Finite Group Databasebeta

← C7×#18315 #13195531

Subgroups — C7×#18315 #13195531

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3461C211yesmaximal, characteristic
2C7×#3232C212yesmaximal, complemented
3C7×#2634C211yesmaximal, characteristic, complemented
4C7×#1617C211yesmaximal, characteristic, complemented
5C7×#1616C211yesmaximal, characteristic
6C7×#1615C212yesmaximal, complemented
7C7×#1611C211yesmaximal, characteristic, complemented
8C7×#1610C211yesmaximal, characteristic
9C7×#1609C212yesmaximal, complemented
10C7×#1521C212yesmaximal, 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—
23C2×C7×#125C2211yescharacteristic
24C4×C7×#41—21no—
25C2×C4×C7×Q8C2211yescharacteristic
26C7×#499—22no—
27C7×#488—22no—
28C7×#484C2212yes—
29C7×#483C2212yes—
30C7×#481C2212yes—
31C7×#480C2212yes—
32C7×#479—22no—
33C7×#478—22no—
34C7×#431C2212yes—
35C7×#400—21no—
36C7×#400C2211yescharacteristic
37C7×#400C2211yescharacteristic
38C7×#398C2211yescharacteristic
39C7×#398C2211yescharacteristic
40C7×#395—21no—
41C7×#387—21no—
42C7×#387—21no—
43C7×#383C2211yescharacteristic
44C7×#328C2212yes—
45C7×#328C2212yes—
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2212yes—
49C7×#324C2211yescharacteristic
50C7×#324C2211yescharacteristic
51#18315C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C22×C7×Q8D411yescharacteristic
54C7×C4⋊Q8—22no—
55C7×C4⋊Q8—22no—
56C7×#126—22no—
57C7×#126—22no—
58C7×#126—22no—
59C7×#126—22no—
60C7×#125—21no—
61C7×#125—22no—
62C7×#125—22no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#125—22no—
66C7×#125D412yes—
67C7×#124—22no—
68C7×#124—21no—
69C7×#124D412yes—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—22no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8C2312yes—
81C7×#121—22no—
82C7×#121—21no—
83C7×#121C2312yes—
84C4×C7×Q8—21no—
85C4×C7×Q8—21no—
86C4×C7×Q8—21no—
87C4×C7×Q8—22no—
88C4×C7×Q8—22no—
89C4×C7×Q8D412yes—
90C4×C7×D4—21no—
91C4×C7×D4—21no—
92C4×C7×D4—21no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4D412yes—
96C7×#117—22no—
97C7×#117—22no—
98C7×#117—21no—
99C7×#117—22no—
100C2×C7×C4⋊C4—21no—
101C2×C7×C4⋊C4—21no—
102C2×C7×C4⋊C4—21no—
103C2×C7×C4⋊C4—21no—
104C2×C7×C4⋊C4D411yescharacteristic
105C2×C7×C4⋊C4C2311yescharacteristic
106C2×C7×C4⋊C4C2311yescharacteristic
107C2×C7×C4⋊C4C2311yescharacteristic
108C2×C7×C4⋊C4C2311yescharacteristic
109C2×C42×C7—22no—
110C2×C42×C7D411yescharacteristic
111C2×C42×C7D411yescharacteristic
112C2×C42×C7C2311yescharacteristic
113C7×#107—22no—
114C7×#107—22no—
115C7×#106—22no—
116C7×#106—22no—
117C7×C4⋊C8—22no—
118C7×#103—22no—
119C7×#103—22no—
120C7×#102—22no—
121C7×#102—22no—
122C7×C22⋊C8C2312yes—
123C7×C22⋊C8C2312yes—
124C4×C7×C8—22no—
125C7×#95—21no—
126C7×#95—21no—
127C7×#95—21no—
128C7×#95—21no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3461C2×C711yescharacteristic
136#3232C2×C712yescomplemented
137#2634C2×C711yescharacteristic, complemented
138#1617C2×C711yescharacteristic, complemented
139#1616C2×C711yescharacteristic
140#1615C2×C712yescomplemented
141#1611C2×C711yescharacteristic, complemented
142#1610C2×C711yescharacteristic
143#1609C2×C712yescomplemented
144#1521C2×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—41no—
153C2×C7×Q8C2×D412yes—
154C2×C7×D4—21no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4C2×D412yes—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C2×C7×C8—22no—
173C2×C7×C8—22no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
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—42no—
192C7×C4⋊C4—41no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—42no—
196C7×C4⋊C4C2×D412yes—
197C7×C4⋊C4C2×D412yes—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C42×C7—22no—
208C42×C7—22no—
209C42×C7—21no—
210C42×C7—22no—
211C42×C7—21no—
212C42×C7—21no—
213C42×C7—21no—
214C42×C7C2×D412yes—
215C42×C7C2×D412yes—
216#544C22×C711yescharacteristic
217#542—22no—
218#541—22no—
219#539—21no—
220#539C22×C712yes—
221#537C22×C711yescharacteristic
222#536—21no—
223#533—22no—
224#529—22no—
225#529—21no—
226#529C22×C712yes—
227C2×#125C22×C711yescharacteristic
228C4×#41—21no—
229C2×C4×Q8C22×C711yescharacteristic
230#499—22no—
231#488—22no—
232#484C22×C712yes—
233#483C22×C712yes—
234#481C22×C712yes—
235#480C22×C712yes—
236#479—22no—
237#478—22no—
238#431C22×C712yes—
239#400—21no—
240#400C22×C711yescharacteristic
241#400C22×C711yescharacteristic
242#398C22×C711yescharacteristic
243#398C22×C711yescharacteristic
244#395—21no—
245#387—21no—
246#387—21no—
247#383C22×C711yescharacteristic
248#328C22×C712yes—
249#328C22×C712yes—
250#327C22×C712yes—
251#327C22×C712yes—
252#324C22×C712yes—
253#324C22×C711yescharacteristic
