Finite Group Databasebeta

← C7×#24304 #13201520

Subgroups — C7×#24304 #13201520

Order 1792.

1150 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#533C211yesmaximal, characteristic, complemented
2C2×C7×#478C212yesmaximal, complemented
3C4×C7×#121C211yesmaximal, characteristic
4C7×#2058C211yesmaximal, characteristic, complemented
5C7×#2057C211yesmaximal, characteristic, complemented
6C7×#2023C211yesmaximal, characteristic, complemented
7C7×#2023C211yesmaximal, characteristic, complemented
8C7×#1989C212yesmaximal, complemented
9C7×#1875C212yesmaximal
10C7×#1854C212yesmaximal, complemented
11C7×#1797C211yesmaximal, characteristic, complemented
12C7×#533—28no—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#125C2212yes—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8—21no—
18C2×C7×#121C2211yescharacteristic
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8—21no—
21C2×C4×C7×D4—21no—
22C2×C4×C7×D4C2211yescharacteristic
23C2×C7×#117C2212yes—
24C22×C42×C7C2211yescharacteristic
25C7×#478—28no—
26C2×C7×#107C2212yes—
27C2×C7×#106C2212yes—
28C2×C7×C4⋊C8C2212yes—
29C2×C7×#102C2212yes—
30C2×C7×#102C2212yes—
31C2×C7×C22⋊C8C2212yes—
32C7×#390C2211yescharacteristic
33C7×#389C2211yescharacteristic
34C7×#388—21no—
35C7×#386—21no—
36C7×#380C2211yescharacteristic
37C4×C7×C4⋊C4C2211yescharacteristic
38C4×C7×C22⋊C4—21no—
39C7×#340C2212yes—
40C7×#340C2212yes—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#336C2211yescharacteristic
44C7×#336C2211yescharacteristic
45#24304C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—21no—
47C22×C7×Q8—21no—
48C22×C7×D4—21no—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7D411yescharacteristic
52C23×C4×C7D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C22×C7×C8C2312yes—
55C22×C7×C8C2312yes—
56C7×C4⋊Q8—24no—
57C7×C4⋊Q8—24no—
58C7×#125—24no—
59C7×#125D418yescomplemented
60C7×C22⋊Q8—28no—
61C7×C22⋊Q8—28no—
62C7×#121—22no—
63C7×#121Q814yescomplemented
64C4×C7×Q8—28no—
65C4×C7×Q8—28no—
66C4×C7×D4—24no—
67C4×C7×D4—44no—
68C7×#117—24no—
69C7×#117—24no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C4⋊C4C2312yes—
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—22no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C7×C22⋊C4—21no—
85C2×C7×C22⋊C4—21no—
86C2×C42×C7—22no—
87C2×C42×C7—24no—
88C2×C42×C7—21no—
89C2×C42×C7D411yescharacteristic
90C2×C42×C7D411yescharacteristic
91C2×C42×C7C2311yescharacteristic
92C2×C42×C7C2311yescharacteristic
93C7×#107—24no—
94C7×#106—24no—
95C7×C4⋊C8—24no—
96C7×#102—24no—
97C7×#102—24no—
98C7×C22⋊C8—24no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C2×#533C2×C711yescharacteristic, complemented
103C2×#478C2×C712yescomplemented
104C4×#121C2×C711yescharacteristic
105#2058C2×C711yescharacteristic, complemented
106#2057C2×C711yescharacteristic, complemented
107#2023C2×C711yescharacteristic, complemented
108#2023C2×C711yescharacteristic, complemented
109#1989C2×C712yescomplemented
110#1875C2×C712yes—
111#1854C2×C712yescomplemented
112#1797C2×C711yescharacteristic, complemented
113C24×C7—21no—
114C24×C7C2×D411yescharacteristic
115C2×C7×Q8—24no—
116C2×C7×Q8—24no—
117C2×C7×Q8—42no—
118C2×C7×Q8—42no—
119C2×C7×Q8—42no—
120C2×C7×Q8—42no—
121C2×C7×D4—24no—
122C2×C7×D4—22no—
123C2×C7×D4—21no—
124C2×C7×D4—22no—
125C2×C7×D4—42no—
126C2×C7×D4—42no—
127C2×C7×D4C2×Q812yes—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×Q812yes—
150C22×C4×C7#4112yes—
151C22×C4×C7C2411yescharacteristic
152C2×C7×C8—22no—
153C2×C7×C8—22no—
154C2×C7×C8—22no—
155C2×C7×C8—22no—
156C2×C7×C8—22no—
