Finite Group Databasebeta

← C7×#18131 #13195347

Subgroups — C7×#18131 #13195347

Order 1792.

932 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3461C211yesmaximal, characteristic
2C7×#3108C212yesmaximal, complemented
3C7×#2399C211yesmaximal, characteristic, complemented
4C7×#1611C211yesmaximal, characteristic, complemented
5C7×#1610C211yesmaximal, characteristic
6C7×#1609C212yesmaximal, complemented
7C7×#1605C211yesmaximal, characteristic, complemented
8C7×#1604C211yesmaximal, characteristic
9C7×#1603C212yesmaximal, complemented
10C7×#1500C212yesmaximal, complemented
11C7×#1461C211yesmaximal, characteristic, complemented
12C7×#544C2211yescharacteristic
13C7×#542—22no—
14C7×#541—22no—
15C7×#539—21no—
16C7×#539C2212yes—
17C7×#537C2211yescharacteristic
18C7×#536—21no—
19C7×#533—22no—
20C7×#529—22no—
21C7×#529—21no—
22C7×#529C2212yes—
23C4×C7×#41—21no—
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×Q8C2211yescharacteristic
26C7×#483C2212yes—
27C7×#480C2212yes—
28C7×#479—22no—
29C7×#464—22no—
30C7×#451C2212yes—
31C7×#448C2212yes—
32C4×C7×Q16—22no—
33C4×C7×SD16—22no—
34C7×#431C2212yes—
35C7×#398—21no—
36C7×#398C2211yescharacteristic
37C7×#398C2211yescharacteristic
38C7×#395—21no—
39C7×#395C2211yescharacteristic
40C7×#395C2211yescharacteristic
41C4×C7×C4⋊C4—21no—
42C4×C7×C4⋊C4—21no—
43C7×#376C2211yescharacteristic
44C7×#328C2212yes—
45C7×#328C2212yes—
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yes—
49C7×#324C2211yescharacteristic
50C7×#324C2211yescharacteristic
51#18131C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C22×C7×Q8D411yescharacteristic
54C22×C7×Q8C2311yescharacteristic
55C2×C7×Q16—22no—
56C2×C7×SD16—22no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8—22no—
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#125—21no—
64C7×#125—22no—
65C7×#125—22no—
66C7×#125—22no—
67C7×#125D412yes—
68C7×#124—22no—
69C7×#124—21no—
70C7×#124D412yes—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—21no—
80C7×C22⋊Q8—22no—
81C7×C22⋊Q8C2312yes—
82C7×#121—22no—
83C7×#121—21no—
84C7×#121C2312yes—
85C4×C7×Q8—21no—
86C4×C7×Q8—21no—
87C4×C7×Q8—21no—
88C4×C7×Q8—22no—
89C4×C7×Q8—22no—
90C4×C7×Q8—22no—
91C4×C7×Q8—22no—
92C4×C7×Q8D412yes—
93C4×C7×D4—21no—
94C4×C7×D4—21no—
95C4×C7×D4—21no—
96C4×C7×D4—22no—
97C4×C7×D4—22no—
98C4×C7×D4D412yes—
99C7×#117—22no—
100C7×#117—22no—
101C7×#117—21no—
102C7×#117—22no—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4D411yescharacteristic
109C2×C7×C4⋊C4C2311yescharacteristic
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C4⋊C4C2311yescharacteristic
112C2×C42×C7—22no—
113C2×C42×C7—21no—
114C2×C42×C7—21no—
115C2×C42×C7D411yescharacteristic
116C2×C42×C7D411yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×#107—22no—
119C7×#106—22no—
120C7×C4⋊C8—22no—
121C7×#103—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×C22⋊C8C2312yes—
126C7×C22⋊C8C2312yes—
127C4×C7×C8—22no—
128C7×#95—21no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3461C2×C711yescharacteristic
136#3108C2×C712yescomplemented
137#2399C2×C711yescharacteristic, complemented
138#1611C2×C711yescharacteristic, complemented
139#1610C2×C711yescharacteristic
140#1609C2×C712yescomplemented
141#1605C2×C711yescharacteristic, complemented
142#1604C2×C711yescharacteristic
143#1603C2×C712yescomplemented
144#1500C2×C712yescomplemented
