Finite Group Databasebeta

← C7×AΣL(1,𝔽64) #72277204

Subgroups — C7×AΣL(1,𝔽64) #72277204

Order 2688.

1254 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×C7×#120⋊C3C211yesmaximal, characteristic, complemented
3C7×#379⋊C3C211yesmaximal, characteristic, complemented
4C7×#2846C311yesmaximal, characteristic, complemented, Hall
5C2×C7×C22⋊A4—21no—
6C2×C7×C22⋊A4C2211yescharacteristic
7C7×D4×A4—41nomaximal
8C7×#120⋊C3—21no—
9C26×C7C2×C311yescharacteristic, complemented
10C2×C7×#120—31no—
11C2×C7×#120—31no—
12C2×C7×#120C2×C311yescharacteristic, complemented
13C7×#379—31no—
14C7×#379—31no—
15C7×#379C2×C311yescharacteristic, complemented
16AΣL(1,𝔽64)C711yesmaximal, characteristic, direct product factor, Hall
17C7×C22⋊A4D411yescharacteristic, complemented
18C22×C7×A4—41no—
19C22×C7×A4—41no—
20C22×C7×A4—82no—
21C4×C7×A4—41no—
22C25×C7—21no—
23C25×C7—31no—
24C25×C7—31no—
25C25×C7—62no—
26C25×C7—62no—
27C25×C7—64no—
28C25×C7—61no—
29C25×C7C22×C311yescharacteristic
30C22×C7×D4—31no—
31C22×C7×D4—31no—
32C22×C7×D4A411yescharacteristic, complemented
33C7×#120—21no—
34C7×#120—62no—
35C7×#120—64no—
36C7×#120—62no—
37C7×#120—61no—
38C2×C7×C22⋊C4—32no—
39C2×C7×C22⋊C4—31no—
40C2×C7×C22⋊C4—31no—
41C2×C7×C22⋊C4—31no—
42C2×C7×C22⋊C4—31no—
43C2×C7×C22⋊C4—31no—
44C22×C22⋊A4C2×C711yescharacteristic, complemented
45C2×#120⋊C3C2×C711yescharacteristic, complemented
46#379⋊C3C2×C711yescharacteristic, complemented
47C2×C7×A4—41no—
48C2×C7×A4—82no—
49C2×C7×A4—81no—
50C2×C7×A4—81no—
51C2×C7×A4—84no—
52C3×C7×D4—161no—
53#2846C3×C711yescharacteristic, complemented, Sylow
54C24×C7—22no—
55C24×C7—31no—
56C24×C7—31no—
57C24×C7—34no—
58C24×C7—32no—
59C24×C7—32no—
60C24×C7—31no—
61C24×C7—61no—
62C24×C7—61no—
63C24×C7—62no—
64C24×C7—64no—
65C24×C7—64no—
66C24×C7—62no—
67C24×C7—64no—
68C24×C7—62no—
69C24×C7—62no—
70C24×C7—64no—
71C24×C7—64no—
72C24×C7—62no—
73C24×C7—62no—
74C24×C7—61no—
75C24×C7—61no—
76C24×C7—62no—
77C24×C7—64no—
78C24×C7—64no—
79C24×C7—64no—
80C24×C7—64no—
81C24×C7—68no—
82C24×C7—68no—
83C24×C7—68no—
84C24×C7—64no—
85C24×C7—68no—
86C24×C7—64no—
87C24×C7—62no—
88C24×C7—64no—
89C24×C7—62no—
90C24×C7C3×D411yescharacteristic, complemented
91C24×C7C2×A411yescharacteristic
92C24×C7C2×A411yescharacteristic, complemented
93C2×C7×D4—61no—
94C2×C7×D4—61no—
95C2×C7×D4—61no—
96C2×C7×D4—62no—
97C2×C7×D4—62no—
98C2×C7×D4—64no—
99C2×C7×D4—62no—
100C2×C7×D4—61no—
101C22×C4×C7—31no—
102C22×C4×C7—31no—
103C22×C4×C7C2×A411yescharacteristic, complemented
104C7×C22⋊C4—62no—
