Finite Group Databasebeta

← C7×#18136 #13195352

Subgroups — C7×#18136 #13195352

Order 1792.

1196 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3459C211yesmaximal, characteristic
2C7×#3004C212yesmaximal, complemented
3C7×#2409C211yesmaximal, characteristic, complemented
4C7×#1603C214yesmaximal, complemented
5C7×#1602C212yesmaximal, complemented
6C7×#1601C212yesmaximal, complemented
7C7×#1513C212yesmaximal, complemented
8C7×#1465C211yesmaximal, characteristic, complemented
9C23×C7×Q8C2211yescharacteristic, complemented
10C2×C7×#137—22no—
11C7×#543C2211yescharacteristic
12C7×#539—22no—
13C7×#539C2212yes—
14C7×#538—22no—
15C7×#538—21no—
16C7×#534—21no—
17C7×#534C2211yescharacteristic
18C7×#529—22no—
19C7×#529—21no—
20C7×#525—22no—
21C7×#525C2212yes—
22C2×C7×C22⋊Q8—21no—
23C2×C4×C7×D4—21no—
24C2×C4×C7×D4C2211yescharacteristic
25C7×#451—22no—
26C7×#451C2214yescomplemented
27C7×#448—22no—
28C7×#448C2214yes—
29C7×#419—22no—
30C7×#407C2212yescomplemented
31C7×#395C2212yescomplemented
32C7×#393C2212yes—
33C7×#391—21no—
34C7×#386—21no—
35C7×#385—21no—
36C7×#380C2211yescharacteristic, complemented
37C7×#328C2214yescomplemented
38C7×#327C2214yes—
39C7×#324C2212yescomplemented
40C7×#323C2212yes—
41#18136C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41D412yes—
43C22×C7×Q8—21no—
44C22×C7×Q8—22no—
45C22×C7×Q8—22no—
46C22×C7×Q8—44no—
47C22×C7×Q8C2312yes—
48C22×C7×D4—21no—
49C22×C7×D4D411yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—21no—
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C7×#137—48no—
55C2×C7×Q16—24no—
56C2×C7×SD16—24no—
57C2×C7×M16—22no—
58C7×#126—22no—
59C7×#126—22no—
60C7×#126—22no—
61C7×#125—21no—
62C7×#125—22no—
63C7×#124—22no—
64C7×#124—21no—
65C7×#123—22no—
66C7×#123—22no—
67C7×#123—22no—
68C7×#123—22no—
69C7×#123—22no—
70C7×#123—22no—
71C7×C22⋊Q8—22no—
72C7×C22⋊Q8—22no—
73C7×C22⋊Q8—21no—
74C7×C22⋊Q8—22no—
75C7×C22⋊Q8—44no—
76C7×C22⋊Q8D412yes—
77C7×C22⋊Q8C2312yes—
78C7×#121—22no—
79C7×#121—22no—
80C7×#121—22no—
81C7×#121—21no—
82C7×#121D412yes—
83C7×#121C2312yes—
84C7×#120—22no—
85C7×#120—22no—
86C7×#120—22no—
87C4×C7×Q8—21no—
88C4×C7×D4—22no—
89C4×C7×D4—22no—
90C4×C7×D4—21no—
91C4×C7×D4—22no—
92C4×C7×D4—22no—
93C4×C7×D4—22no—
94C4×C7×D4—22no—
95C4×C7×D4—22no—
96C7×#117—22no—
97C7×#117D412yes—
98C2×C7×C4⋊C4—21no—
99C2×C7×C4⋊C4—21no—
100C2×C7×C4⋊C4—22no—
101C2×C7×C4⋊C4C2311yescharacteristic
102C2×C7×C22⋊C4—21no—
103C2×C7×C22⋊C4—21no—
104C2×C7×C22⋊C4—21no—
105C2×C7×C22⋊C4—22no—
106C2×C7×C22⋊C4D411yescharacteristic
107C2×C7×C22⋊C4C2311yescharacteristic
108C2×C42×C7—21no—
109C2×C42×C7D411yescharacteristic
110C7×#103—24no—
111C7×#103—24no—
112C7×C22⋊C8—22no—
113C7×C22⋊C8C2314yes—
114C7×#95—22no—
115C7×#95—22no—
116C7×#95C2312yes—
117#3459C2×C711yescharacteristic
118#3004C2×C712yescomplemented
119#2409C2×C711yescharacteristic, complemented
120#1603C2×C714yescomplemented
121#1602C2×C712yescomplemented
122#1601C2×C712yescomplemented
123#1513C2×C712yescomplemented
124#1465C2×C711yescharacteristic, complemented
125C24×C7—21no—
126C24×C7C2×D411yescharacteristic
127C24×C7C2×D411yescharacteristic
128C7×#41—44no—
129C2×C7×Q8—24no—
130C2×C7×Q8—21no—
131C2×C7×Q8—28no—
132C2×C7×Q8—24no—
133C2×C7×Q8—44no—
134C2×C7×Q8—42no—
135C2×C7×Q8—44no—
136C2×C7×Q8—48no—
137C2×C7×Q8—41no—
138C2×C7×Q8—41no—
139C2×C7×Q8C2×D412yes—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—21no—
145C2×C7×D4—22no—
146C2×C7×D4—22no—
147C2×C7×D4—41no—
148C2×C7×D4—42no—
149C2×C7×D4—42no—
150C2×C7×D4C2×D412yes—
151C22×C4×C7—21no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—22no—
156C22×C4×C7—21no—
157C22×C4×C7—22no—
158C22×C4×C7—21no—
159C22×C4×C7—22no—
160C22×C4×C7—42no—