145#1461C2×C711yescharacteristic, complemented
146C7×#41—42no—
147C7×#41—42no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—22no—
151C2×C7×Q8—22no—
152C2×C7×Q8—22no—
153C2×C7×Q8—41no—
154C2×C7×Q8—42no—
155C2×C7×Q8C2×D412yes—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4C2×D412yes—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—21no—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C7×Q16—44no—
175C7×SD16—44no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—42no—
196C7×C4⋊C4—42no—
197C7×C4⋊C4—42no—
198C7×C4⋊C4—42no—
199C7×C4⋊C4C2×D412yes—
200C7×C4⋊C4C2×D412yes—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—22no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C42×C7—24no—
211C42×C7—22no—
212C42×C7—22no—
213C42×C7—21no—
214C42×C7—21no—
215C42×C7—21no—
216C42×C7—22no—
217C42×C7—21no—
218C42×C7—41no—
219C42×C7—41no—
220C42×C7C2×D412yes—
221C42×C7C2×D412yes—
222#544C22×C711yescharacteristic
223#542—22no—
224#541—22no—
225#539—21no—
226#539C22×C712yes—
227#537C22×C711yescharacteristic
228#536—21no—
229#533—22no—
230#529—22no—
231#529—21no—
232#529C22×C712yes—
233C4×#41—21no—
234C2×C4×Q8C22×C711yescharacteristic
235C2×C4×Q8C22×C711yescharacteristic
236#483C22×C712yes—
237#480C22×C712yes—
238#479—22no—
239#464—22no—
240#451C22×C712yes—
241#448C22×C712yes—
242C4×Q16—22no—
243C4×SD16—22no—
244#431C22×C712yes—
245#398—21no—
246#398C22×C711yescharacteristic
247#398C22×C711yescharacteristic
248#395—21no—
249#395C22×C711yescharacteristic
250#395C22×C711yescharacteristic
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253#376C22×C711yescharacteristic
254#328C22×C712yes—
255#328C22×C712yes—
256#327C22×C712yes—
257#327C22×C712yes—
258#324C22×C712yes—
259#324C22×C711yescharacteristic
260#324C22×C711yescharacteristic
261C23×C7—22no—
262C23×C7C22×D411yescharacteristic
263C7×Q8—24no—
264C7×Q8—44no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—42no—
268C7×Q8—81no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—42no—
273C7×D4—81no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—44no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—44no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—82no—
305C2×C4×C7#12011yescharacteristic
306C2×C4×C7#12012yes—
307C2×C4×C7#12011yescharacteristic
308C2×C4×C7#13512yes—
309C2×C4×C7C22×D412yes—
310C7×C8—42no—
311C7×C8—42no—
312C2×#41—21no—
313C22×Q8C7×D411yescharacteristic
314C22×Q8C23×C711yescharacteristic
315C2×Q16—22no—
316C2×SD16—22no—
317C4⋊Q8—22no—
318C4⋊Q8—22no—
319#126—22no—
320#126—22no—
321#126—22no—
322#126—22no—
323#125—21no—
324#125—22no—
325#125—22no—
326#125—22no—
327#125C7×D412yes—
328#124—22no—
329#124—21no—
330#124C7×D412yes—
331#123—22no—
332#123—22no—
333#123—22no—
334#123—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8—22no—
341C22⋊Q8C23×C712yes—
342#121—22no—
343#121—21no—
344#121C23×C712yes—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8C7×D412yes—
353C4×D4—21no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4C7×D412yes—
359#117—22no—
360#117—22no—
361#117—21no—
362#117—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4C7×D411yescharacteristic
369C2×C4⋊C4C23×C711yescharacteristic
370C2×C4⋊C4C23×C711yescharacteristic
371C2×C4⋊C4C23×C711yescharacteristic
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C7×D411yescharacteristic
376C2×C42C7×D411yescharacteristic
