Finite Group Databasebeta

← C7×#18332 #13195548

Subgroups — C7×#18332 #13195548

Order 1792.

1230 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3449C211yesmaximal, characteristic, complemented
2C7×#3213C212yesmaximal, complemented
3C7×#2610C211yesmaximal, characteristic, complemented
4C7×#1604C212yesmaximal, complemented
5C7×#1601C212yesmaximal, complemented
6C7×#1598C212yesmaximal, complemented
7C7×#1595C212yesmaximal, complemented
8C7×#1505C212yesmaximal, complemented
9C7×#1464C211yesmaximal, characteristic, complemented
10C22×C7×#41—21no—
11C7×#537—21no—
12C7×#537C2212yes—
13C7×#536—22no—
14C7×#535—22no—
15C7×#534—21no—
16C7×#534C2212yes—
17C7×#532—21no—
18C7×#532C2212yes—
19C2×C7×#124C2211yescharacteristic, complemented
20C7×#525—24no—
21C2×C7×#117C2211yescharacteristic
22C7×#496—22no—
23C7×#489—22no—
24C7×#463—24no—
25C7×#448C2214yescomplemented
26C7×#447C2214yescomplemented
27C7×#432C2212yescomplemented
28C7×#395C2211yescharacteristic, complemented
29C7×#394C2211yescharacteristic, complemented
30C7×#393C2211yescharacteristic
31C7×#392C2211yescharacteristic
32C7×#390—21no—
33C7×#388—21no—
34C7×#388—21no—
35C7×#386—21no—
36C7×#376C2211yescharacteristic, complemented
37C7×#327C2214yescomplemented
38C7×#327C2214yescomplemented
39C7×#324C2212yescomplemented
40C7×#323C2212yes—
41#18332C711yesmaximal, characteristic, direct product factor, Sylow
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—44no—
47C2×C7×#41—44no—
48C22×C7×Q8—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4—21no—
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—22no—
55C23×C4×C7D411yescharacteristic
56C2×C7×SD16—24no—
57C2×C7×D8—24no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8D412yescomplemented
60C7×#127—21no—
61C7×#127D412yescomplemented
62C7×#124—22no—
63C7×#124—22no—
64C7×#124—22no—
65C7×#124—22no—
66C7×#123—24no—
67C7×#123—24no—
68C7×C22⋊Q8—24no—
69C7×C22⋊Q8—24no—
70C7×#121—24no—
71C7×#121—24no—
72C7×#121—22no—
73C7×#121—22no—
74C7×#121—22no—
75C7×#121C2314yes—
76C7×#120—24no—
77C7×#120—24no—
78C4×C7×D4—22no—
79C4×C7×D4—22no—
80C4×C7×D4—22no—
81C4×C7×D4D414yescomplemented
82C7×#117—22no—
83C7×#117—22no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4D411yescharacteristic
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—21no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—21no—
93C2×C7×C22⋊C4D411yescharacteristic
94C2×C7×C22⋊C4C2311yescharacteristic
95C2×C42×C7—21no—
96C2×C42×C7D411yescharacteristic
97C2×C42×C7C2311yescharacteristic
98C7×C4⋊C8—22no—
99C7×#103—24no—
100C7×#102—24no—
101C7×C22⋊C8C2314yes—
102C7×#97—22no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95C2311yescharacteristic
107C7×#95C2311yescharacteristic
108#3449C2×C711yescharacteristic, complemented
109#3213C2×C712yescomplemented
110#2610C2×C711yescharacteristic, complemented
111#1604C2×C712yescomplemented
112#1601C2×C712yescomplemented
113#1598C2×C712yescomplemented
114#1595C2×C712yescomplemented
115#1505C2×C712yescomplemented
116#1464C2×C711yescharacteristic, complemented
117C24×C7—22no—
118C24×C7—21no—
119C24×C7C2×D411yescharacteristic
120C7×#41—48no—
121C7×#41—48no—
122C7×#41—44no—
123C7×#41—44no—
124C7×#41—44no—
125C7×#41—44no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—44no—
130C2×C7×Q8—81no—
131C2×C7×D4—24no—
132C2×C7×D4—22no—
133C2×C7×D4—24no—
134C2×C7×D4—22no—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—44no—
140C2×C7×D4—42no—
141C2×C7×D4—44no—
142C2×C7×D4—44no—
143C2×C7×D4—42no—
144C2×C7×D4—81no—
145C2×C7×D4—81no—
146C2×C7×D4C2×D414yes—
147C22×C4×C7—22no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—24no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—48no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2411yescharacteristic
