Finite Group Databasebeta

← C7×#24025 #13201241

Subgroups — C7×#24025 #13201241

Order 1792.

1286 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3485C211yesmaximal, characteristic
2C7×#3092C212yesmaximal, complemented
3C7×#2311C211yesmaximal, characteristic, complemented
4C7×#2038C214yesmaximal, complemented
5C7×#1897C212yesmaximal, complemented
6C7×#1856C212yesmaximal, complemented
7C7×#1792C212yesmaximal, complemented
8C7×#1785C211yesmaximal, characteristic, complemented
9C2×C7×H(5,𝔽2)C2211yescharacteristic
10C22×C7×#41C2211yescharacteristic
11C7×#547—22no—
12C7×#547—22no—
13C7×#546—22no—
14C7×#546—22no—
15C7×#536C2212yes—
16C7×#534—22no—
17C7×#525C2212yes—
18C2×C7×#123—22no—
19C7×#518C2211yescharacteristic
20C2×C7×#117C2211yescharacteristic, complemented
21C22×C7×C4⋊C4C2211yescharacteristic, complemented
22C7×#484—22no—
23C7×#483—22no—
24C7×#481—22no—
25C7×#480—22no—
26C7×#428C2214yescomplemented
27C2×C7×#104C2214yes—
28C7×#417C2214yes—
29C2×C7×#101C2212yescomplemented
30C2×C7×#100C2212yes—
31C7×#407C2212yescomplemented
32C7×#391C2212yescomplemented
33C7×#390C2212yes—
34C7×#386—22no—
35C7×#380—22no—
36C7×#339C2214yescomplemented
37#24025C711yesmaximal, characteristic, direct product factor, Sylow
38C7×H(5,𝔽2)—24no—
39C7×H(5,𝔽2)—42no—
40C2×C7×#41—22no—
41C2×C7×#41—24no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—42no—
45C2×C7×#41C2312yes—
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4—22no—
48C22×C7×D4—22no—
49C22×C7×D4D412yes—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C23×C4×C7D412yes—
53C23×C4×C7C2311yescharacteristic
54C2×C7×M16C2314yes—
55C7×C4⋊Q8—22no—
56C7×#124—22no—
57C7×#123—24no—
58C7×#123—24no—
59C7×#123—24no—
60C7×#123—22no—
61C7×#123—24no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8D412yes—
65C7×C22⋊Q8D412yes—
66C7×#121—24no—
67C7×#121—24no—
68C7×#121D412yes—
69C7×#121D412yes—
70C7×#120—24no—
71C7×#120—22no—
72C4×C7×D4—22no—
73C4×C7×D4—22no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C7×#117—42no—
77C7×#117C2312yes—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—22no—
80C2×C7×C4⋊C4—22no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4—21no—
84C2×C7×C4⋊C4C2312yes—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—22no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—21no—
89C2×C7×C22⋊C4—24no—
90C2×C42×C7C2312yes—
91C7×#107—24no—
92C7×#106—24no—
93C7×#104—44no—
94C7×#103—24no—
95C7×#102—24no—
96C7×#101—42no—
97C7×#100—24no—
98C7×C22⋊C8—22no—
99C7×C22⋊C8—22no—
100C7×#95—24no—
101#3485C2×C711yescharacteristic
102#3092C2×C712yescomplemented
103#2311C2×C711yescharacteristic, complemented
104#2038C2×C714yescomplemented
105#1897C2×C712yescomplemented
106#1856C2×C712yescomplemented
107#1792C2×C712yescomplemented
108#1785C2×C711yescharacteristic, complemented
109C24×C7—22no—
110C24×C7C2×D411yescharacteristic
111C24×C7C2×D412yes—
112C7×#41—24no—
113C7×#41—24no—
114C7×#41—48no—
115C7×#41—48no—
116C7×#41—48no—
117C2×C7×Q8—22no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8C2×D412yes—
121C2×C7×D4—24no—
122C2×C7×D4—24no—
123C2×C7×D4—24no—
124C2×C7×D4—22no—
125C2×C7×D4—22no—
126C2×C7×D4—22no—
127C2×C7×D4—22no—
128C2×C7×D4—22no—
129C2×C7×D4—42no—
130C2×C7×D4—48no—
131C2×C7×D4—44no—
132C2×C7×D4—44no—
133C2×C7×D4C2×D412yes—
