Finite Group Databasebeta

← C7×#24003 #13201219

Subgroups — C7×#24003 #13201219

Order 1792.

1500 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C2×C7×#483C212yesmaximal, complemented
3C2×C7×#380C211yesmaximal, characteristic
4C7×#2035C214yesmaximal, complemented
5C7×#1872C212yesmaximal, complemented
6C7×#1853C212yesmaximal, complemented
7C7×#1787C212yesmaximal, complemented
8C7×#1779C211yesmaximal, characteristic
9C23×C7×Q8C2211yescharacteristic
10C24×C4×C7C2211yescharacteristic
11C2×C7×C22⋊Q8—21no—
12C2×C7×C22⋊Q8—21no—
13C2×C7×C22⋊Q8—28no—
14C2×C7×C22⋊Q8C2212yes—
15C2×C7×C22⋊Q8C2212yes—
16C22×C7×C4⋊C4—21no—
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C22×C7×C22⋊C4—21no—
21C22×C7×C22⋊C4—21no—
22C7×#483—28no—
23C2×C7×#106C2214yescomplemented
24C2×C7×#103C2214yescomplemented
25C2×C7×#103C2214yescomplemented
26C2×C7×C22⋊C8C2212yes—
27C2×C7×C22⋊C8C2212yes—
28C2×C7×C22⋊C8C2212yes—
29C7×#380—21no—
30C7×#380—21no—
31C7×#380C2212yes—
32C7×#380C2212yes—
33C2×C7×#95—21no—
34C7×#340C2214yescomplemented
35#24003C711yesmaximal, characteristic, direct product factor, Sylow
36C25×C7D411yescharacteristic
37C22×C7×Q8—24no—
38C22×C7×Q8—21no—
39C22×C7×Q8—24no—
40C22×C7×Q8—42no—
41C22×C7×Q8C2312yes—
42C23×C4×C7—21no—
43C23×C4×C7—21no—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—21no—
47C23×C4×C7—21no—
48C23×C4×C7—42no—
49C23×C4×C7D411yescharacteristic
50C23×C4×C7D411yescharacteristic
51C23×C4×C7D411yescharacteristic
52C23×C4×C7C2311yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C22×C7×C8C2314yes—
56C7×C22⋊Q8—28no—
57C7×C22⋊Q8—28no—
58C7×C22⋊Q8—24no—
59C7×C22⋊Q8—24no—
60C7×C22⋊Q8—24no—
61C7×C22⋊Q8D418yes—
62C2×C7×C4⋊C4—28no—
63C2×C7×C4⋊C4—22no—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—42no—
71C2×C7×C4⋊C4C2312yes—
72C2×C7×C4⋊C4C2312yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—28no—
75C2×C7×C22⋊C4—22no—
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4—22no—
79C2×C7×C22⋊C4—44no—
80C7×#106—28no—
81C7×#103—28no—
82C7×#103—44no—
83C7×C22⋊C8—24no—
84C7×C22⋊C8—24no—
85C7×C22⋊C8—42no—
86C7×#95—22no—
87C7×#95—22no—
88C22×C22⋊Q8C2×C711yescharacteristic
89C2×#483C2×C712yescomplemented
90C2×#380C2×C711yescharacteristic
91#2035C2×C714yescomplemented
92#1872C2×C712yescomplemented
93#1853C2×C712yescomplemented
94#1787C2×C712yescomplemented
95#1779C2×C711yescharacteristic
96C24×C7—22no—
97C24×C7—22no—
98C24×C7—22no—
99C24×C7—22no—
100C24×C7—41no—
101C24×C7—41no—
102C24×C7SD1614yes—
103C24×C7C2×D411yescharacteristic
104C24×C7C2×D411yescharacteristic
105C24×C7C2×D411yescharacteristic
106C2×C7×Q8—24no—
107C2×C7×Q8—22no—
108C2×C7×Q8—24no—
109C2×C7×Q8—24no—
110C2×C7×Q8—416no—
111C2×C7×Q8—42no—
112C2×C7×Q8—42no—
113C2×C7×Q8C2×D414yes—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—24no—
119C22×C4×C7—24no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
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—41no—
134C22×C4×C7—41no—
135C22×C4×C7—41no—
136C22×C4×C7—416no—
137C22×C4×C7—42no—
138C22×C4×C7—42no—
139C22×C4×C7—42no—
140C22×C4×C7—42no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7#4114yes—
152C22×C4×C7C2411yescharacteristic
153C2×C7×C8—24no—
154C2×C7×C8—24no—
155C2×C7×C8—24no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—28no—
158C7×C4⋊C4—28no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—24no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—42no—
