Finite Group Databasebeta

← C7×#18269 #13195485

Subgroups — C7×#18269 #13195485

Order 1792.

924 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3538C211yesmaximal, characteristic
2C7×#3092C211yesmaximal, characteristic
3C2×C7×#484C211yesmaximal, characteristic, complemented
4C7×#2503C211yesmaximal, characteristic, complemented
5C7×#1617C214yesmaximal, complemented
6C7×#1615C214yesmaximal, complemented
7C7×#1508C211yesmaximal, characteristic, complemented
8C7×#1501C211yesmaximal, characteristic
9C7×#1472C211yesmaximal, characteristic, complemented
10C7×#563—22no—
11C7×#552—21no—
12C7×#552—21no—
13C7×#551—22no—
14C7×D4×Q8—21no—
15C7×D4×Q8—21no—
16C7×#544—21no—
17C7×#544C2212yes—
18C7×#541—21no—
19C7×#539—21no—
20C7×#539C2212yes—
21C7×#536—21no—
22C7×#525C2211yescharacteristic
23C2×C7×C22⋊Q8C2211yescharacteristic
24C2×C7×C22⋊Q8C2211yescharacteristic
25C22×C7×C4⋊C4C2211yescharacteristic
26C7×#484—22no—
27C7×#484C2214yes—
28C7×#484C2212yes—
29C7×#483—21no—
30C7×#481C2212yes—
31C7×#480—21no—
32C7×#428—21no—
33C2×C7×#107—21no—
34C7×#417—21no—
35C2×C7×#103—21no—
36C7×#407C2211yescharacteristic
37C2×C7×C22⋊C8C2211yescharacteristic
38C7×#400C2214yes—
39C7×#391—21no—
40C7×#382—21no—
41C7×#381—22no—
42C4×C7×C22⋊C4C2211yescharacteristic
43C7×#328C2212yes—
44C7×#328C2214yes—
45C7×#327C2212yes—
46C7×#324C2214yes—
47#18269C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41D411yescharacteristic
49C22×C7×Q8—21no—
50C22×C7×Q8D411yescharacteristic
51C22×C7×Q8D411yescharacteristic
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C2×C7×M16—21no—
55C22×C7×C8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8—22no—
58C7×C4⋊Q8—22no—
59C7×C4⋊Q8—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#126—22no—
64C7×#125—21no—
65C7×#125—22no—
66C7×#125—22no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#124—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—21no—
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⋊Q8—21no—
81C7×C22⋊Q8—21no—
82C7×C22⋊Q8—21no—
83C7×C22⋊Q8—21no—
84C7×C22⋊Q8—21no—
85C7×C22⋊Q8—21no—
86C7×C22⋊Q8—22no—
87C7×C22⋊Q8D411yescharacteristic
88C7×C22⋊Q8D412yes—
89C7×C22⋊Q8C2312yes—
90C7×C22⋊Q8C2311yescharacteristic
91C7×#121D411yescharacteristic
92C7×#121C2311yescharacteristic
93C7×#120—21no—
94C4×C7×Q8—21no—
95C4×C7×Q8—21no—
96C4×C7×D4—22no—
97C4×C7×D4—21no—
98C4×C7×D4—21no—
99C4×C7×D4—22no—
100C7×#117—21no—
101C7×#117—22no—
102C7×#117—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⋊C4D411yescharacteristic
111C2×C7×C4⋊C4D411yescharacteristic
112C2×C7×C4⋊C4C2312yes—
113C2×C7×C4⋊C4C2312yes—
114C2×C7×C22⋊C4—21no—
115C2×C7×C22⋊C4—21no—
116C2×C7×C22⋊C4D412yes—
117C2×C42×C7—21no—
118C7×#107—22no—
119C7×#107—24no—
120C7×#106—22no—
121C7×#103—22no—
122C7×#103—24no—
123C7×#102—22no—
124C7×C22⋊C8—21no—
125C7×C22⋊C8—21no—
126C7×C22⋊C8C2312yes—
127C7×C22⋊C8C2312yes—
128C7×#95—24no—
129C7×#95—24no—
130C7×#95C2312yes—
131#3538C2×C711yescharacteristic
132#3092C2×C711yescharacteristic
133C2×#484C2×C711yescharacteristic, complemented
134#2503C2×C711yescharacteristic, complemented
135#1617C2×C714yescomplemented
136#1615C2×C714yescomplemented
137#1508C2×C711yescharacteristic, complemented
138#1501C2×C711yescharacteristic
139#1472C2×C711yescharacteristic, complemented
140C24×C7C2×D411yescharacteristic
141C7×#41—42no—
142C2×C7×Q8—22no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—82no—
149C2×C7×Q8C2×D412yes—
150C2×C7×Q8C2×D411yescharacteristic
