Finite Group Databasebeta

← C7×#18960 #13196176

Subgroups — C7×#18960 #13196176

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3555C211yesmaximal, characteristic, complemented
2C7×#3217C211yesmaximal, characteristic
3C7×#3208C211yesmaximal, characteristic, complemented
4C7×#3208C211yesmaximal, characteristic, complemented
5C7×#1628C212yesmaximal, complemented
6C7×#1626C212yesmaximal, complemented
7C7×#1623C212yesmaximal, complemented
8C7×#1622C212yesmaximal, complemented
9C7×#1554C211yesmaximal, characteristic, complemented
10C7×#1497C211yesmaximal, characteristic, complemented
11C7×#1482C211yesmaximal, characteristic, complemented
12C7×#563—22no—
13C7×Q82C2211yescharacteristic, complemented
14C7×#557—21no—
15C7×#557C2211yescharacteristic, complemented
16C7×#556—21no—
17C7×#556—21no—
18C7×#556C2211yescharacteristic, complemented
19C7×#554C2211yescharacteristic, complemented
20C7×#552—21no—
21C7×#544—21no—
22C7×#542—21no—
23C7×#541—21no—
24C7×#541—21no—
25C7×#541—22no—
26C7×#533C2211yescharacteristic
27C7×#533C2211yescharacteristic
28C2×C7×#127C2211yescharacteristic
29C7×#529C2211yescharacteristic
30C7×#496—21no—
31C7×#492—21no—
32C7×#492C2212yes—
33C7×#490—21no—
34C7×#486—21no—
35C7×#486C2212yes—
36C7×#460—22no—
37C7×#460C2212yes—
38C7×#460C2212yes—
39C7×#460C2212yes—
40C7×#459C2212yes—
41C7×#450—21no—
42C7×#450—21no—
43C7×#450—22no—
44C7×#450—21no—
45C7×#447—21no—
46C7×#432C2211yescharacteristic
47C7×#431C2211yescharacteristic
48C7×#431C2211yescharacteristic
49C7×#333C2212yescomplemented
50C7×#332C2212yescomplemented
51C7×Q8⋊C8C2212yescomplemented
52C7×Q8⋊C8C2212yescomplemented
53C7×Q8⋊C8C2212yescomplemented
54C7×D4⋊C8C2212yescomplemented
55#18960C711yesmaximal, characteristic, direct product factor, Sylow
56C22×C7×D4—21no—
57C22×C7×D4—21no—
58C22×C7×D4—21no—
59C2×C7×SD16—22no—
60C2×C7×SD16—22no—
61C2×C7×SD16—24no—
62C2×C7×SD16—22no—
63C2×C7×D8—22no—
64C7×C4⋊Q8—22no—
65C7×C4⋊Q8—21no—
66C7×C4⋊Q8—22no—
67C7×C4⋊Q8—21no—
68C7×C4⋊Q8—21no—
69C7×C4⋊Q8D411yescharacteristic
70C7×C4⋊Q8D411yescharacteristic
71C7×C4⋊Q8C2311yescharacteristic
72C7×C4⋊Q8C2311yescharacteristic
73C7×#127—21no—
74C7×#127—21no—
75C7×#127—21no—
76C7×#127C2312yes—
77C7×#126—22no—
78C7×#126—22no—
79C7×#126—22no—
80C7×#126—22no—
81C7×#126—22no—
82C7×#126—22no—
83C7×#126—21no—
84C7×#125—22no—
85C7×#125—22no—
86C7×#125—22no—
87C7×#125—22no—
88C7×#125—21no—
89C7×#125—22no—
90C7×#125—21no—
91C7×#125—21no—
92C7×#125—21no—
93C7×#125D411yescharacteristic
94C7×#124—22no—
95C7×#124—21no—
96C7×#124D411yescharacteristic
97C7×#123—22no—
98C7×#123—21no—
99C7×C22⋊Q8—22no—
100C7×C22⋊Q8—22no—
101C7×C22⋊Q8—21no—
102C7×C22⋊Q8—21no—
103C7×C22⋊Q8—22no—
104C7×C22⋊Q8—21no—
105C7×C22⋊Q8—21no—
106C7×C22⋊Q8D411yescharacteristic
107C7×C22⋊Q8D411yescharacteristic
108C7×C22⋊Q8D411yescharacteristic
109C7×C22⋊Q8D412yes—
110C7×#121D411yescharacteristic
111C4×C7×Q8—22no—
112C4×C7×Q8—21no—
113C4×C7×Q8—21no—
114C4×C7×Q8—21no—
115C4×C7×Q8—21no—
116C4×C7×Q8—21no—
117C4×C7×Q8—21no—
118C4×C7×Q8—21no—
119C4×C7×Q8D412yes—
120C4×C7×Q8C2311yescharacteristic
121C4×C7×Q8C2311yescharacteristic
122C4×C7×Q8C2311yescharacteristic
123C4×C7×D4—21no—
124C4×C7×D4—21no—
125C4×C7×D4C2311yescharacteristic
126C7×#117—22no—
127C7×#117—21no—