161C22×C4×C7—44no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—82no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D412yes—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2411yescharacteristic
172C7×Q16—48no—
173C7×SD16—48no—
174C7×M16—44no—
175C2×C7×C8—24no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—44no—
187C7×C4⋊C4C2×D412yes—
188C7×C4⋊C4C2×D412yes—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—22no—
196C7×C22⋊C4—22no—
197C7×C22⋊C4—22no—
198C7×C22⋊C4—22no—
199C7×C22⋊C4—22no—
200C7×C22⋊C4—22no—
201C7×C22⋊C4—44no—
202C7×C22⋊C4—42no—
203C42×C7—22no—
204C42×C7—22no—
205C42×C7—21no—
206C42×C7—21no—
207C23×Q8C22×C711yescharacteristic, complemented
208C2×#137—22no—
209#543C22×C711yescharacteristic
210#539—22no—
211#539C22×C712yes—
212#538—22no—
213#538—21no—
214#534—21no—
215#534C22×C711yescharacteristic
216#529—22no—
217#529—21no—
218#525—22no—
219#525C22×C712yes—
220C2×C22⋊Q8—21no—
221C2×C4×D4—21no—
222C2×C4×D4C22×C711yescharacteristic
223#451—22no—
224#451C22×C714yescomplemented
225#448—22no—
226#448C22×C714yes—
227#419—22no—
228#407C22×C712yescomplemented
229#395C22×C712yescomplemented
230#393C22×C712yes—
231#391—21no—
232#386—21no—
233#385—21no—
234#380C22×C711yescharacteristic, complemented
235#328C22×C714yescomplemented
236#327C22×C714yes—
237#324C22×C712yescomplemented
238#323C22×C712yes—
239C23×C7—21no—
240C23×C7—21no—
241C23×C7—21no—
242C23×C7—21no—
243C23×C7—22no—
244C23×C7—41no—
245C23×C7—41no—
246C23×C7—41no—
247C23×C7—41no—
248C23×C7—42no—
249C23×C7—81no—
250C23×C7—81no—
251C23×C7#12012yes—
252C23×C7#13712yescomplemented
253C23×C7C22×D411yescharacteristic
254C7×Q8—28no—
255C7×Q8—42no—
256C7×Q8—44no—
257C7×Q8—44no—
258C7×Q8—44no—
259C7×Q8—81no—
260C7×D4—42no—
261C7×D4—44no—
262C7×D4—44no—
263C7×D4—42no—
264C7×D4—42no—
265C7×D4—82no—
266C7×D4—82no—
267C7×D4—81no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—21no—
272C2×C4×C7—22no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—21no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—42no—
280C2×C4×C7—41no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—48no—
284C2×C4×C7—42no—
285C2×C4×C7—42no—
286C2×C4×C7—44no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—42no—
290C2×C4×C7—42no—
291C2×C4×C7—42no—
292C2×C4×C7—81no—
293C2×C4×C7—81no—
294C2×C4×C7—82no—
295C2×C4×C7—81no—
296C2×C4×C7#12011yescharacteristic
297C2×C4×C7#12011yescharacteristic
298C2×C4×C7C22×D412yes—
299C7×C8—44no—
300C2×#41C7×D412yes—
301C22×Q8—21no—
302C22×Q8—22no—
303C22×Q8—22no—
304C22×Q8—44no—
305C22×Q8C23×C712yes—
306C22×D4—21no—
307C22×D4C7×D411yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310C23×C4C7×D411yescharacteristic
311C23×C4C23×C711yescharacteristic
312#137—48no—
313C2×Q16—24no—
314C2×SD16—24no—
315C2×M16—22no—
316#126—22no—
317#126—22no—
318#126—22no—
319#125—21no—
320#125—22no—
321#124—22no—
322#124—21no—
323#123—22no—
324#123—22no—
325#123—22no—
326#123—22no—
327#123—22no—
328#123—22no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331C22⋊Q8—21no—
332C22⋊Q8—22no—
333C22⋊Q8—44no—
334C22⋊Q8C7×D412yes—
335C22⋊Q8C23×C712yes—
336#121—22no—
337#121—22no—
338#121—22no—
339#121—21no—
340#121C7×D412yes—
341#121C23×C712yes—
342#120—22no—
343#120—22no—
344#120—22no—
345C4×Q8—21no—
346C4×D4—22no—
347C4×D4—22no—
348C4×D4—21no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354#117—22no—
355#117C7×D412yes—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C23×C711yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4C7×D411yescharacteristic
365C2×C22⋊C4C23×C711yescharacteristic
366C2×C42—21no—
367C2×C42C7×D411yescharacteristic
368#103—24no—
369#103—24no—
370C22⋊C8—22no—
