Finite Group Databasebeta

← C7×#25944 #13203160

Subgroups — C7×#25944 #13203160

Order 1792.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×C7×#490C211yesmaximal, characteristic
3C2×C7×#460C211yesmaximal, characteristic
4C7×#2548C211yesmaximal, characteristic
5C7×#2025C212yesmaximal, complemented
6C7×#2023C212yesmaximal, complemented
7C7×#1989C211yesmaximal, characteristic, complemented
8C7×#1985C211yesmaximal, characteristic, complemented
9C7×#1880C212yesmaximal, complemented
10C7×#1880C212yesmaximal, complemented
11C7×#1763C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2212yes—
13C7×#557—24no—
14C7×#557—24no—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#127C2211yescharacteristic
18C2×C7×#125—21no—
19C2×C7×#125—21no—
20C2×C7×#125C2211yescharacteristic
21C2×C7×#125C2211yescharacteristic
22C2×C7×#124—21no—
23C2×C4×C7×Q8—21no—
24C2×C4×C7×Q8C2211yescharacteristic
25C2×C4×C7×Q8C2211yescharacteristic
26C2×C4×C7×Q8C2211yescharacteristic
27C2×C4×C7×Q8C2211yescharacteristic
28C7×#490—24no—
29C7×#460—24no—
30C2×C7×C4⋊C8C2211yescharacteristic
31C2×C7×#103C2212yes—
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C7×#102C2212yes—
35C2×C7×#97C2211yescharacteristic
36C7×#394—21no—
37C7×#394—21no—
38C7×#390—21no—
39C7×#390C2212yes—
40C7×#390C2212yes—
41C4×C7×C4⋊C4C2211yescharacteristic
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45C7×#336C2212yes—
46C7×#336C2212yes—
47#25944C711yesmaximal, characteristic, direct product factor, Sylow
48C22×C7×Q8—21no—
49C22×C7×Q8D411yescharacteristic
50C22×C7×Q8C2311yescharacteristic
51C22×C7×D4—21no—
52C22×C7×D4—21no—
53C22×C7×D4C2312yes—
54C2×C7×SD16—24no—
55C2×C7×SD16—24no—
56C2×C7×SD16—24no—
57C22×C7×C8C2312yes—
58C22×C7×C8C2312yes—
59C7×C4⋊Q8—28no—
60C7×C4⋊Q8—22no—
61C7×C4⋊Q8—22no—
62C7×#127—22no—
63C7×#127—22no—
64C7×#125—28no—
65C7×#125—28no—
66C7×#125—22no—
67C7×#125—22no—
68C7×#125—22no—
69C7×#125D414yes—
70C7×#124—44no—
71C4×C7×Q8—28no—
72C4×C7×Q8—22no—
73C4×C7×Q8—22no—
74C4×C7×Q8—22no—
75C4×C7×Q8—22no—
76C4×C7×Q8—22no—
77C4×C7×Q8—22no—
78C4×C7×Q8—22no—
79C4×C7×Q8D414yes—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—21no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4D411yescharacteristic
89C2×C7×C4⋊C4D411yescharacteristic
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2311yescharacteristic
93C2×C7×C4⋊C4C2311yescharacteristic
94C2×C7×C4⋊C4C2311yescharacteristic
95C2×C7×C22⋊C4—22no—
96C2×C7×C22⋊C4—22no—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—22no—
99C2×C42×C7—21no—
100C2×C42×C7—21no—
101C2×C42×C7—21no—
102C2×C42×C7—21no—
103C2×C42×C7C2311yescharacteristic
104C2×C42×C7C2311yescharacteristic
105C2×C42×C7C2311yescharacteristic
106C7×C4⋊C8—22no—
107C7×#103—24no—
108C7×#102—24no—
109C7×#102—24no—
110C7×#102—24no—
111C7×#97—22no—
112C7×#95—21no—
113C2×#557C2×C711yescharacteristic, complemented
114C2×#490C2×C711yescharacteristic
115C2×#460C2×C711yescharacteristic
116#2548C2×C711yescharacteristic
117#2025C2×C712yescomplemented
118#2023C2×C712yescomplemented
119#1989C2×C711yescharacteristic, complemented
120#1985C2×C711yescharacteristic, complemented
121#1880C2×C712yescomplemented
122#1880C2×C712yescomplemented
123#1763C2×C711yescharacteristic, complemented
124C24×C7—22no—
125C2×C7×Q8—24no—
126C2×C7×Q8—22no—
