Finite Group Databasebeta

← C7×#23987 #13201203

Subgroups — C7×#23987 #13201203

Order 1792.

1514 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3490C211yesmaximal, characteristic, complemented
2C7×#2613C212yesmaximal, complemented
3C7×#2308C211yesmaximal, characteristic, complemented
4C7×#2030C211yesmaximal, characteristic, complemented
5C7×#2029C211yesmaximal, characteristic, complemented
6C7×#2022C211yesmaximal, characteristic, complemented
7C7×#2021C211yesmaximal, characteristic, complemented
8C7×#1898C212yesmaximal, complemented
9C7×#1890C212yesmaximal, complemented
10C7×#1889C212yesmaximal, complemented
11C7×#1819C211yesmaximal, characteristic, complemented
12C23×C7×D4C2211yescharacteristic
13C7×#551—24no—
14C7×#546—24no—
15C7×#543C2211yescharacteristic
16C7×#538—22no—
17C7×#534C2211yescharacteristic
18C7×#525—22no—
19C2×C7×#123C2212yes—
20C2×C7×#120C2212yes—
21C2×C7×#120C2212yes—
22C7×#518C2212yes—
23C22×C7×C22⋊C4C2211yescharacteristic
24C22×C7×C22⋊C4C2211yescharacteristic
25C2×C7×#99C2212yes—
26C2×C7×#99C2212yes—
27C7×#394—24no—
28C7×#392—24no—
29C7×#386—22no—
30C7×#386—22no—
31C7×#381C2211yescharacteristic
32C7×#381C2212yes—
33C7×#381C2211yescharacteristic
34C7×#379C2211yescharacteristic
35C7×#379C2212yes—
36C7×#379C2211yescharacteristic
37C7×#379C2212yes—
38C7×#342C2211yescharacteristic
39C7×#342C2211yescharacteristic
40C7×#342C2212yes—
41C7×#342C2212yes—
42C7×#342C2212yes—
43#23987C711yesmaximal, characteristic, direct product factor, Sylow
44C25×C7C2311yescharacteristic
45C25×C7C2311yescharacteristic
46C2×C7×#41—22no—
47C22×C7×D4—21no—
48C22×C7×D4—21no—
49C22×C7×D4—22no—
50C22×C7×D4—24no—
51C22×C7×D4—22no—
52C22×C7×D4—22no—
53C22×C7×D4—42no—
54C22×C7×D4D414yescomplemented
55C22×C7×D4C2312yes—
56C23×C4×C7—21no—
57C23×C4×C7D412yescomplemented
58C23×C4×C7C2311yescharacteristic
59C7×#126—24no—
60C7×#124—24no—
61C7×#123—24no—
62C7×#123—24no—
63C7×#123—22no—
64C7×#123—24no—
65C7×#123—42no—
66C7×C22⋊Q8—22no—
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8—24no—
69C7×#121—22no—
70C7×#121—24no—
71C7×#121—22no—
72C7×#120—22no—
73C7×#120—22no—
74C7×#120—24no—
75C7×#120—22no—
76C7×#120—24no—
77C7×#120—42no—
78C4×C7×D4—24no—
79C4×C7×D4—24no—
80C4×C7×D4—24no—
81C7×#117—22no—
82C7×#117—22no—
83C2×C7×C4⋊C4—22no—
84C2×C7×C22⋊C4—24no—
85C2×C7×C22⋊C4—22no—
86C2×C7×C22⋊C4—21no—
87C2×C7×C22⋊C4—22no—
88C2×C7×C22⋊C4—22no—
89C2×C7×C22⋊C4—24no—
90C2×C7×C22⋊C4—22no—
91C2×C7×C22⋊C4—22no—
92C2×C7×C22⋊C4—24no—
93C2×C7×C22⋊C4—42no—
94C2×C7×C22⋊C4D412yescomplemented
95C2×C7×C22⋊C4D414yescomplemented
96C2×C7×C22⋊C4C2312yes—
97C2×C7×C22⋊C4C2312yes—
98C2×C7×C22⋊C4C2311yescharacteristic
99C2×C7×C22⋊C4C2311yescharacteristic
100C2×C7×C22⋊C4C2311yescharacteristic
101C2×C7×C22⋊C4C2311yescharacteristic
102C2×C7×C22⋊C4C2312yes—
103C7×#99—42no—
104C7×#99—42no—
105C7×#95—22no—
106C7×#95—22no—
107C7×#95—24no—
108#3490C2×C711yescharacteristic, complemented
109#2613C2×C712yescomplemented