163C7×C4⋊C4—42no—
164C7×C4⋊C4C2×D414yes—
165C7×C22⋊C4—28no—
166C7×C22⋊C4—28no—
167C7×C22⋊C4—44no—
168C7×C22⋊C4—44no—
169C23×Q8C22×C711yescharacteristic
170C24×C4C22×C711yescharacteristic
171C2×C22⋊Q8—21no—
172C2×C22⋊Q8—21no—
173C2×C22⋊Q8—28no—
174C2×C22⋊Q8C22×C712yes—
175C2×C22⋊Q8C22×C712yes—
176C22×C4⋊C4—21no—
177C22×C4⋊C4C22×C711yescharacteristic
178C22×C4⋊C4C22×C711yescharacteristic
179C22×C4⋊C4C22×C711yescharacteristic
180C22×C22⋊C4—21no—
181C22×C22⋊C4—21no—
182#483—28no—
183C2×#106C22×C714yescomplemented
184C2×#103C22×C714yescomplemented
185C2×#103C22×C714yescomplemented
186C2×C22⋊C8C22×C712yes—
187C2×C22⋊C8C22×C712yes—
188C2×C22⋊C8C22×C712yes—
189#380—21no—
190#380—21no—
191#380C22×C712yes—
192#380C22×C712yes—
193C2×#95—21no—
194#340C22×C714yescomplemented
195C23×C7—21no—
196C23×C7—21no—
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7—21no—
202C23×C7—21no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—21no—
206C23×C7—21no—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—28no—
210C23×C7—28no—
211C23×C7—44no—
212C23×C7—44no—
213C23×C7—44no—
214C23×C7—44no—
215C23×C7—44no—
216C23×C7#12012yes—
217C23×C7#12312yes—
218C23×C7C2×SD1612yes—
219C23×C7C2×SD1612yes—
220C23×C7C2×SD1612yes—
221C23×C7#13712yes—
222C23×C7#13712yes—
223C23×C7C22×D411yescharacteristic
224C7×Q8—28no—
225C7×Q8—48no—
226C7×Q8—44no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—28no—
232C2×C4×C7—28no—
233C2×C4×C7—24no—
234C2×C4×C7—24no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—44no—
239C2×C4×C7—44no—
240C2×C4×C7—44no—
241C2×C4×C7—48no—
242C2×C4×C7—48no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7—42no—
254C2×C4×C7—84no—
255C2×C4×C7—84no—
256C2×C4×C7#12012yes—
257C2×C4×C7#12312yes—
258C2×C4×C7C22×D412yes—
259C2×C4×C7C2×#4112yes—
260C2×C4×C7H(5,𝔽2)12yes—
261C7×C8—44no—
262C25C7×D411yescharacteristic
263C22×Q8—24no—
264C22×Q8—21no—
265C22×Q8—24no—
266C22×Q8—42no—
267C22×Q8C23×C712yes—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—24no—
271C23×C4—24no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—42no—
275C23×C4C7×D411yescharacteristic
276C23×C4C7×D411yescharacteristic
277C23×C4C7×D411yescharacteristic
278C23×C4C23×C711yescharacteristic
279C23×C4C23×C711yescharacteristic
280C23×C4C23×C711yescharacteristic
281C22×C8C23×C714yes—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8C7×D418yes—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C23×C712yes—
298C2×C4⋊C4C23×C712yes—
299C2×C4⋊C4C23×C712yes—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—44no—
306#106—28no—
307#103—28no—
308#103—44no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312#95—22no—
313#95—22no—
314C22×C7—22no—
315C22×C7—22no—
316C22×C7—22no—
317C22×C7—22no—
318C22×C7—22no—
319C22×C7—22no—
320C22×C7—22no—
321C22×C7—22no—
322C22×C7—22no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—22no—
326C22×C7—22no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—416no—
330C22×C7—41no—
331C22×C7—41no—
332C22×C7—41no—
333C22×C7—41no—
334C22×C7#44814yes—
335C22×C7#48314yescomplemented
336C22×C7C2×#12011yescharacteristic
337C22×C7C2×#12311yescharacteristic
338C22×C7#53411yescharacteristic
339C22×C7C22×SD1611yescharacteristic
340C22×C7C2×#13711yescharacteristic
341C22×C7C2×#13711yescharacteristic
342C22×C7#57511yescharacteristic
