Finite Group Databasebeta

← C7×#23984 #13201200

Subgroups — C7×#23984 #13201200

Order 1792.

2810 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3490C211yesmaximal, characteristic, complemented
2C7×#2846C212yesmaximal, complemented
3C2×C7×#379C211yesmaximal, characteristic, complemented
4C7×#2029C212yesmaximal, complemented
5C7×#2021C212yesmaximal, complemented
6C7×#1898C212yesmaximal, complemented
7C7×#1889C214yesmaximal, complemented
8C7×#1819C211yesmaximal, characteristic, complemented
9C26×C7C2211yescharacteristic
10C7×#551—24no—
11C7×#546—24no—
12C7×#543C2211yescharacteristic, complemented
13C7×#538—22no—
14C7×#534C2211yescharacteristic, complemented
15C7×#525—22no—
16C2×C7×#120—24no—
17C2×C7×#120C2214yes—
18C2×C7×#120C2212yes—
19C7×#518C2212yescomplemented
20C22×C7×C22⋊C4—22no—
21C22×C7×C22⋊C4C2211yescharacteristic
22C22×C7×C22⋊C4C2211yescharacteristic
23C2×C7×#99C2214yescomplemented
24C7×#379—22no—
25C7×#379—24no—
26C7×#379C2212yes—
27C7×#379C2214yes—
28C7×#379C2212yes—
29C7×#379C2212yes—
30C7×#342C2212yescomplemented
31C7×#342C2214yescomplemented
32C7×#342C2212yescomplemented
33#23984C711yesmaximal, characteristic, direct product factor, Sylow
34C25×C7—21no—
35C25×C7—21no—
36C25×C7—24no—
37C25×C7—24no—
38C25×C7—48no—
39C25×C7—42no—
40C25×C7C2312yes—
41C25×C7C2311yescharacteristic
42C2×C7×#41—22no—
43C22×C7×D4—24no—
44C22×C7×D4D414yescomplemented
45C22×C7×D4C2312yes—
46C23×C4×C7—21no—
47C23×C4×C7D412yescomplemented
48C7×#126—24no—
49C7×#124—24no—
50C7×#123—24no—
51C7×#123—24no—
52C7×#123—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—24no—
56C7×#121—22no—
57C7×#121—24no—
58C7×#121—22no—
59C7×#120—22no—
60C7×#120—22no—
61C7×#120—24no—
62C7×#120—22no—
63C7×#120—28no—
64C7×#120—44no—
65C7×#120—416no—
66C4×C7×D4—24no—
67C4×C7×D4—24no—
68C4×C7×D4—24no—
69C7×#117—22no—
70C7×#117—22no—
71C2×C7×C22⋊C4—24no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4—24no—
76C2×C7×C22⋊C4—24no—
77C2×C7×C22⋊C4—24no—
78C2×C7×C22⋊C4—28no—
79C2×C7×C22⋊C4—42no—
80C2×C7×C22⋊C4—48no—
81C2×C7×C22⋊C4D412yescomplemented
82C2×C7×C22⋊C4D414yescomplemented
83C2×C7×C22⋊C4C2314yes—
84C2×C7×C22⋊C4C2312yes—
85C2×C7×C22⋊C4C2311yescharacteristic
86C2×C7×C22⋊C4C2311yescharacteristic
87C2×C7×C22⋊C4C2312yes—
88C7×#99—44no—
89#3490C2×C711yescharacteristic, complemented
90#2846C2×C712yescomplemented
91C2×#379C2×C711yescharacteristic, complemented
92#2029C2×C712yescomplemented
93#2021C2×C712yescomplemented
94#1898C2×C712yescomplemented
95#1889C2×C714yescomplemented
96#1819C2×C711yescharacteristic, complemented
97C24×C7—28no—
98C24×C7—24no—
99C24×C7—22no—
100C24×C7—24no—
101C24×C7—22no—
102C24×C7—22no—
103C24×C7—216no—
104C24×C7—24no—
105C24×C7—22no—
106C24×C7—24no—
107C24×C7—48no—
108C24×C7—44no—
109C24×C7—416no—
110C24×C7—42no—
111C24×C7—44no—
112C24×C7—48no—
113C24×C7—416no—
114C24×C7—432no—
115C24×C7—42no—
