Finite Group Databasebeta

← C7×#24132 #13201348

Subgroups — C7×#24132 #13201348

Order 1792.

1336 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×#547C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C2×C7×#440C211yesmaximal, characteristic, complemented
4C7×#2536C211yesmaximal, characteristic, complemented
5C7×#2033C212yesmaximal, complemented
6C7×#2000C212yesmaximal, complemented
7C7×#1993C211yesmaximal, characteristic, complemented
8C7×#1955C211yesmaximal, characteristic, complemented
9C7×#1872C212yesmaximal, complemented
10C7×#1852C212yesmaximal, complemented
11C7×#1804C211yesmaximal, characteristic, complemented
12C22×C7×#41—21no—
13C22×C7×SD16C2211yescharacteristic
14C7×#547—24no—
15C7×#547—24no—
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#123—22no—
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8C2212yes—
20C2×C7×#121C2212yes—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2211yescharacteristic
24C22×C7×C4⋊C4C2212yes—
25C7×#477—24no—
26C7×#440—24no—
27C2×C7×#107C2211yescharacteristic
28C2×C7×#107C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#103C2212yes—
31C2×C7×#103C2211yescharacteristic
32C2×C7×#102C2211yescharacteristic
33C2×C7×#102C2212yes—
34C2×C7×C22⋊C8C2212yes—
35C2×C7×C22⋊C8C2212yes—
36C2×C7×#97C2211yescharacteristic
37C7×#398—21no—
38C7×#394—21no—
39C7×#388—22no—
40C7×#382C2211yescharacteristic
41C7×#380C2212yes—
42C7×#340C2212yes—
43C7×#340C2212yes—
44C7×#340C2212yes—
45#24132C711yesmaximal, characteristic, direct product factor, Sylow
46C2×C7×#41—28no—
47C2×C7×#41—48no—
48C22×C7×Q8—21no—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×D4—22no—
51C22×C7×D4D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C23×C4×C7—22no—
54C23×C4×C7—21no—
55C23×C4×C7C2312yes—
56C2×C7×SD16—22no—
57C2×C7×SD16—22no—
58C2×C7×SD16—22no—
59C22×C7×C8C2311yescharacteristic
60C22×C7×C8C2312yes—
61C22×C7×C8C2311yescharacteristic
62C7×C4⋊Q8—22no—
63C7×C4⋊Q8D414yes—
64C7×#123—216no—
65C7×C22⋊Q8—28no—
66C7×C22⋊Q8—24no—
67C7×C22⋊Q8—42no—
68C7×#121—24no—
69C7×#121—24no—
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4D414yes—
76C2×C7×C4⋊C4—22no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4—24no—
79C2×C7×C4⋊C4—24no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4D411yescharacteristic
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C4⋊C4C2311yescharacteristic
85C2×C7×C4⋊C4C2311yescharacteristic
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C22⋊C4—21no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4D412yes—
92C2×C42×C7—21no—
93C2×C42×C7C2311yescharacteristic
94C7×#107—24no—
95C7×#107—22no—
96C7×C4⋊C8—22no—
97C7×#103—24no—
98C7×#103—22no—
99C7×#102—22no—
100C7×#102—42no—
101C7×C22⋊C8—24no—
102C7×C22⋊C8—42no—
103C7×#97—22no—
104C7×#95—22no—
105C7×#95—21no—
106C7×#95—21no—
107C2×#547C2×C711yescharacteristic
108C2×#477C2×C711yescharacteristic, complemented
109C2×#440C2×C711yescharacteristic, complemented
110#2536C2×C711yescharacteristic, complemented
111#2033C2×C712yescomplemented
112#2000C2×C712yescomplemented
113#1993C2×C711yescharacteristic, complemented
114#1955C2×C711yescharacteristic, complemented
115#1872C2×C712yescomplemented
116#1852C2×C712yescomplemented
117#1804C2×C711yescharacteristic, complemented
118C24×C7—21no—
119C24×C7C2×D412yes—
120C7×#41—432no—
121C2×C7×Q8—24no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—44no—
128C2×C7×Q8#4112yescomplemented
129C2×C7×D4—24no—
130C2×C7×D4—28no—
131C2×C7×D4—22no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—44no—
138C2×C7×D4—44no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—28no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—22no—
