Finite Group Databasebeta

← C7×#18312 #13195528

Subgroups — C7×#18312 #13195528

Order 1792.

930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3455C211yesmaximal, characteristic
2C7×#3229C211yesmaximal, characteristic, complemented
3C7×#3226C211yesmaximal, characteristic, complemented
4C7×#2621C211yesmaximal, characteristic, complemented
5C7×#1612C214yesmaximal, complemented
6C7×#1609C214yesmaximal, complemented
7C7×#1518C211yesmaximal, characteristic, complemented
8C7×#1517C211yesmaximal, characteristic, complemented
9C7×#1458C211yesmaximal, characteristic, complemented
10C7×#542—21no—
11C7×#542—21no—
12C7×#541—21no—
13C7×#540—21no—
14C7×#539—21no—
15C7×#539C2212yes—
16C7×#538—21no—
17C7×#538C2212yes—
18C7×#533C2211yescharacteristic
19C7×#532C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C7×#529—22no—
22C7×#529—22no—
23C7×#525—21no—
24C2×C7×C22⋊Q8—21no—
25C4×C7×#41—21no—
26C2×C7×#117C2211yescharacteristic
27C7×#500—21no—
28C7×#498—21no—
29C7×#487—21no—
30C7×#486—21no—
31C7×#483C2214yes—
32C7×#480C2214yes—
33C7×#475—22no—
34C7×#474—22no—
35C7×#431C2211yescharacteristic
36C7×#431C2211yescharacteristic
37C7×#398—21no—
38C7×#398C2212yes—
39C7×#397—21no—
40C7×#397C2212yes—
41C7×#391—21no—
42C7×#382C2211yescharacteristic
43C7×#328C2214yes—
44C7×#327C2214yes—
45C7×#324C2212yes—
46C7×#323C2212yes—
47#18312C711yesmaximal, characteristic, direct product factor, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C23×C4×C7D411yescharacteristic
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8D411yescharacteristic
56C7×C4⋊Q8D411yescharacteristic
57C7×C4⋊Q8D411yescharacteristic
58C7×#127D411yescharacteristic
59C7×#126—22no—
60C7×#126—22no—
61C7×#126—22no—
62C7×#126—22no—
63C7×#125—22no—
64C7×#125—22no—
65C7×#125—21no—
66C7×#124—22no—
67C7×#124—22no—
68C7×#124—21no—
69C7×#123—22no—
70C7×#123—22no—
71C7×#123—22no—
72C7×#123—22no—
73C7×#123—22no—
74C7×C22⋊Q8—21no—
75C7×C22⋊Q8—21no—
76C7×C22⋊Q8—22no—
77C7×C22⋊Q8—22no—
78C7×C22⋊Q8—21no—
79C7×C22⋊Q8—44no—
80C7×C22⋊Q8C2312yes—
81C7×#121—21no—
82C7×#121—21no—
83C7×#121—22no—
84C7×#121—22no—
85C7×#121—21no—
86C7×#121C2312yes—
87C7×#120—22no—
88C4×C7×Q8—21no—
89C4×C7×Q8—21no—
90C4×C7×Q8—21no—
91C4×C7×Q8D412yes—
92C4×C7×D4—21no—
93C4×C7×D4—21no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C4×C7×D4—22no—
97C4×C7×D4—22no—
98C4×C7×D4—21no—
99C4×C7×D4D412yes—
100C7×#117—22no—
101C7×#117—21no—
102C7×#117—21no—
103C7×#117—21no—
104C7×#117—21no—
105C7×#117—21no—
106C2×C7×C4⋊C4—21no—
107C2×C7×C4⋊C4—22no—
108C2×C7×C4⋊C4—21no—
109C2×C7×C4⋊C4D411yescharacteristic
110C2×C7×C4⋊C4C2312yes—
111C2×C7×C4⋊C4C2311yescharacteristic
112C2×C7×C22⋊C4—22no—
113C2×C7×C22⋊C4D411yescharacteristic
114C2×C7×C22⋊C4C2311yescharacteristic
115C2×C42×C7—22no—
116C2×C42×C7D411yescharacteristic
117C2×C42×C7C2311yescharacteristic
118C7×#107—24no—
119C7×#106—24no—
120C7×C4⋊C8—21no—
121C7×C4⋊C8—21no—
122C7×#103—24no—
123C7×#102—24no—
124C7×C22⋊C8C2312yes—
125C7×C22⋊C8C2312yes—
126C4×C7×C8—21no—
127C4×C7×C8—21no—
128C7×#95—21no—
129C7×#95—22no—
130C7×#95—22no—
131C7×#95C2312yes—
132#3455C2×C711yescharacteristic
133#3229C2×C711yescharacteristic, complemented
134#3226C2×C711yescharacteristic, complemented
135#2621C2×C711yescharacteristic, complemented
136#1612C2×C714yescomplemented
137#1609C2×C714yescomplemented
138#1518C2×C711yescharacteristic, complemented
139#1517C2×C711yescharacteristic, complemented
140#1458C2×C711yescharacteristic, complemented
141C24×C7C2×D411yescharacteristic
142C7×#41—42no—
143C7×#41—42no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—41no—
147C2×C7×Q8—41no—
148C2×C7×Q8—42no—
149C2×C7×Q8—42no—
150C2×C7×Q8C2×D412yes—