164C7×SD16—84no—
165C7×D8—84no—
166C2×C7×C8—24no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4C2×D414yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—41no—
182C7×C22⋊C4—41no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—81no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—42no—
190C42×C7C2×D412yes—
191C42×C7C2×D412yes—
192C22×#41—21no—
193#537—21no—
194#537C22×C712yes—
195#536—22no—
196#535—22no—
197#534—21no—
198#534C22×C712yes—
199#532—21no—
200#532C22×C712yes—
201C2×#124C22×C711yescharacteristic, complemented
202#525—24no—
203C2×#117C22×C711yescharacteristic
204#496—22no—
205#489—22no—
206#463—24no—
207#448C22×C714yescomplemented
208#447C22×C714yescomplemented
209#432C22×C712yescomplemented
210#395C22×C711yescharacteristic, complemented
211#394C22×C711yescharacteristic, complemented
212#393C22×C711yescharacteristic
213#392C22×C711yescharacteristic
214#390—21no—
215#388—21no—
216#388—21no—
217#386—21no—
218#376C22×C711yescharacteristic, complemented
219#327C22×C714yescomplemented
220#327C22×C714yescomplemented
221#324C22×C712yescomplemented
222#323C22×C712yes—
223C23×C7—21no—
224C23×C7—21no—
225C23×C7—24no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—44no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—82no—
237C23×C7—82no—
238C23×C7AGL(1,ℤ/8)12yes—
239C23×C7C22×D411yescharacteristic
240C7×Q8—44no—
241C7×Q8—44no—
242C7×Q8—44no—
243C7×D4—44no—
244C7×D4—44no—
245C7×D4—44no—
246C7×D4—44no—
247C7×D4—48no—
248C7×D4—44no—
249C7×D4—44no—
250C7×D4—82no—
251C7×D4—82no—
252C7×D4—82no—
253C7×D4—84no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—22no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—24no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—48no—
267C2×C4×C7—48no—
268C2×C4×C7—41no—
269C2×C4×C7—44no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7#12011yescharacteristic
280C2×C4×C7#12011yescharacteristic
281C2×C4×C7#12012yes—
282C2×C4×C7AGL(1,ℤ/8)12yes—
283C2×C4×C7C22×D412yes—
284C7×C8—44no—
285C2×#41—22no—
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—22no—
289C2×#41—44no—
290C2×#41—44no—
291C22×Q8—21no—
292C22×Q8C23×C711yescharacteristic
293C22×D4—21no—
294C22×D4—22no—
295C22×D4—21no—
296C22×D4C23×C711yescharacteristic
297C23×C4—22no—
298C23×C4C7×D411yescharacteristic
299C2×SD16—24no—
300C2×D8—24no—
301C4⋊Q8—21no—
302C4⋊Q8C7×D412yescomplemented
303#127—21no—
304#127C7×D412yescomplemented
305#124—22no—
306#124—22no—
307#124—22no—
308#124—22no—
309#123—24no—
310#123—24no—
311C22⋊Q8—24no—
312C22⋊Q8—24no—
313#121—24no—
314#121—24no—
315#121—22no—
316#121—22no—
317#121—22no—
318#121C23×C714yes—
319#120—24no—
320#120—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4C7×D414yescomplemented
325#117—22no—
326#117—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4C7×D411yescharacteristic
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4C7×D411yescharacteristic
337C2×C22⋊C4C23×C711yescharacteristic
338C2×C42—21no—
339C2×C42C7×D411yescharacteristic
340C2×C42C23×C711yescharacteristic
341C4⋊C8—22no—
342#103—24no—
343#102—24no—
344C22⋊C8C23×C714yes—
345#97—22no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95C23×C711yescharacteristic
350#95C23×C711yescharacteristic
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—22no—
354C22×C7—42no—
355C22×C7—41no—
356C22×C7—41no—
357C22×C7—44no—
358C22×C7—44no—
359C22×C7—44no—
360C22×C7—81no—
361C22×C7—81no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7—81no—
365C22×C7—88no—
366C22×C7—81no—
367C22×C7—81no—
368C22×C7—81no—
369C22×C7C2×#12011yescharacteristic
370C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
372C4×C7—21no—
373C4×C7—22no—
374C4×C7—21no—
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—42no—