377C2×C42C23×C711yescharacteristic
378#107—22no—
379#106—22no—
380C4⋊C8—22no—
381#103—22no—
382#103—22no—
383#103—22no—
384#102—22no—
385C22⋊C8C23×C712yes—
386C22⋊C8C23×C712yes—
387C4×C8—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95C23×C711yescharacteristic
394#95C23×C711yescharacteristic
395C22×C7—21no—
396C22×C7—21no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—42no—
400C22×C7C2×#12011yescharacteristic
401C22×C7C2×#13511yescharacteristic
402C22×C7#57511yescharacteristic
403C4×C7—22no—
404C4×C7—22no—
405C4×C7—21no—
406C4×C7—22no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—42no—
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—41no—
415C4×C7—42no—
416C4×C7—81no—
417C4×C7#45612yes—
418#41—42no—
419#41—42no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—41no—
426C2×Q8—42no—
427C2×Q8C2×C7×D412yes—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4C2×C7×D412yes—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4C2×C7×D411yescharacteristic
440C22×C4C2×C7×D411yescharacteristic
441C22×C4C2×C7×D411yescharacteristic
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D411yescharacteristic
444C22×C4C2×C7×D411yescharacteristic
445C22×C4C24×C711yescharacteristic
446Q16—44no—
447SD16—44no—
448C2×C8—22no—
449C2×C8—22no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4C2×C7×D412yes—
472C4⋊C4C2×C7×D412yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—21no—
486C42—22no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—41no—
492C42C2×C7×D412yes—
493C42C2×C7×D412yes—
494C2×C7—21no—
495C2×C7—21no—
496C2×C7—82no—
497C2×C7#300211yescharacteristic
498C2×C7C2×#45611yescharacteristic
499C2×C7#302511yescharacteristic
500C23—22no—
501C23C22×C7×D411yescharacteristic
502Q8—24no—
503Q8—44no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—81no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—81no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—82no—
544C2×C4C7×#12011yescharacteristic
545C2×C4C7×#12012yes—
546C2×C4C7×#12011yescharacteristic
547C2×C4C7×#13512yes—
548C2×C4C22×C7×D412yes—
549C8—42no—
550C8—42no—
551C7#1813111yescharacteristic, direct product factor, Sylow
552C22—21no—
553C22—21no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22C2×C7×#12011yescharacteristic
558C22C2×C7×#13511yescharacteristic
559C22C7×#57511yescharacteristic
560C4—22no—
561C4—22no—
562C4—21no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—81no—
574C4C7×#45612yes—
575C2—21no—
576C2—21no—
577C2—82no—
578C2C7×#300211yescharacteristic
579C2C2×C7×#45611yescharacteristic
580C2C7×#302511yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutator, Frattini, cosocleC22×C4C24×C7
socleC22×C7C2×#120
prime radicalC7×#537C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3108C2semidirect
3. C7×#2399C2semidirect
4. C7×#1611C2semidirect
6. C7×#1609C2semidirect
7. C7×#1605C2semidirect
9. C7×#1603C2semidirect
10. C7×#1500C2semidirect
11. C7×#1461C2semidirect
51. #18131C7direct (N × H)
136. #3108C2×C7semidirect
137. #2399C2×C7semidirect
138. #1611C2×C7semidirect
140. #1609C2×C7semidirect
141. #1605C2×C7semidirect
143. #1603C2×C7semidirect
144. #1500C2×C7semidirect
145. #1461C2×C7semidirect
551. C7#18131direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34612
C7×#31082
C7×#23992
C7×#16112
C7×#16102
C7×#16092
C7×#16052
C7×#16042
C7×#16032
C7×#15002
C7×#14612
#181317