Finite Group Databasebeta

← C7×#24057 #13201273

Subgroups — C7×#24057 #13201273

Order 1792.

1392 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic
2C7×#2853C212yesmaximal, complemented
3C7×#2297C211yesmaximal, characteristic, complemented
4C7×#2044C214yesmaximal, complemented
5C7×#1899C212yesmaximal, complemented
6C7×#1867C212yesmaximal, complemented
7C7×#1790C212yesmaximal, complemented
8C7×#1782C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C22×C7×#41C2211yescharacteristic
11C7×#547—24no—
12C7×#546—24no—
13C7×#536—21no—
14C7×#534—21no—
15C7×#534C2212yes—
16C7×#525—21no—
17C2×C7×#123—21no—
18C2×C7×#123C2212yes—
19C7×#518C2211yescharacteristic
20C2×C7×#117C2211yescharacteristic, complemented
21C22×C42×C7C2211yescharacteristic, complemented
22C2×C7×#99C2212yes—
23C2×C7×#99C2214yes—
24C2×C7×#99C2212yescomplemented
25C7×#391—21no—
26C7×#390—21no—
27C7×#388—24no—
28C7×#387—24no—
29C7×#386C2214yes—
30C7×#386C2212yes—
31C7×#386C2212yescomplemented
32C7×#385C2214yescomplemented
33C7×#381C2212yescomplemented
34C4×C7×C22⋊C4—21no—
35C4×C7×C22⋊C4—21no—
36C7×#342C2214yescomplemented
37#24057C711yesmaximal, characteristic, direct product factor, Sylow
38C7×H(5,𝔽2)—24no—
39C7×H(5,𝔽2)—42no—
40C2×C7×#41—24no—
41C2×C7×#41—24no—
42C2×C7×#41—21no—
43C2×C7×#41—22no—
44C2×C7×#41—42no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×D4—21no—
47C22×C7×D4—22no—
48C22×C7×D4D411yescharacteristic
49C22×C7×D4C2311yescharacteristic
50C22×C7×D4C2312yes—
51C22×C7×D4C2311yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7D411yescharacteristic
54C23×C4×C7C2312yes—
55C23×C4×C7C2311yescharacteristic
56C7×C4⋊Q8—22no—
57C7×#124—22no—
58C7×#123—24no—
59C7×#123—22no—
60C7×#123—22no—
61C7×#123—22no—
62C7×#123—22no—
63C7×#123—24no—
64C7×#123—22no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—24no—
67C7×#121—22no—
68C7×#121—24no—
69C7×#121—24no—
70C7×#120—22no—
71C7×#120—22no—
72C7×#120—22no—
73C4×C7×D4—24no—
74C4×C7×D4—24no—
75C7×#117—21no—
76C7×#117—42no—
77C2×C7×C4⋊C4—22no—
78C2×C7×C4⋊C4—21no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4D414yes—
81C2×C7×C22⋊C4—22no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—22no—
84C2×C7×C22⋊C4—22no—
85C2×C7×C22⋊C4—24no—
86C2×C7×C22⋊C4D414yes—
87C2×C7×C22⋊C4C2312yes—
88C2×C7×C22⋊C4C2314yes—
89C2×C7×C22⋊C4C2312yes—
90C2×C42×C7—21no—
91C2×C42×C7—24no—
92C2×C42×C7—21no—
93C2×C42×C7—21no—
94C7×#99—24no—
95C7×#99—44no—
96C7×#99—42no—
97C7×#95—22no—
98C7×#95—22no—
99C7×#95—24no—
100C7×#95—24no—
101C7×#95—24no—
102#3485C2×C711yescharacteristic
103#2853C2×C712yescomplemented
104#2297C2×C711yescharacteristic, complemented
105#2044C2×C714yescomplemented
106#1899C2×C712yescomplemented
107#1867C2×C712yescomplemented
108#1790C2×C712yescomplemented
109#1782C2×C711yescharacteristic, complemented
110C24×C7—21no—
111C24×C7C2×D412yes—
112C24×C7C2×D411yescharacteristic
113C24×C7C2×D411yescharacteristic
114C24×C7C2411yescharacteristic
115C7×#41—416no—
116C7×#41—48no—
117C7×#41—42no—
118C7×#41—42no—
119C2×C7×Q8—21no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—41no—
123C2×C7×D4—22no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—21no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—24no—
133C2×C7×D4—22no—
134C2×C7×D4—24no—
135C2×C7×D4—24no—
136C2×C7×D4—22no—
137C2×C7×D4—41no—
138C2×C7×D4—48no—
139C2×C7×D4—42no—
140C2×C7×D4—42no—
141C2×C7×D4#4114yes—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—24no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—24no—
