Finite Group Databasebeta

← C7×#18331 #13195547

Subgroups — C7×#18331 #13195547

Order 1792.

998 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3460C211yesmaximal, characteristic
2C7×#3196C212yesmaximal, complemented
3C7×#2564C211yesmaximal, characteristic, complemented
4C7×#1618C212yesmaximal, complemented
5C7×#1617C211yesmaximal, characteristic, complemented
6C7×#1616C211yesmaximal, characteristic, complemented
7C7×#1606C212yesmaximal, complemented
8C7×#1605C211yesmaximal, characteristic, complemented
9C7×#1604C211yesmaximal, characteristic, complemented
10C7×#1504C212yesmaximal, complemented
11C7×#1465C211yesmaximal, characteristic, complemented
12C7×#544C2211yescharacteristic
13C7×#542—22no—
14C7×#540—22no—
15C7×#538—21no—
16C7×#538C2212yes—
17C7×#537C2211yescharacteristic
18C7×#535—21no—
19C7×#532—22no—
20C7×#529—22no—
21C7×#525—21no—
22C7×#525C2212yes—
23C2×C7×C22⋊Q8C2211yescharacteristic
24C4×C7×#41—21no—
25C2×C4×C7×D4C2211yescharacteristic
26C7×#485—22no—
27C7×#484C2212yes—
28C7×#481C2212yes—
29C7×#470—22no—
30C7×#468—22no—
31C7×#452—22no—
32C7×#451C2212yes—
33C7×#448C2212yes—
34C7×#407C2212yes—
35C7×#400C2211yescharacteristic
36C7×#399C2211yescharacteristic
37C7×#395C2211yescharacteristic
38C7×#393C2211yescharacteristic
39C7×#387—21no—
40C7×#386—21no—
41C7×#385—21no—
42C7×#380—21no—
43C2×C7×#95C2211yescharacteristic
44C7×#328C2212yes—
45C7×#328C2212yes—
46C7×#327C2212yes—
47C7×#327C2212yes—
48C7×#324C2211yescharacteristic
49C7×#324C2211yescharacteristic
50C7×#323C2212yes—
51#18331C711yesmaximal, characteristic, direct product factor, Sylow
52C2×C7×#41—21no—
53C2×C7×#41—21no—
54C2×C7×#41D412yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4D411yescharacteristic
57C23×C4×C7—21no—
58C23×C4×C7D411yescharacteristic
59C23×C4×C7C2311yescharacteristic
60C2×C7×Q16—22no—
61C2×C7×SD16—22no—
62C2×C7×M16—22no—
63C7×C4⋊Q8—22no—
64C7×#127—22no—
65C7×#126—22no—
66C7×#126—22no—
67C7×#125—22no—
68C7×#124—22no—
69C7×#124—22no—
70C7×#124—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×#123—22no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8—21no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8—22no—
80C7×C22⋊Q8D412yes—
81C7×C22⋊Q8C2312yes—
82C7×#121—21no—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—21no—
89C7×#121D412yes—
90C7×#121C2312yes—
91C7×#120—22no—
92C7×#120—22no—
93C4×C7×Q8—21no—
94C4×C7×Q8—21no—
95C4×C7×D4—21no—
96C4×C7×D4—21no—
97C4×C7×D4—22no—
98C4×C7×D4—22no—
99C4×C7×D4—22no—
100C4×C7×D4—22no—
101C4×C7×D4—22no—
102C4×C7×D4—22no—
103C7×#117—22no—
104C7×#117—21no—
105C7×#117—21no—
106C7×#117D412yes—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4C2311yescharacteristic
111C2×C7×C4⋊C4C2311yescharacteristic
112C2×C7×C22⋊C4—21no—
113C2×C7×C22⋊C4—21no—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4D411yescharacteristic
116C2×C7×C22⋊C4C2311yescharacteristic
117C2×C42×C7—22no—
118C2×C42×C7D411yescharacteristic
119C7×#107—22no—
120C7×#106—22no—
121C7×#103—22no—
122C7×#103—22no—
123C7×#103—22no—
124C7×#102—22no—
125C7×C22⋊C8—22no—
126C7×C22⋊C8C2312yes—
127C7×C22⋊C8C2312yes—
128C7×#95—21no—
129C7×#95—21no—
130C7×#95—21no—
131C7×#95—21no—
132C7×#95—21no—
133C7×#95C2311yescharacteristic
134C7×#95C2311yescharacteristic
135#3460C2×C711yescharacteristic
136#3196C2×C712yescomplemented
137#2564C2×C711yescharacteristic, complemented
138#1618C2×C712yescomplemented
139#1617C2×C711yescharacteristic, complemented
140#1616C2×C711yescharacteristic, complemented
141#1606C2×C712yescomplemented
142#1605C2×C711yescharacteristic, complemented
143#1604C2×C711yescharacteristic, complemented
144#1504C2×C712yescomplemented
145#1465C2×C711yescharacteristic, complemented
146C24×C7C2×D411yescharacteristic
147C24×C7C2×D411yescharacteristic
148C7×#41—42no—
149C7×#41—42no—
150C7×#41—82no—
151C7×#41—82no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—22no—
155C2×C7×Q8—81no—
