Finite Group Databasebeta

← #72277737

Subgroups — #72277737

Order 2688.

1254 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 33 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×#10620796C211yesmaximal, characteristic, complemented
3#13052164C211yesmaximal, characteristic, complemented
4#11741573C311yesmaximal, characteristic, complemented, Hall
5C2×C7×C22⋊A4—21no—
6C2×C7×C22⋊A4C2211yescharacteristic
7A4×C22⋊D7—41nomaximal
8#10620796—21no—
9C26×C7C2×C311yescharacteristic, complemented
10C2×#6516—31no—
11C2×#6516—31no—
12C2×#6516C2×C311yescharacteristic, complemented
13#90019—31no—
14#90019—31no—
15#90019C2×C311yescharacteristic, complemented
16AΣL(1,𝔽64)—71nomaximal, Hall
17C7×C22⋊A4D411yescharacteristic, complemented
18C22×C7×A4—41no—
19C22×C7×A4—82no—
20C2×A4×D7—41no—
21A4×Dic7—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×C22⋊D7—31no—
31C22×C22⋊D7—31no—
32C22×C22⋊D7A411yescharacteristic, complemented
33#6516—21no—
34#6516—62no—
35#6516—64no—
36#6516—62no—
37#6516—61no—
38C2×C22⋊Dic7—32no—
39C2×C22⋊Dic7—31no—
40C2×C22⋊Dic7—31no—
41C2×C22⋊Dic7—31no—
42C2×C22⋊Dic7—31no—
43C2×C22⋊Dic7—31no—
44C22×C22⋊A4D711yescharacteristic, complemented
45C2×#120⋊C3—71no—
46#379⋊C3—71no—
47C2×C7×A4—41no—
48C2×C7×A4—82no—
49C2×C7×A4—81no—
50C2×C7×A4—84no—
51A4×D7—81no—
52C3×C22⋊D7—161no—
53#2846—71noSylow
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
92C23×D7C2×A411yescharacteristic, complemented
93C2×C22⋊D7—61no—
94C2×C22⋊D7—61no—
95C2×C22⋊D7—61no—
96C2×C22⋊D7—62no—
97C2×C22⋊D7—62no—
98C2×C22⋊D7—64no—
99C2×C22⋊D7—62no—
100C2×C22⋊D7—61no—
101C22×Dic7—31no—
102C22×Dic7—31no—
103C22×Dic7C2×A411yescharacteristic, complemented
104C22⋊Dic7—62no—
105C22⋊Dic7—62no—
106C22⋊Dic7—64no—
107C22⋊Dic7—62no—
108C22⋊Dic7—61no—
109C22⋊Dic7—61no—
110C22⋊Dic7—61no—
111C22⋊Dic7—61no—
112C2×C22⋊A4—21no—
113C2×C22⋊A4C2×D711yescharacteristic
114D4×A4—281no—
115#120⋊C3—141no—
116C22×C3×C7—161no—
117C2×C3×D7—161no—
118C7×A4—41no—
119C7×A4—82no—
120C3×Dic7—161no—
121C26C3×D711yescharacteristic, complemented
122C2×#120—71no—
123C2×#120—211no—
124C2×#120—211no—
125#379—71no—
126#379—211no—
127#379—211no—
128C23×C7—24no—
129C23×C7—22no—
130C23×C7—21no—
131C23×C7—34no—
132C23×C7—32no—
133C23×C7—32no—
134C23×C7—32no—
135C23×C7—31no—
136C23×C7—31no—
137C23×C7—31no—
138C23×C7—31no—
139C23×C7—62no—
140C23×C7—62no—
141C23×C7—64no—
142C23×C7—61no—
143C23×C7—61no—
144C23×C7—62no—
145C23×C7—62no—
146C23×C7—62no—
147C23×C7—62no—
148C23×C7—62no—
149C23×C7—64no—
150C23×C7—62no—
151C23×C7—62no—
152C23×C7—64no—
153C23×C7—68no—
154C23×C7—62no—
155C23×C7—64no—
156C23×C7—64no—
157C23×C7—64no—
158C23×C7—68no—
159C23×C7—62no—
160C23×C7—64no—
161C23×C7—64no—
162C23×C7—62no—
163C23×C7—62no—
164C23×C7—64no—
165C23×C7—62no—
166C23×C7—62no—
167C23×C7—64no—
