q8C2312yes—
48C22×C7×D4—21no—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C2×C7×SD16—24no—
55C2×C7×SD16—24no—
56C2×C7×SD16—42no—
57C2×C7×M16—24no—
58C2×C7×M16—24no—
59C2×C7×M16—42no—
60C22×C7×C8C2312yes—
61C22×C7×C8C2312yes—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8—22no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8—24no—
66C7×#121—22no—
67C7×#121D414yes—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—44no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C7×C22⋊C4—21no—
87C2×C42×C7—22no—
88C2×C42×C7—22no—
89C2×C42×C7—21no—
90C2×C42×C7D411yescharacteristic
91C2×C42×C7D411yescharacteristic
92C2×C42×C7C2311yescharacteristic
93C2×C42×C7C2311yescharacteristic
94C7×#107—24no—
95C7×C4⋊C8—24no—
96C7×C4⋊C8—22no—
97C7×C4⋊C8D414yescomplemented
98C7×#103—24no—
99C7×#103—24no—
100C7×#103—42no—
101C7×#102—24no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—22no—
106C7×#95—22no—
107C2×#468C2×C712yes—
108C2×#461C2×C711yescharacteristic, complemented
109C2×#423C2×C711yescharacteristic, complemented
110#2857C2×C711yescharacteristic, complemented
111C4×#121C2×C711yescharacteristic
112#2068C2×C712yescomplemented
113#2055C2×C712yescomplemented
114#1981C2×C711yescharacteristic, complemented
115#1947C2×C711yescharacteristic, complemented
116#1894C2×C712yescomplemented
117#1802C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7C2×D411yescharacteristic
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×D4—24no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—42no—
133C2×C7×D4—42no—
134C2×C7×D4C2×D412yes—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—41no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7#4112yes—
160C22×C4×C7C2411yescharacteristic
161C7×SD16—48no—
162C7×M16—48no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8C2×D414yes—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—44no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4C2×D412yes—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—42no—
188C7×C22⋊C4—42no—
189C42×C7—22no—
190C42×C7—22no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—42no—
194C42×C7C2×D412yes—
195C42×C7#4112yes—
196C22×SD16C22×C712yes—
197C22×M16C22×C712yes—
198C2×C4⋊Q8C22×C711yescharacteristic
199C2×C22⋊Q8C22×C712yes—
200C2×#121C22×C711yescharacteristic
201C2×C4×D4—21no—
202C2×C4×D4C22×C711yescharacteristic
203C22×C42C22×C711yescharacteristic
204#468—28no—
205#461—24no—
206C2×#107C22×C712yes—
207#423—24no—
208C2×C4⋊C8C22×C712yes—
209C2×C4⋊C8C22×C711yescharacteristic
210C2×C4⋊C8C22×C711yescharacteristic
211C2×#103C22×C712yes—
212C2×#103C22×C712yes—
213C2×#103C22×C712yes—
214C2×#102C22×C712yes—
215#391C22×C712yes—
216#390C22×C711yescharacteristic
217#389—21no—
218#389—21no—
219#389C22×C711yescharacteristic
220#388—21no—
221#386—21no—
222#384C22×C711yescharacteristic
223#380—21no—
224#380—21no—
225#380C22×C711yescharacteristic
226C4×C4⋊C4C22×C711yescharacteristic
227C4×C22⋊C4—21no—
228#340C22×C712yes—
229#340C22×C712yes—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7—21no—
233C23×C7—21no—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—22no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7—41no—
240C23×C7—41no—
241C23×C7—41no—
242C23×C7—41no—
243C23×C7#12112yes—
244C23×C7C22×D411yescharacteristic
245C7×Q8—42no—
246C7×Q8—42no—
247C7×Q8—42no—
248C7×Q8—42no—
249C7×D4—22no—
250C7×D4—22no—
251C7×D4—48no—
252C7×D4—41no—
253C7×D4—41no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—21no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—44no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—44no—
272C2×C4×C7—42no—
273C2×C4×C7—41no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—41no—
277C2×C4×C7—42no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—42no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—84no—
285C2×C4×C7#12112yes—
286C2×C4×C7#12112yes—
287C2×C4×C7#12112yes—
288C2×C4×C7AGL(1,ℤ/8)12yes—
289C2×C4×C7#13712yes—
290C2×C4×C7#13712yes—
291C2×C4×C7C22×D411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C7×C8—24no—
298C7×C8—42no—
299C22×Q8C23×C712yes—
300C22×D4—21no—
301C22×D4C23×C711yescharacteristic
302C23×C4—21no—
303C23×C4C7×D411yescharacteristic
304C23×C4C7×D411yescharacteristic
305C23×C4C23×C711yescharacteristic
306C2×SD16—24no—
307C2×SD16—24no—
308C2×SD16—42no—
309C2×M16—24no—
310C2×M16—24no—
311C2×M16—42no—
312C22×C8C23×C712yes—
313C22×C8C23×C712yes—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318#121—22no—
319#121C7×D414yes—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—44no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C4⋊C4C23×C712yes—
