Finite Group Databasebeta

← C7×#14720 #13191936

Subgroups — C7×#14720 #13191936

Order 1792.

2494 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C23×C7×C4⋊C4C211yesmaximal, characteristic
3C2×C7×#395C212yesmaximal, complemented
4C2×C7×#385C212yesmaximal
5C7×#1444C214yesmaximal, complemented
6C7×#1441C212yesmaximal, complemented
7C7×C22⋊#95C211yesmaximal, characteristic
8C7×C22⋊#95C212yesmaximal
9C23×C7×Q8C2211yescharacteristic
10C24×C4×C7C2211yescharacteristic
11C24×C4×C7C2211yescharacteristic
12C24×C4×C7C2211yescharacteristic
13C2×C7×C22⋊Q8—24no—
14C2×C7×C22⋊Q8—28no—
15C22×C7×C4⋊C4—24no—
16C22×C7×C4⋊C4—28no—
17C22×C7×C4⋊C4C2212yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2212yes—
20C22×C7×C4⋊C4C2214yes—
21C22×C7×C4⋊C4C2212yes—
22C22×C7×C22⋊C4C2212yes—
23C22×C7×C22⋊C4C2212yes—
24C7×#395—28no—
25C7×#385—28no—
26C2×C7×#95C2212yes—
27C2×C7×#95C2214yes—
28C2×C7×#95C2214yes—
29C2×C7×#95C2214yes—
30C2×C7×#95C2212yes—
31#14720C711yesmaximal, characteristic, direct product factor, Sylow
32C25×C7C2311yescharacteristic
33C22×C7×Q8—24no—
34C22×C7×Q8—22no—
35C22×C7×Q8—24no—
36C22×C7×Q8—42no—
37C23×C4×C7—22no—
38C23×C4×C7—22no—
39C23×C4×C7—24no—
40C23×C4×C7—24no—
41C23×C4×C7—22no—
42C23×C4×C7—24no—
43C23×C4×C7—24no—
44C23×C4×C7—24no—
45C23×C4×C7—24no—
46C23×C4×C7—42no—
47C23×C4×C7—42no—
48C23×C4×C7D414yes—
49C23×C4×C7Q814yescomplemented
50C23×C4×C7C2312yes—
51C23×C4×C7C2312yes—
52C23×C4×C7C2312yes—
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2314yes—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2312yes—
57C7×C22⋊Q8—232no—
58C2×C7×C4⋊C4—232no—
59C2×C7×C4⋊C4—24no—
60C2×C7×C4⋊C4—24no—
61C2×C7×C4⋊C4—24no—
62C2×C7×C4⋊C4—24no—
63C2×C7×C4⋊C4—24no—
64C2×C7×C4⋊C4—24no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—24no—
67C2×C7×C4⋊C4—24no—
68C2×C7×C4⋊C4—24no—
69C2×C7×C4⋊C4—28no—
70C2×C7×C4⋊C4—28no—
71C2×C7×C4⋊C4—44no—
72C2×C7×C4⋊C4D418yes—
73C2×C7×C22⋊C4—24no—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4D418yes—
78C2×C7×C22⋊C4Q818yescomplemented
79C7×#95—28no—
80C7×#95—24no—
81C7×#95—28no—
82C7×#95—28no—
83C7×#95—42no—
84C22×C22⋊Q8C2×C711yescharacteristic
85C23×C4⋊C4C2×C711yescharacteristic
86C2×#395C2×C712yescomplemented
87C2×#385C2×C712yes—
88#1444C2×C714yescomplemented
89#1441C2×C712yescomplemented
90C22⋊#95C2×C712yes—
91C22⋊#95C2×C711yescharacteristic
92C24×C7—22no—
93C24×C7—22no—
94C24×C7—22no—
95C24×C7—22no—
96C24×C7C2×D414yes—
97C24×C7C2×D412yes—
98C24×C7C2×Q814yes—
99C24×C7C2×Q812yes—
100C24×C7#4112yes—
101C24×C7C2411yescharacteristic
102C2×C7×Q8—28no—
103C2×C7×Q8—416no—
104C2×C7×Q8—44no—
105C2×C7×Q8—44no—
106C22×C4×C7—28no—
107C22×C4×C7—216no—
108C22×C4×C7—28no—
109C22×C4×C7—28no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—216no—
117C22×C4×C7—28no—
118C22×C4×C7—28no—
119C22×C4×C7—24no—
120C22×C4×C7—24no—
121C22×C4×C7—24no—
122C22×C4×C7—24no—
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—24no—
132C22×C4×C7—24no—
133C22×C4×C7—24no—
134C22×C4×C7—24no—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—24no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—28no—
141C22×C4×C7—44no—
142C22×C4×C7—416no—
143C22×C4×C7—416no—
144C22×C4×C7—44no—
145C22×C4×C7—44no—
146C22×C4×C7C2×D414yes—
147C22×C4×C7C2×D414yes—
148C22×C4×C7C2×D412yes—
149C22×C4×C7C2×D414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D414yes—
153C22×C4×C7C2×Q818yes—
154C22×C4×C7C2×Q814yes—
