Finite Group Databasebeta

← C7×#24019 #13201235

Subgroups — C7×#24019 #13201235

Order 1792.

1286 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3486C211yesmaximal, characteristic
2C7×#3092C212yesmaximal, complemented
3C2×C7×#391C211yesmaximal, characteristic, complemented
4C7×#2038C214yesmaximal, complemented
5C7×#1896C212yesmaximal, complemented
6C7×#1857C212yesmaximal, complemented
7C7×#1793C212yesmaximal, complemented
8C7×#1784C211yesmaximal, characteristic, complemented
9C2×C7×2−1+4C2211yescharacteristic
10C23×C7×Q8C2211yescharacteristic
11C7×D4×Q8—22no—
12C7×D4×Q8—22no—
13C7×#548—22no—
14C7×#548—22no—
15C7×#536—22no—
16C7×#536C2212yes—
17C7×#525—22no—
18C7×#525C2212yes—
19C7×#519C2211yescharacteristic
20C22×C7×C4⋊C4C2211yescharacteristic
21C22×C42×C7C2211yescharacteristic
22C7×#484—22no—
23C7×#483—22no—
24C7×#481—22no—
25C7×#480—22no—
26C7×#428C2214yes—
27C2×C7×#104C2214yes—
28C7×#417C2214yes—
29C2×C7×#101C2212yes—
30C2×C7×#101C2212yes—
31C7×#407C2212yes—
32C7×#391—22no—
33C7×#391C2212yes—
34C7×#391C2212yes—
35C2×C7×#95—22no—
36C7×#339C2214yes—
37#24019C711yesmaximal, characteristic, direct product factor, Sylow
38C7×2−1+4—24no—
39C7×2−1+4—42no—
40C2×C7×#41—22no—
41C2×C7×#41—22no—
42C2×C7×#41C2312yes—
43C22×C7×Q8—22no—
44C22×C7×Q8—24no—
45C22×C7×Q8—22no—
46C22×C7×Q8—22no—
47C22×C7×Q8—42no—
48C22×C7×Q8D412yes—
49C22×C7×Q8C2311yescharacteristic
50C22×C7×Q8C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7—21no—
53C23×C4×C7D412yes—
54C23×C4×C7C2311yescharacteristic
55C2×C7×M16C2314yes—
56C7×C4⋊Q8—24no—
57C7×C4⋊Q8—22no—
58C7×#124—24no—
59C7×#124—22no—
60C7×#123—24no—
61C7×#123—22no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—24no—
65C7×C22⋊Q8—24no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8D412yes—
69C7×C22⋊Q8D412yes—
70C7×#121—22no—
71C7×#121—22no—
72C7×#121D412yes—
73C7×#121D412yes—
74C7×#120—22no—
75C4×C7×Q8—22no—
76C4×C7×Q8—22no—
77C4×C7×D4—24no—
78C4×C7×D4—24no—
79C4×C7×D4—22no—
80C4×C7×D4—22no—
81C7×#117—22no—
82C7×#117C2312yes—
83C2×C7×C4⋊C4—22no—
84C2×C7×C4⋊C4—22no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4C2312yes—
87C2×C42×C7—21no—
88C2×C42×C7—42no—
89C2×C42×C7C2312yes—
90C7×#107—24no—
91C7×#106—24no—
92C7×#104—44no—
93C7×#103—24no—
94C7×#102—24no—
95C7×#101—24no—
96C7×#101—42no—
97C7×C22⋊C8—22no—
98C7×C22⋊C8—22no—
99C7×#95—24no—
100C7×#95—24no—
101#3486C2×C711yescharacteristic
102#3092C2×C712yescomplemented
103C2×#391C2×C711yescharacteristic, complemented
104#2038C2×C714yescomplemented
105#1896C2×C712yescomplemented
106#1857C2×C712yescomplemented
107#1793C2×C712yescomplemented
108#1784C2×C711yescharacteristic, complemented
109C24×C7C2×D411yescharacteristic
110C7×#41—24no—
111C7×#41—24no—
112C7×#41—44no—
113C7×#41—48no—
114C7×#41—44no—
115C2×C7×Q8—22no—
116C2×C7×Q8—24no—
117C2×C7×Q8—24no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—42no—
124C2×C7×Q8—48no—
125C2×C7×Q8—48no—
126C2×C7×Q8—48no—
127C2×C7×Q8C2×D412yes—
128C2×C7×Q8#4114yes—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—22no—
132C2×C7×D4—22no—
133C2×C7×D4C2×D412yes—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—24no—
