Finite Group Databasebeta

← C7×#24016 #13201232

Subgroups — C7×#24016 #13201232

Order 1792.

1276 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic
2C2×C7×#485C212yesmaximal, complemented
3C7×#2311C211yesmaximal, characteristic
4C7×#2036C212yesmaximal, complemented
5C7×#2035C212yesmaximal, complemented
6C7×#1872C212yesmaximal, complemented
7C7×#1854C212yesmaximal, complemented
8C7×#1787C212yesmaximal, complemented
9C7×#1780C211yesmaximal, characteristic
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C7×#536—28no—
13C2×C7×C4⋊Q8—21no—
14C2×C7×#124—21no—
15C2×C7×#123—21no—
16C2×C7×#123—21no—
17C2×C7×C22⋊Q8C2212yes—
18C2×C7×C22⋊Q8C2212yes—
19C2×C7×#117C2211yescharacteristic
20C2×C7×#117C2211yescharacteristic
21C22×C7×C4⋊C4C2211yescharacteristic
22C7×#485—28no—
23C2×C7×#107C2212yes—
24C2×C7×#106C2212yes—
25C2×C7×#103C2212yes—
26C2×C7×#103C2212yes—
27C2×C7×#103C2212yes—
28C2×C7×#103C2212yes—
29C2×C7×C22⋊C8C2212yes—
30C2×C7×C22⋊C8C2212yes—
31C2×C7×C22⋊C8C2211yescharacteristic
32C2×C7×C22⋊C8C2211yescharacteristic
33C7×#391—21no—
34C7×#390—21no—
35C7×#386—21no—
36C7×#386—21no—
37C7×#380C2212yes—
38C7×#380C2212yes—
39C7×#340C2212yes—
40C7×#340C2212yes—
41#24016C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—24no—
43C2×C7×#41—24no—
44C2×C7×#41—41no—
45C2×C7×#41—41no—
46C22×C7×Q8—24no—
47C22×C7×Q8—21no—
48C22×C7×Q8—24no—
49C22×C7×Q8—41no—
50C22×C7×Q8—41no—
51C22×C7×Q8D411yescharacteristic
52C22×C7×Q8C2312yes—
53C22×C7×D4D411yescharacteristic
54C22×C7×D4D411yescharacteristic
55C22×C7×D4D411yescharacteristic
56C23×C4×C7—21no—
57C23×C4×C7—21no—
58C23×C4×C7C2311yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C22×C7×C8C2312yes—
62C22×C7×C8C2312yes—
63C7×C4⋊Q8—28no—
64C7×#124—28no—
65C7×#123—28no—
66C7×#123—28no—
67C7×C22⋊Q8—24no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—24no—
70C7×C22⋊Q8D418yes—
71C7×#117—24no—
72C7×#117—24no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4—22no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4—41no—
77C2×C7×C4⋊C4—41no—
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—22no—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C42×C7—21no—
88C2×C42×C7—21no—
89C7×#107—24no—
90C7×#106—24no—
91C7×#103—24no—
92C7×#103—24no—
93C7×#103—42no—
94C7×#103—42no—
95C7×C22⋊C8—24no—
96C7×C22⋊C8—24no—
97C7×C22⋊C8—41no—
98C7×C22⋊C8—41no—
99C7×#95—22no—
100C7×#95—22no—
101C2×#536C2×C711yescharacteristic
102C2×#485C2×C712yescomplemented
103#2311C2×C711yescharacteristic
104#2036C2×C712yescomplemented
105#2035C2×C712yescomplemented
106#1872C2×C712yescomplemented
107#1854C2×C712yescomplemented
108#1787C2×C712yescomplemented
109#1780C2×C711yescharacteristic
110C24×C7C2×D411yescharacteristic
111C24×C7C2×D411yescharacteristic
112C24×C7C2×D411yescharacteristic
113C7×#41—48no—
114C7×#41—48no—
115C2×C7×Q8—24no—
116C2×C7×Q8—22no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—41no—
124C2×C7×Q8—48no—
125C2×C7×Q8—48no—
126C2×C7×Q8—42no—
127C2×C7×Q8—42no—
128C2×C7×Q8C2×D414yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4—22no—
134C2×C7×D4—22no—