128C7×#117—21no—
129C7×#117—21no—
130C7×#117—21no—
131C2×C7×C4⋊C4—21no—
132C2×C42×C7C2311yescharacteristic
133C7×C4⋊C8—21no—
134C7×C4⋊C8C2312yes—
135C7×C4⋊C8C2312yes—
136C7×#102—22no—
137C7×#102—22no—
138C7×#102—24no—
139C7×#102—22no—
140C7×#102—22no—
141C7×C22⋊C8—21no—
142C7×C22⋊C8—21no—
143C7×C22⋊C8—21no—
144C7×#97—21no—
145C4×C7×C8—21no—
146C4×C7×C8C2312yes—
147#3555C2×C711yescharacteristic, complemented
148#3217C2×C711yescharacteristic
149#3208C2×C711yescharacteristic, complemented
150#3208C2×C711yescharacteristic, complemented
151#1628C2×C712yescomplemented
152#1626C2×C712yescomplemented
153#1623C2×C712yescomplemented
154#1622C2×C712yescomplemented
155#1554C2×C711yescharacteristic, complemented
156#1497C2×C711yescharacteristic, complemented
157#1482C2×C711yescharacteristic, complemented
158C24×C7—41no—
159C2×C7×Q8—22no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8C2×D411yescharacteristic
163C2×C7×Q8C2×D411yescharacteristic
164C2×C7×Q8C2×D411yescharacteristic
165C2×C7×Q8C2×D412yes—
166C2×C7×D4—21no—
167C2×C7×D4—22no—
168C2×C7×D4—22no—
169C2×C7×D4—22no—
170C2×C7×D4—41no—
171C2×C7×D4—41no—
172C2×C7×D4—41no—
173C2×C7×D4—82no—
174C2×C7×D4—82no—
175C2×C7×D4—82no—
176C2×C7×D4C2×D411yescharacteristic
177C22×C4×C7—22no—
178C22×C4×C7—21no—
179C22×C4×C7C2×D411yescharacteristic
180C22×C4×C7C2×D411yescharacteristic
181C22×C4×C7C2×D411yescharacteristic
182C7×SD16—82no—
183C7×SD16—82no—
184C7×SD16—82no—
185C7×SD16—84no—
186C7×D8—82no—
187C2×C7×C8—22no—
188C2×C7×C8—22no—
189C2×C7×C8—22no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—22no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4—21no—
203C7×C4⋊C4—21no—
204C7×C4⋊C4—21no—
205C7×C4⋊C4—22no—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—22no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4—21no—
211C7×C4⋊C4—21no—
212C7×C4⋊C4—21no—
213C7×C4⋊C4—21no—
214C7×C4⋊C4C2×D411yescharacteristic
215C7×C4⋊C4C2×D411yescharacteristic
216C7×C4⋊C4C2×D411yescharacteristic
217C7×C4⋊C4C2×D411yescharacteristic
218C7×C4⋊C4C2×D412yes—
219C7×C22⋊C4—22no—
220C7×C22⋊C4—22no—
221C7×C22⋊C4—22no—
222C7×C22⋊C4—21no—
223C7×C22⋊C4—22no—
224C7×C22⋊C4—21no—
225C7×C22⋊C4—21no—
226C7×C22⋊C4—21no—
227C7×C22⋊C4—21no—
228C7×C22⋊C4—21no—
229C7×C22⋊C4—21no—
230C42×C7—22no—
231C42×C7—22no—
232C42×C7—22no—
233C42×C7—22no—
234C42×C7—21no—
235C42×C7—21no—
236C42×C7—21no—
237C42×C7C2×D411yescharacteristic
238C42×C7C2×D411yescharacteristic
239C42×C7C2×D411yescharacteristic
240C42×C7C2411yescharacteristic
241#563—22no—
242Q82C22×C711yescharacteristic, complemented
243#557—21no—
244#557C22×C711yescharacteristic, complemented
245#556—21no—
246#556—21no—
247#556C22×C711yescharacteristic, complemented
248#554C22×C711yescharacteristic, complemented
249#552—21no—
250#544—21no—
251#542—21no—
252#541—21no—
253#541—21no—
254#541—22no—
255#533C22×C711yescharacteristic
256#533C22×C711yescharacteristic
257C2×#127C22×C711yescharacteristic
258#529C22×C711yescharacteristic
259#496—21no—
260#492—21no—
261#492C22×C712yes—
262#490—21no—
263#486—21no—
264#486C22×C712yes—
265#460—22no—
266#460C22×C712yes—
267#460C22×C712yes—
268#460C22×C712yes—
269#459C22×C712yes—
270#450—22no—
271#450—21no—
272#450—21no—
273#450—21no—
274#447—21no—