155C22×C4×C7#4114yes—
156C7×C4⋊C4—216no—
157C7×C4⋊C4—28no—
158C7×C4⋊C4—216no—
159C7×C4⋊C4—216no—
160C7×C4⋊C4—416no—
161C7×C22⋊C4—216no—
162C7×C22⋊C4—216no—
163C23×Q8C22×C711yescharacteristic
164C24×C4C22×C711yescharacteristic
165C24×C4C22×C711yescharacteristic
166C24×C4C22×C711yescharacteristic
167C2×C22⋊Q8—24no—
168C2×C22⋊Q8—28no—
169C22×C4⋊C4—24no—
170C22×C4⋊C4—28no—
171C22×C4⋊C4C22×C712yes—
172C22×C4⋊C4C22×C711yescharacteristic
173C22×C4⋊C4C22×C712yes—
174C22×C4⋊C4C22×C714yes—
175C22×C4⋊C4C22×C712yes—
176C22×C22⋊C4C22×C712yes—
177C22×C22⋊C4C22×C712yes—
178#395—28no—
179#385—28no—
180C2×#95C22×C712yes—
181C2×#95C22×C714yes—
182C2×#95C22×C714yes—
183C2×#95C22×C714yes—
184C2×#95C22×C712yes—
185C23×C7—22no—
186C23×C7—24no—
187C23×C7—22no—
188C23×C7—24no—
189C23×C7—22no—
190C23×C7—22no—
191C23×C7—24no—
192C23×C7—24no—
193C23×C7—28no—
194C23×C7—28no—
195C23×C7—28no—
196C23×C7—28no—
197C23×C7#12014yes—
198C23×C7C22⋊Q818yes—
199C23×C7C22⋊Q814yes—
200C23×C7C4⋊Q814yes—
201C23×C7C4⋊Q818yes—
202C23×C7C22×D411yescharacteristic
203C23×C7C22×D412yes—
204C23×C7C22×D412yes—
205C23×C7C22×Q811yescharacteristic
206C23×C7C22×Q812yes—
207C23×C7C2×#4111yescharacteristic
208C23×C7C2×#4112yes—
209C23×C7H(5,𝔽2)12yes—
210C23×C72−1+412yes—
211C7×Q8—416no—
212C2×C4×C7—232no—
213C2×C4×C7—24no—
214C2×C4×C7—22no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—22no—
221C2×C4×C7—28no—
222C2×C4×C7—24no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—28no—
228C2×C4×C7—22no—
229C2×C4×C7—28no—
230C2×C4×C7—28no—
231C2×C4×C7—28no—
232C2×C4×C7—28no—
233C2×C4×C7—28no—
234C2×C4×C7—416no—
235C2×C4×C7—416no—
236C2×C4×C7—42no—
237C2×C4×C7—42no—
238C2×C4×C7—42no—
239C2×C4×C7—44no—
240C2×C4×C7—42no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—44no—
244C2×C4×C7—44no—
245C2×C4×C7—44no—
246C2×C4×C7—44no—
247C2×C4×C7—44no—
248C2×C4×C7—44no—
249C2×C4×C7—44no—
250C2×C4×C7#12014yes—
251C2×C4×C7C22⋊Q818yescomplemented
252C25C23×C711yescharacteristic
253C22×Q8—24no—
254C22×Q8—22no—
255C22×Q8—24no—
256C22×Q8—42no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—24no—
260C23×C4—24no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—24no—
265C23×C4—24no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4C7×D414yes—
269C23×C4C7×Q814yescomplemented
270C23×C4C23×C712yes—
271C23×C4C23×C712yes—
272C23×C4C23×C712yes—
273C23×C4C23×C711yescharacteristic
274C23×C4C23×C714yes—
275C23×C4C23×C711yescharacteristic
276C23×C4C23×C712yes—
277C22⋊Q8—232no—
278C2×C4⋊C4—232no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C7×D418yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4C7×D418yes—
298C2×C22⋊C4C7×Q818yescomplemented
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—42no—
304C22×C7—24no—
305C22×C7—24no—
306C22×C7—24no—
307C22×C7—24no—
308C22×C7—24no—
309C22×C7—24no—
310C22×C7—216no—
311C22×C7—216no—
312C22×C7#39518yescomplemented
313C22×C7C2×#12011yescharacteristic
314C22×C7C2×#12011yescharacteristic
315C22×C7C2×C22⋊Q811yescharacteristic
316C22×C7C2×C22⋊Q812yes—
317C22×C7C2×C22⋊Q812yes—
318C22×C7#52511yescharacteristic
319C22×C7#52512yes—
320C22×C7C2×C4⋊Q811yescharacteristic
321C22×C7C2×C4⋊Q812yes—
322C22×C7#53411yescharacteristic
323C22×C7#53611yescharacteristic
324C22×C7#53912yes—
325C22×C7#54412yes—
326C22×C7D4214yes—
327C22×C7#54714yes—
328C22×C7D4×Q814yes—
329C22×C7#55414yes—
330C4×C7—28no—
331C4×C7—24no—
332C4×C7—42no—
333C4×C7—44no—
