Finite Group Databasebeta

← C7×#18641 #13195857

Subgroups — C7×#18641 #13195857

Order 1792.

1114 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×#3499C211yesmaximal, characteristic
2C7×#3227C211yesmaximal, characteristic, complemented
3C2×C7×#487C211yesmaximal, characteristic, complemented
4C7×#3079C211yesmaximal, characteristic, complemented
5C7×#1637C214yesmaximal, complemented
6C7×#1635C214yesmaximal, complemented
7C7×#1546C211yesmaximal, characteristic, complemented
8C7×#1492C211yesmaximal, characteristic, complemented
9C7×#1476C211yesmaximal, characteristic, complemented
10C2×C7×2−1+4—21no—
11C7×D4×Q8—21no—
12C7×D4×Q8C2212yes—
13C7×#547—22no—
14C7×#547—21no—
15C7×#547—22no—
16C7×#547C2212yes—
17C7×#536—22no—
18C7×#536—21no—
19C7×#532—21no—
20C7×#532C2211yescharacteristic
21C2×C7×C4⋊Q8—21no—
22C2×C7×C4⋊Q8C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C4×C7×#41C2211yescharacteristic
25C7×#498—21no—
26C7×#487—22no—
27C7×#487C2214yes—
28C7×#486—21no—
29C7×#481—22no—
30C7×#477C2212yes—
31C7×#476C2214yes—
32C7×#476C2212yes—
33C7×#475—21no—
34C7×#474—21no—
35C7×#431C2211yescharacteristic
36C7×#428—21no—
37C7×#423C2211yescharacteristic
38C7×#417—21no—
39C2×C7×#103—22no—
40C2×C4×C7×C8C2211yescharacteristic
41C7×#332C2214yes—
42C7×Q8⋊C8C2212yes—
43C7×D4⋊C8C2212yes—
44C7×D4⋊C8C2214yes—
45#18641C711yesmaximal, characteristic, direct product factor, Sylow
46C7×2−1+4—44no—
47C7×2−1+4—44no—
48C2×C7×#41—22no—
49C2×C7×#41—24no—
50C2×C7×#41—22no—
51C2×C7×#41—21no—
52C2×C7×#41D411yescharacteristic
53C22×C7×Q8—22no—
54C22×C7×Q8—22no—
55C22×C7×Q8—21no—
56C22×C7×Q8—21no—
57C2×C7×M16—21no—
58C22×C7×C8—21no—
59C7×C4⋊Q8—22no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8—21no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—21no—
67C7×C4⋊Q8D412yes—
68C7×C4⋊Q8D411yescharacteristic
69C7×C4⋊Q8C2312yes—
70C7×C4⋊Q8C2312yes—
71C7×#127D411yescharacteristic
72C7×#124—22no—
73C7×#124—21no—
74C7×#123—24no—
75C7×#123—24no—
76C7×#123—24no—
77C7×C22⋊Q8—24no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—24no—
80C7×C22⋊Q8—22no—
81C7×#121—22no—
82C7×#121—22no—
83C7×#121—21no—
84C7×#121D412yes—
85C4×C7×Q8D411yescharacteristic
86C4×C7×Q8C2311yescharacteristic
87C4×C7×D4—22no—
88C4×C7×D4—22no—
89C4×C7×D4—21no—
90C4×C7×D4—21no—
91C4×C7×D4—21no—
92C4×C7×D4D411yescharacteristic
93C4×C7×D4C2311yescharacteristic
94C4×C7×D4C2312yes—
95C7×#117—21no—
96C7×#117D411yescharacteristic
97C2×C7×C4⋊C4—22no—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4—22no—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4D412yes—
103C2×C42×C7—21no—
104C2×C42×C7C2311yescharacteristic
105C7×#107—22no—
106C7×#106—22no—
107C7×#106—24no—
108C7×C4⋊C8—21no—
109C7×C4⋊C8C2312yes—
110C7×C4⋊C8C2312yes—
111C7×#103—22no—
112C7×#103—28no—
113C7×#102—22no—
114C7×#102—24no—
115C7×C22⋊C8—21no—
116C4×C7×C8—21no—
117C4×C7×C8—21no—
118C4×C7×C8C2312yes—
119#3499C2×C711yescharacteristic
120#3227C2×C711yescharacteristic, complemented
121C2×#487C2×C711yescharacteristic, complemented
122#3079C2×C711yescharacteristic, complemented
123#1637C2×C714yescomplemented
124#1635C2×C714yescomplemented
125#1546C2×C711yescharacteristic, complemented
126#1492C2×C711yescharacteristic, complemented
127#1476C2×C711yescharacteristic, complemented
128C7×#41—22no—
129C7×#41—48no—
130C7×#41—416no—
131C7×#41—44no—
132C7×#41—44no—
133C7×#41—42no—
134C7×#41—42no—
135C7×#41SD1614yes—
136C2×C7×Q8—24no—
137C2×C7×Q8—22no—
138C2×C7×Q8—21no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—48no—
142C2×C7×Q8—44no—
143C2×C7×Q8—44no—
144C2×C7×Q8—44no—
145C2×C7×Q8—41no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8C2×D411yescharacteristic