275#432C22×C711yescharacteristic
276#431C22×C711yescharacteristic
277#431C22×C711yescharacteristic
278#333C22×C712yescomplemented
279#332C22×C712yescomplemented
280Q8⋊C8C22×C712yescomplemented
281Q8⋊C8C22×C712yescomplemented
282Q8⋊C8C22×C712yescomplemented
283D4⋊C8C22×C712yescomplemented
284C23×C7—21no—
285C23×C7—42no—
286C23×C7—82no—
287C23×C7—82no—
288C23×C7—82no—
289C23×C7#12011yescharacteristic
290C7×Q8—22no—
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—42no—
294C7×Q8—42no—
295C7×Q8—81no—
296C7×Q8—82no—
297C7×D4—22no—
298C7×D4—44no—
299C7×D4—42no—
300C7×D4—42no—
301C7×D4—42no—
302C7×D4—42no—
303C7×D4—82no—
304C7×D4—82no—
305C7×D4—82no—
306C7×D4—81no—
307C2×C4×C7—22no—
308C2×C4×C7—22no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—22no—
313C2×C4×C7—21no—
314C2×C4×C7—21no—
315C2×C4×C7—21no—
316C2×C4×C7—22no—
317C2×C4×C7—21no—
318C2×C4×C7—41no—
319C2×C4×C7—42no—
320C2×C4×C7—82no—
321C2×C4×C7#12011yescharacteristic
322C2×C4×C7#12011yescharacteristic
323C2×C4×C7#12011yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×D411yescharacteristic
326C2×C4×C7C22×D411yescharacteristic
327C7×C8—42no—
328C7×C8—42no—
329C7×C8—42no—
330C22×D4—21no—
331C22×D4—21no—
332C22×D4—21no—
333C2×SD16—24no—
334C2×SD16—22no—
335C2×SD16—22no—
336C2×SD16—22no—
337C2×D8—22no—
338C4⋊Q8—22no—
339C4⋊Q8—21no—
340C4⋊Q8—22no—
341C4⋊Q8—21no—
342C4⋊Q8—21no—
343C4⋊Q8C7×D411yescharacteristic
344C4⋊Q8C7×D411yescharacteristic
345C4⋊Q8C23×C711yescharacteristic
346C4⋊Q8C23×C711yescharacteristic
347#127—21no—
348#127—21no—
349#127—21no—
350#127C23×C712yes—
351#126—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#126—22no—
357#126—21no—
358#125—22no—
359#125—22no—
360#125—22no—
361#125—22no—
362#125—21no—
363#125—22no—
364#125—21no—
365#125—21no—
366#125—21no—
367#125C7×D411yescharacteristic
368#124—22no—
369#124—21no—
370#124C7×D411yescharacteristic
371#123—22no—
372#123—21no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8C7×D411yescharacteristic
381C22⋊Q8C7×D411yescharacteristic
382C22⋊Q8C7×D411yescharacteristic
383C22⋊Q8C7×D412yes—
384#121C7×D411yescharacteristic
385C4×Q8—22no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—21no—
392C4×Q8—21no—
393C4×Q8C7×D412yes—
394C4×Q8C23×C711yescharacteristic
395C4×Q8C23×C711yescharacteristic
396C4×Q8C23×C711yescharacteristic
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C23×C711yescharacteristic
400#117—22no—
401#117—21no—
402#117—21no—
403#117—21no—
404#117—21no—
405C2×C4⋊C4—21no—
406C2×C42C23×C711yescharacteristic
407C4⋊C8—21no—
408C4⋊C8C23×C712yes—
409C4⋊C8C23×C712yes—
410#102—22no—
411#102—22no—
412#102—24no—
413#102—22no—
414#102—22no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418#97—21no—
419C4×C8—21no—
420C4×C8C23×C712yes—
421C22×C7—21no—
422C22×C7—21no—
423C22×C7—21no—
424C22×C7—41no—
425C22×C7—41no—
426C22×C7—41no—
427C22×C7—82no—
428C22×C7—82no—
429C22×C7—82no—
430C22×C7—164no—
431C22×C7C2×#12011yescharacteristic
432C4×C7—21no—
433C4×C7—21no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—41no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—42no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—41no—
444C4×C7—41no—
445C4×C7#57712yes—
446C4×C7#57712yes—
447C24—41no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×C7×D411yescharacteristic