144C22×C4×C7—24no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—22no—
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7—21no—
153C22×C4×C7—44no—
154C22×C4×C7—44no—
155C22×C4×C7—44no—
156C22×C4×C7C2×D412yes—
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2411yescharacteristic
160C7×SD16—28no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C2×C7×C8—21no—
167C2×C7×C8—22no—
168C2×C7×C8—21no—
169C2×C7×C8#4112yescomplemented
170C7×C4⋊C4—28no—
171C7×C4⋊C4—24no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—22no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—44no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4C2×D412yes—
182C7×C4⋊C4C2×D414yes—
183C7×C22⋊C4—28no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—44no—
188C42×C7—42no—
189C42×C7C2×D412yes—
190C42×C7C2×D412yes—
191C22×#41—21no—
192C22×SD16C22×C711yescharacteristic
193#547—24no—
194#547—24no—
195C2×C4⋊Q8C22×C711yescharacteristic
196C2×#123—22no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8C22×C712yes—
199C2×#121C22×C712yes—
200C2×C4×Q8C22×C711yescharacteristic
201C2×C4×D4C22×C711yescharacteristic
202C2×C4×D4C22×C711yescharacteristic
203C22×C4⋊C4C22×C712yes—
204#477—24no—
205#440—24no—
206C2×#107C22×C712yes—
207C2×#107C22×C711yescharacteristic
208C2×C4⋊C8C22×C711yescharacteristic
209C2×#103C22×C712yes—
210C2×#103C22×C711yescharacteristic
211C2×#102C22×C711yescharacteristic
212C2×#102C22×C712yes—
213C2×C22⋊C8C22×C712yes—
214C2×C22⋊C8C22×C712yes—
215C2×#97C22×C711yescharacteristic
216#398—21no—
217#394—21no—
218#388—22no—
219#382C22×C711yescharacteristic
220#380C22×C712yes—
221#340C22×C712yes—
222#340C22×C712yes—
223#340C22×C712yes—
224C23×C7—22no—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7AGL(1,ℤ/8)14yes—
236C23×C7C22×D411yescharacteristic
237C7×Q8—22no—
238C7×Q8—22no—
239C7×Q8—48no—
240C7×Q8—41no—
241C7×Q8—41no—
242C7×D4—28no—
243C7×D4—22no—
244C7×D4—22no—
245C7×D4—416no—
246C7×D4—41no—
247C7×D4—41no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—48no—
257C2×C4×C7—416no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—42no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—42no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7—42no—
271C2×C4×C7—42no—
272C2×C4×C7#12012yes—
273C2×C4×C7#12012yes—
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×D411yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7H(5,𝔽2)11yescharacteristic
279C2×C4×C72−1+411yescharacteristic
280C7×C8—22no—
281C7×C8—41no—
282C7×C8—42no—
283C2×#41—28no—
284C2×#41—48no—
285C22×Q8—21no—
286C22×Q8C23×C711yescharacteristic
287C22×D4—22no—
288C22×D4C7×D411yescharacteristic
289C22×D4C23×C711yescharacteristic
290C23×C4—22no—
291C23×C4—21no—
292C23×C4C23×C712yes—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×SD16—22no—
296C22×C8C23×C711yescharacteristic
297C22×C8C23×C712yes—
298C22×C8C23×C711yescharacteristic
299C4⋊Q8—22no—
300C4⋊Q8C7×D414yes—
301#123—216no—
302C22⋊Q8—28no—
303C22⋊Q8—24no—
304C22⋊Q8—42no—
305#121—24no—
306#121—24no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4C7×D414yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4C7×D411yescharacteristic
320C2×C4⋊C4C23×C711yescharacteristic
321C2×C4⋊C4C23×C712yes—
322C2×C4⋊C4C23×C711yescharacteristic
323C2×C4⋊C4C23×C712yes—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C7×D412yes—
329C2×C42—21no—
330C2×C42C23×C711yescharacteristic
331#107—22no—
332#107—24no—
333C4⋊C8—22no—
334#103—22no—
335#103—24no—
336#102—22no—
337#102—42no—
338C22⋊C8—24no—
339C22⋊C8—42no—
340#97—22no—
341#95—22no—
342#95—21no—
343#95—21no—
344C22×C7—24no—
345C22×C7—24no—