149C2×C7×D4—21no—
150C2×C7×D4—24no—
151C2×C7×D4—22no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C2×D412yes—
155C2×C7×D4C2×D411yescharacteristic
156C22×C4×C7—21no—
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—41no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C7×M16—42no—
168C2×C7×C8—22no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—42no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—41no—
183C7×C4⋊C4—41no—
184C7×C4⋊C4C2×D414yes—
185C7×C4⋊C4C2×D412yes—
186C7×C4⋊C4C2×D411yescharacteristic
187C7×C4⋊C4C2×D411yescharacteristic
188C7×C22⋊C4—24no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—21no—
191C7×C22⋊C4—21no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—21no—
194C7×C22⋊C4—22no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7C2×D411yescharacteristic
199C42×C7C2×D411yescharacteristic
200C42×C7C2×D411yescharacteristic
201C42×C7C2411yescharacteristic
202C2×2−1+4—21no—
203D4×Q8—21no—
204D4×Q8C22×C712yes—
205#547—22no—
206#547—21no—
207#547—22no—
208#547C22×C712yes—
209#536—22no—
210#536—21no—
211#532—21no—
212#532C22×C711yescharacteristic
213C2×C4⋊Q8—21no—
214C2×C4⋊Q8C22×C711yescharacteristic
215C2×C4⋊Q8C22×C711yescharacteristic
216C4×#41C22×C711yescharacteristic
217#498—21no—
218#487—22no—
219#487C22×C714yes—
220#486—21no—
221#481—22no—
222#477C22×C712yes—
223#476C22×C712yes—
224#476C22×C714yes—
225#475—21no—
226#474—21no—
227#431C22×C711yescharacteristic
228#428—21no—
229#423C22×C711yescharacteristic
230#417—21no—
231C2×#103—22no—
232C2×C4×C8C22×C711yescharacteristic
233#332C22×C714yes—
234Q8⋊C8C22×C712yes—
235D4⋊C8C22×C712yes—
236D4⋊C8C22×C714yes—
237C23×C7—21no—
238C23×C7—22no—
239C23×C7#12011yescharacteristic
240C7×Q8—21no—
241C7×Q8—48no—
242C7×Q8—44no—
243C7×Q8—42no—
244C7×Q8—42no—
245C7×Q8—42no—
246C7×Q8—44no—
247C7×Q8—82no—
248C7×Q8C2×SD1612yes—
249C7×D4—24no—
250C7×D4—21no—
251C7×D4—24no—
252C7×D4—48no—
253C7×D4—44no—
254C7×D4—42no—
255C7×D4—42no—
256C7×D4C2×SD1612yes—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—24no—
260C2×C4×C7—22no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—24no—
264C2×C4×C7—22no—
265C2×C4×C7—48no—
266C2×C4×C7—48no—
267C2×C4×C7—44no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7—42no—
273C2×C4×C7—82no—
274C2×C4×C7#12011yescharacteristic
275C2×C4×C7#12011yescharacteristic
276C2×C4×C7#12011yescharacteristic
277C2×C4×C7C2×SD1612yes—
278C2×C4×C7AGL(1,ℤ/8)12yes—
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C7×C8—42no—
283C7×C8—42no—
284C7×C8—42no—
2852−1+4—44no—
2862−1+4—44no—
287C2×#41—22no—
288C2×#41—24no—
289C2×#41—22no—
290C2×#41—21no—
291C2×#41C7×D411yescharacteristic
292C22×Q8—22no—
293C22×Q8—22no—
294C22×Q8—21no—
295C22×Q8—21no—
296C2×M16—21no—
297C22×C8—21no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8C7×D411yescharacteristic
307C4⋊Q8C7×D412yes—
308C4⋊Q8C23×C712yes—
309C4⋊Q8C23×C712yes—
310#127C7×D411yescharacteristic
311#124—22no—
312#124—21no—
313#123—24no—
314#123—24no—
315#123—24no—
316C22⋊Q8—24no—
317C22⋊Q8—22no—
318C22⋊Q8—24no—
319C22⋊Q8—22no—
320#121—22no—
321#121—22no—
322#121—21no—
323#121C7×D412yes—
324C4×Q8C7×D411yescharacteristic
325C4×Q8C23×C711yescharacteristic
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4C7×D411yescharacteristic
332C4×D4C23×C711yescharacteristic
333C4×D4C23×C712yes—
334#117—21no—
335#117C7×D411yescharacteristic
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C7×D412yes—
342C2×C42—21no—
343C2×C42C23×C711yescharacteristic
344#107—22no—
345#106—22no—
346#106—24no—
347C4⋊C8—21no—
348C4⋊C8C23×C712yes—
349C4⋊C8C23×C712yes—
350#103—22no—
351#103—28no—
352#102—22no—
353#102—24no—