452C2×Q8C2×C7×D411yescharacteristic
453C2×Q8C2×C7×D411yescharacteristic
454C2×Q8C2×C7×D412yes—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—82no—
463C2×D4—82no—
464C2×D4—82no—
465C2×D4C2×C7×D411yescharacteristic
466C22×C4—22no—
467C22×C4—21no—
468C22×C4C2×C7×D411yescharacteristic
469C22×C4C2×C7×D411yescharacteristic
470C22×C4C2×C7×D411yescharacteristic
471SD16—82no—
472SD16—82no—
473SD16—82no—
474SD16—84no—
475D8—82no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4C2×C7×D411yescharacteristic
504C4⋊C4C2×C7×D411yescharacteristic
505C4⋊C4C2×C7×D411yescharacteristic
506C4⋊C4C2×C7×D411yescharacteristic
507C4⋊C4C2×C7×D412yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42C2×C7×D411yescharacteristic
527C42C2×C7×D411yescharacteristic
528C42C2×C7×D411yescharacteristic
529C42C24×C711yescharacteristic
530C2×C7—41no—
531C2×C7—81no—
532C2×C7—162no—
533C2×C7#301011yescharacteristic
534C2×C7#301011yescharacteristic
535C2×C7#302111yescharacteristic
536C23—21no—
537C23—42no—
538C23—82no—
539C23—82no—
540C23—82no—
541C23C7×#12011yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—22no—
545Q8—42no—
546Q8—42no—
547Q8—81no—
548Q8—82no—
549D4—22no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—82no—
556D4—82no—
557D4—82no—
558D4—81no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4C7×#12011yescharacteristic
574C2×C4C7×#12011yescharacteristic
575C2×C4C7×#12011yescharacteristic
576C2×C4C22×C7×D411yescharacteristic
577C2×C4C22×C7×D411yescharacteristic
578C2×C4C22×C7×D411yescharacteristic
579C8—42no—
580C8—42no—
581C8—42no—
582C7#1896011yescharacteristic, direct product factor, Sylow
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—164no—
593C22C2×C7×#12011yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4C7×#57712yes—
608C4C7×#57712yes—
609C2—41no—
610C2—81no—
611C2—162no—
612C2C7×#301011yescharacteristic
613C2C7×#301011yescharacteristic
614C2C7×#302111yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3555C2semidirect
3. C7×#3208C2semidirect
4. C7×#3208C2semidirect
5. C7×#1628C2semidirect
6. C7×#1626C2semidirect
7. C7×#1623C2semidirect
8. C7×#1622C2semidirect
9. C7×#1554C2semidirect
10. C7×#1497C2semidirect
11. C7×#1482C2semidirect
13. C7×Q82C22semidirect
15. C7×#557C22semidirect
18. C7×#556C22semidirect
19. C7×#554C22semidirect
49. C7×#333C22semidirect
50. C7×#332C22semidirect
51. C7×Q8⋊C8C22semidirect
52. C7×Q8⋊C8C22semidirect
53. C7×Q8⋊C8C22semidirect
54. C7×D4⋊C8C22semidirect
55. #18960C7direct (N × H)
147. #3555C2×C7semidirect
149. #3208C2×C7semidirect
150. #3208C2×C7semidirect
151. #1628C2×C7semidirect
152. #1626C2×C7semidirect
153. #1623C2×C7semidirect
154. #1622C2×C7semidirect
155. #1554C2×C7semidirect
156. #1497C2×C7semidirect
157. #1482C2×C7semidirect
242. Q82C22×C7semidirect
244. #557C22×C7semidirect
247. #556C22×C7semidirect
248. #554C22×C7semidirect
278. #333C22×C7semidirect
279. #332C22×C7semidirect
280. Q8⋊C8C22×C7semidirect
281. Q8⋊C8C22×C7semidirect
282. Q8⋊C8C22×C7semidirect
283. D4⋊C8C22×C7semidirect
582. C7#18960direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35552
C7×#32172
C7×#32082
C7×#16282
C7×#16262
C7×#16232
C7×#16222
C7×#15542
C7×#14972
C7×#14822
#189607