135C2×C7×D4—41no—
136C2×C7×D4—41no—
137C2×C7×D4—41no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—82no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D414yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7#4114yes—
162C22×C4×C7C2411yescharacteristic
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C7×C4⋊C4—28no—
170C7×C4⋊C4—28no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4C2×D414yes—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—28no—
180C7×C22⋊C4—28no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—82no—
183C7×C22⋊C4—82no—
184C42×C7—24no—
185C42×C7—42no—
186C22×#41C22×C711yescharacteristic
187C23×Q8C22×C711yescharacteristic
188#536—28no—
189C2×C4⋊Q8—21no—
190C2×#124—21no—
191C2×#123—21no—
192C2×#123—21no—
193C2×C22⋊Q8C22×C712yes—
194C2×C22⋊Q8C22×C712yes—
195C2×#117C22×C711yescharacteristic
196C2×#117C22×C711yescharacteristic
197C22×C4⋊C4C22×C711yescharacteristic
198#485—28no—
199C2×#107C22×C712yes—
200C2×#106C22×C712yes—
201C2×#103C22×C712yes—
202C2×#103C22×C712yes—
203C2×#103C22×C712yes—
204C2×#103C22×C712yes—
205C2×C22⋊C8C22×C712yes—
206C2×C22⋊C8C22×C712yes—
207C2×C22⋊C8C22×C711yescharacteristic
208C2×C22⋊C8C22×C711yescharacteristic
209#391—21no—
210#390—21no—
211#386—21no—
212#386—21no—
213#380C22×C712yes—
214#380C22×C712yes—
215#340C22×C712yes—
216#340C22×C712yes—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—22no—
222C23×C7—22no—
223C23×C7—22no—
224C23×C7—22no—
225C23×C7—21no—
226C23×C7—21no—
227C23×C7—21no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7#12012yes—
231C23×C7#12312yes—
232C23×C7C22×D411yescharacteristic
233C7×Q8—24no—
234C7×Q8—24no—
235C7×Q8—44no—
236C7×Q8—48no—
237C7×Q8—42no—
238C7×Q8—42no—
239C7×D4—44no—
240C7×D4—44no—
241C7×D4—44no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—24no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—48no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—42no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—84no—
269C2×C4×C7#12012yes—
270C2×C4×C7#12312yes—
271C2×C4×C7C22×D412yes—
272C2×C4×C7C2×#4112yes—
273C2×C4×C7H(5,𝔽2)11yescharacteristic
274C2×C4×C7H(5,𝔽2)11yescharacteristic
275C7×C8—42no—
276C7×C8—42no—
277C2×#41—24no—
278C2×#41—24no—
279C2×#41—41no—
280C2×#41—41no—
281C22×Q8—24no—
282C22×Q8—21no—
283C22×Q8—24no—
284C22×Q8—41no—
285C22×Q8—41no—
286C22×Q8C7×D411yescharacteristic
287C22×Q8C23×C712yes—
288C22×D4C7×D411yescharacteristic
289C22×D4C7×D411yescharacteristic
290C22×D4C7×D411yescharacteristic
291C23×C4—21no—
292C23×C4—21no—
293C23×C4C23×C711yescharacteristic
294C23×C4C23×C711yescharacteristic
295C23×C4C23×C711yescharacteristic
296C22×C8C23×C712yes—
297C22×C8C23×C712yes—
298C4⋊Q8—28no—
299#124—28no—
300#123—28no—
301#123—28no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8C7×D418yes—
306#117—24no—
307#117—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—41no—
312C2×C4⋊C4—41no—
313C2×C4⋊C4C23×C712yes—
314C2×C4⋊C4C23×C712yes—
315C2×C4⋊C4C23×C712yes—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—21no—
323C2×C42—21no—
324#107—24no—
325#106—24no—
326#103—24no—
