Finite Group Databasebeta

← C7×#18267 #13195483

Subgroups — C7×#18267 #13195483

Order 1792.

1056 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3536C211yesmaximal, characteristic
2C7×#3092C211yesmaximal, characteristic, complemented
3C2×C7×#484C211yesmaximal, characteristic, complemented
4C7×#2501C211yesmaximal, characteristic, complemented
5C7×#1618C214yesmaximal, complemented
6C7×#1614C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1501C211yesmaximal, characteristic, complemented
9C7×#1471C211yesmaximal, characteristic, complemented
10C7×#562—22no—
11C7×#554—21no—
12C7×#548—21no—
13C7×#547—21no—
14C7×#546—22no—
15C7×#546—21no—
16C7×#543C2212yes—
17C7×#542—21no—
18C7×#539—21no—
19C7×#538C2212yes—
20C7×#537—21no—
21C7×#537—21no—
22C7×#525C2211yescharacteristic
23C2×C7×#123—22no—
24C2×C7×C22⋊Q8C2211yescharacteristic
25C2×C7×#121C2211yescharacteristic
26C22×C7×C4⋊C4C2211yescharacteristic
27C7×#484—22no—
28C7×#484C2212yes—
29C7×#484C2214yes—
30C7×#483—21no—
31C7×#481C2212yes—
32C7×#480—21no—
33C7×#428—21no—
34C2×C7×#107—21no—
35C7×#417—21no—
36C2×C7×#103—21no—
37C7×#407C2211yescharacteristic
38C2×C7×C22⋊C8C2211yescharacteristic
39C7×#399C2214yes—
40C7×#390—21no—
41C7×#384—21no—
42C4×C7×C22⋊C4C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#323C2214yes—
47#18267C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—22no—
49C2×C7×#41—21no—
50C2×C7×#41D411yescharacteristic
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4D412yes—
53C22×C7×D4D411yescharacteristic
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C2×C7×M16—21no—
57C22×C7×C8—21no—
58C7×C4⋊Q8—21no—
59C7×#126—22no—
60C7×#125—22no—
61C7×#124—21no—
62C7×#124—22no—
63C7×#124—22no—
64C7×#123—22no—
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—21no—
68C7×#123—22no—
69C7×#123—48no—
70C7×C22⋊Q8—21no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—22no—
79C7×C22⋊Q8D411yescharacteristic
80C7×C22⋊Q8D412yes—
81C7×C22⋊Q8C2312yes—
82C7×C22⋊Q8C2311yescharacteristic
83C7×#121—21no—
84C7×#121—22no—
85C7×#121—22no—
86C7×#121—22no—
87C7×#121—22no—
88C7×#121—22no—
89C7×#121—21no—
90C7×#121—21no—
91C7×#121—21no—
92C7×#121—21no—
93C7×#121D411yescharacteristic
94C7×#121C2311yescharacteristic
95C7×#120—21no—
96C7×#120—22no—
97C4×C7×Q8—21no—
98C4×C7×D4—22no—
99C4×C7×D4—21no—
100C4×C7×D4—21no—
101C4×C7×D4—21no—
102C4×C7×D4—22no—
103C7×#117D411yescharacteristic
104C2×C7×C4⋊C4—21no—
105C2×C7×C4⋊C4—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—21no—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4—21no—
110C2×C7×C4⋊C4—21no—
111C2×C7×C4⋊C4—24no—
112C2×C7×C4⋊C4D411yescharacteristic
113C2×C7×C4⋊C4D411yescharacteristic
114C2×C7×C4⋊C4C2312yes—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4—21no—
117C2×C7×C22⋊C4—22no—
118C2×C7×C22⋊C4—24no—
119C2×C7×C22⋊C4C2312yes—
120C2×C42×C7—21no—
121C7×#107—22no—
122C7×#107—24no—
123C7×#106—22no—
124C7×#103—22no—
125C7×#103—24no—
126C7×#102—22no—
127C7×C22⋊C8—21no—
128C7×C22⋊C8—21no—
129C7×C22⋊C8C2312yes—
130C7×C22⋊C8C2312yes—
131C7×#95C2312yes—
132#3536C2×C711yescharacteristic
133#3092C2×C711yescharacteristic, complemented
134C2×#484C2×C711yescharacteristic, complemented
135#2501C2×C711yescharacteristic, complemented
136#1618C2×C714yescomplemented
137#1614C2×C714yescomplemented
138#1508C2×C711yescharacteristic, complemented
139#1501C2×C711yescharacteristic, complemented
140#1471C2×C711yescharacteristic, complemented
141C24×C7C2×D411yescharacteristic
142C24×C7C2×D412yes—
143C7×#41—42no—
144C7×#41—84no—
145C7×#41—82no—
146C2×C7×Q8—22no—
147C2×C7×Q8—21no—
148C2×C7×Q8—42no—
149C2×C7×Q8C2×D411yescharacteristic
150C2×C7×Q8C2×D412yes—
151C2×C7×D4—21no—
152C2×C7×D4—22no—
153C2×C7×D4—21no—
154C2×C7×D4—22no—
155C2×C7×D4—22no—
156C2×C7×D4—21no—