346C22×C7—24no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—82no—
355C22×C7—82no—
356C22×C7C2×#12011yescharacteristic
357C22×C7#54611yescharacteristic
358C22×C7#54711yescharacteristic
359C22×C7C2×AGL(1,ℤ/8)12yes—
360C22×C7#57812yes—
361C4×C7—22no—
362C4×C7—22no—
363C4×C7—42no—
364C4×C7—42no—
365C4×C7—42no—
366C4×C7—44no—
367C4×C7—81no—
368C4×C7—82no—
369C4×C7#52511yescharacteristic
370C4×C7#53711yescharacteristic
371C4×C7#54811yescharacteristic
372C4×C7#55011yescharacteristic
373C24—21no—
374C24C2×C7×D412yes—
375#41—432no—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—44no—
383C2×Q8C7×#4112yescomplemented
384C2×D4—24no—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4C2×C7×D412yes—
395C22×C4—28no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4C2×C7×D412yes—
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C24×C711yescharacteristic
415SD16—28no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8—21no—
424C2×C8C7×#4112yescomplemented
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4C2×C7×D412yes—
437C4⋊C4C2×C7×D414yes—
438C22⋊C4—28no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C42—42no—
444C42C2×C7×D412yes—
445C42C2×C7×D412yes—
446C2×C7—44no—
447C2×C7—82no—
448C2×C7#240711yescharacteristic
449C2×C7#300311yescharacteristic
450C2×C7#301111yescharacteristic
451C2×C7#330812yes—
452C2×C7#334312yes—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23C7×AGL(1,ℤ/8)14yes—
465C23C22×C7×D411yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—48no—
469Q8—41no—
470Q8—41no—
471D4—28no—
472D4—22no—
473D4—22no—
474D4—416no—
475D4—41no—
476D4—41no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—416no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4C7×#12012yes—
502C2×C4C7×#12012yes—
503C2×C4C22×C7×D411yescharacteristic
504C2×C4C22×C7×D411yescharacteristic
505C2×C4C2×C7×#4111yescharacteristic
506C2×C4C2×C7×#4111yescharacteristic
507C2×C4C7×H(5,𝔽2)11yescharacteristic
508C2×C4C7×2−1+411yescharacteristic
509C8—22no—
510C8—41no—
511C8—42no—
512C7#2413211yescharacteristic, direct product factor, Sylow
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22—82no—
525C22C2×C7×#12011yescharacteristic
526C22C7×#54611yescharacteristic
527C22C7×#54711yescharacteristic
528C22C2×C7×AGL(1,ℤ/8)12yes—
529C22C7×#57812yes—
530C4—22no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—44no—
536C4—81no—
537C4—82no—
538C4C7×#52511yescharacteristic
539C4C7×#53711yescharacteristic
540C4C7×#54811yescharacteristic
541C4C7×#55011yescharacteristic
542C2—44no—
543C2—82no—
544C2C7×#240711yescharacteristic
545C2C7×#300311yescharacteristic
546C2C7×#301111yescharacteristic
547C2C7×#330812yes—
548C2C7×#334312yes—
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC2×C7×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C7×#477C2semidirect
3. C2×C7×#440C2semidirect
4. C7×#2536C2semidirect
5. C7×#2033C2semidirect
6. C7×#2000C2semidirect
7. C7×#1993C2semidirect
8. C7×#1955C2semidirect
9. C7×#1872C2semidirect
10. C7×#1852C2semidirect
11. C7×#1804C2semidirect
45. #24132C7direct (N × H)
108. C2×#477C2×C7semidirect
109. C2×#440C2×C7semidirect
110. #2536C2×C7semidirect
111. #2033C2×C7semidirect
112. #2000C2×C7semidirect
113. #1993C2×C7semidirect
114. #1955C2×C7semidirect
115. #1872C2×C7semidirect
116. #1852C2×C7semidirect
117. #1804C2×C7semidirect
128. C2×C7×Q8#41semidirect
169. C2×C7×C8#41semidirect
383. C2×Q8C7×#41semidirect
424. C2×C8C7×#41semidirect
512. C7#24132direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C2×C7×#4772
C2×C7×#4402
C7×#25362
C7×#20332
C7×#20002
C7×#19932
C7×#19552
C7×#18722
C7×#18522
C7×#18042
#241327