137C22×C4×C7—24no—
138C22×C4×C7—22no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—24no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—42no—
145C22×C4×C7—42no—
146C22×C4×C7—42no—
147C22×C4×C7—44no—
148C22×C4×C7—41no—
149C22×C4×C7—41no—
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2411yescharacteristic
156C7×M16—24no—
157C7×M16—24no—
158C2×C7×C8—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—24no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—24no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4C2×D412yes—
168C7×C4⋊C4C2×D412yes—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—24no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C42×C7—24no—
177C42×C7—22no—
178C42×C7—24no—
179C42×C7—42no—
180C2×2−1+4C22×C711yescharacteristic
181C23×Q8C22×C711yescharacteristic
182D4×Q8—22no—
183D4×Q8—22no—
184#548—22no—
185#548—22no—
186#536—22no—
187#536C22×C712yes—
188#525—22no—
189#525C22×C712yes—
190#519C22×C711yescharacteristic
191C22×C4⋊C4C22×C711yescharacteristic
192C22×C42C22×C711yescharacteristic
193#484—22no—
194#483—22no—
195#481—22no—
196#480—22no—
197#428C22×C714yes—
198C2×#104C22×C714yes—
199#417C22×C714yes—
200C2×#101C22×C712yes—
201C2×#101C22×C712yes—
202#407C22×C712yes—
203#391—22no—
204#391C22×C712yes—
205#391C22×C712yes—
206C2×#95—22no—
207#339C22×C714yes—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7#12011yescharacteristic
214C23×C7#12011yescharacteristic
215C23×C7C22×D411yescharacteristic
216C7×Q8—28no—
217C7×Q8—24no—
218C7×Q8—48no—
219C7×Q8—44no—
220C7×Q8—44no—
221C7×D4—24no—
222C7×D4—44no—
223C7×D4—42no—
224C7×D4—42no—
225C7×D4—44no—
226C7×D4—44no—
227C2×C4×C7—22no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—22no—
231C2×C4×C7—24no—
232C2×C4×C7—24no—
233C2×C4×C7—24no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—44no—
241C2×C4×C7—44no—
242C2×C4×C7—48no—
243C2×C4×C7—48no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—44no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—82no—
250C2×C4×C7—82no—
251C2×C4×C7#12012yes—
252C2×C4×C7#12314yes—
253C2×C4×C7C22×D412yes—
254C2×C4×C7C2×#4112yes—
255C2×C4×C7H(5,𝔽2)12yes—
256C7×C8—44no—
2572−1+4—24no—
2582−1+4—42no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C23×C712yes—
262C22×Q8—22no—
263C22×Q8—24no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—42no—
267C22×Q8C7×D412yes—
268C22×Q8C23×C711yescharacteristic
269C22×Q8C23×C711yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C7×D412yes—
273C23×C4C23×C711yescharacteristic
274C2×M16C23×C714yes—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277#124—24no—
278#124—22no—
279#123—24no—
280#123—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8C7×D412yes—
288C22⋊Q8C7×D412yes—
289#121—22no—
290#121—22no—
291#121C7×D412yes—
292#121C7×D412yes—
293#120—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300#117—22no—
301#117C23×C712yes—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4C23×C712yes—
306C2×C42—21no—
307C2×C42—42no—
308C2×C42C23×C712yes—
309#107—24no—
310#106—24no—
311#104—44no—
312#103—24no—
313#102—24no—
314#101—24no—
315#101—42no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318#95—24no—