168C23×C7—64no—
169C23×C7—68no—
170C23×C7—64no—
171C23×C7—64no—
172C23×C7—68no—
173C23×C7—64no—
174C23×C7—64no—
175C23×C7—68no—
176C23×C7—64no—
177C23×C7—68no—
178C23×C7—64no—
179C23×C7—64no—
180C23×C7—68no—
181C23×C7—64no—
182C23×C7—68no—
183C23×C7—68no—
184C23×C7—64no—
185C23×C7—68no—
186C23×C7—68no—
187C23×C7—68no—
188C23×C7—68no—
189C23×C7—64no—
190C23×C7—62no—
191C23×C7—61no—
192C23×C7C22×A411yescharacteristic
193C22×D7—21no—
194C22×D7—61no—
195C22×D7—61no—
196C22⋊D7—41no—
197C22⋊D7—121no—
198C22⋊D7—122no—
199C22⋊D7—122no—
200C22⋊D7—124no—
201C2×Dic7—61no—
202C2×Dic7—62no—
203C2×Dic7—61no—
204C2×Dic7—61no—
205C2×Dic7—61no—
206C2×Dic7—61no—
207C22⋊A4C22⋊D711yescharacteristic, complemented
208C22×A4—41no—
209C22×A4—82no—
210C22×A4—281no—
211C4×A4—281no—
212C2×C3×C7—161no—
213C2×C3×C7—321no—
214C3×D7—321no—
215C25—21no—
216C25—31no—
217C25—31no—
218C25—62no—
219C25—62no—
220C25—64no—
221C25—61no—
222C25C2×C3×D711yescharacteristic
223C22×D4—71no—
224C22×D4—211no—
225C22×D4—211no—
226#120—141no—
227#120—422no—
228#120—424no—
229#120—422no—
230#120—421no—
231C2×C22⋊C4—211no—
232C2×C22⋊C4—211no—
233C2×C22⋊C4—212no—
234C2×C22⋊C4—211no—
235C2×C22⋊C4—211no—
236C2×C22⋊C4—211no—
237C22×C7—22no—
238C22×C7—31no—
239C22×C7—31no—
240C22×C7—34no—
241C22×C7—32no—
242C22×C7—32no—
243C22×C7—31no—
244C22×C7—61no—
245C22×C7—61no—
246C22×C7—64no—
247C22×C7—64no—
248C22×C7—61no—
249C22×C7—61no—
250C22×C7—64no—
251C22×C7—62no—
252C22×C7—64no—
253C22×C7—62no—
254C22×C7—64no—
255C22×C7—62no—
256C22×C7—62no—
257C22×C7—64no—
258C22×C7—62no—
259C22×C7—62no—
260C22×C7—62no—
261C22×C7—64no—
262C22×C7—64no—
263C22×C7—64no—
264C22×C7—64no—
265C22×C7—68no—
266C22×C7—68no—
267C22×C7—64no—
268C22×C7—68no—
269C22×C7—68no—
270C22×C7—64no—
271C22×C7—62no—
272C22×C7—62no—
273C22×C7#120⋊C311yescharacteristic, complemented
274C22×C7D4×A411yescharacteristic
275C2×D7—41no—
276C2×D7—121no—
277C2×D7—121no—
278Dic7—41no—
279Dic7—121no—
280Dic7—121no—
281C2×A4—41no—
282C2×A4—82no—
283C2×A4—81no—
284C2×A4—84no—
285C2×A4—561no—
286C3×D4—1121no—
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—
324C24—71no—
325C24C3×C22⋊D711yescharacteristic, complemented
326C24A4×D711yescharacteristic
327C2×D4—421no—
328C2×D4—421no—
329C2×D4—421no—
330C2×D4—422no—
331C2×D4—422no—
332C2×D4—422no—
333C2×D4—424no—
334C2×D4—421no—
335C22×C4—71no—
336C22×C4—211no—
337C22×C4—211no—
338C22⋊C4—422no—
339C22⋊C4—422no—
340C22⋊C4—424no—
341C22⋊C4—422no—
342C22⋊C4—421no—
343C22⋊C4—421no—
344C22⋊C4—421no—
345C22⋊C4—421no—
346C2×C7—21no—
347C2×C7—31no—
348C2×C7—31no—
349C2×C7—62no—
350C2×C7—64no—
351C2×C7—62no—
352C2×C7—61no—
353C2×C7C2×#120⋊C311yescharacteristic
354D7—81no—
355C22×C3—161no—
356C22×C3—1121no—
357A4—41no—
358A4—82no—
359C3×C4—1121no—
360C23—24no—
361C23—22no—
362C23—21no—