105C7×C22⋊C4—62no—
106C7×C22⋊C4—64no—
107C7×C22⋊C4—62no—
108C7×C22⋊C4—61no—
109C7×C22⋊C4—61no—
110C7×C22⋊C4—61no—
111C7×C22⋊C4—61no—
112C2×C22⋊A4—21no—
113C2×C22⋊A4C22×C711yescharacteristic
114D4×A4—41no—
115#120⋊C3—21no—
116C22×C3×C7—161no—
117C22×C3×C7—161no—
118C7×A4—41no—
119C7×A4—82no—
120C3×C4×C7—161no—
121C26C2×C3×C711yescharacteristic, complemented
122C2×#120—31no—
123C2×#120—31no—
124C2×#120C2×C3×C711yescharacteristic, complemented
125#379—31no—
126#379—31no—
127#379C2×C3×C711yescharacteristic, complemented
128C23×C7—24no—
129C23×C7—22no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—34no—
133C23×C7—32no—
134C23×C7—32no—
135C23×C7—32no—
136C23×C7—31no—
137C23×C7—31no—
138C23×C7—31no—
139C23×C7—31no—
140C23×C7—62no—
141C23×C7—62no—
142C23×C7—64no—
143C23×C7—61no—
144C23×C7—61no—
145C23×C7—61no—
146C23×C7—62no—
147C23×C7—62no—
148C23×C7—62no—
149C23×C7—62no—
150C23×C7—62no—
151C23×C7—64no—
152C23×C7—62no—
153C23×C7—62no—
154C23×C7—64no—
155C23×C7—68no—
156C23×C7—62no—
157C23×C7—64no—
158C23×C7—64no—
159C23×C7—64no—
160C23×C7—68no—
161C23×C7—62no—
162C23×C7—64no—
163C23×C7—61no—
164C23×C7—64no—
165C23×C7—62no—
166C23×C7—62no—
167C23×C7—64no—
168C23×C7—62no—
169C23×C7—62no—
170C23×C7—64no—
171C23×C7—64no—
172C23×C7—68no—
173C23×C7—64no—
174C23×C7—64no—
175C23×C7—68no—
176C23×C7—64no—
177C23×C7—64no—
178C23×C7—68no—
179C23×C7—64no—
180C23×C7—68no—
181C23×C7—64no—
182C23×C7—64no—
183C23×C7—68no—
184C23×C7—64no—
185C23×C7—68no—
186C23×C7—68no—
187C23×C7—64no—
188C23×C7—68no—
189C23×C7—68no—
190C23×C7—68no—
191C23×C7—68no—
192C23×C7—64no—
193C23×C7—62no—
194C23×C7—61no—
195C23×C7C22×A411yescharacteristic
196C7×D4—41no—
197C7×D4—121no—
198C7×D4—122no—
199C7×D4—122no—
200C7×D4—124no—
201C2×C4×C7—61no—
202C2×C4×C7—62no—
203C2×C4×C7—61no—
204C2×C4×C7—61no—
205C2×C4×C7—61no—
206C2×C4×C7—61no—
207C22⋊A4C7×D411yescharacteristic, complemented
208C22×A4—41no—
209C22×A4—41no—
210C22×A4—82no—
211C4×A4—41no—
212C2×C3×C7—161no—
213C2×C3×C7—321no—
214C2×C3×C7—321no—
215C25—21no—
216C25—31no—
217C25—31no—
218C25—62no—
219C25—62no—
220C25—64no—
221C25—61no—
222C25C22×C3×C711yescharacteristic
223C22×D4—31no—
224C22×D4—31no—
225C22×D4C7×A411yescharacteristic, complemented
226#120—21no—
227#120—62no—
228#120—64no—
229#120—62no—
230#120—61no—
231C2×C22⋊C4—32no—
232C2×C22⋊C4—31no—
233C2×C22⋊C4—31no—
234C2×C22⋊C4—31no—
235C2×C22⋊C4—31no—
236C2×C22⋊C4—31no—
237C22×C7—22no—
238C22×C7—31no—
239C22×C7—31no—
240C22×C7—34no—
241C22×C7—32no—
242C22×C7—32no—
