Finite Group Databasebeta

← #34346258

Subgroups — #34346258

Order 20224.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766267C211yesmaximal, characteristic
2#23764962C211yesmaximal, characteristic, complemented
3#23764901C211yesmaximal, characteristic, complemented
4#23764891C211yesmaximal, characteristic, complemented
5#23764735C211yesmaximal, characteristic, complemented
6#23764358C211yesmaximal, characteristic, complemented
7#23764161C211yesmaximal, characteristic, complemented
8#23763989C211yesmaximal, characteristic, complemented
9#23763968C211yesmaximal, characteristic, complemented
10#23763599C211yesmaximal, characteristic
11#23763585C211yesmaximal, characteristic
12#23763488C211yesmaximal, characteristic
13#23763431C211yesmaximal, characteristic
14#23763121C211yesmaximal, characteristic
15C79×#2441C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579117—21no—
18C22×C4⋊Dic79C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579092C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C22⋊(C79⋊C8)C2211yescharacteristic
26#19098461C2211yescharacteristic
27#19098461C2211yescharacteristic
28#19098448—21no—
29#19098448—21no—
30#19098424—21no—
31#19098423C2211yescharacteristic
32C4×C4⋊Dic79C2211yescharacteristic
33C4×C4⋊Dic79C2211yescharacteristic
34#19098325C2211yescharacteristic
35#19098325C2211yescharacteristic
36#19098325C2211yescharacteristic
37#19098325C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#125×C79C2211yescharacteristic
49C2×#117×C79C2211yescharacteristic
50#399×C79—21no—
51#396×C79—21no—
52#389×C79—21no—
53#389×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55#387×C79—21no—
56#384×C79C2211yescharacteristic
57#380×C79C2211yescharacteristic
58C4×C22⋊C4×C79C2211yescharacteristic
59C23×Dic79—21no—
60C2×#11929608—22no—
61C2×#11929608—22no—
62C2×#11929608—41no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579117—44no—
68C2×C4⋊Dic79—21no—
69C2×C4⋊Dic79—22no—
70C2×C4⋊Dic79Q814yes—
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×C4⋊Dic79C2311yescharacteristic
74C2×C4⋊Dic79C2311yescharacteristic
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×C22⋊Dic79—21no—
78C2×C4×Dic79—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094—22no—
82#14579092—41no—
83#14579092—41no—
84C4⋊(C79⋊C8)—22no—
85C4⋊(C79⋊C8)—41no—
86C22⋊(C79⋊C8)—22no—
87C79⋊#95—21no—
88C79⋊#95—21no—
89C79⋊#95—21no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C23×C4×C79C2311yescharacteristic
93#125×C79—22no—
94#125×C79—41no—
95#117×C79—22no—
96#117×C79—41no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79—21no—
102C2×C4⋊C4×C79—21no—
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C4⋊C4×C79C2311yescharacteristic
106C2×C4⋊C4×C79C2311yescharacteristic
107C2×C22⋊C4×C79—21no—
108C2×C22⋊C4×C79—21no—
109C2×C22⋊C4×C79D411yescharacteristic
110C2×C22⋊C4×C79D411yescharacteristic
111C2×C42×C79—21no—
112C2×C42×C79C2311yescharacteristic
113C2×C42×C79C2311yescharacteristic
114#95×C79—21no—
115#95×C79—21no—
116#95×C79—21no—
117#95×C79D411yescharacteristic
118#95×C79D411yescharacteristic
119C22×Dic79—24no—
120C22×Dic79—21no—
121C22×Dic79—21no—
122C22×Dic79—21no—
123C22×Dic79—21no—
124C22×Dic79—42no—
125C22×Dic79—42no—
126#11929608—44no—
127C2×C79⋊C8—21no—
128C2×C79⋊C8—21no—
129C2×C79⋊C8—21no—
130C2×C79⋊C8—21no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8#4112yes—
139C4⋊Dic79—24no—
140C4⋊Dic79—21no—
141C4⋊Dic79—21no—
142C4⋊Dic79—21no—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79C2×Q812yes—
147C4⋊Dic79C2×Q812yes—
148Dic79⋊C4—42no—
149Dic79⋊C4—42no—
150C22⋊Dic79—42no—
151C4×Dic79—42no—
152C4×Dic79—42no—
153C24×C79C2×D411yescharacteristic
154C22×C4×C79—21no—
155C22×C4×C79—21no—
156C22×C4×C79—21no—
157C22×C4×C79—21no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2×Q812yes—
168C22×C4×C79#4112yes—
169C22×C4×C79C2411yescharacteristic
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—21no—