254#324C22×C711yescharacteristic
255C23×C7—22no—
256C23×C7C22×D411yescharacteristic
257C7×Q8—44no—
258C7×Q8—42no—
259C7×Q8—42no—
260C7×Q8—81no—
261C7×D4—42no—
262C7×D4—42no—
263C7×D4—42no—
264C7×D4—42no—
265C7×D4—81no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—44no—
281C2×C4×C7—44no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—82no—
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12011yescharacteristic
298C2×C4×C7#12012yes—
299C2×C4×C7#13512yes—
300C2×C4×C7C22×D412yes—
301C7×C8—42no—
302C7×C8—42no—
303C2×#41—21no—
304C22×Q8C7×D411yescharacteristic
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307#126—22no—
308#126—22no—
309#126—22no—
310#126—22no—
311#125—21no—
312#125—22no—
313#125—22no—
314#125—22no—
315#125—22no—
316#125—22no—
317#125C7×D412yes—
318#124—22no—
319#124—21no—
320#124C7×D412yes—
321#123—22no—
322#123—22no—
323#123—22no—
324#123—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—21no—
330C22⋊Q8—22no—
331C22⋊Q8C23×C712yes—
332#121—22no—
333#121—21no—
334#121C23×C712yes—
335C4×Q8—21no—
336C4×Q8—21no—
337C4×Q8—21no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8C7×D412yes—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—22no—
345C4×D4—22no—
346C4×D4C7×D412yes—
347#117—22no—
348#117—22no—
349#117—21no—
350#117—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4C7×D411yescharacteristic
356C2×C4⋊C4C23×C711yescharacteristic
357C2×C4⋊C4C23×C711yescharacteristic
358C2×C4⋊C4C23×C711yescharacteristic
359C2×C4⋊C4C23×C711yescharacteristic
360C2×C42—22no—
361C2×C42C7×D411yescharacteristic
362C2×C42C7×D411yescharacteristic
363C2×C42C23×C711yescharacteristic
364#107—22no—
365#107—22no—
366#106—22no—
367#106—22no—
368C4⋊C8—22no—
369#103—22no—
370#103—22no—
371#102—22no—
372#102—22no—
373C22⋊C8C23×C712yes—
374C22⋊C8C23×C712yes—
375C4×C8—22no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95C23×C711yescharacteristic
385#95C23×C711yescharacteristic
386C22×C7—21no—
387C22×C7—21no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7C2×#12011yescharacteristic
392C22×C7C2×#13511yescharacteristic
393C22×C7#57511yescharacteristic
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—21no—
397C4×C7—21no—
398C4×C7—22no—
399C4×C7—41no—
400C4×C7—42no—
401C4×C7—42no—
402C4×C7—41no—
403C4×C7—41no—
404C4×C7—42no—
405C4×C7—81no—
406C4×C7—81no—
407C4×C7—81no—
408C4×C7—81no—
409#41—42no—
410#41—42no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—41no—
416C2×Q8C2×C7×D412yes—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4C2×C7×D412yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C2×C7×D411yescharacteristic
430C22×C4C2×C7×D411yescharacteristic
431C22×C4C2×C7×D411yescharacteristic
432C22×C4C2×C7×D411yescharacteristic
433C22×C4C2×C7×D411yescharacteristic
434C22×C4C24×C711yescharacteristic
435C2×C8—22no—
436C2×C8—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4C2×C7×D412yes—
460C4⋊C4C2×C7×D412yes—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—22no—
471C42—22no—
472C42—21no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42C2×C7×D412yes—
478C42C2×C7×D412yes—
479C2×C7—21no—
480C2×C7—21no—
481C2×C7—82no—
482C2×C7#300211yescharacteristic
483C2×C7#302011yescharacteristic
484C2×C7#302811yescharacteristic
485C23—22no—
486C23C22×C7×D411yescharacteristic
487Q8—44no—
488Q8—42no—
489Q8—42no—
490Q8—81no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—81no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4C7×#12011yescharacteristic
527C2×C4C7×#12011yescharacteristic
528C2×C4C7×#12012yes—
529C2×C4C7×#13512yes—
530C2×C4C22×C7×D412yes—
531C8—42no—
532C8—42no—
533C7#1831511yescharacteristic, direct product factor, Sylow
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×C7×#12011yescharacteristic
540C22C2×C7×#13511yescharacteristic
541C22C7×#57511yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2C7×#300211yescharacteristic
561C2C7×#302011yescharacteristic
562C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#537C22
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×#3232C2semidirect
3. C7×#2634C2semidirect
4. C7×#1617C2semidirect
6. C7×#1615C2semidirect
7. C7×#1611C2semidirect
9. C7×#1609C2semidirect
10. C7×#1521C2semidirect
11. C7×#1461C2semidirect
51. #18315C7direct (N × H)
136. #3232C2×C7semidirect
137. #2634C2×C7semidirect
138. #1617C2×C7semidirect
140. #1615C2×C7semidirect
141. #1611C2×C7semidirect
143. #1609C2×C7semidirect
144. #1521C2×C7semidirect
145. #1461C2×C7semidirect
533. C7#18315direct (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×#32322
C7×#26342
C7×#16172
C7×#16162
C7×#16152
C7×#16112
C7×#16102
C7×#16092
C7×#15212
C7×#14612
#183157