Finite Group Databasebeta

← C7×#18756 #13195972

Subgroups — C7×#18756 #13195972

Order 1792.

1224 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic, complemented
2C7×#3226C211yesmaximal, characteristic, complemented
3C7×#3100C211yesmaximal, characteristic, complemented
4C7×#3036C211yesmaximal, characteristic, complemented
5C7×#1647C214yesmaximal, complemented
6C7×#1625C214yesmaximal, complemented
7C7×#1546C211yesmaximal, characteristic, complemented
8C7×#1486C211yesmaximal, characteristic, complemented
9C7×#1478C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C2×C7×AGL(1,ℤ/8)—21no—
12C7×#547—22no—
13C7×#547—22no—
14C7×#547—22no—
15C7×#547C2214yes—
16C7×#532C2211yescharacteristic
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×#121—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C22×C7×C4⋊C4—21no—
24C7×#500C2212yescomplemented
25C7×#486C2212yescomplemented
26C7×#480—22no—
27C7×#480—21no—
28C7×#474—22no—
29C7×#474C2212yescomplemented
30C7×#463—22no—
31C7×#461C2212yescomplemented
32C7×#459C2212yescomplemented
33C7×#459C2212yescomplemented
34C7×#447—21no—
35C4×C7×D8—22no—
36C7×#431C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#423C2211yescharacteristic
39C7×#417—21no—
40C7×#332C2214yescomplemented
41C7×D4⋊C8C2214yescomplemented
42C7×D4⋊C8C2214yescomplemented
43#18756C711yesmaximal, characteristic, direct product factor, Sylow
44C2×C7×#41—24no—
45C2×C7×#41—24no—
46C2×C7×#41—21no—
47C2×C7×#41—44no—
48C2×C7×#41—44no—
49C22×C7×Q8—21no—
50C22×C7×D4—22no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4D411yescharacteristic
53C23×C4×C7—22no—
54C23×C4×C7—21no—
55C23×C4×C7—21no—
56C7×AGL(1,ℤ/8)—44no—
57C2×C7×SD16—22no—
58C2×C7×D8—22no—
59C2×C7×D8—22no—
60C2×C7×M16—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8D412yes—
64C7×C4⋊Q8C2311yescharacteristic
65C7×C4⋊Q8C2312yes—
66C7×#127C2311yescharacteristic
67C7×#124—21no—
68C7×#123—28no—
69C7×#123—28no—
70C7×C22⋊Q8—24no—
71C7×C22⋊Q8—24no—
72C7×#121—24no—
73C7×#121—24no—
74C7×#121—21no—
75C7×#121—21no—
76C4×C7×D4—21no—
77C4×C7×D4—21no—
78C4×C7×D4—21no—
79C4×C7×D4—21no—
80C4×C7×D4—21no—
81C4×C7×D4—21no—
82C4×C7×D4D412yes—
83C4×C7×D4D412yes—
84C4×C7×D4C2312yes—
85C4×C7×D4C2312yes—
86C2×C7×C4⋊C4—22no—
87C2×C7×C4⋊C4—24no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—24no—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4D412yes—
92C2×C7×C4⋊C4D411yescharacteristic
93C2×C7×C22⋊C4—22no—
94C2×C7×C22⋊C4—21no—
95C2×C7×C22⋊C4—21no—
96C2×C42×C7C2311yescharacteristic
97C7×#107—24no—
98C7×#107—22no—
99C7×C4⋊C8—21no—
100C7×C4⋊C8—21no—
101C7×C4⋊C8C2312yes—
102C7×C4⋊C8C2312yes—
103C7×#103—22no—
104C7×#102—22no—
105C7×#102—24no—
106C7×#102—22no—
107C7×#102—22no—
108C7×C22⋊C8—21no—
109C7×C22⋊C8—21no—
110C4×C7×C8—21no—
111C4×C7×C8C2312yes—
112C2×#547C2×C711yescharacteristic, complemented
113#3226C2×C711yescharacteristic, complemented
114#3100C2×C711yescharacteristic, complemented
115#3036C2×C711yescharacteristic, complemented
116#1647C2×C714yescomplemented
117#1625C2×C714yescomplemented
118#1546C2×C711yescharacteristic, complemented
119#1486C2×C711yescharacteristic, complemented
120#1478C2×C711yescharacteristic, complemented
121C24×C7—21no—
122C24×C7—21no—
123C7×#41—48no—
124C7×#41—416no—
125C7×#41—48no—
126C7×#41—42no—
127C7×#41—42no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—21no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×D4—28no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—42no—
141C2×C7×D4—48no—
142C2×C7×D4—41no—
143C2×C7×D4D814yescomplemented
144C2×C7×D4C2×D412yes—
145C2×C7×D4C2×D412yes—
146C22×C4×C7—28no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—44no—
155C22×C4×C7—48no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—41no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C7×SD16—44no—