154C22×C4×C7—22no—
155C22×C4×C7—44no—
156C22×C4×C7—41no—
157C22×C4×C7—41no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D414yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D414yes—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—48no—
170C7×C4⋊C4—42no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—24no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—24no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—24no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—44no—
182C7×C22⋊C4—44no—
183C7×C22⋊C4—48no—
184C42×C7—22no—
185C42×C7—42no—
186C42×C7—42no—
187C42×C7—42no—
188C2×H(5,𝔽2)C22×C711yescharacteristic
189C22×#41C22×C711yescharacteristic
190#547—24no—
191#546—24no—
192#536—21no—
193#534—21no—
194#534C22×C712yes—
195#525—21no—
196C2×#123—21no—
197C2×#123C22×C712yes—
198#518C22×C711yescharacteristic
199C2×#117C22×C711yescharacteristic, complemented
200C22×C42C22×C711yescharacteristic, complemented
201C2×#99C22×C712yes—
202C2×#99C22×C714yes—
203C2×#99C22×C712yescomplemented
204#391—21no—
205#390—21no—
206#388—24no—
207#387—24no—
208#386C22×C714yes—
209#386C22×C712yes—
210#386C22×C712yescomplemented
211#385C22×C714yescomplemented
212#381C22×C712yescomplemented
213C4×C22⋊C4—21no—
214C4×C22⋊C4—21no—
215#342C22×C714yescomplemented
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—22no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—24no—
223C23×C7—42no—
224C23×C7—42no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7—44no—
228C23×C7—82no—
229C23×C7#12312yes—
230C23×C7C22×D411yescharacteristic
231C23×C7C22×D412yes—
232C23×C7C2×#4112yes—
233C23×C7H(5,𝔽2)12yes—
234C7×Q8—44no—
235C7×Q8—42no—
236C7×D4—28no—
237C7×D4—44no—
238C7×D4—42no—
239C7×D4—44no—
240C7×D4—42no—
241C7×D4—42no—
242C7×D4—44no—
243C7×D4—44no—
244C7×D4—84no—
245C2×C4×C7—21no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—28no—
252C2×C4×C7—24no—
253C2×C4×C7—28no—
254C2×C4×C7—24no—
255C2×C4×C7—21no—
256C2×C4×C7—22no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—24no—
260C2×C4×C7—44no—
261C2×C4×C7—44no—
262C2×C4×C7—44no—
263C2×C4×C7—44no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—44no—
268C2×C4×C7—44no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—44no—
272C2×C4×C7—42no—
273C2×C4×C7—44no—
274C2×C4×C7—82no—
275C2×C4×C7#12012yes—
276C2×C4×C7#12012yes—
277C2×C4×C7#12312yes—
278H(5,𝔽2)—24no—
279H(5,𝔽2)—42no—
280C2×#41—24no—
281C2×#41—21no—
282C2×#41—24no—
283C2×#41—22no—
284C2×#41—42no—
285C22×Q8C7×D411yescharacteristic
286C22×D4—21no—
287C22×D4—22no—
288C22×D4C7×D411yescharacteristic
289C22×D4C23×C712yes—
290C22×D4C23×C711yescharacteristic
291C22×D4C23×C711yescharacteristic
292C23×C4C7×D411yescharacteristic
293C23×C4C7×D411yescharacteristic
294C23×C4C23×C711yescharacteristic
295C23×C4C23×C712yes—
296C4⋊Q8—22no—
297#124—22no—
298#123—24no—
299#123—22no—
300#123—22no—
301#123—22no—
302#123—22no—
303#123—24no—
304#123—22no—
305C22⋊Q8—22no—
306C22⋊Q8—24no—
307#121—22no—
308#121—24no—
309#121—24no—
310#120—22no—
311#120—22no—
312#120—22no—
313C4×D4—24no—
314C4×D4—24no—
315#117—21no—
316#117—42no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C7×D414yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C7×D414yes—
327C2×C22⋊C4C23×C712yes—
328C2×C22⋊C4C23×C714yes—
329C2×C22⋊C4C23×C712yes—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—24no—
333C2×C42—21no—
334#99—24no—
335#99—44no—
336#99—42no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342C22×C7—24no—