127C2×C7×Q8—22no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8—41no—
131C2×C7×Q8—42no—
132C2×C7×Q8—42no—
133C2×C7×Q8C2×D412yes—
134C2×C7×Q8#4114yes—
135C2×C7×D4—22no—
136C2×C7×D4—22no—
137C2×C7×D4—22no—
138C2×C7×D4—22no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—42no—
142C2×C7×D4—42no—
143C2×C7×D4—42no—
144C2×C7×D4—42no—
145C2×C7×D4—82no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—22no—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C7×SD16—48no—
159C2×C7×C8—22no—
160C2×C7×C8—22no—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—24no—
168C7×C4⋊C4—24no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—22no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—22no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4C2×D412yes—
182C7×C4⋊C4C2×D412yes—
183C7×C4⋊C4C2×D412yes—
184C7×C22⋊C4—44no—
185C7×C22⋊C4—44no—
186C7×C22⋊C4—82no—
187C7×C22⋊C4—82no—
188C42×C7—24no—
189C42×C7—24no—
190C42×C7—24no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—42no—
196C42×C7C2×D412yes—
197C42×C7C2×D412yes—
198C22×SD16C22×C712yes—
199#557—24no—
200#557—24no—
201C2×C4⋊Q8—21no—
202C2×C4⋊Q8C22×C711yescharacteristic
203C2×#127C22×C711yescharacteristic
204C2×#125—21no—
205C2×#125—21no—
206C2×#125C22×C711yescharacteristic
207C2×#125C22×C711yescharacteristic
208C2×#124—21no—
209C2×C4×Q8—21no—
210C2×C4×Q8C22×C711yescharacteristic
211C2×C4×Q8C22×C711yescharacteristic
212C2×C4×Q8C22×C711yescharacteristic
213C2×C4×Q8C22×C711yescharacteristic
214#490—24no—
215#460—24no—
216C2×C4⋊C8C22×C711yescharacteristic
217C2×#103C22×C712yes—
218C2×#102C22×C712yes—
219C2×#102C22×C712yes—
220C2×#102C22×C712yes—
221C2×#97C22×C711yescharacteristic
222#394—21no—
223#394—21no—
224#390—21no—
225#390C22×C712yes—
226#390C22×C712yes—
227C4×C4⋊C4C22×C711yescharacteristic
228#340C22×C712yes—
229#340C22×C712yes—
230#340C22×C712yes—
231#336C22×C712yes—
232#336C22×C712yes—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7—42no—
238C23×C7—42no—
239C23×C7—42no—
240C23×C7C22×D411yescharacteristic
241C7×Q8—22no—
242C7×Q8—24no—
243C7×Q8—44no—
244C7×Q8—48no—
245C7×Q8#12414yescomplemented
246C7×D4—42no—
247C7×D4—42no—
248C7×D4—42no—
249C7×D4—42no—
250C7×D4—84no—
251C7×D4—84no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—24no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—42no—
276C2×C4×C7—42no—
277C2×C4×C7#12012yes—
278C2×C4×C7#12012yes—
279C2×C4×C7AGL(1,ℤ/8)12yes—
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4112yes—
283C2×C4×C72−1+412yes—
284C7×C8—42no—
285C7×C8—42no—
286C22×Q8—21no—
287C22×Q8C7×D411yescharacteristic
288C22×Q8C23×C711yescharacteristic
289C22×D4—21no—
290C22×D4—21no—
291C22×D4C23×C712yes—
292C2×SD16—24no—
293C2×SD16—24no—
294C2×SD16—24no—
295C22×C8C23×C712yes—
296C22×C8C23×C712yes—
297C4⋊Q8—28no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300#127—22no—
301#127—22no—
302#125—28no—
303#125—28no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125C7×D414yes—
308#124—44no—
309C4×Q8—28no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8C7×D414yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C7×D411yescharacteristic
327C2×C4⋊C4C7×D411yescharacteristic
328C2×C4⋊C4C7×D411yescharacteristic
329C2×C4⋊C4C23×C712yes—
330C2×C4⋊C4C23×C711yescharacteristic
331C2×C4⋊C4C23×C711yescharacteristic