177C4⋊C4×C79—42no—
178C4⋊C4×C79—42no—
179C4⋊C4×C79—42no—
180C4⋊C4×C79—42no—
181C4⋊C4×C79—81no—
182C4⋊C4×C79—81no—
183C4⋊C4×C79#4112yes—
184C22⋊C4×C79—22no—
185C22⋊C4×C79—41no—
186C22⋊C4×C79—42no—
187C22⋊C4×C79—81no—
188C42×C79—21no—
189C42×C79—21no—
190C42×C79—21no—
191C42×C79—21no—
192C42×C79—42no—
193C23×C79—21no—
194C23×C79—21no—
195C23×C79—21no—
196C23×C79—21no—
197C23×C79—21no—
198C23×C79—21no—
199C23×C79C22⋊Q812yes—
200C23×C79C22×D411yescharacteristic
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—21no—
205C2×C4×C79—22no—
206C2×C4×C79—22no—
207C2×C4×C79—22no—
208C2×C4×C79—22no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—22no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—21no—
217C2×C4×C79—21no—
218C2×C4×C79—21no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79—41no—
227C2×C4×C79—41no—
228C2×C4×C79—41no—
229C2×C4×C79—41no—
230C2×C4×C79C22⋊Q812yes—
231C2×C4×C79AGL(1,ℤ/8)12yes—
232C2×C4×C79C22×Q811yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×C4×C79C2×#4111yescharacteristic
235C2×C4×C79C2×#4111yescharacteristic
236C2×C4×C79H(5,𝔽2)11yescharacteristic
237C2×C4×C792−1+411yescharacteristic
238C2×Dic79—22no—
239C2×Dic79—22no—
240C2×Dic79—48no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243C2×Dic79—41no—
244C2×Dic79—41no—
245C2×Dic79—41no—
246C2×Dic79—41no—
247C2×Dic79—41no—
248C2×Dic79—41no—
249C2×Dic79—41no—
250C2×Dic79—41no—
251C79⋊C8—22no—
252C79⋊C8—41no—
253C79⋊C8—41no—
254C79⋊C8—41no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—41no—
259C22×C79—41no—
260C22×C79—41no—
261C22×C79—41no—
262C22×C79C2×C22⋊Q811yescharacteristic
263C22×C79#54611yescharacteristic
264C22×C79#54711yescharacteristic
265C22×C79C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C79#57511yescharacteristic
267C22×C79#57711yescharacteristic
268C22×C79#57811yescharacteristic
269C4×C79—22no—
270C4×C79—22no—
271C4×C79—41no—
272C4×C79—42no—
273C4×C79—41no—
274C4×C79—41no—
275C4×C79—81no—
276C4×C79—81no—
277C4×C79—81no—
278C4×C79#53311yescharacteristic
279C4×C79#54411yescharacteristic
280C4×C79#55111yescharacteristic
281C4×C79#55211yescharacteristic
282Dic79—42no—
283Dic79—42no—
284Dic79—81no—
285Dic79—81no—
286#26014—791nomaximal, Sylow
287C2×C79—42no—
288C2×C79#243811yescharacteristic
289C2×C79#307711yescharacteristic
290C2×C79#308211yescharacteristic
291C2×C79#331711yescharacteristic
292C2×C79#333011yescharacteristic
293C2×C79#334511yescharacteristic
294C2×C79#335211yescharacteristic
295#2441D7911yescharacteristic
296#2344—791no—
297#2100—791no—
298#2088—791no—
299#2086—791no—
300#2059—791no—
301#1989—791no—
302#1948—791no—
303#1921—791no—
304#1918—791no—
305#1854—791no—
306#1851—791no—
307#1824—791no—
308#1809—791no—
309#1736—791no—
310C79#2601411yescharacteristic, complemented, Sylow
311C22×M16—791no—
312C2×#125C2×D7911yescharacteristic
313C2×#117—1581no—
314C2×#117C2×D7911yescharacteristic
315C22×C4⋊C4—791no—
316C2×#107—791no—
317C2×#107—791no—
318C2×#106—791no—
319C2×#106—791no—
320C2×C4⋊C8—791no—
321C2×C4⋊C8—791no—
322C2×C22⋊C8—791no—
323#399—21no—
324#396—21no—
325#389—21no—
326#389—791no—
327#389—791no—
328#389C2×D7911yescharacteristic
329#389C2×D7911yescharacteristic
330#387—21no—
331#387—1581no—
332#387—1581no—
333#384C2×D7911yescharacteristic
334#381—1581no—
335#380—791no—
336#380C2×D7911yescharacteristic
337C4×C4⋊C4—791no—
338C4×C4⋊C4—791no—
339C4×C22⋊C4C2×D7911yescharacteristic
340#340—791no—
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#340—791no—
350#340—791no—
351#340—791no—
352#336—791no—
353#336—791no—
354C23×C4—1581no—
355C23×C4C22×D7911yescharacteristic
356C2×M16—3161no—
357C2×M16—1582no—
358C2×M16—1582no—
359C22×C8—791no—
360C22×C8—791no—
361C22×C8—791no—
362C22×C8—791no—
363#125—22no—
364#125—41no—
365#117—22no—
366#117—41no—
367#117—3164no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—794no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—791no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4—1581no—