134C2×C7×D4#4114yes—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—24no—
139C22×C4×C7—22no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—42no—
147C22×C4×C7—42no—
148C22×C4×C7—82no—
149C22×C4×C7C2×D412yes—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D412yes—
152C22×C4×C7C2×D412yes—
153C22×C4×C7C2411yescharacteristic
154C7×M16—24no—
155C7×M16—24no—
156C2×C7×C8—24no—
157C7×C4⋊C4—24no—
158C7×C4⋊C4—24no—
159C7×C4⋊C4—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—24no—
162C7×C4⋊C4—42no—
163C7×C4⋊C4—41no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4C2×D412yes—
166C7×C4⋊C4C2×D412yes—
167C7×C22⋊C4—24no—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—24no—
170C7×C22⋊C4—22no—
171C7×C22⋊C4—22no—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—84no—
177C42×C7—22no—
178C42×C7—24no—
179C2×H(5,𝔽2)C22×C711yescharacteristic
180C22×#41C22×C711yescharacteristic
181#547—22no—
182#547—22no—
183#546—22no—
184#546—22no—
185#536C22×C712yes—
186#534—22no—
187#525C22×C712yes—
188C2×#123—22no—
189#518C22×C711yescharacteristic
190C2×#117C22×C711yescharacteristic, complemented
191C22×C4⋊C4C22×C711yescharacteristic, complemented
192#484—22no—
193#483—22no—
194#481—22no—
195#480—22no—
196#428C22×C714yescomplemented
197C2×#104C22×C714yes—
198#417C22×C714yes—
199C2×#101C22×C712yescomplemented
200C2×#100C22×C712yes—
201#407C22×C712yescomplemented
202#391C22×C712yescomplemented
203#390C22×C712yes—
204#386—22no—
205#380—22no—
206#339C22×C714yescomplemented
207C23×C7—22no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7—44no—
217C23×C7—82no—
218C23×C7—82no—
219C23×C7#12012yes—
220C23×C7#12314yes—
221C23×C7C22×D411yescharacteristic
222C7×Q8—24no—
223C7×Q8—44no—
224C7×D4—28no—
225C7×D4—24no—
226C7×D4—48no—
227C7×D4—48no—
228C7×D4—42no—
229C7×D4—42no—
230C7×D4—82no—
231C7×D4—82no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—44no—
243C2×C4×C7—44no—
244C2×C4×C7—44no—
245C2×C4×C7—44no—
246C2×C4×C7—42no—
247C2×C4×C7—44no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—82no—
252C2×C4×C7—82no—
253C2×C4×C7—84no—
254C2×C4×C7#12011yescharacteristic
255C2×C4×C7#12011yescharacteristic
256C2×C4×C7C22×D412yes—
257C2×C4×C7C2×#4112yes—
258C2×C4×C7H(5,𝔽2)12yes—
259C7×C8—44no—
260H(5,𝔽2)—24no—
261H(5,𝔽2)—42no—
262C2×#41—22no—
263C2×#41—24no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—42no—
267C2×#41C23×C712yes—
268C22×Q8C23×C711yescharacteristic
269C22×D4—22no—
270C22×D4—22no—
271C22×D4C7×D412yes—
272C22×D4C23×C711yescharacteristic
273C23×C4—21no—
274C23×C4C7×D412yes—
275C23×C4C23×C711yescharacteristic
276C2×M16C23×C714yes—
277C4⋊Q8—22no—
278#124—22no—
279#123—24no—
280#123—24no—
281#123—24no—
282#123—22no—
283#123—24no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8C7×D412yes—
287C22⋊Q8C7×D412yes—
288#121—24no—
289#121—24no—
290#121C7×D412yes—
291#121C7×D412yes—
292#120—24no—
293#120—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298#117—42no—
299#117C23×C712yes—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C23×C712yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C42C23×C712yes—
313#107—24no—
314#106—24no—
315#104—44no—
316#103—24no—
317#102—24no—
318#101—42no—
319#100—24no—
320C22⋊C8—22no—
321C22⋊C8—22no—
322#95—24no—
323C22×C7—21no—
324C22×C7—21no—
325C22×C7—22no—
326C22×C7—24no—
327C22×C7—22no—
328C22×C7—22no—
