Finite Group Databasebeta

← C7×#18313 #13195529

Subgroups — C7×#18313 #13195529

Order 1792.

910 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3455C211yesmaximal, characteristic
2C7×#3230C211yesmaximal, characteristic, complemented
3C7×#3227C211yesmaximal, characteristic, complemented
4C7×#2628C211yesmaximal, characteristic, complemented
5C7×#1618C214yesmaximal, complemented
6C7×#1615C214yesmaximal, complemented
7C7×#1518C211yesmaximal, characteristic, complemented
8C7×#1517C211yesmaximal, characteristic, complemented
9C7×#1458C211yesmaximal, characteristic, complemented
10C7×#542—21no—
11C7×#542—21no—
12C7×#541—21no—
13C7×#540—21no—
14C7×#539—21no—
15C7×#539C2212yes—
16C7×#538—21no—
17C7×#538C2212yes—
18C7×#533C2211yescharacteristic
19C7×#532C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C7×#529—22no—
22C7×#529—22no—
23C7×#525—21no—
24C4×C7×#41—21no—
25C2×C7×#117C2211yescharacteristic
26C7×#500—21no—
27C7×#498—21no—
28C7×#487—21no—
29C7×#486—21no—
30C7×#484C2214yes—
31C7×#481C2214yes—
32C7×#477—22no—
33C7×#476—22no—
34C7×#431C2211yescharacteristic
35C7×#431C2211yescharacteristic
36C7×#400C2212yes—
37C7×#399C2212yes—
38C7×#395—21no—
39C7×#393—21no—
40C7×#389—21no—
41C7×#382C2211yescharacteristic
42C7×#380—21no—
43C7×#328C2214yes—
44C7×#327C2214yes—
45C7×#324C2212yes—
46C7×#323C2212yes—
47#18313C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C23×C4×C7D411yescharacteristic
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8D411yescharacteristic
56C7×C4⋊Q8D411yescharacteristic
57C7×C4⋊Q8D411yescharacteristic
58C7×#127D411yescharacteristic
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#125—21no—
66C7×#124—22no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8C2312yes—
80C7×#121—21no—
81C7×#121—21no—
82C7×#121—22no—
83C7×#121—22no—
84C7×#121—21no—
85C7×#121C2312yes—
86C7×#120—22no—
87C4×C7×Q8—21no—
88C4×C7×Q8—21no—
89C4×C7×Q8—21no—
90C4×C7×Q8D412yes—
91C4×C7×D4—21no—
92C4×C7×D4—21no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—22no—
97C4×C7×D4—21no—
98C4×C7×D4D412yes—
99C7×#117—22no—
100C7×#117—21no—
101C7×#117—21no—
102C7×#117—21no—
103C7×#117—21no—
104C7×#117—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—22no—
108C2×C7×C4⋊C4D411yescharacteristic
109C2×C7×C4⋊C4C2312yes—
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C22⋊C4—22no—
112C2×C7×C22⋊C4D411yescharacteristic
113C2×C7×C22⋊C4C2311yescharacteristic
114C2×C42×C7—22no—
115C2×C42×C7D411yescharacteristic
116C2×C42×C7C2311yescharacteristic
117C7×#107—24no—
118C7×#106—24no—
119C7×C4⋊C8—21no—
120C7×C4⋊C8—21no—
121C7×#103—24no—
122C7×#102—24no—
123C7×C22⋊C8C2312yes—
124C7×C22⋊C8C2312yes—
125C4×C7×C8—21no—
126C4×C7×C8—21no—
127C7×#95—22no—
128C7×#95—21no—
129C7×#95—22no—
130C7×#95C2312yes—
131#3455C2×C711yescharacteristic
132#3230C2×C711yescharacteristic, complemented
133#3227C2×C711yescharacteristic, complemented
134#2628C2×C711yescharacteristic, complemented
135#1618C2×C714yescomplemented
136#1615C2×C714yescomplemented
137#1518C2×C711yescharacteristic, complemented
138#1517C2×C711yescharacteristic, complemented
139#1458C2×C711yescharacteristic, complemented
140C24×C7C2×D411yescharacteristic
141C7×#41—42no—
142C7×#41—42no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—41no—
146C2×C7×Q8—41no—
147C2×C7×Q8—82no—
148C2×C7×Q8C2×D412yes—
149C2×C7×D4—22no—
150C2×C7×D4—22no—
151C2×C7×D4—22no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C2×D412yes—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—22no—
