Finite Group Databasebeta

← C7×#24115 #13201331

Subgroups — C7×#24115 #13201331

Order 1792.

1078 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3491C211yesmaximal, characteristic
2C7×#2623C212yesmaximal, complemented
3C7×#2317C211yesmaximal, characteristic, complemented
4C7×#2062C212yesmaximal, complemented
5C7×#2044C212yesmaximal
6C7×#1899C212yesmaximal, complemented
7C7×#1867C214yesmaximal, complemented
8C7×#1820C211yesmaximal, characteristic, complemented
9C7×#552—22no—
10C7×#552—22no—
11C7×#551—22no—
12C7×#551—22no—
13C7×#543C2211yescharacteristic
14C7×#539—21no—
15C7×#539—21no—
16C7×#534C2211yescharacteristic
17C7×#529—21no—
18C7×#529—21no—
19C2×C7×#123C2212yes—
20C7×#518C2212yes—
21C2×C7×#117C2211yescharacteristic
22C2×C7×#117C2211yescharacteristic
23C22×C7×C4⋊C4C2211yescharacteristic
24C2×C7×#99C2214yes—
25C7×#400—22no—
26C7×#399—22no—
27C7×#398—22no—
28C7×#397—22no—
29C7×#389—21no—
30C7×#389—21no—
31C7×#386C2214yes—
32C7×#385C2212yes—
33C7×#385C2214yes—
34C7×#385C2212yes—
35C7×#384—21no—
36C7×#382—21no—
37C7×#381C2212yes—
38C7×#342C2212yes—
39C7×#342C2214yes—
40C7×#342C2212yes—
41#24115C711yesmaximal, characteristic, direct product factor, Sylow
42C22×C7×D4C2312yes—
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C7×#126—22no—
46C7×#126—22no—
47C7×#126—22no—
48C7×#126—22no—
49C7×#125—21no—
50C7×#125—21no—
51C7×#125—22no—
52C7×#125—22no—
53C7×#124—21no—
54C7×#124—21no—
55C7×#123—22no—
56C7×#123—22no—
57C7×#123—22no—
58C7×#123—22no—
59C7×#123—22no—
60C7×#123—22no—
61C7×#123—22no—
62C7×C22⋊Q8—22no—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—21no—
65C7×C22⋊Q8—21no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×#121—21no—
69C7×#121—21no—
70C7×#120—22no—
71C4×C7×Q8—21no—
72C4×C7×Q8—21no—
73C4×C7×D4—21no—
74C4×C7×D4—21no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4—22no—
78C4×C7×D4—22no—
79C7×#117—22no—
80C7×#117—22no—
81C7×#117—22no—
82C7×#117—22no—
83C7×#117—22no—
84C7×#117—24no—
85C2×C7×C4⋊C4—21no—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—21no—
88C2×C7×C4⋊C4—22no—
89C2×C7×C4⋊C4—22no—
90C2×C7×C4⋊C4—22no—
91C2×C7×C4⋊C4—22no—
92C2×C7×C4⋊C4—24no—
93C2×C7×C4⋊C4D411yescharacteristic
94C2×C7×C4⋊C4D411yescharacteristic
95C2×C7×C4⋊C4D412yes—
96C2×C7×C4⋊C4D412yes—
97C2×C7×C22⋊C4—22no—
98C2×C7×C22⋊C4—24no—
99C2×C7×C22⋊C4D412yes—
100C2×C7×C22⋊C4D412yes—
101C2×C7×C22⋊C4C2314yes—
102C2×C7×C22⋊C4C2311yescharacteristic
103C2×C7×C22⋊C4C2311yescharacteristic
104C2×C7×C22⋊C4C2312yes—
105C2×C7×C22⋊C4C2312yes—
106C2×C42×C7—21no—
107C2×C42×C7D411yescharacteristic
108C2×C42×C7D411yescharacteristic
109C7×#99—28no—
110C7×#95—22no—
111C7×#95—22no—
112C7×#95—22no—
113C7×#95—22no—
114C7×#95—24no—
115C7×#95—24no—
116#3491C2×C711yescharacteristic
117#2623C2×C712yescomplemented
118#2317C2×C711yescharacteristic, complemented
119#2062C2×C712yescomplemented
120#2044C2×C712yes—
121#1899C2×C712yescomplemented
122#1867C2×C714yescomplemented
123#1820C2×C711yescharacteristic, complemented
124C24×C7C2×D412yes—
125C24×C7C2411yescharacteristic
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4—24no—