157C2×C7×C8—22no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—24no—
160C7×C4⋊C4—28no—
161C7×C4⋊C4—28no—
162C7×C4⋊C4—22no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—42no—
168C7×C4⋊C4C2×D414yes—
169C7×C4⋊C4C2×D414yes—
170C7×C4⋊C4C2×Q812yes—
171C7×C22⋊C4—28no—
172C7×C22⋊C4—24no—
173C7×C22⋊C4—42no—
174C7×C22⋊C4—42no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—42no—
177C42×C7—28no—
178C42×C7—22no—
179C42×C7—22no—
180C42×C7—22no—
181C42×C7—42no—
182C42×C7C2×D414yes—
183#533—28no—
184C2×C4⋊Q8C22×C711yescharacteristic
185C2×C4⋊Q8C22×C711yescharacteristic
186C2×#125C22×C712yes—
187C2×C22⋊Q8—21no—
188C2×C22⋊Q8—21no—
189C2×#121C22×C711yescharacteristic
190C2×C4×Q8—21no—
191C2×C4×Q8—21no—
192C2×C4×D4—21no—
193C2×C4×D4C22×C711yescharacteristic
194C2×#117C22×C712yes—
195C22×C42C22×C711yescharacteristic
196#478—28no—
197C2×#107C22×C712yes—
198C2×#106C22×C712yes—
199C2×C4⋊C8C22×C712yes—
200C2×#102C22×C712yes—
201C2×#102C22×C712yes—
202C2×C22⋊C8C22×C712yes—
203#390C22×C711yescharacteristic
204#389C22×C711yescharacteristic
205#388—21no—
206#386—21no—
207#380C22×C711yescharacteristic
208C4×C4⋊C4C22×C711yescharacteristic
209C4×C22⋊C4—21no—
210#340C22×C712yes—
211#340C22×C712yes—
212#340C22×C712yes—
213#340C22×C712yes—
214#336C22×C711yescharacteristic
215#336C22×C711yescharacteristic
216C23×C7—21no—
217C23×C7—21no—
218C23×C7—21no—
219C23×C7—21no—
220C23×C7—21no—
221C23×C7—21no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7C22⋊Q812yes—
230C23×C7C22×D411yescharacteristic
231C7×Q8—48no—
232C7×Q8—48no—
233C7×D4—24no—
234C7×D4—48no—
235C7×D4—42no—
236C2×C4×C7—21no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—21no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—21no—
245C2×C4×C7—21no—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—24no—
249C2×C4×C7—24no—
250C2×C4×C7—22no—
251C2×C4×C7—44no—
252C2×C4×C7—44no—
253C2×C4×C7—42no—
254C2×C4×C7—41no—
255C2×C4×C7—42no—
256C2×C4×C7—41no—
257C2×C4×C7—42no—
258C2×C4×C7—42no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7—41no—
265C2×C4×C7—84no—
266C2×C4×C7#12012yes—
267C2×C4×C7#12012yes—
268C2×C4×C7C22⋊Q812yes—
269C2×C4×C7#13512yes—
270C2×C4×C7AGL(1,ℤ/8)12yes—
271C2×C4×C7C22×D412yes—
272C2×C4×C7C22×Q811yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C72−1+412yes—
275C7×C8—42no—
276C7×C8—42no—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×D4—21no—
280C22×D4C23×C711yescharacteristic
281C23×C4—21no—
282C23×C4C7×D411yescharacteristic
283C23×C4C7×D411yescharacteristic
284C23×C4C23×C711yescharacteristic
285C22×C8C23×C712yes—
286C22×C8C23×C712yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289#125—24no—
290#125C7×D418yescomplemented
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293#121—22no—
294#121C7×Q814yescomplemented
295C4×Q8—28no—
296C4×Q8—28no—
297C4×D4—24no—
298C4×D4—44no—
299#117—24no—
300#117—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C23×C711yescharacteristic
307C2×C4⋊C4C23×C711yescharacteristic
308C2×C4⋊C4C23×C712yes—
309C2×C4⋊C4C23×C712yes—
310C2×C4⋊C4C23×C711yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42C7×D411yescharacteristic
321C2×C42C7×D411yescharacteristic
322C2×C42C23×C711yescharacteristic
323C2×C42C23×C711yescharacteristic
324#107—24no—
325#106—24no—
326C4⋊C8—24no—
327#102—24no—
328#102—24no—
329C22⋊C8—24no—
330#95—21no—
331#95—21no—
332#95—21no—
333C22×C7—22no—
334C22×C7—22no—
335C22×C7—22no—
336C22×C7—41no—
337C22×C7—41no—
338C22×C7—41no—
339C22×C7—41no—
340C22×C7—42no—
341C22×C7—42no—
342C22×C7—42no—
343C22×C7—81no—
344C22×C7—81no—
