Finite Group Databasebeta

← C7×#18431 #13195647

Subgroups — C7×#18431 #13195647

Order 1792.

928 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3559C211yesmaximal, characteristic
2C7×#3218C211yesmaximal, characteristic, complemented
3C7×#3206C211yesmaximal, characteristic, complemented
4C7×#2660C211yesmaximal, characteristic, complemented
5C7×#1604C214yesmaximal, complemented
6C7×#1603C214yesmaximal, complemented
7C7×#1505C211yesmaximal, characteristic, complemented
8C7×#1500C211yesmaximal, characteristic, complemented
9C7×#1470C211yesmaximal, characteristic, complemented
10C7×#563—21no—
11C7×#562—21no—
12C7×#554—21no—
13C7×#553—21no—
14C7×#553—21no—
15C7×#547—22no—
16C7×#547—21no—
17C7×#544—21no—
18C7×#540—21no—
19C7×#539—22no—
20C7×#539C2212yes—
21C7×#537C2212yes—
22C7×#532C2211yescharacteristic
23C2×C7×C4⋊Q8C2211yescharacteristic
24C7×#529C2211yescharacteristic
25C2×C7×#125C2211yescharacteristic
26C7×#497—21no—
27C7×#492—21no—
28C7×#491—21no—
29C7×#487—21no—
30C7×#462—21no—
31C7×#461—22no—
32C7×#461—21no—
33C7×#451C2212yes—
34C7×#448C2214yes—
35C7×#448C2212yes—
36C7×#432C2211yescharacteristic
37C7×#431C2211yescharacteristic
38C7×#398—22no—
39C7×#395C2214yes—
40C7×#384—21no—
41C7×#382—21no—
42C7×#376C2211yescharacteristic
43C7×#328C2212yes—
44C7×#327C2212yes—
45C7×#327C2214yes—
46C7×#324C2214yes—
47#18431C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C2×C7×#41—22no—
50C22×C7×Q8C2312yes—
51C2×C7×Q16—22no—
52C2×C7×SD16—24no—
53C2×C7×SD16—22no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—22no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8D411yescharacteristic
61C7×#127D411yescharacteristic
62C7×#126—22no—
63C7×#126—22no—
64C7×#126—21no—
65C7×#125—21no—
66C7×#125—22no—
67C7×#125—21no—
68C7×#125—21no—
69C7×#125—21no—
70C7×#125—21no—
71C7×#125D411yescharacteristic
72C7×#124—21no—
73C7×#124D411yescharacteristic
74C7×#123—22no—
75C7×#123—22no—
76C7×#123—22no—
77C7×#123—22no—
78C7×#123—22no—
79C7×#123—21no—
80C7×#123—22no—
81C7×C22⋊Q8—22no—
82C7×C22⋊Q8—22no—
83C7×C22⋊Q8—22no—
84C7×C22⋊Q8—22no—
85C7×C22⋊Q8—22no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8C2311yescharacteristic
88C7×#121—21no—
89C7×#121—22no—
90C7×#121—21no—
91C7×#121—22no—
92C7×#121—22no—
93C7×#121—21no—
94C7×#121—22no—
95C7×#121C2311yescharacteristic
96C7×#121C2312yes—
97C4×C7×Q8D411yescharacteristic
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—22no—
103C4×C7×D4—21no—
104C4×C7×D4D411yescharacteristic
105C4×C7×D4D412yes—
106C7×#117—21no—
107C7×#117—21no—
108C2×C7×C4⋊C4—22no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4—21no—
112C2×C7×C4⋊C4—24no—
113C2×C7×C4⋊C4D411yescharacteristic
114C2×C7×C4⋊C4D411yescharacteristic
115C2×C7×C4⋊C4D412yes—
116C2×C7×C4⋊C4C2312yes—
117C2×C42×C7—21no—
118C2×C42×C7C2311yescharacteristic
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×#103—24no—
122C7×#103—22no—
123C7×#103—22no—
124C7×C22⋊C8C2312yes—
125C7×C22⋊C8C2312yes—
126C7×#97—21no—
127C4×C7×C8—21no—
128C7×#95—21no—
129C7×#95—24no—
130C7×#95C2312yes—
131#3559C2×C711yescharacteristic
132#3218C2×C711yescharacteristic, complemented
133#3206C2×C711yescharacteristic, complemented
134#2660C2×C711yescharacteristic, complemented
135#1604C2×C714yescomplemented
136#1603C2×C714yescomplemented
137#1505C2×C711yescharacteristic, complemented
138#1500C2×C711yescharacteristic, complemented
139#1470C2×C711yescharacteristic, complemented
140C7×#41—82no—