156C2×C7×Q8C2×D412yes—
157C2×C7×D4—21no—
158C2×C7×D4—22no—
159C2×C7×D4—22no—
160C2×C7×D4—22no—
161C2×C7×D4—22no—
162C2×C7×D4—21no—
163C2×C7×D4—22no—
164C2×C7×D4—22no—
165C2×C7×D4—22no—
166C2×C7×D4—41no—
167C2×C7×D4C2×D412yes—
168C22×C4×C7—22no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7—21no—
177C22×C4×C7—42no—
178C22×C4×C7—42no—
179C22×C4×C7—81no—
180C22×C4×C7—81no—
181C22×C4×C7—81no—
182C22×C4×C7C2×D411yescharacteristic
183C22×C4×C7C2×D411yescharacteristic
184C22×C4×C7C2×D411yescharacteristic
185C22×C4×C7C2×D411yescharacteristic
186C22×C4×C7C2×D412yes—
187C22×C4×C7C2×D412yes—
188C22×C4×C7C2411yescharacteristic
189C7×Q16—82no—
190C7×SD16—82no—
191C7×M16—82no—
192C2×C7×C8—22no—
193C2×C7×C8—22no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—22no—
201C7×C4⋊C4—22no—
202C7×C4⋊C4—22no—
203C7×C4⋊C4—22no—
204C7×C4⋊C4—42no—
205C7×C4⋊C4—41no—
206C7×C4⋊C4—41no—
207C7×C4⋊C4—81no—
208C7×C4⋊C4C2×D412yes—
209C7×C4⋊C4C2×D412yes—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—22no—
213C7×C22⋊C4—22no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—22no—
216C7×C22⋊C4—22no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—42no—
222C7×C22⋊C4—42no—
223C7×C22⋊C4—81no—
224C42×C7—22no—
225C42×C7—22no—
226C42×C7—21no—
227C42×C7—22no—
228C42×C7—21no—
229#544C22×C711yescharacteristic
230#542—22no—
231#540—22no—
232#538—21no—
233#538C22×C712yes—
234#537C22×C711yescharacteristic
235#535—21no—
236#532—22no—
237#529—22no—
238#525—21no—
239#525C22×C712yes—
240C2×C22⋊Q8C22×C711yescharacteristic
241C4×#41—21no—
242C2×C4×D4C22×C711yescharacteristic
243#485—22no—
244#484C22×C712yes—
245#481C22×C712yes—
246#470—22no—
247#468—22no—
248#452—22no—
249#451C22×C712yes—
250#448C22×C712yes—
251#407C22×C712yes—
252#400C22×C711yescharacteristic
253#399C22×C711yescharacteristic
254#395C22×C711yescharacteristic
255#393C22×C711yescharacteristic
256#387—21no—
257#386—21no—
258#385—21no—
259#380—21no—
260C2×#95C22×C711yescharacteristic
261#328C22×C712yes—
262#328C22×C712yes—
263#327C22×C712yes—
264#327C22×C712yes—
265#324C22×C711yescharacteristic
266#324C22×C711yescharacteristic
267#323C22×C712yes—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—22no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7#12012yes—
279C23×C7C22×D411yescharacteristic
280C7×Q8—42no—
281C7×Q8—42no—
282C7×Q8—42no—
283C7×Q8—42no—
284C7×Q8—81no—
285C7×D4—44no—
286C7×D4—42no—
287C7×D4—42no—
288C7×D4—42no—
289C7×D4—42no—
290C7×D4—82no—
291C7×D4—82no—
292C7×D4—82no—
293C7×D4—81no—
294C7×D4—82no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
302C2×C4×C7—21no—
303C2×C4×C7—42no—
304C2×C4×C7—41no—
305C2×C4×C7—42no—
306C2×C4×C7—41no—
307C2×C4×C7—42no—
308C2×C4×C7—44no—
309C2×C4×C7—44no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—41no—
322C2×C4×C7—41no—
323C2×C4×C7—81no—
324C2×C4×C7—82no—
325C2×C4×C7—84no—
326C2×C4×C7#12011yescharacteristic
327C2×C4×C7#12011yescharacteristic
328C2×C4×C7#13712yes—
329C2×C4×C7C22×D412yes—
330C7×C8—42no—
331C7×C8—42no—
332C2×#41—21no—
333C2×#41—21no—
334C2×#41C7×D412yes—
335C22×Q8C23×C711yescharacteristic
336C22×D4C7×D411yescharacteristic
337C23×C4—21no—
338C23×C4C7×D411yescharacteristic
339C23×C4C23×C711yescharacteristic
340C2×Q16—22no—
341C2×SD16—22no—
342C2×M16—22no—
343C4⋊Q8—22no—
344#127—22no—
345#126—22no—
346#126—22no—
347#125—22no—
348#124—22no—
349#124—22no—
350#124—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8C7×D412yes—
361C22⋊Q8C23×C712yes—
362#121—21no—
363#121—22no—
364#121—22no—
365#121—22no—
366#121—22no—
367#121—22no—
368#121—21no—
369#121C7×D412yes—
370#121C23×C712yes—
371#120—22no—
372#120—22no—
373C4×Q8—21no—
374C4×Q8—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—22no—
383#117—22no—
384#117—21no—
385#117—21no—
386#117C7×D412yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4C23×C711yescharacteristic
391C2×C4⋊C4C23×C711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4C7×D411yescharacteristic