131C2×C7×D4—24no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—22no—
142C22×C4×C7—82no—
143C22×C4×C7C2×D412yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D412yes—
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—22no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—22no—
164C7×C4⋊C4—22no—
165C7×C4⋊C4—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—41no—
170C7×C4⋊C4—41no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—82no—
173C7×C4⋊C4—82no—
174C7×C4⋊C4—84no—
175C7×C22⋊C4—22no—
176C7×C22⋊C4—22no—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—22no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—84no—
191C7×C22⋊C4#4114yes—
192C42×C7—24no—
193C42×C7—22no—
194C42×C7—22no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199#552—22no—
200#552—22no—
201#551—22no—
202#551—22no—
203#543C22×C711yescharacteristic
204#539—21no—
205#539—21no—
206#534C22×C711yescharacteristic
207#529—21no—
208#529—21no—
209C2×#123C22×C712yes—
210#518C22×C712yes—
211C2×#117C22×C711yescharacteristic
212C2×#117C22×C711yescharacteristic
213C22×C4⋊C4C22×C711yescharacteristic
214C2×#99C22×C714yes—
215#400—22no—
216#399—22no—
217#398—22no—
218#397—22no—
219#389—21no—
220#389—21no—
221#386C22×C714yes—
222#385C22×C712yes—
223#385C22×C712yes—
224#385C22×C714yes—
225#384—21no—
226#382—21no—
227#381C22×C712yes—
228#342C22×C712yes—
229#342C22×C714yes—
230#342C22×C712yes—
231C23×C7—22no—
232C23×C7—24no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7C22×D411yescharacteristic
238C23×C7C22×D412yes—
239C23×C7C2×#4112yes—
240C23×C72−1+412yes—
241C7×Q8—81no—
242C7×Q8—81no—
243C7×D4—42no—
244C7×D4—42no—
245C7×D4—44no—
246C7×D4—81no—
247C7×D4—81no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—24no—
253C2×C4×C7—24no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—42no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—42no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—42no—
270C2×C4×C7—41no—
271C2×C4×C7—44no—
272C2×C4×C7—44no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7—44no—
276C2×C4×C7—44no—
277C2×C4×C7—42no—
278C2×C4×C7—44no—
279C2×C4×C7—82no—
280C2×C4×C7—82no—
281C2×C4×C7#12011yescharacteristic
282C2×C4×C7#12011yescharacteristic
283C2×C4×C7#12012yes—
284C22×D4C23×C712yes—
285C23×C4C23×C711yescharacteristic
286C23×C4C23×C712yes—
287#126—22no—
288#126—22no—
289#126—22no—
290#126—22no—
291#125—21no—
292#125—21no—
293#125—22no—
294#125—22no—
295#124—21no—
296#124—21no—
297#123—22no—
298#123—22no—
299#123—22no—
300#123—22no—
301#123—22no—
302#123—22no—
303#123—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310#121—21no—
311#121—21no—
312#120—22no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—24no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C7×D411yescharacteristic
336C2×C4⋊C4C7×D411yescharacteristic
337C2×C4⋊C4C7×D412yes—
338C2×C4⋊C4C7×D412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C7×D412yes—
342C2×C22⋊C4C7×D412yes—
343C2×C22⋊C4C23×C714yes—
344C2×C22⋊C4C23×C711yescharacteristic
345C2×C22⋊C4C23×C711yescharacteristic
346C2×C22⋊C4C23×C712yes—
347C2×C22⋊C4C23×C712yes—
348C2×C42—21no—
349C2×C42C7×D411yescharacteristic