331C2×C4⋊C4C23×C711yescharacteristic
332C2×C4⋊C4C23×C711yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42C7×D411yescharacteristic
343C2×C42C7×D411yescharacteristic
344C2×C42C23×C711yescharacteristic
345C2×C42C23×C711yescharacteristic
346#107—24no—
347C4⋊C8—24no—
348C4⋊C8—22no—
349C4⋊C8C7×D414yescomplemented
350#103—24no—
351#103—24no—
352#103—42no—
353#102—24no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—22no—
359C22×C7—22no—
360C22×C7—22no—
361C22×C7—22no—
362C22×C7—41no—
363C22×C7—41no—
364C22×C7—41no—
365C22×C7—41no—
366C22×C7—42no—
367C22×C7—42no—
368C22×C7—42no—
369C22×C7—81no—
370C22×C7—81no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7C2×#12111yescharacteristic
374C22×C7C2×#12111yescharacteristic
375C22×C7#53711yescharacteristic
376C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
377C22×C7C2×#13711yescharacteristic
378C22×C7C2×#13711yescharacteristic
379C22×C7#57511yescharacteristic
380C4×C7—22no—
381C4×C7—22no—
382C4×C7—22no—
383C4×C7—42no—
384C4×C7—41no—
385C4×C7—82no—
386C4×C7—81no—
387C4×C7—82no—
388C4×C7—81no—
389C4×C7#46814yescomplemented
390C4×C7D4211yescharacteristic
391C4×C7#54711yescharacteristic
392C4×C7#54811yescharacteristic
393C4×C7#55311yescharacteristic
394C24—21no—
395C24C2×C7×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×C7×D412yes—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C2×C7×D411yescharacteristic
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D412yes—
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C2×C7×D411yescharacteristic
435C22×C4C7×#4112yes—
436C22×C4C24×C711yescharacteristic
437SD16—48no—
438M16—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8C2×C7×D414yes—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4C2×C7×D412yes—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C42—22no—
466C42—22no—
467C42—21no—
468C42—21no—
469C42—42no—
470C42C2×C7×D412yes—
471C42C7×#4112yes—
472C2×C7—42no—
473C2×C7—82no—
474C2×C7#241611yescharacteristic
475C2×C7#303611yescharacteristic
476C2×C7#303911yescharacteristic
477C2×C7C2×#46811yescharacteristic
478C2×C7#305611yescharacteristic
479C2×C7#325211yescharacteristic
480C2×C7#325411yescharacteristic
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C7×#12112yes—
495C23C22×C7×D411yescharacteristic
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
500D4—22no—
501D4—22no—
502D4—48no—
503D4—41no—
504D4—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4C7×#12112yes—
537C2×C4C7×#12112yes—
538C2×C4C7×#12112yes—
539C2×C4C7×AGL(1,ℤ/8)12yes—
540C2×C4C7×#13712yes—
541C2×C4C7×#13712yes—
542C2×C4C22×C7×D411yescharacteristic
543C2×C4C22×C7×D411yescharacteristic
544C2×C4C2×C7×#4111yescharacteristic
545C2×C4C2×C7×#4111yescharacteristic
546C2×C4C7×H(5,𝔽2)11yescharacteristic
547C2×C4C7×2−1+411yescharacteristic
548C8—24no—
549C8—42no—
550C7#2475911yescharacteristic, direct product factor, Sylow
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×C7×#12111yescharacteristic
566C22C2×C7×#12111yescharacteristic
567C22C7×#53711yescharacteristic
568C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
569C22C2×C7×#13711yescharacteristic
570C22C2×C7×#13711yescharacteristic
571C22C7×#57511yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—41no—
577C4—82no—
578C4—81no—
579C4—82no—
580C4—81no—
581C4C7×#46814yescomplemented
582C4C7×D4211yescharacteristic
583C4C7×#54711yescharacteristic
584C4C7×#54811yescharacteristic
585C4C7×#55311yescharacteristic
586C2—42no—
587C2—82no—
588C2C7×#241611yescharacteristic
589C2C7×#303611yescharacteristic
590C2C7×#303911yescharacteristic
591C2C2×C7×#46811yescharacteristic
592C2C7×#305611yescharacteristic
593C2C7×#325211yescharacteristic
594C2C7×#325411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#461C2semidirect
3. C2×C7×#423C2semidirect
4. C7×#2857C2semidirect
6. C7×#2068C2semidirect
7. C7×#2055C2semidirect
8. C7×#1981C2semidirect
9. C7×#1947C2semidirect
10. C7×#1894C2semidirect
11. C7×#1802C2semidirect
46. #24759C7direct (N × H)
97. C7×C4⋊C8D4semidirect
108. C2×#461C2×C7semidirect
109. C2×#423C2×C7semidirect
110. #2857C2×C7semidirect
112. #2068C2×C7semidirect
113. #2055C2×C7semidirect
114. #1981C2×C7semidirect
115. #1947C2×C7semidirect
116. #1894C2×C7semidirect
117. #1802C2×C7semidirect
349. C4⋊C8C7×D4semidirect
389. C4×C7#468semidirect
550. C7#24759direct (N × H)
581. C4C7×#468semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4682
C2×C7×#4612
C2×C7×#4232
C7×#28572
C4×C7×#1212
C7×#20682
C7×#20552
C7×#19812
C7×#19472
C7×#18942
C7×#18022
#247597