163C7×D8—44no—
164C7×D8—44no—
165C7×M16—42no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C7×C4⋊C4—28no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4C2×D412yes—
178C7×C4⋊C4C2×D412yes—
179C7×C4⋊C4C2×D414yes—
180C7×C22⋊C4—28no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—41no—
186C7×C22⋊C4—41no—
187C42×C7C2×D411yescharacteristic
188C42×C7C2×D411yescharacteristic
189C42×C7C2×D411yescharacteristic
190C42×C7C2411yescharacteristic
191C22×#41—21no—
192C2×AGL(1,ℤ/8)—21no—
193#547—22no—
194#547—22no—
195#547—22no—
196#547C22×C714yes—
197#532C22×C711yescharacteristic
198C2×C4⋊Q8C22×C711yescharacteristic
199C2×#123—22no—
200C2×C22⋊Q8—21no—
201C2×#121—21no—
202C2×C4×D4C22×C711yescharacteristic
203C2×C4×D4C22×C711yescharacteristic
204C22×C4⋊C4—21no—
205#500C22×C712yescomplemented
206#486C22×C712yescomplemented
207#480—21no—
208#480—22no—
209#474—22no—
210#474C22×C712yescomplemented
211#463—22no—
212#461C22×C712yescomplemented
213#459C22×C712yescomplemented
214#459C22×C712yescomplemented
215#447—21no—
216C4×D8—22no—
217#431C22×C711yescharacteristic
218#431C22×C711yescharacteristic
219#423C22×C711yescharacteristic
220#417—21no—
221#332C22×C714yescomplemented
222D4⋊C8C22×C714yescomplemented
223D4⋊C8C22×C714yescomplemented
224C23×C7—22no—
225C23×C7—24no—
226C23×C7—22no—
227C23×C7—42no—
228C23×C7—42no—
229C23×C7—42no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—41no—
233C23×C7#12011yescharacteristic
234C7×Q8—42no—
235C7×Q8—42no—
236C7×Q8—44no—
237C7×Q8—41no—
238C7×Q8—41no—
239C7×D4—22no—
240C7×D4—22no—
241C7×D4—22no—
242C7×D4—24no—
243C7×D4—42no—
244C7×D4—42no—
245C7×D4—416no—
246C7×D4—41no—
247C7×D4—41no—
248C7×D4—42no—
249C7×D4—82no—
250C7×D4C2×D814yes—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—41no—
256C2×C4×C7—44no—
257C2×C4×C7—42no—
258C2×C4×C7—416no—
259C2×C4×C7—42no—
260C2×C4×C7—44no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—44no—
264C2×C4×C7—44no—
265C2×C4×C7—41no—
266C2×C4×C7—82no—
267C2×C4×C7#12011yescharacteristic
268C2×C4×C7#12011yescharacteristic
269C2×C4×C7#12011yescharacteristic
270C2×C4×C7C2×D812yes—
271C2×C4×C7AGL(1,ℤ/8)12yes—
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C7×C8—42no—
276C7×C8—42no—
277C7×C8—42no—
278C2×#41—24no—
279C2×#41—24no—
280C2×#41—21no—
281C2×#41—44no—
282C2×#41—44no—
283C22×Q8—21no—
284C22×D4—22no—
285C22×D4C7×D411yescharacteristic
286C22×D4C7×D411yescharacteristic
287C23×C4—22no—
288C23×C4—21no—
289C23×C4—21no—
290AGL(1,ℤ/8)—44no—
291C2×SD16—22no—
292C2×D8—22no—
293C2×D8—22no—
294C2×M16—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8C7×D412yes—
298C4⋊Q8C23×C711yescharacteristic
299C4⋊Q8C23×C712yes—
300#127C23×C711yescharacteristic
301#124—21no—
302#123—28no—
303#123—28no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306#121—24no—
307#121—24no—
308#121—21no—
309#121—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4C7×D412yes—
317C4×D4C7×D412yes—
318C4×D4C23×C712yes—
319C4×D4C23×C712yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C7×D411yescharacteristic
325C2×C4⋊C4C7×D411yescharacteristic
326C2×C4⋊C4C7×D412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C42C23×C711yescharacteristic
331#107—22no—
332#107—24no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8C23×C712yes—
336C4⋊C8C23×C712yes—
337#103—22no—
338#102—22no—
339#102—22no—
340#102—22no—
341#102—24no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C4×C8—21no—
345C4×C8C23×C712yes—
346C22×C7—21no—
347C22×C7—21no—
348C22×C7—24no—
349C22×C7—48no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—84no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7#44712yes—
360C22×C7C2×#12011yescharacteristic
361C4×C7—21no—
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—42no—
