Finite Group Databasebeta

← C7×#18279 #13195495

Subgroups — C7×#18279 #13195495

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3563C211yesmaximal, characteristic
2C7×#3113C211yesmaximal, characteristic, complemented
3C7×#3108C211yesmaximal, characteristic
4C7×#2584C211yesmaximal, characteristic, complemented
5C7×#1611C212yesmaximal, complemented
6C7×#1609C212yesmaximal
7C7×#1605C212yesmaximal, complemented
8C7×#1603C212yesmaximal
9C7×#1509C211yesmaximal, characteristic
10C7×#1503C211yesmaximal, characteristic, complemented
11C7×#1470C211yesmaximal, characteristic, complemented
12C7×#564—21no—
13C7×#563—21no—
14C7×#557—21no—
15C7×#554—21no—
16C7×#552—21no—
17C7×#552—22no—
18C7×#552—21no—
19C7×#544—21no—
20C7×#544C2212yes—
21C7×#541—21no—
22C7×#541—22no—
23C7×#539C2212yes—
24C7×#533C2211yescharacteristic
25C7×#529C2211yescharacteristic
26C2×C7×#125C2211yescharacteristic
27C2×C4×C7×Q8C2211yescharacteristic
28C7×#483C2212yes—
29C7×#483C2211yescharacteristic
30C7×#480C2211yescharacteristic
31C7×#479—21no—
32C7×#475—21no—
33C7×#464—21no—
34C7×#462—21no—
35C7×#451C2212yes—
36C7×#451C2211yescharacteristic
37C7×#448C2211yescharacteristic
38C7×#441—22no—
39C4×C7×Q16—21no—
40C4×C7×SD16—21no—
41C7×#432C2211yescharacteristic
42C7×#431C2211yescharacteristic
43C7×#398C2212yes—
44C7×#395C2212yes—
45C7×#389—21no—
46C7×#389—22no—
47C7×#389—21no—
48C4×C7×C4⋊C4C2211yescharacteristic
49C7×#328C2212yes—
50C7×#328C2212yes—
51C7×#328C2212yes—
52C7×#327C2212yes—
53C7×#324C2212yes—
54C7×#324C2212yes—
55#18279C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×Q8C2311yescharacteristic
57C2×C7×Q16—21no—
58C2×C7×Q16—22no—
59C2×C7×SD16—21no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—22no—
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8D411yescharacteristic
64C7×C4⋊Q8D411yescharacteristic
65C7×#126—22no—
66C7×#126—22no—
67C7×#126—22no—
68C7×#126—22no—
69C7×#126—21no—
70C7×#125—22no—
71C7×#125—22no—
72C7×#125—22no—
73C7×#125—22no—
74C7×#125—21no—
75C7×#125—22no—
76C7×#125—22no—
77C7×#125—22no—
78C7×#125—22no—
79C7×#125—21no—
80C7×#125—21no—
81C7×#125—21no—
82C7×#125—21no—
83C7×#125—21no—
84C7×#125D411yescharacteristic
85C7×#124D411yescharacteristic
86C7×#123—22no—
87C7×#123—21no—
88C7×#123—22no—
89C7×C22⋊Q8—21no—
90C7×C22⋊Q8—21no—
91C7×C22⋊Q8—22no—
92C7×C22⋊Q8—22no—
93C7×C22⋊Q8—22no—
94C7×C22⋊Q8—21no—
95C7×C22⋊Q8—22no—
96C7×C22⋊Q8C2311yescharacteristic
97C7×C22⋊Q8C2312yes—
98C7×#121C2311yescharacteristic
99C4×C7×Q8—21no—
100C4×C7×Q8—22no—
101C4×C7×Q8—21no—
102C4×C7×Q8—21no—
103C4×C7×Q8—21no—
104C4×C7×Q8—21no—
105C4×C7×Q8—21no—
106C4×C7×Q8—21no—
107C4×C7×Q8D411yescharacteristic
108C4×C7×Q8D412yes—
109C4×C7×D4—21no—
110C4×C7×D4—22no—
111C4×C7×D4D411yescharacteristic
112C7×#117—22no—
113C7×#117—21no—
114C7×#117—21no—
115C7×#117—22no—
116C7×#117—21no—
117C2×C7×C4⋊C4—21no—
118C2×C7×C4⋊C4—21no—
119C2×C7×C4⋊C4—21no—
120C2×C7×C4⋊C4—22no—
121C2×C7×C4⋊C4—22no—
122C2×C7×C4⋊C4D411yescharacteristic
123C2×C7×C4⋊C4D412yes—
124C2×C7×C4⋊C4D411yescharacteristic
125C2×C7×C4⋊C4C2311yescharacteristic
126C2×C7×C4⋊C4C2312yes—
127C2×C42×C7—21no—
128C2×C42×C7—21no—
129C2×C42×C7C2311yescharacteristic
130C7×#107—21no—
131C7×#106—22no—
132C7×#106—21no—
133C7×C4⋊C8—21no—
134C7×C4⋊C8—21no—
135C7×#103—21no—
136C7×#103—22no—
137C7×#103—22no—
138C7×#103—21no—
139C7×#103—21no—
140C7×#102—21no—
141C7×C22⋊C8C2311yescharacteristic
142C7×C22⋊C8C2312yes—
143C7×C22⋊C8C2311yescharacteristic
144C7×#97—21no—
145C4×C7×C8—21no—
146C7×#95—22no—
147C7×#95—22no—
148C7×#95C2312yes—
149#3563C2×C711yescharacteristic
150#3113C2×C711yescharacteristic, complemented
151#3108C2×C711yescharacteristic
152#2584C2×C711yescharacteristic, complemented
153#1611C2×C712yescomplemented
154#1609C2×C712yes—
155#1605C2×C712yescomplemented
156#1603C2×C712yes—
157#1509C2×C711yescharacteristic
158#1503C2×C711yescharacteristic, complemented
159#1470C2×C711yescharacteristic, complemented