334C4×C7—44no—
335C4×C7—44no—
336C4×C7—48no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C2×C7×D414yes—
342C24C2×C7×D412yes—
343C24C2×C7×Q814yes—
344C24C2×C7×Q812yes—
345C24C7×#4112yes—
346C24C24×C711yescharacteristic
347C2×Q8—28no—
348C2×Q8—416no—
349C2×Q8—44no—
350C2×Q8—44no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C2×C7×D414yes—
392C22×C4C2×C7×D414yes—
393C22×C4C2×C7×D412yes—
394C22×C4C2×C7×D414yes—
395C22×C4C2×C7×D414yes—
396C22×C4C2×C7×D412yes—
397C22×C4C2×C7×D414yes—
398C22×C4C2×C7×Q818yes—
399C22×C4C2×C7×Q814yes—
400C22×C4C7×#4114yes—
401C4⋊C4—216no—
402C4⋊C4—28no—
403C4⋊C4—216no—
404C4⋊C4—216no—
405C4⋊C4—416no—
406C22⋊C4—216no—
407C22⋊C4—216no—
408C2×C7—28no—
409C2×C7C2×#39511yescharacteristic
410C2×C7#240311yescharacteristic
411C2×C7#240611yescharacteristic
412C2×C7#243411yescharacteristic
413C2×C7#244014yes—
414C2×C7#244914yes—
415C2×C7#258311yescharacteristic
416C2×C7#262711yescharacteristic
417C2×C7#284811yescharacteristic
418C23—22no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—28no—
427C23—28no—
428C23—28no—
429C23—28no—
430C23C7×#12014yes—
431C23C7×C22⋊Q818yes—
432C23C7×C22⋊Q814yes—
433C23C7×C4⋊Q814yes—
434C23C7×C4⋊Q818yes—
435C23C22×C7×D411yescharacteristic
436C23C22×C7×D412yes—
437C23C22×C7×D412yes—
438C23C22×C7×Q811yescharacteristic
439C23C22×C7×Q812yes—
440C23C2×C7×#4111yescharacteristic
441C23C2×C7×#4112yes—
442C23C7×H(5,𝔽2)12yes—
443C23C7×2−1+412yes—
444Q8—416no—
445C2×C4—232no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—416no—
468C2×C4—416no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4C7×#12014yes—
484C2×C4C7×C22⋊Q818yescomplemented
485C7#1472011yescharacteristic, direct product factor, Sylow
486C22—24no—
487C22—24no—
488C22—24no—
489C22—216no—
490C22—216no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22C7×#39518yescomplemented
495C22C2×C7×#12011yescharacteristic
496C22C2×C7×#12011yescharacteristic
497C22C2×C7×C22⋊Q812yes—
498C22C2×C7×C22⋊Q811yescharacteristic
499C22C2×C7×C22⋊Q812yes—
500C22C7×#52511yescharacteristic
501C22C7×#52512yes—
502C22C2×C7×C4⋊Q811yescharacteristic
503C22C2×C7×C4⋊Q812yes—
504C22C7×#53411yescharacteristic
505C22C7×#53611yescharacteristic
506C22C7×#53912yes—
507C22C7×#54412yes—
508C22C7×D4214yes—
509C22C7×#54714yes—
510C22C7×D4×Q814yes—
511C22C7×#55414yes—
512C4—24no—
513C4—28no—
514C4—42no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—48no—
519C2—28no—
520C2C2×C7×#39511yescharacteristic
521C2C7×#240311yescharacteristic
522C2C7×#240611yescharacteristic
523C2C7×#243411yescharacteristic
524C2C7×#244014yes—
525C2C7×#244914yes—
526C2C7×#258311yescharacteristic
527C2C7×#262711yescharacteristic
528C2C7×#284811yescharacteristic
subgroupG/H
center, socleC24×C7C24
commutator, Frattini, cosocleC24C24×C7
prime radicalC25×C7C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C7×#395C2semidirect
5. C7×#1444C2semidirect
6. C7×#1441C2semidirect
31. #14720C7direct (N × H)
49. C23×C4×C7Q8semidirect
78. C2×C7×C22⋊C4Q8semidirect
86. C2×#395C2×C7semidirect
88. #1444C2×C7semidirect
89. #1441C2×C7semidirect
251. C2×C4×C7C22⋊Q8semidirect
269. C23×C4C7×Q8semidirect
298. C2×C22⋊C4C7×Q8semidirect
312. C22×C7#395semidirect
484. C2×C4C7×C22⋊Q8semidirect
485. C7#14720direct (N × H)
494. C22C7×#395semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C23×C7×C4⋊C42
C2×C7×#3952
C2×C7×#3852
C7×#14442
C7×#14412
C7×C22⋊#952
#147207