371C22⋊C8C23×C714yes—
372#95—22no—
373#95—22no—
374#95C23×C712yes—
375C22×C7—21no—
376C22×C7—21no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—81no—
390C22×C7—84no—
391C22×C7C2×#12011yescharacteristic
392C22×C7C2×#13711yescharacteristic
393C22×C7#57511yescharacteristic
394C4×C7—22no—
395C4×C7—21no—
396C4×C7—41no—
397C4×C7—42no—
398C4×C7—41no—
399C4×C7—42no—
400C4×C7—44no—
401C4×C7—41no—
402C4×C7—42no—
403C4×C7—82no—
404C4×C7B(4,𝔽2)12yes—
405C24—21no—
406C24C2×C7×D411yescharacteristic
407C24C2×C7×D411yescharacteristic
408#41—44no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—28no—
412C2×Q8—24no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—48no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8C2×C7×D412yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—41no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C2×C7×D412yes—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—82no—
445C22×C4C2×C7×D411yescharacteristic
446C22×C4C2×C7×D411yescharacteristic
447C22×C4C2×C7×D412yes—
448C22×C4C2×C7×D411yescharacteristic
449C22×C4C2×C7×D411yescharacteristic
450C22×C4C2×C7×D412yes—
451C22×C4C24×C711yescharacteristic
452Q16—48no—
453SD16—48no—
454M16—44no—
455C2×C8—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4C2×C7×D412yes—
468C4⋊C4C2×C7×D412yes—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—21no—
486C42—21no—
487C2×C7—21no—
488C2×C7—21no—
489C2×C7—42no—
490C2×C7—42no—
491C2×C7—82no—
492C2×C7#300711yescharacteristic
493C2×C7#302211yescharacteristic
494C2×C7C2×B(4,𝔽2)11yescharacteristic
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—22no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—81no—
506C23—81no—
507C23C7×#12012yes—
508C23C7×#13712yescomplemented
509C23C22×C7×D411yescharacteristic
510Q8—28no—
511Q8—42no—
512Q8—44no—
513Q8—44no—
514Q8—44no—
515Q8—81no—
516D4—42no—
517D4—44no—
518D4—44no—
519D4—42no—
520D4—42no—
521D4—82no—
522D4—82no—
523D4—81no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4C7×#12011yescharacteristic
553C2×C4C7×#12011yescharacteristic
554C2×C4C22×C7×D412yes—
555C8—44no—
556C7#1813611yescharacteristic, direct product factor, Sylow
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—84no—
573C22C2×C7×#12011yescharacteristic
574C22C2×C7×#13711yescharacteristic
575C22C7×#57511yescharacteristic
576C4—22no—
577C4—21no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—44no—
583C4—41no—
584C4—42no—
585C4—82no—
586C4C7×B(4,𝔽2)12yes—
587C2—21no—
588C2—21no—
589C2—42no—
590C2—42no—
591C2—82no—
592C2C7×#300711yescharacteristic
593C2C7×#302211yescharacteristic
594C2C2×C7×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3459C2
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×#3004C2semidirect
3. C7×#2409C2semidirect
4. C7×#1603C2semidirect
5. C7×#1602C2semidirect
6. C7×#1601C2semidirect
7. C7×#1513C2semidirect
8. C7×#1465C2semidirect
9. C23×C7×Q8C22semidirect
26. C7×#451C22semidirect
30. C7×#407C22semidirect
31. C7×#395C22semidirect
36. C7×#380C22semidirect
37. C7×#328C22semidirect
39. C7×#324C22semidirect
41. #18136C7direct (N × H)
118. #3004C2×C7semidirect
119. #2409C2×C7semidirect
120. #1603C2×C7semidirect
121. #1602C2×C7semidirect
122. #1601C2×C7semidirect
123. #1513C2×C7semidirect
124. #1465C2×C7semidirect
207. C23×Q8C22×C7semidirect
224. #451C22×C7semidirect
228. #407C22×C7semidirect
229. #395C22×C7semidirect
234. #380C22×C7semidirect
235. #328C22×C7semidirect
237. #324C22×C7semidirect
252. C23×C7#137semidirect
508. C23C7×#137semidirect
556. C7#18136direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34592
C7×#30042
C7×#24092
C7×#16032
C7×#16022
C7×#16012
C7×#15132
C7×#14652
#181367