243C22×C7—31no—
244C22×C7—41no—
245C22×C7—61no—
246C22×C7—61no—
247C22×C7—64no—
248C22×C7—64no—
249C22×C7—61no—
250C22×C7—61no—
251C22×C7—64no—
252C22×C7—62no—
253C22×C7—64no—
254C22×C7—62no—
255C22×C7—64no—
256C22×C7—62no—
257C22×C7—62no—
258C22×C7—64no—
259C22×C7—62no—
260C22×C7—62no—
261C22×C7—62no—
262C22×C7—64no—
263C22×C7—64no—
264C22×C7—64no—
265C22×C7—64no—
266C22×C7—68no—
267C22×C7—68no—
268C22×C7—64no—
269C22×C7—68no—
270C22×C7—68no—
271C22×C7—64no—
272C22×C7—62no—
273C22×C7—62no—
274C22×C7—121no—
275C22×C7—121no—
276C22×C7#120⋊C311yescharacteristic, complemented
277C22×C7D4×A411yescharacteristic
278C4×C7—41no—
279C4×C7—121no—
280C4×C7—121no—
281C2×A4—41no—
282C2×A4—82no—
283C2×A4—81no—
284C2×A4—81no—
285C2×A4—84no—
286C3×D4—161no—
287C3×C7—161noHall
288C24—22no—
289C24—31no—
290C24—31no—
291C24—34no—
292C24—32no—
293C24—32no—
294C24—31no—
295C24—62no—
296C24—61no—
297C24—61no—
298C24—64no—
299C24—64no—
300C24—62no—
301C24—62no—
302C24—61no—
303C24—61no—
304C24—62no—
305C24—64no—
306C24—62no—
307C24—64no—
308C24—64no—
309C24—62no—
310C24—62no—
311C24—64no—
312C24—62no—
313C24—64no—
314C24—62no—
315C24—64no—
316C24—64no—
317C24—64no—
318C24—64no—
319C24—68no—
320C24—68no—
321C24—68no—
322C24—64no—
323C24—68no—
324C24C3×C7×D411yescharacteristic, complemented
325C24C2×C7×A411yescharacteristic
326C24C2×C7×A411yescharacteristic, complemented
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—62no—
333C2×D4—64no—
334C2×D4—61no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4C2×C7×A411yescharacteristic, complemented
338C22⋊C4—62no—
339C22⋊C4—62no—
340C22⋊C4—64no—
341C22⋊C4—62no—
342C22⋊C4—61no—
343C22⋊C4—61no—
344C22⋊C4—61no—
345C22⋊C4—61no—
346C2×C7—21no—
347C2×C7—31no—
348C2×C7—31no—
349C2×C7—62no—
350C2×C7—64no—
351C2×C7—62no—
352C2×C7—61no—
353C2×C7—81no—
354C2×C7C2×#120⋊C311yescharacteristic
355C22×C3—161no—
356C22×C3—161no—
357A4—41no—
358A4—82no—
359C3×C4—161no—
360C23—24no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—32no—
365C23—34no—
366C23—32no—
367C23—32no—
368C23—31no—
369C23—31no—
370C23—31no—
371C23—31no—
372C23—62no—
373C23—62no—
374C23—64no—
375C23—61no—
376C23—61no—
377C23—62no—
378C23—61no—
379C23—62no—
380C23—62no—
381C23—62no—
382C23—62no—
383C23—64no—
384C23—62no—
385C23—62no—
386C23—64no—
387C23—68no—
388C23—62no—
389C23—64no—
390C23—64no—
391C23—64no—
392C23—68no—
393C23—62no—
394C23—64no—
395C23—61no—
396C23—64no—
397C23—62no—
398C23—62no—
399C23—64no—
400C23—62no—
401C23—62no—
402C23—64no—
403C23—64no—