363C23—32no—
364C23—34no—
365C23—32no—
366C23—32no—
367C23—31no—
368C23—31no—
369C23—31no—
370C23—31no—
371C23—62no—
372C23—62no—
373C23—64no—
374C23—61no—
375C23—62no—
376C23—61no—
377C23—62no—
378C23—62no—
379C23—62no—
380C23—62no—
381C23—64no—
382C23—62no—
383C23—62no—
384C23—64no—
385C23—68no—
386C23—62no—
387C23—64no—
388C23—64no—
389C23—64no—
390C23—68no—
391C23—62no—
392C23—64no—
393C23—64no—
394C23—62no—
395C23—62no—
396C23—64no—
397C23—62no—
398C23—62no—
399C23—64no—
400C23—64no—
401C23—68no—
402C23—64no—
403C23—64no—
404C23—68no—
405C23—64no—
406C23—64no—
407C23—68no—
408C23—64no—
409C23—68no—
410C23—64no—
411C23—68no—
412C23—64no—
413C23—64no—
414C23—68no—
415C23—68no—
416C23—64no—
417C23—68no—
418C23—68no—
419C23—68no—
420C23—68no—
421C23—64no—
422C23—61no—
423C23—62no—
424C23—141no—
425C23—421no—
426C23—421no—
427C23C2×A4×D711yescharacteristic
428D4—281no—
429D4—841no—
430D4—842no—
431D4—842no—
432D4—844no—
433C2×C4—421no—
434C2×C4—421no—
435C2×C4—422no—
436C2×C4—421no—
437C2×C4—421no—
438C2×C4—421no—
439C7AΣL(1,𝔽64)11yescharacteristic, complemented, Sylow
440C2×C3—161no—
441C2×C3—321no—
442C2×C3—2241no—
443C22—22no—
444C22—31no—
445C22—31no—
446C22—32no—
447C22—32no—
448C22—34no—
449C22—31no—
450C22—61no—
451C22—61no—
452C22—64no—
453C22—64no—
454C22—61no—
455C22—61no—
456C22—62no—
457C22—64no—
458C22—62no—
459C22—62no—
460C22—64no—
461C22—62no—
462C22—64no—
463C22—64no—
464C22—62no—
465C22—62no—
466C22—62no—
467C22—64no—
468C22—64no—
469C22—64no—
470C22—64no—
471C22—68no—
472C22—68no—
473C22—64no—
474C22—68no—
475C22—68no—
476C22—64no—
477C22—62no—
478C22—62no—
479C22—281no—
480C22—841no—
481C22—841no—
482C22#1062079611yescharacteristic, complemented
483C22A4×C22⋊D711yescharacteristic
484C4—281no—
485C4—841no—
486C4—841no—
487C3—161noSylow
488C2—21no—
489C2—31no—
490C2—31no—
491C2—62no—
492C2—64no—
493C2—62no—
494C2—61no—
495C2—561no—
496C2C2×#1062079611yescharacteristic
subgroupG/H
centerC2C2×#10620796
commutator, cosocleC25×C7C22×C3
FrattiniC23C2×A4×D7
FittingC26×C7C2×C3
socleC23×C7C22×A4
hypercenterC22#10620796
hypocenterC24×C7C3×D4
LCSG ▷ C25×C7 ▷ C24×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C7×C22⋊A4C2semidirect
2. C2×#10620796C2semidirect
3. #13052164C2semidirect
4. #11741573C3semidirect
9. C26×C7C2×C3semidirect
12. C2×#6516C2×C3semidirect
15. #90019C2×C3semidirect
17. C7×C22⋊A4D4semidirect
32. C22×C22⋊D7A4semidirect
44. C22×C22⋊A4D7semidirect
90. C24×C7C3×D4semidirect
92. C23×D7C2×A4semidirect
103. C22×Dic7C2×A4semidirect
121. C26C3×D7semidirect
207. C22⋊A4C22⋊D7semidirect
273. C22×C7#120⋊C3semidirect
325. C24C3×C22⋊D7semidirect
439. C7AΣL(1,𝔽64)semidirect
482. C22#10620796semidirect

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×#106207962
#130521642
#117415733
A4×C22⋊D74
AΣL(1,𝔽64)7