381C2×C4⋊C4—1581no—
382C2×C4⋊C4—1582no—
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C4⋊C4C22×D7911yescharacteristic
385C2×C4⋊C4C22×D7911yescharacteristic
386C2×C4⋊C4C22×D7911yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—1581no—
390C2×C22⋊C4C22⋊D7911yescharacteristic
391C2×C22⋊C4C22⋊D7911yescharacteristic
392C2×C42—21no—
393C2×C42—1581no—
394C2×C42—1581no—
395C2×C42C22×D7911yescharacteristic
396C2×C42C22×D7911yescharacteristic
397#107—1582no—
398#107—1582no—
399#106—3161no—
400#106—3161no—
401C4⋊C8—3161no—
402C4⋊C8—1582no—
403C22⋊C8—1582no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1581no—
408#95—1581no—
409#95—1581no—
410#95—1581no—
411#95—1581no—
412#95C22⋊D7911yescharacteristic
413#95C22⋊D7911yescharacteristic
414C24C2×C22⋊D7911yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—3162no—
424C22×C4—3162no—
425C22×C4—1581no—
426C22×C4—1581no—
427C22×C4—1584no—
428C22×C4—1581no—
429C22×C4—1581no—
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C2×C22⋊D7911yescharacteristic
434C22×C4C2×C22⋊D7911yescharacteristic
435C22×C4Q8×D7912yes—
436C22×C4#1192962512yes—
437C22×C4C23×D7911yescharacteristic
438M16—3164no—
439C2×C8—792no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C2×C8—1581no—
447C2×C8—1581no—
448C2×C8—1581no—
449C2×C8—1581no—
450C2×C8—1581no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—3162no—
465C4⋊C4—3162no—
466C4⋊C4—792no—
467C4⋊C4—792no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1581no—
471C4⋊C4—1581no—
472C4⋊C4—1581no—
473C4⋊C4—1581no—
474C4⋊C4—1584no—
475C4⋊C4#1192962312yes—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—3162no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—3162no—
487C42—3162no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#1457913712yes—
495C23C22×C22⋊D7911yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—3168no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—3161no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3161no—
535C2×C4—3161no—
536C2×C4—1582no—
537C2×C4—1582no—
538C2×C4#1457913712yes—
539C2×C4#1457918612yes—
540C2×C4C2×Q8×D7911yescharacteristic
541C2×C4#41×D7911yescharacteristic
542C2×C4C2×#1192962311yescharacteristic
543C2×C4C2×#1192962511yescharacteristic
544C2×C4#1457920611yescharacteristic
545C2×C4#1457920811yescharacteristic
546C8—3161no—
547C8—3161no—
548C8—3161no—
549C8—1582no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#1457913711yescharacteristic
558C22#1909920711yescharacteristic
559C22#1909921211yescharacteristic
560C22C2×#1457918611yescharacteristic
561C22#1909936111yescharacteristic
562C22#1909937411yescharacteristic
563C22#1909937811yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—42no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—3162no—
574C4—3162no—
575C4—6321no—
576C4—6321no—
577C4#1909911511yescharacteristic
578C4#1909919211yescharacteristic
579C4#1909924011yescharacteristic
580C4#1909925511yescharacteristic
581C2—42no—
582C2#2376707211yescharacteristic
583C2#2377319311yescharacteristic
584C2#2377323611yescharacteristic
585C2#2377552611yescharacteristic
586C2#2377570511yescharacteristic
587C2#2377586411yescharacteristic
588C2#2377596911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2441C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2441D79
hypocenterC79#26014
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #23764962C2semidirect
3. #23764901C2semidirect
4. #23764891C2semidirect
5. #23764735C2semidirect
6. #23764358C2semidirect
7. #23764161C2semidirect
8. #23763989C2semidirect
9. #23763968C2semidirect
310. C79#26014semidirect

Maximal subgroups

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

subgroupindex
#237662672
#237649622
#237649012
#237648912
#237647352
#237643582
#237641612
#237639892
#237639682
#237635992
#237635852
#237634882
#237634312
#237631212
C79×#24412
#2601479