Finite Group Databasebeta

← C7×#19050 #13196266

Subgroups — C7×#19050 #13196266

Order 1792.

1026 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3545C211yesmaximal, characteristic, complemented
2C7×#3217C212yesmaximal, complemented
3C2×C7×#496C211yesmaximal, characteristic, complemented
4C7×#1657C212yesmaximal, complemented
5C7×#1656C214yesmaximal, complemented
6C7×#1655C212yesmaximal, complemented
7C7×#1533C211yesmaximal, characteristic, complemented
8C7×#1532C212yesmaximal, complemented
9C22×C7×SD16—21no—
10C22×C7×D8—21no—
11C7×#563C2211yescharacteristic, complemented
12C7×#562C2212yescomplemented
13C7×H(3,𝔽4)C2211yescharacteristic, complemented
14C7×#555—21no—
15C7×#551—21no—
16C7×#551—22no—
17C7×#546—22no—
18C7×#542—22no—
19C7×#538—22no—
20C7×#536—21no—
21C7×#534—21no—
22C2×C7×#127C2211yescharacteristic, complemented
23C7×#529C2212yes—
24C2×C7×#124C2211yescharacteristic
25C7×#496—21no—
26C7×#496—21no—
27C7×#496C2212yes—
28C7×#496C2214yescomplemented
29C7×#496C2212yescomplemented
30C7×#490C2214yescomplemented
31C7×#460—22no—
32C7×#459—22no—
33C7×#450—22no—
34C7×#447—22no—
35C7×#432C2212yescomplemented
36C2×C7×#97C2211yescharacteristic, complemented
37C7×#330C2214yescomplemented
38C7×#329C2214yescomplemented
39C7×#329C2212yescomplemented
40C7×#329C2212yescomplemented
41#19050C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—21no—
43C22×C7×Q8D411yescharacteristic
44C22×C7×D4—21no—
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4D411yescharacteristic
48C2×C7×SD16—22no—
49C2×C7×SD16—22no—
50C2×C7×SD16—24no—
51C2×C7×SD16—44no—
52C2×C7×D8—22no—
53C2×C7×D8—22no—
54C2×C7×D8—24no—
55C2×C7×D8—44no—
56C22×C7×C8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×#127—21no—
60C7×#127—22no—
61C7×#127C2312yes—
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—22no—
65C7×#125C2312yes—
66C7×#124—21no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#124D411yescharacteristic
72C7×#124D411yescharacteristic
73C7×#124C2311yescharacteristic
74C7×#124C2311yescharacteristic
75C7×#124C2312yes—
76C7×#123—21no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—21no—
80C7×#123—22no—
81C7×#123—22no—
82C7×#123—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8D412yes—
88C7×#121—22no—
89C7×#121—22no—
90C7×#121—22no—
91C7×#121—22no—
92C7×#121D412yes—
93C7×#120—21no—
94C7×#120—22no—
95C7×#120—21no—
96C7×#120—22no—
97C4×C7×Q8D412yes—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4D412yes—
102C7×#117—21no—
103C7×#117—22no—
104C7×#117—22no—
105C2×C7×C22⋊C4—21no—
106C2×C7×C22⋊C4—22no—
107C2×C7×C22⋊C4—22no—
108C2×C42×C7C2311yescharacteristic
109C7×C4⋊C8—22no—
110C7×#102—24no—
111C7×#102—24no—
112C7×C22⋊C8—22no—
113C7×#97—21no—
114C7×#97C2314yes—
115C7×#97C2312yes—
116#3545C2×C711yescharacteristic, complemented
117#3217C2×C712yescomplemented
118C2×#496C2×C711yescharacteristic, complemented
119#1657C2×C712yescomplemented
120#1656C2×C714yescomplemented
121#1655C2×C712yescomplemented
122#1533C2×C711yescharacteristic, complemented
123#1532C2×C712yescomplemented
124C24×C7—21no—
125C24×C7—21no—
126C24×C7—41no—
127C7×#41—82no—
128C2×C7×Q8—21no—
129C2×C7×Q8—21no—
130C2×C7×Q8—42no—
131C2×C7×Q8C2×D412yes—
132C2×C7×Q8C2×D411yescharacteristic
133C2×C7×Q8C2×D411yescharacteristic
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—21no—
138C2×C7×D4—22no—
139C2×C7×D4—21no—
140C2×C7×D4—24no—
141C2×C7×D4—22no—
142C2×C7×D4—42no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4—42no—
146C2×C7×D4—41no—
147C2×C7×D4—84no—
148C2×C7×D4—82no—
149C2×C7×D4C2×D412yes—
150C2×C7×D4C2×D411yescharacteristic
