Finite Group Databasebeta

← C7×#24002 #13201218

Subgroups — C7×#24002 #13201218

Order 1792.

1844 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#121C211yesmaximal, characteristic
2C2×C7×#480C212yesmaximal, complemented
3C2×C7×#380C211yesmaximal, characteristic, complemented
4C7×#2033C214yesmaximal, complemented
5C7×#1874C212yesmaximal, complemented
6C7×#1852C212yesmaximal, complemented
7C7×#1786C212yesmaximal, complemented
8C7×#1779C211yesmaximal, characteristic, complemented
9C23×C7×D4—21no—
10C23×C7×D4C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C2×C7×#121—21no—
13C2×C7×#121—21no—
14C2×C7×#121—28no—
15C2×C7×#121C2212yes—
16C2×C7×#121C2212yes—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic, complemented
19C22×C7×C4⋊C4C2211yescharacteristic, complemented
20C22×C7×C22⋊C4—21no—
21C22×C7×C22⋊C4—21no—
22C7×#480—28no—
23C2×C7×#107C2214yescomplemented
24C2×C7×#102C2214yescomplemented
25C2×C7×#102C2214yescomplemented
26C2×C7×C22⋊C8C2212yes—
27C2×C7×C22⋊C8C2212yescomplemented
28C2×C7×C22⋊C8C2212yescomplemented
29C7×#380—21no—
30C7×#380—21no—
31C7×#380C2212yes—
32C7×#380C2212yescomplemented
33C2×C7×#95—21no—
34C7×#340C2214yescomplemented
35#24002C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7—21no—
37C25×C7D411yescharacteristic
38C22×C7×D4—28no—
39C22×C7×D4—22no—
40C22×C7×D4—22no—
41C22×C7×D4—24no—
42C22×C7×D4—21no—
43C22×C7×D4—24no—
44C22×C7×D4—42no—
45C22×C7×D4—44no—
46C22×C7×D4—44no—
47C22×C7×D4C2312yes—
48C23×C4×C7—21no—
49C23×C4×C7—24no—
50C23×C4×C7—24no—
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7—42no—
54C23×C4×C7D411yescharacteristic
55C23×C4×C7D411yescharacteristic
56C23×C4×C7D411yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C22×C7×C8C2314yes—
61C7×#121—28no—
62C7×#121—28no—
63C7×#121—24no—
64C7×#121—24no—
65C7×#121—24no—
66C7×#121D418yes—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—24no—
69C2×C7×C4⋊C4—24no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—42no—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C4⋊C4C2312yes—
76C2×C7×C22⋊C4—28no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—22no—
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—44no—
82C7×#107—28no—
83C7×#102—28no—
84C7×#102—44no—
85C7×C22⋊C8—24no—
86C7×C22⋊C8—24no—
87C7×C22⋊C8—42no—
88C7×#95—22no—
89C7×#95—22no—
90C22×#121C2×C711yescharacteristic
91C2×#480C2×C712yescomplemented
92C2×#380C2×C711yescharacteristic, complemented
93#2033C2×C714yescomplemented
94#1874C2×C712yescomplemented
95#1852C2×C712yescomplemented
96#1786C2×C712yescomplemented
97#1779C2×C711yescharacteristic, complemented
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—22no—
101C24×C7—22no—
102C24×C7—24no—
103C24×C7—22no—
104C24×C7—41no—
105C24×C7—41no—
106C24×C7—42no—
107C24×C7—42no—
108C24×C7—42no—
109C24×C7—42no—
110C24×C7—42no—
111C24×C7—42no—
112C24×C7D814yes—
113C24×C7C2×D411yescharacteristic
114C24×C7C2×D411yescharacteristic
115C24×C7C2×D411yescharacteristic
116C2×C7×D4—24no—
117C2×C7×D4—28no—
118C2×C7×D4—28no—
119C2×C7×D4—22no—
120C2×C7×D4—24no—
121C2×C7×D4—24no—
122C2×C7×D4—416no—
123C2×C7×D4—42no—
124C2×C7×D4—432no—
125C2×C7×D4—44no—
126C2×C7×D4—44no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—42no—
130C2×C7×D4C2×D414yes—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—24no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
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—41no—
149C22×C4×C7—41no—
150C22×C4×C7—41no—
151C22×C4×C7—416no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D414yes—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7#4114yes—
165C22×C4×C7C2411yescharacteristic
166C2×C7×C8—24no—
167C2×C7×C8—24no—
168C2×C7×C8—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4C2×D414yes—
176C7×C22⋊C4—28no—
177C7×C22⋊C4—28no—