110#2308C2×C711yescharacteristic, complemented
111#2030C2×C711yescharacteristic, complemented
112#2029C2×C711yescharacteristic, complemented
113#2022C2×C711yescharacteristic, complemented
114#2021C2×C711yescharacteristic, complemented
115#1898C2×C712yescomplemented
116#1890C2×C712yescomplemented
117#1889C2×C712yescomplemented
118#1819C2×C711yescharacteristic, complemented
119C24×C7—21no—
120C24×C7—21no—
121C24×C7—22no—
122C24×C7—21no—
123C24×C7—22no—
124C24×C7—21no—
125C24×C7—22no—
126C24×C7—82no—
127C24×C7—82no—
128C24×C7C2×D412yes—
129C24×C7C2×D414yes—
130C24×C7C2×D411yescharacteristic
131C24×C7C2×D411yescharacteristic
132C24×C7#4112yescomplemented
133C24×C7C2411yescharacteristic
134C7×#41—84no—
135C2×C7×Q8—22no—
136C2×C7×D4—24no—
137C2×C7×D4—24no—
138C2×C7×D4—24no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—22no—
146C2×C7×D4—24no—
147C2×C7×D4—48no—
148C2×C7×D4—48no—
149C2×C7×D4—42no—
150C2×C7×D4—42no—
151C2×C7×D4—42no—
152C2×C7×D4—42no—
153C2×C7×D4—44no—
154C2×C7×D4—84no—
155C2×C7×D4—84no—
156C22×C4×C7—22no—
157C22×C4×C7—21no—
158C22×C4×C7—22no—
159C22×C4×C7—21no—
160C22×C4×C7—22no—
161C22×C4×C7—24no—
162C22×C4×C7—22no—
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C22×C4×C7—22no—
166C22×C4×C7—22no—
167C22×C4×C7—42no—
168C22×C4×C7—82no—
169C22×C4×C7—82no—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D412yes—
172C22×C4×C7C2×D414yes—
173C22×C4×C7C2×D412yes—
174C22×C4×C7#4112yescomplemented
175C7×C4⋊C4—24no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—24no—
179C7×C4⋊C4—44no—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—24no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—22no—
189C7×C22⋊C4—22no—
190C7×C22⋊C4—22no—
191C7×C22⋊C4—22no—
192C7×C22⋊C4—22no—
193C7×C22⋊C4—22no—
194C7×C22⋊C4—22no—
195C7×C22⋊C4—44no—
196C7×C22⋊C4—44no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—82no—
199C7×C22⋊C4—82no—
200C7×C22⋊C4—84no—
201C7×C22⋊C4—84no—
202C42×C7—24no—
203C23×D4C22×C711yescharacteristic
204#551—24no—
205#546—24no—
206#543C22×C711yescharacteristic
207#538—22no—
208#534C22×C711yescharacteristic
209#525—22no—
210C2×#123C22×C712yes—
211C2×#120C22×C712yes—
212C2×#120C22×C712yes—
213#518C22×C712yes—
214C22×C22⋊C4C22×C711yescharacteristic
215C22×C22⋊C4C22×C711yescharacteristic
216C2×#99C22×C712yes—
217C2×#99C22×C712yes—
218#394—24no—
219#392—24no—
220#386—22no—
221#386—22no—
222#381C22×C711yescharacteristic
223#381C22×C711yescharacteristic
224#381C22×C712yes—
225#379C22×C711yescharacteristic
226#379C22×C711yescharacteristic
227#379C22×C712yes—
228#379C22×C712yes—
229#342C22×C712yes—
230#342C22×C711yescharacteristic
231#342C22×C711yescharacteristic
232#342C22×C712yes—
233#342C22×C712yes—
234C23×C7—21no—
235C23×C7—21no—
236C23×C7—21no—
237C23×C7—21no—
238C23×C7—24no—
239C23×C7—24no—
240C23×C7—22no—
241C23×C7—22no—
242C23×C7—24no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—42no—
246C23×C7—42no—
247C23×C7—42no—
248C23×C7—44no—