343C4×C7—22no—
344C4×C7—22no—
345C4×C7—41no—
346C4×C7—44no—
347C4×C7—44no—
348C4×C7—82no—
349C4×C7—82no—
350C4×C7#54614yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—41no—
356C24—41no—
357C24C7×SD1614yes—
358C24C2×C7×D411yescharacteristic
359C24C2×C7×D411yescharacteristic
360C24C2×C7×D411yescharacteristic
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—416no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8C2×C7×D414yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—416no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4C2×C7×D411yescharacteristic
403C22×C4C2×C7×D411yescharacteristic
404C22×C4C2×C7×D414yes—
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C7×#4114yes—
407C22×C4C24×C711yescharacteristic
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C4⋊C4—24no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4C2×C7×D414yes—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C2×C7—24no—
425C2×C7—24no—
426C2×C7—42no—
427C2×C7#240511yescharacteristic
428C2×C7C2×#44811yescharacteristic
429C2×C7#300711yescharacteristic
430C2×C7C2×#48311yescharacteristic
431C2×C7#309511yescharacteristic
432C2×C7#318711yescharacteristic
433C2×C7#319511yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—28no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23C7×#12012yes—
456C23C7×#12312yes—
457C23C2×C7×SD1612yes—
458C23C2×C7×SD1612yes—
459C23C2×C7×SD1612yes—
460C23C7×#13712yes—
461C23C7×#13712yes—
462C23C22×C7×D411yescharacteristic
463Q8—28no—
464Q8—48no—
465Q8—44no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4C7×#12012yes—
496C2×C4C7×#12312yes—
497C2×C4C22×C7×D412yes—
498C2×C4C2×C7×#4112yes—
499C2×C4C7×H(5,𝔽2)12yes—
500C8—44no—
501C7#2400311yescharacteristic, direct product factor, Sylow
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—416no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22C7×#44814yes—
523C22C7×#48314yescomplemented
524C22C2×C7×#12011yescharacteristic
525C22C2×C7×#12311yescharacteristic
526C22C7×#53411yescharacteristic
527C22C22×C7×SD1611yescharacteristic
528C22C2×C7×#13711yescharacteristic
529C22C2×C7×#13711yescharacteristic
530C22C7×#57511yescharacteristic
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—44no—
536C4—82no—
537C4—82no—
538C4C7×#54614yes—
539C2—24no—
540C2—24no—
541C2—42no—
542C2C7×#240511yescharacteristic
543C2C2×C7×#44811yescharacteristic
544C2C7×#300711yescharacteristic
545C2C2×C7×#48311yescharacteristic
546C2C7×#309511yescharacteristic
547C2C7×#318711yescharacteristic
548C2C7×#319511yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC25×C7D4
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×#483C2semidirect
4. C7×#2035C2semidirect
5. C7×#1872C2semidirect
6. C7×#1853C2semidirect
7. C7×#1787C2semidirect
23. C2×C7×#106C22semidirect
24. C2×C7×#103C22semidirect
25. C2×C7×#103C22semidirect
34. C7×#340C22semidirect
35. #24003C7direct (N × H)
89. C2×#483C2×C7semidirect
91. #2035C2×C7semidirect
92. #1872C2×C7semidirect
93. #1853C2×C7semidirect
94. #1787C2×C7semidirect
183. C2×#106C22×C7semidirect
184. C2×#103C22×C7semidirect
185. C2×#103C22×C7semidirect
194. #340C22×C7semidirect
335. C22×C7#483semidirect
501. C7#24003direct (N × H)
523. C22C7×#483semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C2×C7×#4832
C2×C7×#3802
C7×#20352
C7×#18722
C7×#18532
C7×#17872
C7×#17792
#240037