161C22×C4×C7—22no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C2×C7×C8—22no—
171C2×C7×C8—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
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—41no—
189C7×C4⋊C4—41no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—82no—
192C7×C4⋊C4C2×D412yes—
193C7×C4⋊C4C2×D412yes—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—21no—
202C7×C22⋊C4—21no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—21no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—82no—
208C42×C7—22no—
209C42×C7—22no—
210C42×C7—21no—
211C42×C7—22no—
212C42×C7—21no—
213C42×C7C2×D411yescharacteristic
214C42×C7C2×D411yescharacteristic
215C42×C7C2×D411yescharacteristic
216C42×C7C2×D411yescharacteristic
217#542—21no—
218#542—21no—
219#541—21no—
220#540—21no—
221#539—21no—
222#539C22×C712yes—
223#538—21no—
224#538C22×C712yes—
225#533C22×C711yescharacteristic
226#532C22×C711yescharacteristic
227C2×C4⋊Q8C22×C711yescharacteristic
228#529—22no—
229#529—22no—
230#525—21no—
231C4×#41—21no—
232C2×#117C22×C711yescharacteristic
233#500—21no—
234#498—21no—
235#487—21no—
236#486—21no—
237#484C22×C714yes—
238#481C22×C714yes—
239#477—22no—
240#476—22no—
241#431C22×C711yescharacteristic
242#431C22×C711yescharacteristic
243#400C22×C712yes—
244#399C22×C712yes—
245#395—21no—
246#393—21no—
247#389—21no—
248#382C22×C711yescharacteristic
249#380—21no—
250#328C22×C714yes—
251#327C22×C714yes—
252#324C22×C712yes—
253#323C22×C712yes—
254C23×C7—21no—
255C23×C7—22no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7C22×D411yescharacteristic
261C7×Q8—42no—
262C7×Q8—42no—
263C7×Q8—81no—
264C7×Q8—82no—
265C7×Q8—82no—
266C7×D4—42no—
267C7×D4—42no—
268C7×D4—42no—
269C7×D4—42no—
270C7×D4—81no—
271C7×D4—82no—
272C7×D4—82no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—42no—
284C2×C4×C7—42no—
285C2×C4×C7—44no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—82no—
295C2×C4×C7—82no—
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12011yescharacteristic
298C2×C4×C7#12011yescharacteristic
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7#13512yes—
301C2×C4×C7#13712yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C7×C8—42no—
305C7×C8—42no—
306C2×#41—21no—
307C22×Q8—21no—
308C23×C4C7×D411yescharacteristic
309C4⋊Q8—21no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8C7×D411yescharacteristic
314C4⋊Q8C7×D411yescharacteristic
315C4⋊Q8C7×D411yescharacteristic
316#127C7×D411yescharacteristic
317#126—22no—
318#126—22no—
319#126—22no—
320#126—22no—
321#125—22no—
322#125—22no—
323#125—21no—
324#124—22no—
325#124—22no—
326#124—21no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—21no—
337C22⋊Q8C23×C712yes—
338#121—21no—
339#121—21no—
340#121—22no—
341#121—22no—
342#121—21no—
343#121C23×C712yes—
344#120—22no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—21no—
348C4×Q8C7×D412yes—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4C7×D412yes—
357#117—22no—
358#117—21no—
359#117—21no—
360#117—21no—
361#117—21no—
362#117—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C7×D411yescharacteristic
367C2×C4⋊C4C23×C712yes—
368C2×C4⋊C4C23×C711yescharacteristic
369C2×C22⋊C4—22no—
370C2×C22⋊C4C7×D411yescharacteristic
371C2×C22⋊C4C23×C711yescharacteristic
372C2×C42—22no—
373C2×C42C7×D411yescharacteristic
374C2×C42C23×C711yescharacteristic
375#107—24no—
376#106—24no—
377C4⋊C8—21no—
378C4⋊C8—21no—
379#103—24no—
380#102—24no—
381C22⋊C8C23×C712yes—
382C22⋊C8C23×C712yes—