151C2×C7×D4C2×D411yescharacteristic
152C22×C4×C7—21no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D412yes—
159C7×SD16—44no—
160C7×SD16—44no—
161C7×SD16—84no—
162C7×D8—44no—
163C7×D8—44no—
164C7×D8—84no—
165C2×C7×C8—24no—
166C2×C7×C8—22no—
167C2×C7×C8—24no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4C2×D412yes—
176C7×C4⋊C4C2×D412yes—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—21no—
179C7×C22⋊C4—21no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—22no—
195C42×C7C2×D411yescharacteristic
196C42×C7C2×D412yes—
197C42×C7C2411yescharacteristic
198C22×SD16—21no—
199C22×D8—21no—
200#563C22×C711yescharacteristic, complemented
201#562C22×C712yescomplemented
202H(3,𝔽4)C22×C711yescharacteristic, complemented
203#555—21no—
204#551—21no—
205#551—22no—
206#546—22no—
207#542—22no—
208#538—22no—
209#536—21no—
210#534—21no—
211C2×#127C22×C711yescharacteristic, complemented
212#529C22×C712yes—
213C2×#124C22×C711yescharacteristic
214#496—21no—
215#496—21no—
216#496C22×C712yes—
217#496C22×C712yescomplemented
218#496C22×C714yescomplemented
219#490C22×C714yescomplemented
220#460—22no—
221#459—22no—
222#450—22no—
223#447—22no—
224#432C22×C712yescomplemented
225C2×#97C22×C711yescharacteristic, complemented
226#330C22×C714yescomplemented
227#329C22×C714yescomplemented
228#329C22×C712yescomplemented
229#329C22×C712yescomplemented
230C23×C7—21no—
231C23×C7—22no—
232C23×C7—21no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—82no—
236C23×C7—84no—
237C23×C7—81no—
238C23×C7—82no—
239C23×C7—82no—
240C23×C7#12011yescharacteristic
241C7×Q8—41no—
242C7×Q8—41no—
243C7×Q8—42no—
244C7×Q8—81no—
245C7×Q8—81no—
246C7×D4—41no—
247C7×D4—41no—
248C7×D4—42no—
249C7×D4—42no—
250C7×D4—44no—
251C7×D4—42no—
252C7×D4—44no—
253C7×D4—44no—
254C7×D4—81no—
255C7×D4—82no—
256C7×D4—84no—
257C7×D4—82no—
258C7×D4—82no—
259C7×D4—82no—
260C7×D4—81no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—82no—
270C2×C4×C7—82no—
271C2×C4×C7—81no—
272C2×C4×C7—82no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12011yescharacteristic
275C2×C4×C7C22×D412yes—
276C2×C4×C7C22×D411yescharacteristic
277C7×C8—24no—
278C7×C8—44no—
279C2×#41—21no—
280C22×Q8C7×D411yescharacteristic
281C22×D4—21no—
282C22×D4—21no—
283C22×D4—22no—
284C22×D4C7×D411yescharacteristic
285C2×SD16—22no—
286C2×SD16—22no—
287C2×SD16—24no—
288C2×SD16—44no—
289C2×D8—22no—
290C2×D8—22no—
291C2×D8—24no—
292C2×D8—44no—
293C22×C8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296#127—22no—
297#127—21no—
298#127C23×C712yes—
299#126—22no—
300#126—22no—
301#126—22no—
302#125C23×C712yes—
303#124—21no—
304#124—22no—
305#124—21no—
306#124—22no—
307#124—22no—
308#124C7×D411yescharacteristic
309#124C7×D411yescharacteristic
310#124C23×C711yescharacteristic
311#124C23×C711yescharacteristic
312#124C23×C712yes—
313#123—21no—
314#123—22no—
315#123—22no—
316#123—21no—
317#123—22no—
318#123—22no—
319#123—22no—
320C22⋊Q8—22no—
321C22⋊Q8—22no—
322C22⋊Q8—22no—
323C22⋊Q8—22no—
324C22⋊Q8C7×D412yes—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121C7×D412yes—
330#120—21no—
331#120—22no—
332#120—21no—
333#120—22no—
334C4×Q8C7×D412yes—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4C7×D412yes—
339#117—21no—
340#117—22no—
341#117—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C42C23×C711yescharacteristic
346C4⋊C8—22no—
347#102—24no—
348#102—24no—
349C22⋊C8—22no—
350#97—21no—
351#97C23×C714yes—
352#97C23×C712yes—
353C22×C7—22no—
354C22×C7—42no—
355C22×C7—41no—
356C22×C7—41no—
357C22×C7—42no—
358C22×C7—42no—
359C22×C7—42no—