151C2×C7×D4—22no—
152C2×C7×D4—22no—
153C2×C7×D4—22no—
154C2×C7×D4—21no—
155C2×C7×D4—21no—
156C2×C7×D4C2×D412yes—
157C22×C4×C7—22no—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—22no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7—22no—
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C2×C7×C8—22no—
173C2×C7×C8—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—22no—
189C7×C4⋊C4—22no—
190C7×C4⋊C4—44no—
191C7×C4⋊C4—41no—
192C7×C4⋊C4—41no—
193C7×C4⋊C4—42no—
194C7×C4⋊C4C2×D412yes—
195C7×C4⋊C4C2×D412yes—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—22no—
202C7×C22⋊C4—22no—
203C7×C22⋊C4—21no—
204C7×C22⋊C4—21no—
205C7×C22⋊C4—22no—
206C7×C22⋊C4—21no—
207C7×C22⋊C4—21no—
208C7×C22⋊C4—22no—
209C7×C22⋊C4—44no—
210C42×C7—22no—
211C42×C7—22no—
212C42×C7—21no—
213C42×C7—22no—
214C42×C7—21no—
215C42×C7C2×D411yescharacteristic
216C42×C7C2×D411yescharacteristic
217C42×C7C2×D411yescharacteristic
218C42×C7C2×D411yescharacteristic
219#542—21no—
220#542—21no—
221#541—21no—
222#540—21no—
223#539—21no—
224#539C22×C712yes—
225#538—21no—
226#538C22×C712yes—
227#533C22×C711yescharacteristic
228#532C22×C711yescharacteristic
229C2×C4⋊Q8C22×C711yescharacteristic
230#529—22no—
231#529—22no—
232#525—21no—
233C2×C22⋊Q8—21no—
234C4×#41—21no—
235C2×#117C22×C711yescharacteristic
236#500—21no—
237#498—21no—
238#487—21no—
239#486—21no—
240#483C22×C714yes—
241#480C22×C714yes—
242#475—22no—
243#474—22no—
244#431C22×C711yescharacteristic
245#431C22×C711yescharacteristic
246#398—21no—
247#398C22×C712yes—
248#397—21no—
249#397C22×C712yes—
250#391—21no—
251#382C22×C711yescharacteristic
252#328C22×C714yes—
253#327C22×C714yes—
254#324C22×C712yes—
255#323C22×C712yes—
256C23×C7—21no—
257C23×C7—22no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7C22×D411yescharacteristic
263C7×Q8—42no—
264C7×Q8—42no—
265C7×Q8—81no—
266C7×Q8—82no—
267C7×Q8—82no—
268C7×D4—42no—
269C7×D4—42no—
270C7×D4—42no—
271C7×D4—42no—
272C7×D4—81no—
273C7×D4—82no—
274C7×D4—82no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—42no—
286C2×C4×C7—42no—
287C2×C4×C7—44no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—42no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—82no—
297C2×C4×C7—82no—
298C2×C4×C7#12011yescharacteristic
299C2×C4×C7#12011yescharacteristic
300C2×C4×C7#12011yescharacteristic
301C2×C4×C7#12011yescharacteristic
302C2×C4×C7#13512yes—
303C2×C4×C7AGL(1,ℤ/8)12yes—
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×D411yescharacteristic
306C7×C8—42no—
307C7×C8—42no—
308C2×#41—21no—
309C22×Q8—21no—
310C23×C4C7×D411yescharacteristic
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8C7×D411yescharacteristic
316C4⋊Q8C7×D411yescharacteristic
317C4⋊Q8C7×D411yescharacteristic
318#127C7×D411yescharacteristic
319#126—22no—
320#126—22no—
321#126—22no—
322#126—22no—
323#125—22no—
324#125—22no—
325#125—21no—
326#124—22no—
327#124—22no—
328#124—21no—
329#123—22no—
330#123—22no—
331#123—22no—
332#123—22no—
333#123—22no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—44no—
340C22⋊Q8C23×C712yes—
341#121—21no—
342#121—21no—
343#121—22no—
344#121—22no—
345#121—21no—
346#121C23×C712yes—
347#120—22no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×Q8—21no—
351C4×Q8C7×D412yes—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—21no—
359C4×D4C7×D412yes—
360#117—22no—
361#117—21no—
362#117—21no—
363#117—21no—
364#117—21no—
365#117—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4C7×D411yescharacteristic
370C2×C4⋊C4C23×C712yes—
371C2×C4⋊C4C23×C711yescharacteristic
372C2×C22⋊C4—22no—
373C2×C22⋊C4C7×D411yescharacteristic
374C2×C22⋊C4C23×C711yescharacteristic
375C2×C42—22no—
376C2×C42C7×D411yescharacteristic