350C2×C42C7×D411yescharacteristic
351#99—28no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358C22×C7—21no—
359C22×C7—22no—
360C22×C7—21no—
361C22×C7—22no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—42no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—84no—
368C22×C7C2×#12011yescharacteristic
369C22×C7#52511yescharacteristic
370C22×C7#53611yescharacteristic
371C22×C7#54714yes—
372C4×C7—22no—
373C4×C7—21no—
374C4×C7—21no—
375C4×C7—44no—
376C4×C7—42no—
377C4×C7—41no—
378C4×C7—42no—
379C4×C7—41no—
380C4×C7—41no—
381C4×C7—42no—
382C4×C7—41no—
383C4×C7—84no—
384C4×C7—82no—
385C24C2×C7×D412yes—
386C24C24×C711yescharacteristic
387C2×Q8—21no—
388C2×Q8—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—21no—
394C2×D4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—82no—
404C22×C4C2×C7×D412yes—
405C22×C4C2×C7×D412yes—
406C22×C4C2×C7×D412yes—
407C22×C4C2×C7×D411yescharacteristic
408C22×C4C2×C7×D411yescharacteristic
409C22×C4C2×C7×D411yescharacteristic
410C22×C4C2×C7×D411yescharacteristic
411C22×C4C2×C7×D412yes—
412C22×C4C2×C7×D411yescharacteristic
413C22×C4C2×C7×D411yescharacteristic
414C22×C4C2×C7×D412yes—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—84no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—84no—
452C22⋊C4C7×#4114yes—
453C42—24no—
454C42—22no—
455C42—22no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C2×C7—21no—
461C2×C7—24no—
462C2×C7—21no—
463C2×C7—82no—
464C2×C7#240611yescharacteristic
465C2×C7#302711yescharacteristic
466C2×C7#302811yescharacteristic
467C23—22no—
468C23—24no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23C22×C7×D411yescharacteristic
474C23C22×C7×D412yes—
475C23C2×C7×#4112yes—
476C23C7×2−1+412yes—
477Q8—81no—
478Q8—81no—
479D4—42no—
480D4—42no—
481D4—44no—
482D4—81no—
483D4—81no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4C7×#12011yescharacteristic
518C2×C4C7×#12011yescharacteristic
519C2×C4C7×#12012yes—
520C7#2411511yescharacteristic, direct product factor, Sylow
521C22—21no—
522C22—22no—
523C22—21no—
524C22—22no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—84no—
531C22C2×C7×#12011yescharacteristic
532C22C7×#52511yescharacteristic
533C22C7×#53611yescharacteristic
534C22C7×#54714yes—
535C4—22no—
536C4—21no—
537C4—21no—
538C4—44no—
539C4—42no—
540C4—41no—
541C4—42no—
542C4—41no—
543C4—41no—
544C4—42no—
545C4—41no—
546C4—84no—
547C4—82no—
548C2—21no—
549C2—24no—
550C2—21no—
551C2—82no—
552C2C7×#240611yescharacteristic
553C2C7×#302711yescharacteristic
554C2C7×#302811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
prime radicalC7×#534C22
LCSG ▷ C24 ▷ 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×#2623C2semidirect
3. C7×#2317C2semidirect
4. C7×#2062C2semidirect
6. C7×#1899C2semidirect
7. C7×#1867C2semidirect
8. C7×#1820C2semidirect
41. #24115C7direct (N × H)
117. #2623C2×C7semidirect
118. #2317C2×C7semidirect
119. #2062C2×C7semidirect
121. #1899C2×C7semidirect
122. #1867C2×C7semidirect
123. #1820C2×C7semidirect
520. C7#24115direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34912
C7×#26232
C7×#23172
C7×#20622
C7×#20442
C7×#18992
C7×#18672
C7×#18202
#241157