404C23—68no—
405C23—64no—
406C23—64no—
407C23—68no—
408C23—64no—
409C23—64no—
410C23—68no—
411C23—64no—
412C23—68no—
413C23—64no—
414C23—68no—
415C23—64no—
416C23—64no—
417C23—68no—
418C23—68no—
419C23—64no—
420C23—68no—
421C23—68no—
422C23—68no—
423C23—68no—
424C23—64no—
425C23—61no—
426C23—62no—
427C23C22×C7×A411yescharacteristic
428D4—41no—
429D4—121no—
430D4—122no—
431D4—122no—
432D4—124no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C7AΣL(1,𝔽64)11yescharacteristic, direct product factor, Sylow
440C2×C3—161no—
441C2×C3—321no—
442C2×C3—321no—
443C22—22no—
444C22—31no—
445C22—31no—
446C22—32no—
447C22—32no—
448C22—34no—
449C22—31no—
450C22—41no—
451C22—61no—
452C22—61no—
453C22—64no—
454C22—64no—
455C22—61no—
456C22—61no—
457C22—62no—
458C22—64no—
459C22—62no—
460C22—62no—
461C22—64no—
462C22—62no—
463C22—64no—
464C22—64no—
465C22—62no—
466C22—62no—
467C22—62no—
468C22—64no—
469C22—64no—
470C22—64no—
471C22—64no—
472C22—68no—
473C22—68no—
474C22—64no—
475C22—68no—
476C22—68no—
477C22—64no—
478C22—62no—
479C22—62no—
480C22—121no—
481C22—121no—
482C22C7×#120⋊C311yescharacteristic, complemented
483C22C7×D4×A411yescharacteristic
484C4—41no—
485C4—121no—
486C4—121no—
487C3—161noSylow
488C2—21no—
489C2—31no—
490C2—31no—
491C2—62no—
492C2—64no—
493C2—62no—
494C2—61no—
495C2—81no—
496C2C2×C7×#120⋊C311yescharacteristic
subgroupG/H
centerC2×C7C2×#120⋊C3
commutator, cosocleC25C22×C3×C7
FrattiniC23C22×C7×A4
FittingC7×#2846C3
socleC23×C7C22×A4
hypercenterC22×C7#120⋊C3
hypocenterC24C3×C7×D4
LCSG ▷ C25 ▷ C24
UCSC1 ◁ C2×C7 ◁ C22×C7

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊A4C2semidirect
2. C2×C7×#120⋊C3C2semidirect
3. C7×#379⋊C3C2semidirect
4. C7×#2846C3semidirect
9. C26×C7C2×C3semidirect
12. C2×C7×#120C2×C3semidirect
15. C7×#379C2×C3semidirect
16. AΣL(1,𝔽64)C7direct (N × H)
17. C7×C22⋊A4D4semidirect
32. C22×C7×D4A4semidirect
44. C22×C22⋊A4C2×C7semidirect
45. C2×#120⋊C3C2×C7semidirect
46. #379⋊C3C2×C7semidirect
53. #2846C3×C7semidirect
90. C24×C7C3×D4semidirect
92. C24×C7C2×A4semidirect
103. C22×C4×C7C2×A4semidirect
121. C26C2×C3×C7semidirect
124. C2×#120C2×C3×C7semidirect
127. #379C2×C3×C7semidirect
207. C22⋊A4C7×D4semidirect
225. C22×D4C7×A4semidirect
276. C22×C7#120⋊C3semidirect
324. C24C3×C7×D4semidirect
326. C24C2×C7×A4semidirect
337. C22×C4C2×C7×A4semidirect
439. C7AΣL(1,𝔽64)direct (N × H)
482. C22C7×#120⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊A42
C2×C7×#120⋊C32
C7×#379⋊C32
C7×#28463
C7×D4×A44
AΣL(1,𝔽64)7