396C2×C22⋊C4C23×C711yescharacteristic
397C2×C42—22no—
398C2×C42C7×D411yescharacteristic
399#107—22no—
400#106—22no—
401#103—22no—
402#103—22no—
403#103—22no—
404#102—22no—
405C22⋊C8—22no—
406C22⋊C8C23×C712yes—
407C22⋊C8C23×C712yes—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95C23×C711yescharacteristic
414#95C23×C711yescharacteristic
415C22×C7—21no—
416C22×C7—21no—
417C22×C7—22no—
418C22×C7—22no—
419C22×C7—22no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—41no—
425C22×C7—42no—
426C22×C7—41no—
427C22×C7—41no—
428C22×C7—42no—
429C22×C7—81no—
430C22×C7C2×#12011yescharacteristic
431C22×C7C2×#13711yescharacteristic
432C22×C7#57511yescharacteristic
433C4×C7—22no—
434C4×C7—22no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—42no—
438C4×C7—42no—
439C4×C7—41no—
440C4×C7—42no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7—81no—
446C24C2×C7×D411yescharacteristic
447C24C2×C7×D411yescharacteristic
448#41—42no—
449#41—42no—
450#41—82no—
451#41—82no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—22no—
455C2×Q8—81no—
456C2×Q8C2×C7×D412yes—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—41no—
467C2×D4C2×C7×D412yes—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4C2×C7×D411yescharacteristic
483C22×C4C2×C7×D411yescharacteristic
484C22×C4C2×C7×D411yescharacteristic
485C22×C4C2×C7×D411yescharacteristic
486C22×C4C2×C7×D412yes—
487C22×C4C2×C7×D412yes—
488C22×C4C24×C711yescharacteristic
489Q16—82no—
490SD16—82no—
491M16—82no—
492C2×C8—22no—
493C2×C8—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—42no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—81no—
508C4⋊C4C2×C7×D412yes—
509C4⋊C4C2×C7×D412yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—81no—
524C42—22no—
525C42—22no—
526C42—21no—
527C42—22no—
528C42—21no—
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—42no—
532C2×C7—82no—
533C2×C7—81no—
534C2×C7#300711yescharacteristic
535C2×C7#301811yescharacteristic
536C2×C7#302611yescharacteristic
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—21no—
542C23—21no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C7×#12012yes—
548C23C22×C7×D411yescharacteristic
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—81no—
554D4—44no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—82no—
561D4—82no—
562D4—81no—
563D4—82no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—82no—
594C2×C4—84no—
595C2×C4C7×#12011yescharacteristic
596C2×C4C7×#12011yescharacteristic
597C2×C4C7×#13712yes—
598C2×C4C22×C7×D412yes—
599C8—42no—
600C8—42no—
601C7#1833111yescharacteristic, direct product factor, Sylow
602C22—21no—
603C22—21no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—41no—
615C22—42no—
616C22—81no—
617C22C2×C7×#12011yescharacteristic
618C22C2×C7×#13711yescharacteristic
619C22C7×#57511yescharacteristic
620C4—22no—
621C4—22no—
622C4—21no—
623C4—21no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—21no—
634C2—21no—
635C2—42no—
636C2—81no—
637C2—82no—
638C2C7×#300711yescharacteristic
639C2C7×#301811yescharacteristic
640C2C7×#302611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3460C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3196C2semidirect
3. C7×#2564C2semidirect
4. C7×#1618C2semidirect
5. C7×#1617C2semidirect
6. C7×#1616C2semidirect
7. C7×#1606C2semidirect
8. C7×#1605C2semidirect
9. C7×#1604C2semidirect
10. C7×#1504C2semidirect
11. C7×#1465C2semidirect
51. #18331C7direct (N × H)
136. #3196C2×C7semidirect
137. #2564C2×C7semidirect
138. #1618C2×C7semidirect
139. #1617C2×C7semidirect
140. #1616C2×C7semidirect
141. #1606C2×C7semidirect
142. #1605C2×C7semidirect
143. #1604C2×C7semidirect
144. #1504C2×C7semidirect
145. #1465C2×C7semidirect
601. C7#18331direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34602
C7×#31962
C7×#25642
C7×#16182
C7×#16172
C7×#16162
C7×#16062
C7×#16052
C7×#16042
C7×#15042
C7×#14652
#183317