160C2×C7×Q8—22no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—42no—
165C2×C7×Q8C2×D412yes—
166C2×C7×Q8C2×D411yescharacteristic
167C2×C7×D4—21no—
168C2×C7×D4C2×D411yescharacteristic
169C22×C4×C7—22no—
170C22×C4×C7—21no—
171C22×C4×C7—22no—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2×D412yes—
178C22×C4×C7C2411yescharacteristic
179C7×Q16—42no—
180C7×Q16—44no—
181C7×SD16—42no—
182C2×C7×C8—21no—
183C2×C7×C8—21no—
184C2×C7×C8—22no—
185C7×C4⋊C4—24no—
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—21no—
195C7×C4⋊C4—21no—
196C7×C4⋊C4—21no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—22no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—21no—
206C7×C4⋊C4—22no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—22no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—22no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—44no—
214C7×C4⋊C4—41no—
215C7×C4⋊C4—41no—
216C7×C4⋊C4—82no—
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D412yes—
219C7×C4⋊C4C2×D411yescharacteristic
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—22no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C42×C7—22no—
230C42×C7—22no—
231C42×C7—22no—
232C42×C7—22no—
233C42×C7—21no—
234C42×C7—41no—
235C42×C7—41no—
236C42×C7—81no—
237C42×C7C2×D411yescharacteristic
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2×D411yescharacteristic
241#564—21no—
242#563—21no—
243#557—21no—
244#554—21no—
245#552—21no—
246#552—22no—
247#552—21no—
248#544—21no—
249#544C22×C712yes—
250#541—21no—
251#541—22no—
252#539C22×C712yes—
253#533C22×C711yescharacteristic
254#529C22×C711yescharacteristic
255C2×#125C22×C711yescharacteristic
256C2×C4×Q8C22×C711yescharacteristic
257#483C22×C711yescharacteristic
258#483C22×C712yes—
259#480C22×C711yescharacteristic
260#479—21no—
261#475—21no—
262#464—21no—
263#462—21no—
264#451C22×C711yescharacteristic
265#451C22×C712yes—
266#448C22×C711yescharacteristic
267#441—22no—
268C4×Q16—21no—
269C4×SD16—21no—
270#432C22×C711yescharacteristic
271#431C22×C711yescharacteristic
272#398C22×C712yes—
273#395C22×C712yes—
274#389—21no—
275#389—22no—
276#389—21no—
277C4×C4⋊C4C22×C711yescharacteristic
278#328C22×C712yes—
279#328C22×C712yes—
280#328C22×C712yes—
281#327C22×C712yes—
282#324C22×C712yes—
283#324C22×C712yes—
284C23×C7—21no—
285C23×C7C22×D411yescharacteristic
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8—41no—
289C7×Q8—42no—
290C7×Q8—41no—
291C7×Q8—41no—
292C7×Q8—42no—
293C7×Q8—82no—
294C7×D4—41no—
295C7×D4—41no—
296C7×D4—41no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—24no—
301C2×C4×C7—22no—
302C2×C4×C7—22no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—22no—
306C2×C4×C7—21no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—22no—
310C2×C4×C7—22no—
311C2×C4×C7—41no—
312C2×C4×C7—42no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—42no—
316C2×C4×C7—42no—
317C2×C4×C7—42no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—82no—
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7#12011yescharacteristic
325C2×C4×C7#12011yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C2×C4×C7C22×D411yescharacteristic
328C7×C8—41no—
329C7×C8—41no—
330C7×C8—42no—
331C22×Q8C23×C711yescharacteristic
332C2×Q16—21no—
333C2×Q16—22no—
334C2×SD16—21no—
335C4⋊Q8—21no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C4⋊Q8C7×D411yescharacteristic
339C4⋊Q8C7×D411yescharacteristic
340#126—22no—
341#126—22no—
342#126—22no—
343#126—22no—
344#126—21no—
345#125—22no—
346#125—22no—
347#125—22no—
348#125—22no—
349#125—21no—
350#125—22no—
351#125—22no—
352#125—22no—
353#125—22no—
354#125—21no—
355#125—21no—
356#125—21no—
357#125—21no—
358#125—21no—
359#125C7×D411yescharacteristic
360#124C7×D411yescharacteristic
361#123—22no—
362#123—21no—
363#123—22no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—22no—
367C22⋊Q8—22no—
368C22⋊Q8—22no—