157C2×C7×D4—22no—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—42no—
161C2×C7×D4—44no—
162C2×C7×D4C2×D411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—22no—
168C22×C4×C7—21no—
169C22×C4×C7—21no—
170C22×C4×C7—22no—
171C22×C4×C7—42no—
172C22×C4×C7—44no—
173C22×C4×C7—41no—
174C22×C4×C7—41no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D411yescharacteristic
178C22×C4×C7C2×D412yes—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2411yescharacteristic
182C7×M16—42no—
183C2×C7×C8—22no—
184C2×C7×C8—22no—
185C2×C7×C8—42no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—21no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4—48no—
200C7×C4⋊C4—41no—
201C7×C4⋊C4—41no—
202C7×C4⋊C4C2×D411yescharacteristic
203C7×C4⋊C4C2×D412yes—
204C7×C4⋊C4C2×D411yescharacteristic
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—24no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—22no—
211C7×C22⋊C4—22no—
212C7×C22⋊C4—21no—
213C7×C22⋊C4—21no—
214C7×C22⋊C4—22no—
215C7×C22⋊C4—21no—
216C7×C22⋊C4—21no—
217C7×C22⋊C4—22no—
218C7×C22⋊C4—22no—
219C7×C22⋊C4—21no—
220C7×C22⋊C4—48no—
221C42×C7—22no—
222C42×C7—21no—
223C42×C7—42no—
224#562—22no—
225#554—21no—
226#548—21no—
227#547—21no—
228#546—22no—
229#546—21no—
230#543C22×C712yes—
231#542—21no—
232#539—21no—
233#538C22×C712yes—
234#537—21no—
235#537—21no—
236#525C22×C711yescharacteristic
237C2×#123—22no—
238C2×C22⋊Q8C22×C711yescharacteristic
239C2×#121C22×C711yescharacteristic
240C22×C4⋊C4C22×C711yescharacteristic
241#484—22no—
242#484C22×C714yes—
243#484C22×C712yes—
244#483—21no—
245#481C22×C712yes—
246#480—21no—
247#428—21no—
248C2×#107—21no—
249#417—21no—
250C2×#103—21no—
251#407C22×C711yescharacteristic
252C2×C22⋊C8C22×C711yescharacteristic
253#399C22×C714yes—
254#390—21no—
255#384—21no—
256C4×C22⋊C4C22×C711yescharacteristic
257#328C22×C712yes—
258#328C22×C714yes—
259#327C22×C712yes—
260#323C22×C714yes—
261C23×C7—22no—
262C23×C7—22no—
263C23×C7—22no—
264C23×C7—21no—
265C23×C7—24no—
266C23×C7—41no—
267C23×C7—42no—
268C23×C7—42no—
269C23×C7#12011yescharacteristic
270C23×C7#12011yescharacteristic
271C23×C7C22×D411yescharacteristic
272C7×Q8—42no—
273C7×Q8—41no—
274C7×Q8—42no—
275C7×Q8—82no—
276C7×D4—41no—
277C7×D4—42no—
278C7×D4—44no—
279C7×D4—42no—
280C7×D4—42no—
281C7×D4—81no—
282C7×D4—82no—
283C7×D4—81no—
284C7×D4—82no—
285C7×D4—82no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—24no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—42no—
300C2×C4×C7—42no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—42no—
304C2×C4×C7—44no—
305C2×C4×C7—42no—
306C2×C4×C7—81no—
307C2×C4×C7—82no—
308C2×C4×C7—82no—
309C2×C4×C7—82no—
310C2×C4×C7—82no—
311C2×C4×C7—82no—
312C2×C4×C7#12011yescharacteristic
313C2×C4×C7#12011yescharacteristic
314C2×C4×C7C22×D411yescharacteristic
315C2×C4×C7C22×D411yescharacteristic
316C7×C8—42no—
317C7×C8—42no—
318C2×#41—22no—
319C2×#41—21no—
320C2×#41C7×D411yescharacteristic
321C22×Q8C7×D411yescharacteristic
322C22×D4C7×D412yes—
323C22×D4C7×D411yescharacteristic
324C23×C4—21no—
325C23×C4C23×C711yescharacteristic
326C2×M16—21no—
327C22×C8—21no—
328C4⋊Q8—21no—
329#126—22no—
330#125—22no—
331#124—21no—
332#124—22no—
333#124—22no—
334#123—22no—
335#123—22no—
336#123—22no—
337#123—21no—
338#123—22no—
339#123—48no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—22no—
347C22⋊Q8—22no—
348C22⋊Q8—22no—
349C22⋊Q8C7×D412yes—
350C22⋊Q8C7×D411yescharacteristic
351C22⋊Q8C23×C712yes—
352C22⋊Q8C23×C711yescharacteristic
353#121—21no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—22no—
359#121—21no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121C7×D411yescharacteristic
364#121C23×C711yescharacteristic
365#120—21no—
366#120—22no—
367C4×Q8—21no—
368C4×D4—22no—
369C4×D4—21no—