378C4×C7—44no—
379C4×C7—41no—
380C4×C7—41no—
381C4×C7—81no—
382C4×C7—81no—
383C24—22no—
384C24—21no—
385C24C2×C7×D411yescharacteristic
386#41—48no—
387#41—48no—
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—44no—
396C2×Q8—81no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4C2×C7×D414yes—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—48no—
424C22×C4C2×C7×D411yescharacteristic
425C22×C4C2×C7×D411yescharacteristic
426C22×C4C2×C7×D411yescharacteristic
427C22×C4C2×C7×D411yescharacteristic
428C22×C4C2×C7×D411yescharacteristic
429C22×C4C24×C711yescharacteristic
430SD16—84no—
431D8—84no—
432C2×C8—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—81no—
439C4⋊C4C2×C7×D414yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—22no—
443C22⋊C4—24no—
444C22⋊C4—22no—
445C22⋊C4—24no—
446C22⋊C4—42no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C42—21no—
454C42—21no—
455C42—42no—
456C42C2×C7×D412yes—
457C42C2×C7×D412yes—
458C2×C7—21no—
459C2×C7—21no—
460C2×C7—42no—
461C2×C7—84no—
462C2×C7—161no—
463C2×C7#300311yescharacteristic
464C2×C7#301811yescharacteristic
465C2×C7#302711yescharacteristic
466C23—21no—
467C23—21no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—44no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—82no—
480C23—82no—
481C23C7×AGL(1,ℤ/8)12yes—
482C23C22×C7×D411yescharacteristic
483Q8—44no—
484Q8—44no—
485Q8—44no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—48no—
491D4—44no—
492D4—44no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—84no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4C7×#12011yescharacteristic
523C2×C4C7×#12011yescharacteristic
524C2×C4C7×#12012yes—
525C2×C4C7×AGL(1,ℤ/8)12yes—
526C2×C4C22×C7×D412yes—
527C8—44no—
528C7#1833211yescharacteristic, direct product factor, Sylow
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—88no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×C7×#12011yescharacteristic
548C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
549C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
550C4—21no—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—44no—
557C4—41no—
558C4—41no—
559C4—81no—
560C4—81no—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—84no—
565C2—161no—
566C2C7×#300311yescharacteristic
567C2C7×#301811yescharacteristic
568C2C7×#302711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
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
1. C7×#3449C2semidirect
2. C7×#3213C2semidirect
3. C7×#2610C2semidirect
4. C7×#1604C2semidirect
5. C7×#1601C2semidirect
6. C7×#1598C2semidirect
7. C7×#1595C2semidirect
8. C7×#1505C2semidirect
9. C7×#1464C2semidirect
19. C2×C7×#124C22semidirect
25. C7×#448C22semidirect
26. C7×#447C22semidirect
27. C7×#432C22semidirect
28. C7×#395C22semidirect
29. C7×#394C22semidirect
36. C7×#376C22semidirect
37. C7×#327C22semidirect
38. C7×#327C22semidirect
39. C7×#324C22semidirect
41. #18332C7direct (N × H)
59. C7×C4⋊Q8D4semidirect
61. C7×#127D4semidirect
81. C4×C7×D4D4semidirect
108. #3449C2×C7semidirect
109. #3213C2×C7semidirect
110. #2610C2×C7semidirect
111. #1604C2×C7semidirect
112. #1601C2×C7semidirect
113. #1598C2×C7semidirect
114. #1595C2×C7semidirect
115. #1505C2×C7semidirect
116. #1464C2×C7semidirect
201. C2×#124C22×C7semidirect
207. #448C22×C7semidirect
208. #447C22×C7semidirect
209. #432C22×C7semidirect
210. #395C22×C7semidirect
211. #394C22×C7semidirect
218. #376C22×C7semidirect
219. #327C22×C7semidirect
220. #327C22×C7semidirect
221. #324C22×C7semidirect
302. C4⋊Q8C7×D4semidirect
304. #127C7×D4semidirect
324. C4×D4C7×D4semidirect
528. C7#18332direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34492
C7×#32132
C7×#26102
C7×#16042
C7×#16012
C7×#15982
C7×#15952
C7×#15052
C7×#14642
#183327