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C2×D411yescharacteristic
155C22×C4×C7—21no—
156C22×C4×C7—22no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—22no—
161C22×C4×C7—42no—
162C22×C4×C7—44no—
163C22×C4×C7—41no—
164C22×C4×C7—41no—
165C22×C4×C7C2×D412yes—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2411yescharacteristic
173C7×M16—42no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8—42no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—22no—
191C7×C4⋊C4—22no—
192C7×C4⋊C4—22no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—22no—
195C7×C4⋊C4—22no—
196C7×C4⋊C4—41no—
197C7×C4⋊C4—41no—
198C7×C4⋊C4C2×D411yescharacteristic
199C7×C4⋊C4C2×D412yes—
200C7×C4⋊C4C2×D411yescharacteristic
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—22no—
204C7×C22⋊C4—22no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—22no—
207C7×C22⋊C4—21no—
208C7×C22⋊C4—21no—
209C7×C22⋊C4—22no—
210C7×C22⋊C4—21no—
211C7×C22⋊C4—21no—
212C7×C22⋊C4—21no—
213C7×C22⋊C4—42no—
214C42×C7—22no—
215C42×C7—22no—
216C42×C7—22no—
217C42×C7—21no—
218C42×C7—42no—
219#563—22no—
220#552—21no—
221#552—21no—
222#551—22no—
223D4×Q8—21no—
224D4×Q8—21no—
225#544—21no—
226#544C22×C712yes—
227#541—21no—
228#539—21no—
229#539C22×C712yes—
230#536—21no—
231#525C22×C711yescharacteristic
232C2×C22⋊Q8C22×C711yescharacteristic
233C2×C22⋊Q8C22×C711yescharacteristic
234C22×C4⋊C4C22×C711yescharacteristic
235#484—22no—
236#484C22×C714yes—
237#484C22×C712yes—
238#483—21no—
239#481C22×C712yes—
240#480—21no—
241#428—21no—
242C2×#107—21no—
243#417—21no—
244C2×#103—21no—
245#407C22×C711yescharacteristic
246C2×C22⋊C8C22×C711yescharacteristic
247#400C22×C714yes—
248#391—21no—
249#382—21no—
250#381—22no—
251C4×C22⋊C4C22×C711yescharacteristic
252#328C22×C714yes—
253#328C22×C712yes—
254#327C22×C712yes—
255#324C22×C714yes—
256C23×C7—22no—
257C23×C7—22no—
258C23×C7—21no—
259C23×C7—41no—
260C23×C7#12011yescharacteristic
261C23×C7#12011yescharacteristic
262C23×C7C22×D411yescharacteristic
263C7×Q8—42no—
264C7×Q8—41no—
265C7×Q8—42no—
266C7×Q8—42no—
267C7×Q8—81no—
268C7×Q8—82no—
269C7×D4—41no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—81no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—42no—
288C2×C4×C7—42no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—44no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—81no—
296C2×C4×C7—82no—
297C2×C4×C7—82no—
298C2×C4×C7—82no—
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7#12011yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×D411yescharacteristic
303C7×C8—42no—
304C7×C8—42no—
305C2×#41C7×D411yescharacteristic
306C22×Q8—21no—
307C22×Q8C7×D411yescharacteristic
308C22×Q8C7×D411yescharacteristic
309C23×C4—21no—
310C23×C4C23×C711yescharacteristic
311C2×M16—21no—
312C22×C8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317#126—22no—
318#126—22no—
319#126—22no—
320#126—22no—
321#125—21no—
322#125—22no—
323#125—22no—
324#125—22no—
325#125—22no—
326#124—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—21no—
331#123—22no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—22no—
344C22⋊Q8C7×D412yes—
345C22⋊Q8C7×D411yescharacteristic
346C22⋊Q8C23×C712yes—
347C22⋊Q8C23×C711yescharacteristic
348#121C7×D411yescharacteristic
349#121C23×C711yescharacteristic
350#120—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×D4—22no—
354C4×D4—21no—
355C4×D4—21no—
356C4×D4—22no—
357#117—21no—
358#117—22no—
359#117—22no—
360#117C7×D411yescharacteristic
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4C7×D411yescharacteristic