178C7×C22⋊C4—44no—
179C7×C22⋊C4—44no—
180C23×D4—21no—
181C23×D4C22×C711yescharacteristic
182C24×C4C22×C711yescharacteristic
183C2×#121—21no—
184C2×#121—21no—
185C2×#121—28no—
186C2×#121C22×C712yes—
187C2×#121C22×C712yes—
188C22×C4⋊C4C22×C711yescharacteristic
189C22×C4⋊C4C22×C711yescharacteristic, complemented
190C22×C4⋊C4C22×C711yescharacteristic, complemented
191C22×C22⋊C4—21no—
192C22×C22⋊C4—21no—
193#480—28no—
194C2×#107C22×C714yescomplemented
195C2×#102C22×C714yescomplemented
196C2×#102C22×C714yescomplemented
197C2×C22⋊C8C22×C712yes—
198C2×C22⋊C8C22×C712yescomplemented
199C2×C22⋊C8C22×C712yescomplemented
200#380—21no—
201#380—21no—
202#380C22×C712yes—
203#380C22×C712yescomplemented
204C2×#95—21no—
205#340C22×C714yescomplemented
206C23×C7—21no—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—21no—
211C23×C7—21no—
212C23×C7—21no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—28no—
221C23×C7—28no—
222C23×C7—24no—
223C23×C7—44no—
224C23×C7—44no—
225C23×C7—44no—
226C23×C7—44no—
227C23×C7—44no—
228C23×C7—48no—
229C23×C7—48no—
230C23×C7—48no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—84no—
238C23×C7—84no—
239C23×C7—84no—
240C23×C7—84no—
241C23×C7#12012yes—
242C23×C7#12312yes—
243C23×C7C2×D812yes—
244C23×C7C2×D812yes—
245C23×C7C2×D812yes—
246C23×C7AGL(1,ℤ/8)12yes—
247C23×C7#13712yes—
248C23×C7C22×D411yescharacteristic
249C7×D4—28no—
250C7×D4—48no—
251C7×D4—416no—
252C7×D4—416no—
253C7×D4—44no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—28no—
259C2×C4×C7—28no—
260C2×C4×C7—21no—
261C2×C4×C7—24no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—44no—
265C2×C4×C7—44no—
266C2×C4×C7—44no—
267C2×C4×C7—48no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—44no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7—84no—
278C2×C4×C7—84no—
279C2×C4×C7#12012yes—
280C2×C4×C7#12312yes—
281C2×C4×C7C22×D412yes—
282C2×C4×C7C2×#4112yes—
283C2×C4×C7H(5,𝔽2)12yes—
284C7×C8—44no—
285C25—21no—
286C25C7×D411yescharacteristic
287C22×D4—28no—
288C22×D4—22no—
289C22×D4—22no—
290C22×D4—24no—
291C22×D4—21no—
292C22×D4—24no—
293C22×D4—42no—
294C22×D4—44no—
295C22×D4—44no—
296C22×D4C23×C712yes—
297C23×C4—21no—
298C23×C4—24no—
299C23×C4—24no—
300C23×C4—21no—
301C23×C4—21no—
302C23×C4—42no—
303C23×C4C7×D411yescharacteristic
304C23×C4C7×D411yescharacteristic
305C23×C4C7×D411yescharacteristic
306C23×C4C23×C711yescharacteristic
307C23×C4C23×C711yescharacteristic
308C23×C4C23×C711yescharacteristic
309C22×C8C23×C714yes—
310#121—28no—
311#121—28no—
312#121—24no—
313#121—24no—
314#121—24no—
315#121C7×D418yes—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4C23×C712yes—
323C2×C4⋊C4C23×C712yes—
324C2×C4⋊C4C23×C712yes—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—44no—
331#107—28no—
332#102—28no—
333#102—44no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336C22⋊C8—42no—
337#95—22no—
338#95—22no—
339C22×C7—22no—
340C22×C7—22no—
341C22×C7—22no—
342C22×C7—22no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—22no—
346C22×C7—22no—
347C22×C7—22no—
348C22×C7—22no—
349C22×C7—22no—
350C22×C7—22no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—416no—
355C22×C7—41no—
356C22×C7—41no—
357C22×C7—41no—
358C22×C7—41no—
359C22×C7—44no—
360C22×C7—44no—
361C22×C7—44no—
362C22×C7—816no—
363C22×C7—82no—
364C22×C7—82no—
365C22×C7—82no—
366C22×C7—82no—
367C22×C7#44714yes—
368C22×C7#48014yescomplemented
369C22×C7C2×#12011yescharacteristic
370C22×C7C2×#12311yescharacteristic
371C22×C7#53411yescharacteristic
372C22×C7C22×D811yescharacteristic
373C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C7C2×#13711yescharacteristic
375C22×C7#57511yescharacteristic
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—41no—
379C4×C7—44no—
380C4×C7—82no—
381C4×C7—82no—
382C4×C7#54614yes—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24—22no—