345C22×C7—81no—
346C22×C7—81no—
347C22×C7C2×#12011yescharacteristic
348C22×C7C2×C22⋊Q811yescharacteristic
349C22×C7#53611yescharacteristic
350C22×C7C2×#13511yescharacteristic
351C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C7#57511yescharacteristic
353C22×C7#57511yescharacteristic
354C4×C7—22no—
355C4×C7—22no—
356C4×C7—22no—
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7—44no—
361C4×C7—44no—
362C4×C7—81no—
363C4×C7—81no—
364C4×C7#54812yes—
365C4×C7D4×Q812yes—
366C24—21no—
367C24C2×C7×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4C2×C7×Q812yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4C2×C7×D411yescharacteristic
398C22×C4C2×C7×D411yescharacteristic
399C22×C4C2×C7×D411yescharacteristic
400C22×C4C2×C7×D411yescharacteristic
401C22×C4C2×C7×D411yescharacteristic
402C22×C4C2×C7×Q812yes—
403C22×C4C7×#4112yes—
404C22×C4C24×C711yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—42no—
421C4⋊C4C2×C7×D414yes—
422C4⋊C4C2×C7×D414yes—
423C4⋊C4C2×C7×Q812yes—
424C22⋊C4—28no—
425C22⋊C4—24no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C42—28no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—42no—
435C42C2×C7×D414yes—
436C2×C7—42no—
437C2×C7—82no—
438C2×C7#240911yescharacteristic
439C2×C7#300211yescharacteristic
440C2×C7#300611yescharacteristic
441C2×C7#307611yescharacteristic
442C2×C7#307711yescharacteristic
443C2×C7#322811yescharacteristic
444C2×C7#324011yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C7×C22⋊Q812yes—
459C23C22×C7×D411yescharacteristic
460Q8—48no—
461Q8—48no—
462D4—24no—
463D4—48no—
464D4—42no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—84no—
495C2×C4C7×#12012yes—
496C2×C4C7×#12012yes—
497C2×C4C7×C22⋊Q812yes—
498C2×C4C7×#13512yes—
499C2×C4C7×AGL(1,ℤ/8)12yes—
500C2×C4C22×C7×D412yes—
501C2×C4C22×C7×Q811yescharacteristic
502C2×C4C2×C7×#4111yescharacteristic
503C2×C4C7×2−1+412yes—
504C8—42no—
505C8—42no—
506C7#2430411yescharacteristic, direct product factor, Sylow
507C22—22no—
508C22—22no—
509C22—22no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22C2×C7×#12011yescharacteristic
522C22C2×C7×C22⋊Q811yescharacteristic
523C22C7×#53611yescharacteristic
524C22C2×C7×#13511yescharacteristic
525C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
526C22C7×#57511yescharacteristic
527C22C7×#57511yescharacteristic
528C4—22no—
529C4—22no—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—44no—
534C4—44no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4C7×#54812yes—
539C4C7×D4×Q812yes—
540C2—42no—
541C2—82no—
542C2C7×#240911yescharacteristic
543C2C7×#300211yescharacteristic
544C2C7×#300611yescharacteristic
545C2C7×#307611yescharacteristic
546C2C7×#307711yescharacteristic
547C2C7×#322811yescharacteristic
548C2C7×#324011yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#533C2semidirect
2. C2×C7×#478C2semidirect
4. C7×#2058C2semidirect
5. C7×#2057C2semidirect
6. C7×#2023C2semidirect
7. C7×#2023C2semidirect
8. C7×#1989C2semidirect
10. C7×#1854C2semidirect
11. C7×#1797C2semidirect
45. #24304C7direct (N × H)
59. C7×#125D4semidirect
63. C7×#121Q8semidirect
102. C2×#533C2×C7semidirect
103. C2×#478C2×C7semidirect
105. #2058C2×C7semidirect
106. #2057C2×C7semidirect
107. #2023C2×C7semidirect
108. #2023C2×C7semidirect
109. #1989C2×C7semidirect
111. #1854C2×C7semidirect
112. #1797C2×C7semidirect
290. #125C7×D4semidirect
294. #121C7×Q8semidirect
506. C7#24304direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5332
C2×C7×#4782
C4×C7×#1212
C7×#20582
C7×#20572
C7×#20232
C7×#19892
C7×#18752
C7×#18542
C7×#17972
#243047