383C4×C8—21no—
384C4×C8—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95C23×C712yes—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—42no—
392C22×C7—41no—
393C22×C7—42no—
394C22×C7—42no—
395C22×C7—41no—
396C22×C7—42no—
397C22×C7—41no—
398C22×C7—81no—
399C22×C7C2×#12011yescharacteristic
400C22×C7C2×#13511yescharacteristic
401C22×C7C2×#13711yescharacteristic
402C4×C7—21no—
403C4×C7—21no—
404C4×C7—21no—
405C4×C7—21no—
406C4×C7—41no—
407C4×C7—42no—
408C4×C7—41no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7—42no—
412C4×C7—82no—
413C4×C7—82no—
414C4×C7#45212yes—
415C4×C7#45212yes—
416C24C2×C7×D411yescharacteristic
417#41—42no—
418#41—42no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—82no—
424C2×Q8C2×C7×D412yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4C2×C7×D412yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4C2×C7×D411yescharacteristic
441C22×C4C2×C7×D411yescharacteristic
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D411yescharacteristic
444C22×C4C2×C7×D411yescharacteristic
445C22×C4C24×C711yescharacteristic
446C2×C8—22no—
447C2×C8—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—82no—
468C4⋊C4C2×C7×D412yes—
469C4⋊C4C2×C7×D412yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—22no—
483C22⋊C4—82no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—22no—
488C42—21no—
489C42C2×C7×D411yescharacteristic
490C42C2×C7×D411yescharacteristic
491C42C2×C7×D411yescharacteristic
492C42C2×C7×D411yescharacteristic
493C2×C7—21no—
494C2×C7—21no—
495C2×C7—82no—
496C2×C7—81no—
497C2×C7C2×#45211yescharacteristic
498C2×C7#302111yescharacteristic
499C2×C7#302811yescharacteristic
500C23—21no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—41no—
505C23—41no—
506C23C22×C7×D411yescharacteristic
507Q8—42no—
508Q8—42no—
509Q8—81no—
510Q8—82no—
511Q8—82no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—81no—
517D4—82no—
518D4—82no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4C7×#12011yescharacteristic
543C2×C4C7×#12011yescharacteristic
544C2×C4C7×#12011yescharacteristic
545C2×C4C7×#12011yescharacteristic
546C2×C4C7×#13512yes—
547C2×C4C7×#13712yes—
548C2×C4C22×C7×D411yescharacteristic
549C2×C4C22×C7×D411yescharacteristic
550C8—42no—
551C8—42no—
552C7#1831311yescharacteristic, direct product factor, Sylow
553C22—21no—
554C22—21no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—41no—
562C22—81no—
563C22C2×C7×#12011yescharacteristic
564C22C2×C7×#13511yescharacteristic
565C22C2×C7×#13711yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—41no—
571C4—42no—
572C4—41no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4C7×#45212yes—
579C4C7×#45212yes—
580C2—21no—
581C2—21no—
582C2—81no—
583C2—82no—
584C2C2×C7×#45211yescharacteristic
585C2C7×#302111yescharacteristic
586C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3455C2
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×#3230C2semidirect
3. C7×#3227C2semidirect
4. C7×#2628C2semidirect
5. C7×#1618C2semidirect
6. C7×#1615C2semidirect
7. C7×#1518C2semidirect
8. C7×#1517C2semidirect
9. C7×#1458C2semidirect
47. #18313C7direct (N × H)
132. #3230C2×C7semidirect
133. #3227C2×C7semidirect
134. #2628C2×C7semidirect
135. #1618C2×C7semidirect
136. #1615C2×C7semidirect
137. #1518C2×C7semidirect
138. #1517C2×C7semidirect
139. #1458C2×C7semidirect
552. C7#18313direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34552
C7×#32302
C7×#32272
C7×#26282
C7×#16182
C7×#16152
C7×#15182
C7×#15172
C7×#14582
#183137