354C22⋊C8—21no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8C23×C712yes—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7—22no—
361C22×C7—41no—
362C22×C7—41no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7#45012yes—
367C22×C7C2×#12011yescharacteristic
368C4×C7—22no—
369C4×C7—21no—
370C4×C7—21no—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7—41no—
374C4×C7—41no—
375C4×C7—82no—
376C4×C7#45012yes—
377C4×C7#45412yes—
378C4×C7C22×SD1611yescharacteristic
379C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
380C4×C7#57812yes—
381#41—22no—
382#41—48no—
383#41—416no—
384#41—44no—
385#41—44no—
386#41—42no—
387#41—42no—
388#41C7×SD1614yes—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—48no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—41no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C2×C7×D411yescharacteristic
402C2×D4—21no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×C7×D412yes—
408C2×D4C2×C7×D411yescharacteristic
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D411yescharacteristic
419C22×C4C2×C7×D411yescharacteristic
420M16—42no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—42no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4C2×C7×D414yes—
438C4⋊C4C2×C7×D412yes—
439C4⋊C4C2×C7×D411yescharacteristic
440C4⋊C4C2×C7×D411yescharacteristic
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C22⋊C4—21no—
447C22⋊C4—22no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42C2×C7×D411yescharacteristic
452C42C2×C7×D411yescharacteristic
453C42C2×C7×D411yescharacteristic
454C42C24×C711yescharacteristic
455C2×C7—22no—
456C2×C7—42no—
457C2×C7—82no—
458C2×C7C2×#45011yescharacteristic
459C2×C7#301111yescharacteristic
460C2×C7C2×#45411yescharacteristic
461C23—21no—
462C23—22no—
463C23C7×#12011yescharacteristic
464Q8—21no—
465Q8—48no—
466Q8—44no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470Q8—44no—
471Q8—82no—
472Q8C2×C7×SD1612yes—
473D4—24no—
474D4—21no—
475D4—24no—
476D4—48no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4C2×C7×SD1612yes—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—82no—
498C2×C4C7×#12011yescharacteristic
499C2×C4C7×#12011yescharacteristic
500C2×C4C7×#12011yescharacteristic
501C2×C4C2×C7×SD1612yes—
502C2×C4C7×AGL(1,ℤ/8)12yes—
503C2×C4C22×C7×D411yescharacteristic
504C2×C4C22×C7×D411yescharacteristic
505C2×C4C22×C7×D411yescharacteristic
506C8—42no—
507C8—42no—
508C8—42no—
509C7#1864111yescharacteristic, direct product factor, Sylow
510C22—21no—
511C22—21no—
512C22—22no—
513C22—41no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22C7×#45012yes—
519C22C2×C7×#12011yescharacteristic
520C4—22no—
521C4—21no—
522C4—21no—
523C4—44no—
524C4—42no—
525C4—41no—
526C4—41no—
527C4—82no—
528C4C7×#45012yes—
529C4C7×#45412yes—
530C4C22×C7×SD1611yescharacteristic
531C4C2×C7×AGL(1,ℤ/8)11yescharacteristic
532C4C7×#57812yes—
533C2—22no—
534C2—42no—
535C2—82no—
536C2C2×C7×#45011yescharacteristic
537C2C7×#301111yescharacteristic
538C2C2×C7×#45411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3499C2
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
2. C7×#3227C2semidirect
3. C2×C7×#487C2semidirect
4. C7×#3079C2semidirect
5. C7×#1637C2semidirect
6. C7×#1635C2semidirect
7. C7×#1546C2semidirect
8. C7×#1492C2semidirect
9. C7×#1476C2semidirect
45. #18641C7direct (N × H)
120. #3227C2×C7semidirect
121. C2×#487C2×C7semidirect
122. #3079C2×C7semidirect
123. #1637C2×C7semidirect
124. #1635C2×C7semidirect
125. #1546C2×C7semidirect
126. #1492C2×C7semidirect
127. #1476C2×C7semidirect
509. C7#18641direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34992
C7×#32272
C2×C7×#4872
C7×#30792
C7×#16372
C7×#16352
C7×#15462
C7×#14922
C7×#14762
#186417