370C4×D4—21no—
371C4×D4—21no—
372C4×D4—22no—
373#117C7×D411yescharacteristic
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4C7×D411yescharacteristic
383C2×C4⋊C4C7×D411yescharacteristic
384C2×C4⋊C4C23×C712yes—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4C23×C712yes—
390C2×C42—21no—
391#107—24no—
392#107—22no—
393#106—22no—
394#103—24no—
395#103—22no—
396#102—22no—
397C22⋊C8—21no—
398C22⋊C8—21no—
399C22⋊C8C23×C712yes—
400C22⋊C8C23×C712yes—
401#95C23×C712yes—
402C22×C7—21no—
403C22×C7—21no—
404C22×C7—21no—
405C22×C7—21no—
406C22×C7—21no—
407C22×C7—21no—
408C22×C7—21no—
409C22×C7—44no—
410C22×C7—41no—
411C22×C7—42no—
412C22×C7—41no—
413C22×C7—42no—
414C22×C7—41no—
415C22×C7—42no—
416C22×C7—84no—
417C22×C7#45312yes—
418C22×C7C2×#12011yescharacteristic
419C22×C7#57812yes—
420C4×C7—21no—
421C4×C7—21no—
422C4×C7—21no—
423C4×C7—44no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—42no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—82no—
431C4×C7#45812yes—
432C24C2×C7×D411yescharacteristic
433C24C2×C7×D412yes—
434#41—42no—
435#41—84no—
436#41—82no—
437C2×Q8—22no—
438C2×Q8—21no—
439C2×Q8—42no—
440C2×Q8C2×C7×D411yescharacteristic
441C2×Q8C2×C7×D412yes—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4C2×C7×D411yescharacteristic
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4C2×C7×D411yescharacteristic
467C22×C4C2×C7×D411yescharacteristic
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D412yes—
470C22×C4C2×C7×D411yescharacteristic
471C22×C4C2×C7×D411yescharacteristic
472C22×C4C24×C711yescharacteristic
473M16—42no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—42no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—48no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4C2×C7×D411yescharacteristic
494C4⋊C4C2×C7×D412yes—
495C4⋊C4C2×C7×D411yescharacteristic
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—48no—
512C42—22no—
513C42—21no—
514C42—42no—
515C2×C7—22no—
516C2×C7—22no—
517C2×C7—41no—
518C2×C7—81no—
519C2×C7—82no—
520C2×C7#301111yescharacteristic
521C2×C7C2×#45311yescharacteristic
522C2×C7C2×#45811yescharacteristic
523C23—22no—
524C23—22no—
525C23—22no—
526C23—21no—
527C23—24no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23C7×#12011yescharacteristic
532C23C7×#12011yescharacteristic
533C23C22×C7×D411yescharacteristic
534Q8—42no—
535Q8—41no—
536Q8—42no—
537Q8—82no—
538D4—41no—
539D4—42no—
540D4—44no—
541D4—42no—
542D4—42no—
543D4—81no—
544D4—82no—
545D4—81no—
546D4—82no—
547D4—82no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C22×C7×D411yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C8—42no—
579C8—42no—
580C7#1826711yescharacteristic, direct product factor, Sylow
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—84no—
596C22C7×#45312yes—
597C22C2×C7×#12011yescharacteristic
598C22C7×#57812yes—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—82no—
610C4C7×#45812yes—
611C2—22no—
612C2—22no—
613C2—41no—
614C2—82no—
615C2—81no—
616C2C7×#301111yescharacteristic
617C2C2×C7×#45311yescharacteristic
618C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3536C2
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×#3092C2semidirect
3. C2×C7×#484C2semidirect
4. C7×#2501C2semidirect
5. C7×#1618C2semidirect
6. C7×#1614C2semidirect
7. C7×#1508C2semidirect
8. C7×#1501C2semidirect
9. C7×#1471C2semidirect
47. #18267C7direct (N × H)
133. #3092C2×C7semidirect
134. C2×#484C2×C7semidirect
135. #2501C2×C7semidirect
136. #1618C2×C7semidirect
137. #1614C2×C7semidirect
138. #1508C2×C7semidirect
139. #1501C2×C7semidirect
140. #1471C2×C7semidirect
580. C7#18267direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35362
C7×#30922
C2×C7×#4842
C7×#25012
C7×#16182
C7×#16142
C7×#15082
C7×#15012
C7×#14712
#182677