116C24×C7—48no—
117C24×C7—432no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7C2×D411yescharacteristic
121C24×C7C2×D414yes—
122C24×C7C2×D412yes—
123C24×C7C2×D411yescharacteristic
124C24×C7#4114yescomplemented
125C24×C7C2411yescharacteristic
126C7×#41—84no—
127C2×C7×Q8—22no—
128C2×C7×D4—24no—
129C2×C7×D4—22no—
130C2×C7×D4—22no—
131C2×C7×D4—24no—
132C2×C7×D4—24no—
133C2×C7×D4—22no—
134C2×C7×D4—24no—
135C2×C7×D4—48no—
136C2×C7×D4—48no—
137C2×C7×D4—416no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—24no—
143C22×C4×C7—44no—
144C22×C4×C7—44no—
145C22×C4×C7—42no—
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D412yes—
148C22×C4×C7C2×D414yes—
149C22×C4×C7C2×D412yes—
150C7×C4⋊C4—24no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4—22no—
153C7×C4⋊C4—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—24no—
156C7×C22⋊C4—24no—
157C7×C22⋊C4—24no—
158C7×C22⋊C4—24no—
159C7×C22⋊C4—22no—
160C7×C22⋊C4—22no—
161C7×C22⋊C4—22no—
162C7×C22⋊C4—22no—
163C7×C22⋊C4—24no—
164C7×C22⋊C4—24no—
165C7×C22⋊C4—24no—
166C7×C22⋊C4—48no—
167C7×C22⋊C4—48no—
168C7×C22⋊C4—48no—
169C7×C22⋊C4—42no—
170C7×C22⋊C4—416no—
171C7×C22⋊C4—48no—
172C42×C7—24no—
173C26C22×C711yescharacteristic
174#551—24no—
175#546—24no—
176#543C22×C711yescharacteristic, complemented
177#538—22no—
178#534C22×C711yescharacteristic, complemented
179#525—22no—
180C2×#120—24no—
181C2×#120C22×C714yes—
182C2×#120C22×C712yes—
183#518C22×C712yescomplemented
184C22×C22⋊C4—22no—
185C22×C22⋊C4C22×C711yescharacteristic
186C22×C22⋊C4C22×C711yescharacteristic
187C2×#99C22×C714yescomplemented
188#379—24no—
189#379—22no—
190#379C22×C712yes—
191#379C22×C714yes—
192#379C22×C712yes—
193#379C22×C712yes—
194#342C22×C712yescomplemented
195#342C22×C712yescomplemented
196#342C22×C714yescomplemented
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—24no—
200C23×C7—28no—
201C23×C7—216no—
202C23×C7—216no—
203C23×C7—21no—
204C23×C7—24no—
205C23×C7—21no—
206C23×C7—28no—
207C23×C7—28no—
208C23×C7—28no—
209C23×C7—22no—
210C23×C7—24no—
211C23×C7—42no—
212C23×C7—44no—
213C23×C7—48no—
214C23×C7—42no—
215C23×C7—48no—
216C23×C7—416no—
217C23×C7—48no—
218C23×C7—48no—
219C23×C7—48no—
220C23×C7—44no—
221C23×C7—48no—
222C23×C7—48no—
223C23×C7—416no—
224C23×C7—464no—
225C23×C7—48no—
226C23×C7—416no—
227C23×C7—48no—
228C23×C7—416no—
229C23×C7—44no—
230C23×C7—464no—
231C23×C7—416no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—44no—
235C23×C7—48no—
236C23×C7#12012yes—
237C23×C7#12012yes—
238C23×C7C22×D411yescharacteristic
239C23×C7C22×D412yes—
240C23×C7C2×#4112yes—
241C23×C7H(5,𝔽2)12yes—
242C7×Q8—82no—
243C7×D4—42no—
244C7×D4—42no—
245C7×D4—44no—
246C7×D4—48no—
247C7×D4—84no—
248C7×D4—82no—
249C7×D4—816no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—22no—
254C2×C4×C7—24no—
255C2×C4×C7—44no—
256C2×C4×C7—42no—
257C2×C4×C7—42no—
258C2×C4×C7—44no—
259C2×C4×C7—44no—