369C22⋊Q8—21no—
370C22⋊Q8—22no—
371C22⋊Q8C23×C711yescharacteristic
372C22⋊Q8C23×C712yes—
373#121C23×C711yescharacteristic
374C4×Q8—21no—
375C4×Q8—22no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—21no—
381C4×Q8—21no—
382C4×Q8C7×D411yescharacteristic
383C4×Q8C7×D412yes—
384C4×D4—21no—
385C4×D4—22no—
386C4×D4C7×D411yescharacteristic
387#117—22no—
388#117—21no—
389#117—21no—
390#117—22no—
391#117—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4C7×D411yescharacteristic
398C2×C4⋊C4C7×D412yes—
399C2×C4⋊C4C7×D411yescharacteristic
400C2×C4⋊C4C23×C711yescharacteristic
401C2×C4⋊C4C23×C712yes—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42C23×C711yescharacteristic
405#107—21no—
406#106—21no—
407#106—22no—
408C4⋊C8—21no—
409C4⋊C8—21no—
410#103—21no—
411#103—22no—
412#103—21no—
413#103—21no—
414#103—22no—
415#102—21no—
416C22⋊C8C23×C712yes—
417C22⋊C8C23×C711yescharacteristic
418C22⋊C8C23×C711yescharacteristic
419#97—21no—
420C4×C8—21no—
421#95—22no—
422#95—22no—
423#95C23×C712yes—
424C22×C7—21no—
425C22×C7—21no—
426C22×C7—41no—
427C22×C7—41no—
428C22×C7—41no—
429C22×C7C2×#12011yescharacteristic
430C22×C7#57711yescharacteristic
431C22×C7#57811yescharacteristic
432C4×C7—21no—
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—44no—
438C4×C7—42no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—42no—
442C4×C7—42no—
443C4×C7—42no—
444C4×C7—42no—
445C4×C7—41no—
446C4×C7—81no—
447C4×C7#45612yes—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—42no—
453C2×Q8C2×C7×D412yes—
454C2×Q8C2×C7×D411yescharacteristic
455C2×D4—21no—
456C2×D4C2×C7×D411yescharacteristic
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4C2×C7×D412yes—
463C22×C4C2×C7×D411yescharacteristic
464C22×C4C2×C7×D411yescharacteristic
465C22×C4C2×C7×D412yes—
466C22×C4C24×C711yescharacteristic
467Q16—42no—
468Q16—44no—
469SD16—42no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—44no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—82no—
505C4⋊C4C2×C7×D411yescharacteristic
506C4⋊C4C2×C7×D412yes—
507C4⋊C4C2×C7×D411yescharacteristic
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—41no—
523C42—41no—
524C42—81no—
525C42C2×C7×D411yescharacteristic
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C2×C7—21no—
530C2×C7—21no—
531C2×C7—81no—
532C2×C7#301311yescharacteristic
533C2×C7C2×#45611yescharacteristic
534C2×C7#302511yescharacteristic
535C23—21no—
536C23C22×C7×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—42no—
541Q8—41no—
542Q8—41no—
543Q8—42no—
544Q8—82no—
545D4—41no—
546D4—41no—
547D4—41no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—82no—
573C2×C4C7×#12011yescharacteristic
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C7×#12011yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C2×C4C22×C7×D411yescharacteristic
579C8—41no—
580C8—41no—
581C8—42no—
582C7#1827911yescharacteristic, direct product factor, Sylow
583C22—21no—
584C22—21no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C2×C7×#12011yescharacteristic
589C22C7×#57711yescharacteristic
590C22C7×#57811yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—41no—
605C4—81no—
606C4C7×#45612yes—
607C2—21no—
608C2—21no—
609C2—81no—
610C2C7×#301311yescharacteristic
611C2C2×C7×#45611yescharacteristic
612C2C7×#302511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#121C23
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×#3113C2semidirect
4. C7×#2584C2semidirect
5. C7×#1611C2semidirect
7. C7×#1605C2semidirect
10. C7×#1503C2semidirect
11. C7×#1470C2semidirect
55. #18279C7direct (N × H)
150. #3113C2×C7semidirect
152. #2584C2×C7semidirect
153. #1611C2×C7semidirect
155. #1605C2×C7semidirect
158. #1503C2×C7semidirect
159. #1470C2×C7semidirect
582. C7#18279direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35632
C7×#31132
C7×#31082
C7×#25842
C7×#16112
C7×#16092
C7×#16052
C7×#16032
C7×#15092
C7×#15032
C7×#14702
#182797