141C7×#41—84no—
142C2×C7×Q8—22no—
143C2×C7×Q8—21no—
144C2×C7×Q8—22no—
145C2×C7×Q8—22no—
146C2×C7×Q8—82no—
147C2×C7×Q8C2×D411yescharacteristic
148C2×C7×D4—21no—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C2×D411yescharacteristic
155C2×C7×D4C2×D412yes—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2411yescharacteristic
168C7×Q16—82no—
169C7×SD16—84no—
170C7×SD16—82no—
171C2×C7×C8—22no—
172C2×C7×C8—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—41no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—41no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4—84no—
195C7×C4⋊C4C2×D412yes—
196C7×C4⋊C4C2×D411yescharacteristic
197C7×C4⋊C4C2×D411yescharacteristic
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—22no—
210C42×C7—24no—
211C42×C7—22no—
212C42×C7—21no—
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215C42×C7C2×D411yescharacteristic
216C42×C7C2×D411yescharacteristic
217#563—21no—
218#562—21no—
219#554—21no—
220#553—21no—
221#553—21no—
222#547—22no—
223#547—21no—
224#544—21no—
225#540—21no—
226#539—22no—
227#539C22×C712yes—
228#537C22×C712yes—
229#532C22×C711yescharacteristic
230C2×C4⋊Q8C22×C711yescharacteristic
231#529C22×C711yescharacteristic
232C2×#125C22×C711yescharacteristic
233#497—21no—
234#492—21no—
235#491—21no—
236#487—21no—
237#462—21no—
238#461—22no—
239#461—21no—
240#451C22×C712yes—
241#448C22×C714yes—
242#448C22×C712yes—
243#432C22×C711yescharacteristic
244#431C22×C711yescharacteristic
245#398—22no—
246#395C22×C714yes—
247#384—21no—
248#382—21no—
249#376C22×C711yescharacteristic
250#328C22×C712yes—
251#327C22×C714yes—
252#327C22×C712yes—
253#324C22×C714yes—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7C22×D411yescharacteristic
257C7×Q8—41no—
258C7×Q8—44no—
259C7×Q8—42no—
260C7×Q8—42no—
261C7×Q8—81no—
262C7×Q8—82no—
263C7×D4—42no—
264C7×D4—41no—
265C7×D4—44no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—81no—
269C7×D4—82no—
270C7×D4—82no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—24no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—41no—
282C2×C4×C7—42no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—44no—
287C2×C4×C7—42no—
288C2×C4×C7—82no—
289C2×C4×C7—82no—
290C2×C4×C7—82no—
291C2×C4×C7—82no—
292C2×C4×C7#12011yescharacteristic
293C2×C4×C7#12011yescharacteristic
294C2×C4×C7#12011yescharacteristic
295C2×C4×C7#12011yescharacteristic
296C2×C4×C7C22×D411yescharacteristic
297C2×C4×C7C22×D411yescharacteristic
298C7×C8—42no—
299C7×C8—42no—
300C2×#41—21no—
301C2×#41—22no—
302C22×Q8C23×C712yes—
303C2×Q16—22no—
304C2×SD16—24no—
305C2×SD16—22no—
306C4⋊Q8—21no—
307C4⋊Q8—22no—
308C4⋊Q8—21no—
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8C7×D411yescharacteristic
313#127C7×D411yescharacteristic
314#126—22no—
315#126—22no—
316#126—21no—
317#125—21no—
318#125—22no—
319#125—21no—
320#125—21no—
321#125—21no—
322#125—21no—
323#125C7×D411yescharacteristic
324#124—21no—
325#124C7×D411yescharacteristic
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—21no—
332#123—22no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8C23×C711yescharacteristic
340#121—21no—
341#121—22no—
342#121—21no—
343#121—22no—
344#121—22no—
345#121—21no—
346#121—22no—
347#121C23×C711yescharacteristic
348#121C23×C712yes—
349C4×Q8C7×D411yescharacteristic
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4C7×D411yescharacteristic
357C4×D4C7×D412yes—
358#117—21no—