249C23×C7—44no—
250C23×C7—42no—
251C23×C7—42no—
252C23×C7—44no—
253C23×C7—44no—
254C23×C7—44no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—42no—
258C23×C7—82no—
259C23×C7—82no—
260C23×C7—82no—
261C23×C7—82no—
262C23×C7—84no—
263C23×C7—84no—
264C23×C7#12012yes—
265C23×C7#12012yes—
266C23×C7C22×D411yescharacteristic
267C23×C7C22×D412yes—
268C23×C7C2×#4111yescharacteristic
269C23×C7C2×#4111yescharacteristic
270C23×C7H(5,𝔽2)12yes—
271C7×Q8—82no—
272C7×D4—48no—
273C7×D4—48no—
274C7×D4—48no—
275C7×D4—42no—
276C7×D4—42no—
277C7×D4—42no—
278C7×D4—42no—
279C7×D4—84no—
280C7×D4—82no—
281C7×D4—88no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—24no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—24no—
289C2×C4×C7—22no—
290C2×C4×C7—44no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7—42no—
296C2×C4×C7—44no—
297C2×C4×C7—44no—
298C2×C4×C7—42no—
299C2×C4×C7—42no—
300C2×C4×C7—44no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—42no—
304C2×C4×C7—42no—
305C2×C4×C7—82no—
306C2×C4×C7—84no—
307C2×C4×C7—84no—
308C25C23×C711yescharacteristic
309C25C23×C711yescharacteristic
310C2×#41—22no—
311C22×D4—21no—
312C22×D4—21no—
313C22×D4—22no—
314C22×D4—24no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—42no—
318C22×D4C7×D414yescomplemented
319C22×D4C23×C712yes—
320C23×C4—21no—
321C23×C4C7×D412yescomplemented
322C23×C4C23×C711yescharacteristic
323#126—24no—
324#124—24no—
325#123—24no—
326#123—24no—
327#123—22no—
328#123—24no—
329#123—42no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C22⋊Q8—24no—
333#121—22no—
334#121—24no—
335#121—22no—
336#120—22no—
337#120—22no—
338#120—24no—
339#120—22no—
340#120—24no—
341#120—42no—
342C4×D4—24no—
343C4×D4—24no—
344C4×D4—24no—
345#117—22no—
346#117—22no—
347C2×C4⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4C7×D412yescomplemented
359C2×C22⋊C4C7×D414yescomplemented
360C2×C22⋊C4C23×C711yescharacteristic
361C2×C22⋊C4C23×C711yescharacteristic
362C2×C22⋊C4C23×C712yes—
363C2×C22⋊C4C23×C712yes—
364C2×C22⋊C4C23×C711yescharacteristic
365C2×C22⋊C4C23×C711yescharacteristic
366C2×C22⋊C4C23×C712yes—
367#99—42no—
368#99—42no—
369#95—22no—
370#95—22no—
371#95—24no—
372C22×C7—22no—
373C22×C7—24no—
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—24no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7—42no—
382C22×C7—42no—
383C22×C7—42no—
384C22×C7—42no—
385C22×C7—42no—
386C22×C7—44no—
387C22×C7—44no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—44no—
393C22×C7—84no—
394C22×C7—84no—
395C22×C7—82no—
396C22×C7—82no—
397C22×C7—82no—
398C22×C7—84no—
399C22×C7—82no—
400C22×C7C2×#12011yescharacteristic
401C22×C7#52511yescharacteristic
402C22×C7#53511yescharacteristic
403C22×C7#54612yes—
404C22×C7#54612yes—
405C4×C7—42no—
406C4×C7—42no—
407C4×C7—42no—
408C4×C7—42no—
409C4×C7—44no—
410C4×C7—82no—
411C4×C7—81no—
412C4×C7—81no—
413C4×C7—82no—
414C24—21no—
415C24—21no—
416C24—22no—
417C24—21no—
418C24—22no—
419C24—21no—