327#103—24no—
328#103—42no—
329#103—42no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—41no—
333C22⋊C8—41no—
334#95—22no—
335#95—22no—
336C22×C7—24no—
337C22×C7—22no—
338C22×C7—22no—
339C22×C7—24no—
340C22×C7—42no—
341C22×C7—41no—
342C22×C7—41no—
343C22×C7—41no—
344C22×C7—41no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—41no—
348C22×C7—41no—
349C22×C7—41no—
350C22×C7—42no—
351C22×C7C2×#12011yescharacteristic
352C22×C7C2×#12311yescharacteristic
353C22×C7#53411yescharacteristic
354C22×C7#57712yes—
355C22×C7#57812yes—
356C4×C7—22no—
357C4×C7—22no—
358C4×C7—41no—
359C4×C7—44no—
360C4×C7—44no—
361C4×C7—82no—
362C4×C7—82no—
363C4×C7#54612yes—
364C4×C7#54612yes—
365C24C2×C7×D411yescharacteristic
366C24C2×C7×D411yescharacteristic
367C24C2×C7×D411yescharacteristic
368#41—48no—
369#41—48no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8C2×C7×D414yes—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C2×C7×D414yes—
415C22×C4C2×C7×D411yescharacteristic
416C22×C4C7×#4114yes—
417C22×C4C24×C711yescharacteristic
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4C2×C7×D414yes—
433C22⋊C4—24no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—42no—
437C22⋊C4—82no—
438C22⋊C4—82no—
439C42—24no—
440C42—42no—
441C2×C7—42no—
442C2×C7—42no—
443C2×C7—81no—
444C2×C7#240511yescharacteristic
445C2×C7#301312yes—
446C2×C7#309712yes—
447C2×C7#320011yescharacteristic
448C2×C7#320211yescharacteristic
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23C7×#12012yes—
463C23C7×#12312yes—
464C23C22×C7×D411yescharacteristic
465Q8—24no—
466Q8—24no—
467Q8—44no—
468Q8—48no—
469Q8—42no—
470Q8—42no—
471D4—44no—
472D4—44no—
473D4—44no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—84no—
501C2×C4C7×#12012yes—
502C2×C4C7×#12312yes—
503C2×C4C22×C7×D412yes—
504C2×C4C2×C7×#4112yes—
505C2×C4C7×H(5,𝔽2)11yescharacteristic
506C2×C4C7×H(5,𝔽2)11yescharacteristic
507C8—42no—
508C8—42no—
509C7#2401611yescharacteristic, direct product factor, Sylow
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—42no—
519C22—42no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22C2×C7×#12011yescharacteristic
526C22C2×C7×#12311yescharacteristic
527C22C7×#53411yescharacteristic
528C22C7×#57712yes—
529C22C7×#57812yes—
530C4—22no—
531C4—22no—
532C4—41no—
533C4—44no—
534C4—44no—
535C4—82no—
536C4—82no—
537C4C7×#54612yes—
538C4C7×#54612yes—
539C2—42no—
540C2—42no—
541C2—81no—
542C2C7×#240511yescharacteristic
543C2C7×#301312yes—
544C2C7×#309712yes—
545C2C7×#320011yescharacteristic
546C2C7×#320211yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#485C2semidirect
4. C7×#2036C2semidirect
5. C7×#2035C2semidirect
6. C7×#1872C2semidirect
7. C7×#1854C2semidirect
8. C7×#1787C2semidirect
41. #24016C7direct (N × H)
102. C2×#485C2×C7semidirect
104. #2036C2×C7semidirect
105. #2035C2×C7semidirect
106. #1872C2×C7semidirect
107. #1854C2×C7semidirect
108. #1787C2×C7semidirect
509. C7#24016direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5362
C2×C7×#4852
C7×#23112
C7×#20362
C7×#20352
C7×#18722
C7×#18542
C7×#17872
C7×#17802
#240167