359#117—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4C7×D411yescharacteristic
366C2×C4⋊C4C7×D412yes—
367C2×C4⋊C4C7×D411yescharacteristic
368C2×C4⋊C4C23×C712yes—
369C2×C42—21no—
370C2×C42C23×C711yescharacteristic
371C4⋊C8—21no—
372C4⋊C8—21no—
373#103—24no—
374#103—22no—
375#103—22no—
376C22⋊C8C23×C712yes—
377C22⋊C8C23×C712yes—
378#97—21no—
379C4×C8—21no—
380#95—21no—
381#95—24no—
382#95C23×C712yes—
383C22×C7—21no—
384C22×C7—21no—
385C22×C7—42no—
386C22×C7—41no—
387C22×C7—41no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—41no—
391C22×C7C2×#12011yescharacteristic
392C22×C7#57712yes—
393C4×C7—21no—
394C4×C7—21no—
395C4×C7—21no—
396C4×C7—21no—
397C4×C7—21no—
398C4×C7—44no—
399C4×C7—42no—
400C4×C7—42no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—82no—
406C4×C7#45612yes—
407#41—82no—
408#41—84no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—82no—
414C2×Q8C2×C7×D411yescharacteristic
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C7×D411yescharacteristic
422C2×D4C2×C7×D412yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4C2×C7×D412yes—
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D412yes—
434C22×C4C24×C711yescharacteristic
435Q16—82no—
436SD16—84no—
437SD16—82no—
438C2×C8—22no—
439C2×C8—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—84no—
462C4⋊C4C2×C7×D412yes—
463C4⋊C4C2×C7×D411yescharacteristic
464C4⋊C4C2×C7×D411yescharacteristic
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—21no—
476C22⋊C4—22no—
477C42—24no—
478C42—22no—
479C42—21no—
480C42C2×C7×D411yescharacteristic
481C42C2×C7×D411yescharacteristic
482C42C2×C7×D411yescharacteristic
483C42C2×C7×D411yescharacteristic
484C2×C7—22no—
485C2×C7—81no—
486C2×C7—82no—
487C2×C7#300911yescharacteristic
488C2×C7C2×#45611yescharacteristic
489C2×C7#302711yescharacteristic
490C23—21no—
491C23—22no—
492C23C22×C7×D411yescharacteristic
493Q8—41no—
494Q8—44no—
495Q8—42no—
496Q8—42no—
497Q8—81no—
498Q8—82no—
499D4—42no—
500D4—41no—
501D4—44no—
502D4—42no—
503D4—42no—
504D4—81no—
505D4—82no—
506D4—82no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4C7×#12011yescharacteristic
529C2×C4C7×#12011yescharacteristic
530C2×C4C7×#12011yescharacteristic
531C2×C4C7×#12011yescharacteristic
532C2×C4C22×C7×D411yescharacteristic
533C2×C4C22×C7×D411yescharacteristic
534C8—42no—
535C8—42no—
536C7#1843111yescharacteristic, direct product factor, Sylow
537C22—21no—
538C22—21no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—41no—
545C22C2×C7×#12011yescharacteristic
546C22C7×#57712yes—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—41no—
559C4—82no—
560C4C7×#45612yes—
561C2—22no—
562C2—81no—
563C2—82no—
564C2C7×#300911yescharacteristic
565C2C2×C7×#45611yescharacteristic
566C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3559C2
LCSG ▷ C22×C4 ▷ 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×#3218C2semidirect
3. C7×#3206C2semidirect
4. C7×#2660C2semidirect
5. C7×#1604C2semidirect
6. C7×#1603C2semidirect
7. C7×#1505C2semidirect
8. C7×#1500C2semidirect
9. C7×#1470C2semidirect
47. #18431C7direct (N × H)
132. #3218C2×C7semidirect
133. #3206C2×C7semidirect
134. #2660C2×C7semidirect
135. #1604C2×C7semidirect
136. #1603C2×C7semidirect
137. #1505C2×C7semidirect
138. #1500C2×C7semidirect
139. #1470C2×C7semidirect
536. C7#18431direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35592
C7×#32182
C7×#32062
C7×#26602
C7×#16042
C7×#16032
C7×#15052
C7×#15002
C7×#14702
#184317