368C2×C4⋊C4C7×D411yescharacteristic
369C2×C4⋊C4C23×C712yes—
370C2×C4⋊C4C23×C712yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4C7×D412yes—
374C2×C42—21no—
375#107—24no—
376#107—22no—
377#106—22no—
378#103—24no—
379#103—22no—
380#102—22no—
381C22⋊C8—21no—
382C22⋊C8—21no—
383C22⋊C8C23×C712yes—
384C22⋊C8C23×C712yes—
385#95—24no—
386#95—24no—
387#95C23×C712yes—
388C22×C7—21no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—21no—
395C22×C7—44no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7#45412yes—
400C22×C7C2×#12011yescharacteristic
401C22×C7#57812yes—
402C4×C7—21no—
403C4×C7—21no—
404C4×C7—21no—
405C4×C7—42no—
406C4×C7—41no—
407C4×C7—42no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—42no—
412C4×C7—82no—
413C4×C7—82no—
414C4×C7#45812yes—
415C24C2×C7×D411yescharacteristic
416#41—42no—
417C2×Q8—22no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—82no—
424C2×Q8C2×C7×D412yes—
425C2×Q8C2×C7×D411yescharacteristic
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×C7×D411yescharacteristic
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4C2×C7×D412yes—
441C22×C4C2×C7×D411yescharacteristic
442C22×C4C2×C7×D411yescharacteristic
443C22×C4C2×C7×D411yescharacteristic
444C22×C4C2×C7×D412yes—
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C24×C711yescharacteristic
448M16—42no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—42no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4C2×C7×D411yescharacteristic
474C4⋊C4C2×C7×D412yes—
475C4⋊C4C2×C7×D411yescharacteristic
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—22no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—42no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—42no—
494C2×C7—22no—
495C2×C7—22no—
496C2×C7—41no—
497C2×C7—81no—
498C2×C7#301111yescharacteristic
499C2×C7C2×#45411yescharacteristic
500C2×C7C2×#45811yescharacteristic
501C23—22no—
502C23—22no—
503C23—21no—
504C23—41no—
505C23C7×#12011yescharacteristic
506C23C7×#12011yescharacteristic
507C23C22×C7×D411yescharacteristic
508Q8—42no—
509Q8—41no—
510Q8—42no—
511Q8—42no—
512Q8—81no—
513Q8—82no—
514D4—41no—
515D4—42no—
516D4—42no—
517D4—81no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4C7×#12011yescharacteristic
545C2×C4C7×#12011yescharacteristic
546C2×C4C22×C7×D411yescharacteristic
547C2×C4C22×C7×D411yescharacteristic
548C8—42no—
549C8—42no—
550C7#1826911yescharacteristic, direct product factor, Sylow
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—44no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22C7×#45412yes—
563C22C2×C7×#12011yescharacteristic
564C22C7×#57812yes—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4C7×#45812yes—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—81no—
582C2C7×#301111yescharacteristic
583C2C2×C7×#45411yescharacteristic
584C2C2×C7×#45811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#525C22
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
3. C2×C7×#484C2semidirect
4. C7×#2503C2semidirect
5. C7×#1617C2semidirect
6. C7×#1615C2semidirect
7. C7×#1508C2semidirect
9. C7×#1472C2semidirect
47. #18269C7direct (N × H)
133. C2×#484C2×C7semidirect
134. #2503C2×C7semidirect
135. #1617C2×C7semidirect
136. #1615C2×C7semidirect
137. #1508C2×C7semidirect
139. #1472C2×C7semidirect
550. C7#18269direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35382
C7×#30922
C2×C7×#4842
C7×#25032
C7×#16172
C7×#16152
C7×#15082
C7×#15012
C7×#14722
#182697