360C22×C7—42no—
361C22×C7—82no—
362C22×C7—84no—
363C22×C7—162no—
364C22×C7—164no—
365C22×C7—162no—
366C22×C7#45312yes—
367C22×C7C2×#12011yescharacteristic
368C4×C7—21no—
369C4×C7—21no—
370C4×C7—22no—
371C4×C7—22no—
372C4×C7—41no—
373C4×C7—41no—
374C4×C7—42no—
375C4×C7—42no—
376C4×C7—42no—
377C24—21no—
378C24—21no—
379C24—41no—
380#41—82no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8C2×C7×D412yes—
385C2×Q8C2×C7×D411yescharacteristic
386C2×Q8C2×C7×D411yescharacteristic
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—41no—
400C2×D4—84no—
401C2×D4—82no—
402C2×D4C2×C7×D412yes—
403C2×D4C2×C7×D411yescharacteristic
404C2×D4C2×C7×D411yescharacteristic
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D412yes—
412SD16—44no—
413SD16—44no—
414SD16—84no—
415D8—44no—
416D8—44no—
417D8—84no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4C2×C7×D412yes—
429C4⋊C4C2×C7×D412yes—
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—21no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C42—21no—
446C42—21no—
447C42—22no—
448C42C2×C7×D411yescharacteristic
449C42C2×C7×D412yes—
450C42C24×C711yescharacteristic
451C2×C7—22no—
452C2×C7—82no—
453C2×C7—81no—
454C2×C7—81no—
455C2×C7—162no—
456C2×C7C2×#45311yescharacteristic
457C2×C7#302112yes—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—82no—
464C23—84no—
465C23—81no—
466C23—82no—
467C23—82no—
468C23C7×#12011yescharacteristic
469Q8—41no—
470Q8—41no—
471Q8—42no—
472Q8—81no—
473Q8—81no—
474D4—41no—
475D4—41no—
476D4—42no—
477D4—42no—
478D4—44no—
479D4—42no—
480D4—44no—
481D4—44no—
482D4—81no—
483D4—82no—
484D4—84no—
485D4—82no—
486D4—82no—
487D4—82no—
488D4—81no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4C7×#12012yes—
502C2×C4C7×#12011yescharacteristic
503C2×C4C22×C7×D412yes—
504C2×C4C22×C7×D411yescharacteristic
505C8—24no—
506C8—44no—
507C7#1905011yescharacteristic, direct product factor, Sylow
508C22—22no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—84no—
518C22—162no—
519C22—164no—
520C22—162no—
521C22C7×#45312yes—
522C22C2×C7×#12011yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—42no—
532C2—22no—
533C2—81no—
534C2—82no—
535C2—81no—
536C2—162no—
537C2C2×C7×#45311yescharacteristic
538C2C7×#302112yes—
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. C7×#3545C2semidirect
2. C7×#3217C2semidirect
3. C2×C7×#496C2semidirect
4. C7×#1657C2semidirect
5. C7×#1656C2semidirect
6. C7×#1655C2semidirect
7. C7×#1533C2semidirect
8. C7×#1532C2semidirect
11. C7×#563C22semidirect
12. C7×#562C22semidirect
13. C7×H(3,𝔽4)C22semidirect
22. C2×C7×#127C22semidirect
28. C7×#496C22semidirect
29. C7×#496C22semidirect
30. C7×#490C22semidirect
35. C7×#432C22semidirect
36. C2×C7×#97C22semidirect
37. C7×#330C22semidirect
38. C7×#329C22semidirect
39. C7×#329C22semidirect
40. C7×#329C22semidirect
41. #19050C7direct (N × H)
116. #3545C2×C7semidirect
117. #3217C2×C7semidirect
118. C2×#496C2×C7semidirect
119. #1657C2×C7semidirect
120. #1656C2×C7semidirect
121. #1655C2×C7semidirect
122. #1533C2×C7semidirect
123. #1532C2×C7semidirect
200. #563C22×C7semidirect
201. #562C22×C7semidirect
202. H(3,𝔽4)C22×C7semidirect
211. C2×#127C22×C7semidirect
217. #496C22×C7semidirect
218. #496C22×C7semidirect
219. #490C22×C7semidirect
224. #432C22×C7semidirect
225. C2×#97C22×C7semidirect
226. #330C22×C7semidirect
227. #329C22×C7semidirect
228. #329C22×C7semidirect
229. #329C22×C7semidirect
507. C7#19050direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35452
C7×#32172
C2×C7×#4962
C7×#16572
C7×#16562
C7×#16552
C7×#15332
C7×#15322
#190507