260C2×C4×C7—48no—
261C2×C4×C7—42no—
262C2×C4×C7—44no—
263C2×C4×C7—82no—
264C2×C4×C7—84no—
265C2×C4×C7—84no—
266C25—21no—
267C25—24no—
268C25—21no—
269C25—24no—
270C25—48no—
271C25—42no—
272C25C23×C712yes—
273C25C23×C711yescharacteristic
274C2×#41—22no—
275C22×D4—24no—
276C22×D4C7×D414yescomplemented
277C22×D4C23×C712yes—
278C23×C4—21no—
279C23×C4C7×D412yescomplemented
280#126—24no—
281#124—24no—
282#123—24no—
283#123—24no—
284#123—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8—24no—
288#121—22no—
289#121—24no—
290#121—22no—
291#120—22no—
292#120—22no—
293#120—24no—
294#120—22no—
295#120—28no—
296#120—44no—
297#120—416no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301#117—22no—
302#117—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4C7×D412yescomplemented
314C2×C22⋊C4C7×D414yescomplemented
315C2×C22⋊C4C23×C712yes—
316C2×C22⋊C4C23×C714yes—
317C2×C22⋊C4C23×C711yescharacteristic
318C2×C22⋊C4C23×C711yescharacteristic
319C2×C22⋊C4C23×C712yes—
320#99—44no—
321C22×C7—24no—
322C22×C7—24no—
323C22×C7—22no—
324C22×C7—22no—
325C22×C7—24no—
326C22×C7—22no—
327C22×C7—216no—
328C22×C7—28no—
329C22×C7—22no—
330C22×C7—24no—
331C22×C7—42no—
332C22×C7—42no—
333C22×C7—416no—
334C22×C7—48no—
335C22×C7—42no—
336C22×C7—416no—
337C22×C7—48no—
338C22×C7—42no—
339C22×C7—44no—
340C22×C7—432no—
341C22×C7—432no—
342C22×C7—44no—
343C22×C7—48no—
344C22×C7—42no—
345C22×C7—44no—
346C22×C7—84no—
347C22×C7B(4,𝔽2)14yes—
348C22×C7C2×#12011yescharacteristic
349C22×C7#52511yescharacteristic
350C22×C7#53511yescharacteristic
351C22×C7#54614yes—
352C4×C7—42no—
353C4×C7—42no—
354C4×C7—42no—
355C4×C7—44no—
356C4×C7—82no—
357C4×C7—84no—
358C24—28no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—216no—
365C24—24no—
366C24—22no—
367C24—24no—
368C24—44no—
369C24—48no—
370C24—416no—
371C24—42no—
372C24—44no—
373C24—48no—
374C24—416no—
375C24—432no—
376C24—42no—
377C24—48no—
378C24—432no—
379C24—42no—
380C24—42no—
381C24C2×C7×D411yescharacteristic
382C24C2×C7×D414yes—
383C24C2×C7×D412yes—
384C24C2×C7×D411yescharacteristic
385C24C7×#4114yescomplemented
386C24C24×C711yescharacteristic
387#41—84no—
388C2×Q8—22no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—24no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—416no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4C2×C7×D412yes—
408C22×C4C2×C7×D412yes—
409C22×C4C2×C7×D414yes—
410C22×C4C2×C7×D412yes—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—42no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C42—24no—
434C2×C7—21no—
435C2×C7—24no—
436C2×C7—21no—
437C2×C7—24no—
438C2×C7—48no—
439C2×C7—42no—
440C2×C7—82no—
441C2×C7#240411yescharacteristic
442C2×C7C2×B(4,𝔽2)12yes—
443C23—21no—
444C23—21no—
445C23—24no—
446C23—28no—
447C23—216no—
448C23—216no—
449C23—21no—
450C23—24no—
451C23—21no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—22no—