389C24—41no—
390C24—41no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24C7×D814yes—
398C24C2×C7×D411yescharacteristic
399C24C2×C7×D411yescharacteristic
400C24C2×C7×D411yescharacteristic
401C2×D4—24no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—416no—
408C2×D4—42no—
409C2×D4—432no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4C2×C7×D414yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—416no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C2×C7×D414yes—
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C7×#4114yes—
450C22×C4C24×C711yescharacteristic
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4C2×C7×D414yes—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C2×C7—24no—
466C2×C7—24no—
467C2×C7—42no—
468C2×C7—84no—
469C2×C7#240511yescharacteristic
470C2×C7C2×#44711yescharacteristic
471C2×C7#300711yescharacteristic
472C2×C7C2×#48011yescharacteristic
473C2×C7#309411yescharacteristic
474C2×C7#318611yescharacteristic
475C2×C7#319311yescharacteristic
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—28no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—28no—
492C23—24no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23C7×#12012yes—
512C23C7×#12312yes—
513C23C2×C7×D812yes—
514C23C2×C7×D812yes—
515C23C2×C7×D812yes—
516C23C7×AGL(1,ℤ/8)12yes—
517C23C7×#13712yes—
518C23C22×C7×D411yescharacteristic
519D4—28no—
520D4—48no—
521D4—416no—
522D4—416no—
523D4—44no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4C7×#12012yes—
550C2×C4C7×#12312yes—
551C2×C4C22×C7×D412yes—
552C2×C4C2×C7×#4112yes—
553C2×C4C7×H(5,𝔽2)12yes—
554C8—44no—
555C7#2400211yescharacteristic, direct product factor, Sylow
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—416no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—816no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22C7×#44714yes—
585C22C7×#48014yescomplemented
586C22C2×C7×#12011yescharacteristic
587C22C2×C7×#12311yescharacteristic
588C22C7×#53411yescharacteristic
589C22C22×C7×D811yescharacteristic
590C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
591C22C2×C7×#13711yescharacteristic
592C22C7×#57511yescharacteristic
593C4—22no—
594C4—22no—
595C4—41no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4C7×#54614yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—84no—
604C2C7×#240511yescharacteristic
605C2C2×C7×#44711yescharacteristic
606C2C7×#300711yescharacteristic
607C2C2×C7×#48011yescharacteristic
608C2C7×#309411yescharacteristic
609C2C7×#318611yescharacteristic
610C2C7×#319311yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C24×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#480C2semidirect
3. C2×C7×#380C2semidirect
4. C7×#2033C2semidirect
5. C7×#1874C2semidirect
6. C7×#1852C2semidirect
7. C7×#1786C2semidirect
8. C7×#1779C2semidirect
18. C22×C7×C4⋊C4C22semidirect
19. C22×C7×C4⋊C4C22semidirect
23. C2×C7×#107C22semidirect
24. C2×C7×#102C22semidirect
25. C2×C7×#102C22semidirect
27. C2×C7×C22⋊C8C22semidirect
28. C2×C7×C22⋊C8C22semidirect
32. C7×#380C22semidirect
34. C7×#340C22semidirect
35. #24002C7direct (N × H)
91. C2×#480C2×C7semidirect
92. C2×#380C2×C7semidirect
93. #2033C2×C7semidirect
94. #1874C2×C7semidirect
95. #1852C2×C7semidirect
96. #1786C2×C7semidirect
97. #1779C2×C7semidirect
189. C22×C4⋊C4C22×C7semidirect
190. C22×C4⋊C4C22×C7semidirect
194. C2×#107C22×C7semidirect
195. C2×#102C22×C7semidirect
196. C2×#102C22×C7semidirect
198. C2×C22⋊C8C22×C7semidirect
199. C2×C22⋊C8C22×C7semidirect
203. #380C22×C7semidirect
205. #340C22×C7semidirect
368. C22×C7#480semidirect
555. C7#24002direct (N × H)
585. C22C7×#480semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1212
C2×C7×#4802
C2×C7×#3802
C7×#20332
C7×#18742
C7×#18522
C7×#17862
C7×#17792
#240027