420C24—22no—
421C24—82no—
422C24—82no—
423C24C2×C7×D412yes—
424C24C2×C7×D414yes—
425C24C2×C7×D411yescharacteristic
426C24C2×C7×D411yescharacteristic
427C24C7×#4112yescomplemented
428C24C24×C711yescharacteristic
429#41—84no—
430C2×Q8—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—24no—
442C2×D4—48no—
443C2×D4—48no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4—84no—
450C2×D4—84no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—42no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C2×C7×D412yes—
466C22×C4C2×C7×D412yes—
467C22×C4C2×C7×D414yes—
468C22×C4C2×C7×D412yes—
469C22×C4C7×#4112yescomplemented
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—24no—
498C2×C7—21no—
499C2×C7—22no—
500C2×C7—22no—
501C2×C7—21no—
502C2×C7—42no—
503C2×C7—42no—
504C2×C7—82no—
505C2×C7—82no—
506C2×C7#240411yescharacteristic
507C2×C7#302612yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—84no—
537C23—84no—
538C23C7×#12012yes—
539C23C7×#12012yes—
540C23C22×C7×D411yescharacteristic
541C23C22×C7×D412yes—
542C23C2×C7×#4111yescharacteristic
543C23C2×C7×#4111yescharacteristic
544C23C7×H(5,𝔽2)12yes—
545Q8—82no—
546D4—48no—
547D4—48no—
548D4—48no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—84no—
554D4—82no—
555D4—88no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—84no—
582C7#2398711yescharacteristic, direct product factor, Sylow
583C22—22no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—84no—
610C22—82no—
611C22C2×C7×#12011yescharacteristic
612C22C7×#52511yescharacteristic
613C22C7×#53511yescharacteristic
614C22C7×#54612yes—
615C22C7×#54612yes—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4—82no—
625C2—21no—
626C2—22no—
627C2—22no—
628C2—21no—
629C2—42no—
630C2—42no—
631C2—82no—
632C2—82no—
633C2C7×#240411yescharacteristic
634C2C7×#302612yes—
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×#2613C2semidirect
3. C7×#2308C2semidirect
4. C7×#2030C2semidirect
5. C7×#2029C2semidirect
6. C7×#2022C2semidirect
7. C7×#2021C2semidirect
8. C7×#1898C2semidirect
9. C7×#1890C2semidirect
10. C7×#1889C2semidirect
11. C7×#1819C2semidirect
43. #23987C7direct (N × H)
54. C22×C7×D4D4semidirect
57. C23×C4×C7D4semidirect
94. C2×C7×C22⋊C4D4semidirect
95. C2×C7×C22⋊C4D4semidirect
108. #3490C2×C7semidirect
109. #2613C2×C7semidirect
110. #2308C2×C7semidirect
111. #2030C2×C7semidirect
112. #2029C2×C7semidirect
113. #2022C2×C7semidirect
114. #2021C2×C7semidirect
115. #1898C2×C7semidirect
116. #1890C2×C7semidirect
117. #1889C2×C7semidirect
118. #1819C2×C7semidirect
132. C24×C7#41semidirect
174. C22×C4×C7#41semidirect
318. C22×D4C7×D4semidirect
321. C23×C4C7×D4semidirect
358. C2×C22⋊C4C7×D4semidirect
359. C2×C22⋊C4C7×D4semidirect
427. C24C7×#41semidirect
469. C22×C4C7×#41semidirect
582. C7#23987direct (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×#26132
C7×#23082
C7×#20302
C7×#20292
C7×#20222
C7×#20212
C7×#18982
C7×#18902
C7×#18892
C7×#18192
#239877