456C23—24no—
457C23—42no—
458C23—44no—
459C23—48no—
460C23—42no—
461C23—48no—
462C23—416no—
463C23—48no—
464C23—48no—
465C23—48no—
466C23—44no—
467C23—48no—
468C23—48no—
469C23—416no—
470C23—464no—
471C23—48no—
472C23—416no—
473C23—48no—
474C23—416no—
475C23—44no—
476C23—464no—
477C23—416no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—48no—
482C23C7×#12012yes—
483C23C7×#12012yes—
484C23C22×C7×D411yescharacteristic
485C23C22×C7×D412yes—
486C23C2×C7×#4112yes—
487C23C7×H(5,𝔽2)12yes—
488Q8—82no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—48no—
493D4—84no—
494D4—82no—
495D4—816no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—82no—
510C2×C4—84no—
511C2×C4—84no—
512C7#2398411yescharacteristic, direct product factor, Sylow
513C22—24no—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—24no—
518C22—22no—
519C22—216no—
520C22—28no—
521C22—22no—
522C22—24no—
523C22—42no—
524C22—42no—
525C22—416no—
526C22—48no—
527C22—42no—
528C22—416no—
529C22—48no—
530C22—42no—
531C22—44no—
532C22—432no—
533C22—432no—
534C22—44no—
535C22—48no—
536C22—42no—
537C22—44no—
538C22—84no—
539C22C7×B(4,𝔽2)14yes—
540C22C2×C7×#12011yescharacteristic
541C22C7×#52511yescharacteristic
542C22C7×#53511yescharacteristic
543C22C7×#54614yes—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—44no—
548C4—82no—
549C4—84no—
550C2—21no—
551C2—24no—
552C2—21no—
553C2—24no—
554C2—48no—
555C2—42no—
556C2—82no—
557C2C7×#240411yescharacteristic
558C2C2×C7×B(4,𝔽2)12yes—
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC24C24×C7
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C25×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3490C2semidirect
2. C7×#2846C2semidirect
3. C2×C7×#379C2semidirect
4. C7×#2029C2semidirect
5. C7×#2021C2semidirect
6. C7×#1898C2semidirect
7. C7×#1889C2semidirect
8. C7×#1819C2semidirect
12. C7×#543C22semidirect
14. C7×#534C22semidirect
19. C7×#518C22semidirect
23. C2×C7×#99C22semidirect
30. C7×#342C22semidirect
31. C7×#342C22semidirect
32. C7×#342C22semidirect
33. #23984C7direct (N × H)
44. C22×C7×D4D4semidirect
47. C23×C4×C7D4semidirect
81. C2×C7×C22⋊C4D4semidirect
82. C2×C7×C22⋊C4D4semidirect
89. #3490C2×C7semidirect
90. #2846C2×C7semidirect
91. C2×#379C2×C7semidirect
92. #2029C2×C7semidirect
93. #2021C2×C7semidirect
94. #1898C2×C7semidirect
95. #1889C2×C7semidirect
96. #1819C2×C7semidirect
124. C24×C7#41semidirect
176. #543C22×C7semidirect
178. #534C22×C7semidirect
183. #518C22×C7semidirect
187. C2×#99C22×C7semidirect
194. #342C22×C7semidirect
195. #342C22×C7semidirect
196. #342C22×C7semidirect
276. C22×D4C7×D4semidirect
279. C23×C4C7×D4semidirect
313. C2×C22⋊C4C7×D4semidirect
314. C2×C22⋊C4C7×D4semidirect
385. C24C7×#41semidirect
512. C7#23984direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#34902
C7×#28462
C2×C7×#3792
C7×#20292
C7×#20212
C7×#18982
C7×#18892
C7×#18192
#239847