343C22×C7—22no—
344C22×C7—22no—
345C22×C7—42no—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—41no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—44no—
354C22×C7—81no—
355C22×C7—81no—
356C22×C7—84no—
357C22×C7—162no—
358C22×C7C2×#12011yescharacteristic
359C22×C7C2×#12311yescharacteristic
360C22×C7#53411yescharacteristic
361C22×C7#54614yes—
362C4×C7—22no—
363C4×C7—22no—
364C4×C7—22no—
365C4×C7—42no—
366C4×C7—41no—
367C4×C7—44no—
368C4×C7—44no—
369C4×C7—82no—
370C4×C7—84no—
371C24—21no—
372C24C2×C7×D412yes—
373C24C2×C7×D411yescharacteristic
374C24C2×C7×D411yescharacteristic
375C24C24×C711yescharacteristic
376#41—416no—
377#41—48no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—41no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4C7×#4114yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—44no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4C2×C7×D412yes—
420C22×C4C2×C7×D411yescharacteristic
421C22×C4C2×C7×D414yes—
422C22×C4C2×C7×D411yescharacteristic
423C22×C4C2×C7×D411yescharacteristic
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D414yes—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C22⋊C4—22no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C42—22no—
446C42—42no—
447C42—42no—
448C42—42no—
449C2×C7—21no—
450C2×C7—24no—
451C2×C7—21no—
452C2×C7—42no—
453C2×C7—81no—
454C2×C7—82no—
455C2×C7#240511yescharacteristic
456C2×C7#302512yes—
457C23—21no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—82no—
470C23C7×#12312yes—
471C23C22×C7×D411yescharacteristic
472C23C22×C7×D412yes—
473C23C2×C7×#4112yes—
474C23C7×H(5,𝔽2)12yes—
475Q8—44no—
476Q8—42no—
477D4—28no—
478D4—44no—
479D4—42no—
480D4—44no—
481D4—42no—
482D4—42no—
483D4—44no—
484D4—44no—
485D4—84no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—82no—
516C2×C4C7×#12012yes—
517C2×C4C7×#12012yes—
518C2×C4C7×#12312yes—
519C7#2405711yescharacteristic, direct product factor, Sylow
520C22—24no—
521C22—22no—
522C22—22no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—41no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—81no—
533C22—81no—
534C22—84no—
535C22—162no—
536C22C2×C7×#12011yescharacteristic
537C22C2×C7×#12311yescharacteristic
538C22C7×#53411yescharacteristic
539C22C7×#54614yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—41no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—84no—
549C2—21no—
550C2—24no—
551C2—21no—
552C2—42no—
553C2—81no—
554C2—82no—
555C2C7×#240511yescharacteristic
556C2C7×#302512yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#3485C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2853C2semidirect
3. C7×#2297C2semidirect
4. C7×#2044C2semidirect
5. C7×#1899C2semidirect
6. C7×#1867C2semidirect
7. C7×#1790C2semidirect
8. C7×#1782C2semidirect
20. C2×C7×#117C22semidirect
21. C22×C42×C7C22semidirect
24. C2×C7×#99C22semidirect
31. C7×#386C22semidirect
32. C7×#385C22semidirect
33. C7×#381C22semidirect
36. C7×#342C22semidirect
37. #24057C7direct (N × H)
103. #2853C2×C7semidirect
104. #2297C2×C7semidirect
105. #2044C2×C7semidirect
106. #1899C2×C7semidirect
107. #1867C2×C7semidirect
108. #1790C2×C7semidirect
109. #1782C2×C7semidirect
199. C2×#117C22×C7semidirect
200. C22×C42C22×C7semidirect
203. C2×#99C22×C7semidirect
210. #386C22×C7semidirect
211. #385C22×C7semidirect
212. #381C22×C7semidirect
215. #342C22×C7semidirect
519. C7#24057direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C7×#28532
C7×#22972
C7×#20442
C7×#18992
C7×#18672
C7×#17902
C7×#17822
#240577