319#95—24no—
320C22×C7—24no—
321C22×C7—24no—
322C22×C7—21no—
323C22×C7—21no—
324C22×C7—21no—
325C22×C7—21no—
326C22×C7—21no—
327C22×C7—21no—
328C22×C7—42no—
329C22×C7—42no—
330C22×C7—42no—
331C22×C7#45412yes—
332C22×C7#45612yes—
333C22×C7C2×#12011yescharacteristic
334C22×C7C2×#12311yescharacteristic
335C22×C7#53411yescharacteristic
336C4×C7—22no—
337C4×C7—24no—
338C4×C7—44no—
339C4×C7—42no—
340C4×C7—42no—
341C4×C7—42no—
342C4×C7—42no—
343C4×C7—82no—
344C4×C7#54614yes—
345C24C2×C7×D411yescharacteristic
346#41—24no—
347#41—24no—
348#41—44no—
349#41—48no—
350#41—44no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—42no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8C2×C7×D412yes—
364C2×Q8C7×#4114yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4C2×C7×D412yes—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4C2×C7×D412yes—
387C22×C4C2×C7×D411yescharacteristic
388C22×C4C2×C7×D412yes—
389C22×C4C2×C7×D412yes—
390C22×C4C2×C7×D412yes—
391C22×C4C24×C711yescharacteristic
392M16—24no—
393M16—24no—
394C2×C8—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—42no—
403C4⋊C4C2×C7×D412yes—
404C4⋊C4C2×C7×D412yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—24no—
413C42—22no—
414C42—24no—
415C42—42no—
416C2×C7—21no—
417C2×C7—21no—
418C2×C7—22no—
419C2×C7—22no—
420C2×C7—82no—
421C2×C7#240511yescharacteristic
422C2×C7C2×#45411yescharacteristic
423C2×C7C2×#45611yescharacteristic
424C23—21no—
425C23—21no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23C7×#12011yescharacteristic
430C23C7×#12011yescharacteristic
431C23C22×C7×D411yescharacteristic
432Q8—28no—
433Q8—24no—
434Q8—48no—
435Q8—44no—
436Q8—44no—
437D4—24no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—44no—
442D4—44no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4C7×#12012yes—
468C2×C4C7×#12314yes—
469C2×C4C22×C7×D412yes—
470C2×C4C2×C7×#4112yes—
471C2×C4C7×H(5,𝔽2)12yes—
472C8—44no—
473C7#2401911yescharacteristic, direct product factor, Sylow
474C22—24no—
475C22—24no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22C7×#45412yes—
486C22C7×#45612yes—
487C22C2×C7×#12011yescharacteristic
488C22C2×C7×#12311yescharacteristic
489C22C7×#53411yescharacteristic
490C4—22no—
491C4—24no—
492C4—44no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—82no—
498C4C7×#54614yes—
499C2—21no—
500C2—21no—
501C2—22no—
502C2—22no—
503C2—82no—
504C2C7×#240511yescharacteristic
505C2C2×C7×#45411yescharacteristic
506C2C2×C7×#45611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3486C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3092C2semidirect
3. C2×C7×#391C2semidirect
4. C7×#2038C2semidirect
5. C7×#1896C2semidirect
6. C7×#1857C2semidirect
7. C7×#1793C2semidirect
8. C7×#1784C2semidirect
37. #24019C7direct (N × H)
102. #3092C2×C7semidirect
103. C2×#391C2×C7semidirect
104. #2038C2×C7semidirect
105. #1896C2×C7semidirect
106. #1857C2×C7semidirect
107. #1793C2×C7semidirect
108. #1784C2×C7semidirect
473. C7#24019direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34862
C7×#30922
C2×C7×#3912
C7×#20382
C7×#18962
C7×#18572
C7×#17932
C7×#17842
#240197