332C2×C4⋊C4C23×C711yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42C23×C711yescharacteristic
342C2×C42C23×C711yescharacteristic
343C2×C42C23×C711yescharacteristic
344C4⋊C8—22no—
345#103—24no—
346#102—24no—
347#102—24no—
348#102—24no—
349#97—22no—
350#95—21no—
351C22×C7—82no—
352C22×C7—82no—
353C22×C7—82no—
354C22×C7—82no—
355C22×C7—82no—
356C22×C7—82no—
357C22×C7—82no—
358C22×C7C2×#12011yescharacteristic
359C22×C7#54812yes—
360C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C7#57511yescharacteristic
362C22×C7#57712yes—
363C4×C7—24no—
364C4×C7—22no—
365C4×C7—22no—
366C4×C7—22no—
367C4×C7—41no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—41no—
371C4×C7—41no—
372C4×C7—42no—
373C4×C7—82no—
374C4×C7C2×#12411yescharacteristic
375C4×C7#53611yescharacteristic
376C4×C7#54812yes—
377C24—22no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8C2×C7×D412yes—
387C2×Q8C7×#4114yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—82no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4C2×C7×D411yescharacteristic
405C22×C4C2×C7×D411yescharacteristic
406C22×C4C2×C7×D411yescharacteristic
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C24×C711yescharacteristic
411SD16—48no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4C2×C7×D412yes—
435C4⋊C4C2×C7×D412yes—
436C4⋊C4C2×C7×D412yes—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—42no—
449C42C2×C7×D412yes—
450C42C2×C7×D412yes—
451C2×C7—162no—
452C2×C7#243611yescharacteristic
453C2×C7#300611yescharacteristic
454C2×C7#301011yescharacteristic
455C2×C7#336812yes—
456C2×C7#337112yes—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C22×C7×D411yescharacteristic
465Q8—22no—
466Q8—24no—
467Q8—44no—
468Q8—48no—
469Q8C7×#12414yescomplemented
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—84no—
475D4—84no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4C7×#12012yes—
502C2×C4C7×#12012yes—
503C2×C4C7×AGL(1,ℤ/8)12yes—
504C2×C4C22×C7×D411yescharacteristic
505C2×C4C22×C7×D411yescharacteristic
506C2×C4C2×C7×#4112yes—
507C2×C4C7×2−1+412yes—
508C8—42no—
509C8—42no—
510C7#2594411yescharacteristic, direct product factor, Sylow
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22C2×C7×#12011yescharacteristic
519C22C7×#54812yes—
520C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
521C22C7×#57511yescharacteristic
522C22C7×#57712yes—
523C4—24no—
524C4—22no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—82no—
534C4C2×C7×#12411yescharacteristic
535C4C7×#53611yescharacteristic
536C4C7×#54812yes—
537C2—162no—
538C2C7×#243611yescharacteristic
539C2C7×#300611yescharacteristic
540C2C7×#301011yescharacteristic
541C2C7×#336812yes—
542C2C7×#337112yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
5. C7×#2025C2semidirect
6. C7×#2023C2semidirect
7. C7×#1989C2semidirect
8. C7×#1985C2semidirect
9. C7×#1880C2semidirect
10. C7×#1880C2semidirect
11. C7×#1763C2semidirect
47. #25944C7direct (N × H)
113. C2×#557C2×C7semidirect
117. #2025C2×C7semidirect
118. #2023C2×C7semidirect
119. #1989C2×C7semidirect
120. #1985C2×C7semidirect
121. #1880C2×C7semidirect
122. #1880C2×C7semidirect
123. #1763C2×C7semidirect
245. C7×Q8#124semidirect
469. Q8C7×#124semidirect
510. C7#25944direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×C7×#4902
C2×C7×#4602
C7×#25482
C7×#20252
C7×#20232
C7×#19892
C7×#19852
C7×#18802
C7×#17632
#259447