365C4×C7—44no—
366C4×C7—42no—
367C4×C7—81no—
368C4×C7#44712yes—
369C4×C7C22×D811yescharacteristic
370C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
371C4×C7#57512yes—
372C24—21no—
373C24—21no—
374#41—48no—
375#41—416no—
376#41—48no—
377#41—42no—
378#41—42no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×D4—28no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—42no—
392C2×D4—48no—
393C2×D4—41no—
394C2×D4C7×D814yescomplemented
395C2×D4C2×C7×D412yes—
396C2×D4C2×C7×D412yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D411yescharacteristic
412C22×C4C2×C7×D411yescharacteristic
413SD16—44no—
414D8—44no—
415D8—44no—
416M16—42no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C4⋊C4—28no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4C2×C7×D412yes—
429C4⋊C4C2×C7×D412yes—
430C4⋊C4C2×C7×D414yes—
431C22⋊C4—28no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C42C2×C7×D411yescharacteristic
439C42C2×C7×D411yescharacteristic
440C42C2×C7×D411yescharacteristic
441C42C24×C711yescharacteristic
442C2×C7—22no—
443C2×C7—44no—
444C2×C7—82no—
445C2×C7—161no—
446C2×C7C2×#44711yescharacteristic
447C2×C7#300511yescharacteristic
448C2×C7#302011yescharacteristic
449C23—22no—
450C23—24no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23C7×#12011yescharacteristic
459Q8—42no—
460Q8—42no—
461Q8—44no—
462Q8—41no—
463Q8—41no—
464D4—22no—
465D4—22no—
466D4—22no—
467D4—24no—
468D4—42no—
469D4—42no—
470D4—416no—
471D4—41no—
472D4—41no—
473D4—42no—
474D4—82no—
475D4C2×C7×D814yes—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—416no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4C7×#12011yescharacteristic
493C2×C4C7×#12011yescharacteristic
494C2×C4C7×#12011yescharacteristic
495C2×C4C2×C7×D812yes—
496C2×C4C7×AGL(1,ℤ/8)12yes—
497C2×C4C22×C7×D411yescharacteristic
498C2×C4C22×C7×D411yescharacteristic
499C2×C4C22×C7×D411yescharacteristic
500C8—42no—
501C8—42no—
502C8—42no—
503C7#1875611yescharacteristic, direct product factor, Sylow
504C22—21no—
505C22—21no—
506C22—24no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—84no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22C7×#44712yes—
518C22C2×C7×#12011yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—44no—
524C4—42no—
525C4—81no—
526C4C7×#44712yes—
527C4C22×C7×D811yescharacteristic
528C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
529C4C7×#57512yes—
530C2—22no—
531C2—44no—
532C2—82no—
533C2—161no—
534C2C2×C7×#44711yescharacteristic
535C2C7×#300511yescharacteristic
536C2C7×#302011yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#547C2semidirect
2. C7×#3226C2semidirect
3. C7×#3100C2semidirect
4. C7×#3036C2semidirect
5. C7×#1647C2semidirect
6. C7×#1625C2semidirect
7. C7×#1546C2semidirect
8. C7×#1486C2semidirect
9. C7×#1478C2semidirect
24. C7×#500C22semidirect
25. C7×#486C22semidirect
29. C7×#474C22semidirect
31. C7×#461C22semidirect
32. C7×#459C22semidirect
33. C7×#459C22semidirect
40. C7×#332C22semidirect
41. C7×D4⋊C8C22semidirect
42. C7×D4⋊C8C22semidirect
43. #18756C7direct (N × H)
112. C2×#547C2×C7semidirect
113. #3226C2×C7semidirect
114. #3100C2×C7semidirect
115. #3036C2×C7semidirect
116. #1647C2×C7semidirect
117. #1625C2×C7semidirect
118. #1546C2×C7semidirect
119. #1486C2×C7semidirect
120. #1478C2×C7semidirect
143. C2×C7×D4D8semidirect
205. #500C22×C7semidirect
206. #486C22×C7semidirect
210. #474C22×C7semidirect
212. #461C22×C7semidirect
213. #459C22×C7semidirect
214. #459C22×C7semidirect
221. #332C22×C7semidirect
222. D4⋊C8C22×C7semidirect
223. D4⋊C8C22×C7semidirect
394. C2×D4C7×D8semidirect
503. C7#18756direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C7×#32262
C7×#31002
C7×#30362
C7×#16472
C7×#16252
C7×#15462
C7×#14862
C7×#14782
#187567