329C22×C7—44no—
330C22×C7—42no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—42no—
334C22×C7—42no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7—82no—
338C22×C7—164no—
339C22×C7C2×#12011yescharacteristic
340C22×C7C2×#12311yescharacteristic
341C22×C7#53411yescharacteristic
342C4×C7—21no—
343C4×C7—21no—
344C4×C7—42no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7—42no—
348C4×C7—82no—
349C4×C7—82no—
350C4×C7#54614yes—
351C24—22no—
352C24C2×C7×D411yescharacteristic
353C24C2×C7×D412yes—
354#41—24no—
355#41—24no—
356#41—48no—
357#41—48no—
358#41—48no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8C2×C7×D412yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—48no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4C2×C7×D412yes—
376C2×D4C7×#4114yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—82no—
391C22×C4C2×C7×D412yes—
392C22×C4C2×C7×D411yescharacteristic
393C22×C4C2×C7×D412yes—
394C22×C4C2×C7×D412yes—
395C22×C4C24×C711yescharacteristic
396M16—24no—
397M16—24no—
398C2×C8—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C7×D412yes—
408C4⋊C4C2×C7×D412yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—42no—
418C22⋊C4—84no—
419C42—22no—
420C42—24no—
421C2×C7—21no—
422C2×C7—21no—
423C2×C7—42no—
424C2×C7—44no—
425C2×C7—82no—
426C2×C7#240511yescharacteristic
427C2×C7#302111yescharacteristic
428C2×C7#302211yescharacteristic
429C23—22no—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—82no—
440C23—82no—
441C23C7×#12012yes—
442C23C7×#12314yes—
443C23C22×C7×D411yescharacteristic
444Q8—24no—
445Q8—44no—
446D4—28no—
447D4—24no—
448D4—48no—
449D4—48no—
450D4—42no—
451D4—42no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4C7×#12011yescharacteristic
477C2×C4C7×#12011yescharacteristic
478C2×C4C22×C7×D412yes—
479C2×C4C2×C7×#4112yes—
480C2×C4C7×H(5,𝔽2)12yes—
481C8—44no—
482C7#2402511yescharacteristic, direct product factor, Sylow
483C22—21no—
484C22—21no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—164no—
499C22C2×C7×#12011yescharacteristic
500C22C2×C7×#12311yescharacteristic
501C22C7×#53411yescharacteristic
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4C7×#54614yes—
511C2—21no—
512C2—21no—
513C2—42no—
514C2—44no—
515C2—82no—
516C2C7×#240511yescharacteristic
517C2C7×#302111yescharacteristic
518C2C7×#302211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC7×#3485C2
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. C7×#2311C2semidirect
4. C7×#2038C2semidirect
5. C7×#1897C2semidirect
6. C7×#1856C2semidirect
7. C7×#1792C2semidirect
8. C7×#1785C2semidirect
20. C2×C7×#117C22semidirect
21. C22×C7×C4⋊C4C22semidirect
26. C7×#428C22semidirect
29. C2×C7×#101C22semidirect
31. C7×#407C22semidirect
32. C7×#391C22semidirect
36. C7×#339C22semidirect
37. #24025C7direct (N × H)
102. #3092C2×C7semidirect
103. #2311C2×C7semidirect
104. #2038C2×C7semidirect
105. #1897C2×C7semidirect
106. #1856C2×C7semidirect
107. #1792C2×C7semidirect
108. #1785C2×C7semidirect
190. C2×#117C22×C7semidirect
191. C22×C4⋊C4C22×C7semidirect
196. #428C22×C7semidirect
199. C2×#101C22×C7semidirect
201. #407C22×C7semidirect
202. #391C22×C7semidirect
206. #339C22×C7semidirect
482. C7#24025direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34852
C7×#30922
C7×#23112
C7×#20382
C7×#18972
C7×#18562
C7×#17922
C7×#17852
#240257