377C2×C42C23×C711yescharacteristic
378#107—24no—
379#106—24no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382#103—24no—
383#102—24no—
384C22⋊C8C23×C712yes—
385C22⋊C8C23×C712yes—
386C4×C8—21no—
387C4×C8—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95C23×C712yes—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—42no—
395C22×C7—41no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—41no—
399C22×C7—42no—
400C22×C7—41no—
401C22×C7—81no—
402C22×C7C2×#12011yescharacteristic
403C22×C7C2×#13511yescharacteristic
404C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
405C4×C7—21no—
406C4×C7—21no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—41no—
412C4×C7—42no—
413C4×C7—41no—
414C4×C7—42no—
415C4×C7—82no—
416C4×C7—82no—
417C4×C7#44912yes—
418C4×C7#44912yes—
419C24C2×C7×D411yescharacteristic
420#41—42no—
421#41—42no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—41no—
425C2×Q8—41no—
426C2×Q8—42no—
427C2×Q8—42no—
428C2×Q8C2×C7×D412yes—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C7×D412yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4C2×C7×D411yescharacteristic
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C2×C7×D411yescharacteristic
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C24×C711yescharacteristic
450C2×C8—22no—
451C2×C8—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—44no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4C2×C7×D412yes—
473C4⋊C4C2×C7×D412yes—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—22no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—21no—
491C42—22no—
492C42—21no—
493C42C2×C7×D411yescharacteristic
494C42C2×C7×D411yescharacteristic
495C42C2×C7×D411yescharacteristic
496C42C2×C7×D411yescharacteristic
497C2×C7—21no—
498C2×C7—21no—
499C2×C7—82no—
500C2×C7—81no—
501C2×C7C2×#44911yescharacteristic
502C2×C7#302211yescharacteristic
503C2×C7#302811yescharacteristic
504C23—21no—
505C23—22no—
506C23—21no—
507C23—21no—
508C23—41no—
509C23—41no—
510C23C22×C7×D411yescharacteristic
511Q8—42no—
512Q8—42no—
513Q8—81no—
514Q8—82no—
515Q8—82no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—81no—
521D4—82no—
522D4—82no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4C7×#12011yescharacteristic
547C2×C4C7×#12011yescharacteristic
548C2×C4C7×#12011yescharacteristic
549C2×C4C7×#12011yescharacteristic
550C2×C4C7×#13512yes—
551C2×C4C7×AGL(1,ℤ/8)12yes—
552C2×C4C22×C7×D411yescharacteristic
553C2×C4C22×C7×D411yescharacteristic
554C8—42no—
555C8—42no—
556C7#1831211yescharacteristic, direct product factor, Sylow
557C22—21no—
558C22—21no—
559C22—42no—
560C22—41no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—81no—
567C22C2×C7×#12011yescharacteristic
568C22C2×C7×#13511yescharacteristic
569C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4C7×#44912yes—
583C4C7×#44912yes—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C2×C7×#44911yescharacteristic
589C2C7×#302211yescharacteristic
590C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3455C2
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×#3229C2semidirect
3. C7×#3226C2semidirect
4. C7×#2621C2semidirect
5. C7×#1612C2semidirect
6. C7×#1609C2semidirect
7. C7×#1518C2semidirect
8. C7×#1517C2semidirect
9. C7×#1458C2semidirect
47. #18312C7direct (N × H)
133. #3229C2×C7semidirect
134. #3226C2×C7semidirect
135. #2621C2×C7semidirect
136. #1612C2×C7semidirect
137. #1609C2×C7semidirect
138. #1518C2×C7semidirect
139. #1517C2×C7semidirect
140. #1458C2×C7semidirect
556. C7#18312direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34552
C7×#32292
C7×#32262
C7×#26212
C7×#16122
C7×#16092
C7×#15182
C7×#15172
C7×#14582
#183127