Finite Group Databasebeta

← #34298536

Subgroups — #34298536

Order 20224.

1266 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C8×D79C211yesmaximal, characteristic
2C2×#19098420C211yesmaximal, characteristic
3#380×D79C211yesmaximal, characteristic
4#23763590C211yesmaximal, characteristic, complemented
5#23763589C212yesmaximal, complemented
6#23763588C211yesmaximal, characteristic, complemented
7#23763587C211yesmaximal, characteristic, complemented
8#23763586C212yesmaximal, complemented
9#23763428C212yesmaximal, complemented
10#23763426C212yesmaximal, complemented
11C79×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D79C2211yescharacteristic
13C22×C8×D79C2212yes—
14C22×C4⋊Dic79C2211yescharacteristic
15C2×C4⋊C4×D79C2211yescharacteristic
16C2×C4⋊C4×D79C2211yescharacteristic
17C22×(C2×D79)⋊C4—21no—
18C2×C22⋊C4×D79—21no—
19C2×#14579095C2212yescomplemented
20C2×#14579094C2212yescomplemented
21C2×#14579094C2212yescomplemented
22C2×C22⋊(C79⋊C8)C2211yescharacteristic
23C2×(C2×D79)⋊C8C2212yes—
24C22⋊C8×D79C418yescomplemented
25#19098423C2211yescharacteristic
26#19098422—21no—
27#19098421—21no—
28#19098420—21no—
29#19098420—21no—
30#19098420C2211yescharacteristic
31#19098420C2211yescharacteristic
32#19098420C2212yes—
33#19098420C2212yes—
34C2×C79⋊#95—21no—
35#95×D79—21no—
36#19098326C2212yescomplemented
37#19098325C2212yescomplemented
38#19098325C2212yescomplemented
39C22×C4⋊C4×C79C2211yescharacteristic
40C2×#107×C79C2212yescomplemented
41C2×C22⋊C8×C79C2211yescharacteristic
42#380×C79C2211yescharacteristic
43#340×C79C2212yescomplemented
44C24×D79D411yescharacteristic
45C23×Dic79—21no—
46C23×Dic79D411yescharacteristic
47C22×C4×D79—21no—
48C22×C4×D79—21no—
49C22×C4×D79—42no—
50C22×C4×D79—42no—
51C22×C4×D79C2×C414yes—
52C22×C4×D79D411yescharacteristic
53C22×C4×D79D411yescharacteristic
54C22×C4×D79Q814yes—
55C22×C4×D79C2311yescharacteristic
56C22×C4×D79C2311yescharacteristic
57C22×C79⋊C8C2312yes—
58C2×C8×D79—42no—
59C2×C8×D79—42no—
60C2×C8×D79C2×C418yescomplemented
61C2×C4⋊Dic79—21no—
62C2×C4⋊Dic79—42no—
63C2×C4⋊Dic79C2312yes—
64C2×C4⋊Dic79C2311yescharacteristic
65C2×C4⋊Dic79C2311yescharacteristic
66C2×Dic79⋊C4—21no—
67C2×Dic79⋊C4—21no—
68C4⋊C4×D79—42no—
69C4⋊C4×D79—42no—
70C2×C22⋊Dic79—21no—
71C2×(C2×D79)⋊C4—22no—
72C2×(C2×D79)⋊C4—22no—
73C2×(C2×D79)⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—44no—
76C22⋊C4×D79—44no—
77#14579095—42no—
78#14579094—42no—
79#14579094—42no—
80C22⋊(C79⋊C8)C2×C414yescomplemented
81(C2×D79)⋊C8C2×C418yescomplemented
82C79⋊#95—21no—
83C79⋊#95—21no—
84C79⋊#95—21no—
85C79⋊#95—22no—
86C79⋊#95—22no—
87C23×C4×C79—21no—
88C23×C4×C79C2311yescharacteristic
89C22×C8×C79C2312yes—
90C2×C4⋊C4×C79—21no—
91C2×C4⋊C4×C79—42no—
92C2×C4⋊C4×C79C2311yescharacteristic
93C2×C4⋊C4×C79C2312yes—
94C2×C4⋊C4×C79C2311yescharacteristic
95C2×C22⋊C4×C79—21no—
96#107×C79—42no—
97C22⋊C8×C79C2×C414yescomplemented
98#95×C79—21no—
99C23×D79—41no—
100C23×D79—41no—
101C23×D79—41no—
102C23×D79—41no—
103C23×D79C4⋊C414yes—
104C23×D79D814yes—
105C23×D79Q1614yes—
106C23×D79C2×D411yescharacteristic
107C23×D79C2×D411yescharacteristic
108C22×Dic79—22no—
109C22×Dic79—21no—
110C22×Dic79—21no—
111C22×Dic79—41no—
112C22×Dic79—41no—
113C22×Dic79—42no—
114C22×Dic79—42no—
115C22×Dic79—42no—
116C22×Dic79C4⋊C414yes—
117C22×Dic79C2×D411yescharacteristic
118C22×Dic79C2×D411yescharacteristic
119C2×C4×D79—22no—
120C2×C4×D79—22no—
121C2×C4×D79—22no—
122C2×C4×D79—22no—
123C2×C4×D79—41no—
124C2×C4×D79—41no—
125C2×C4×D79—41no—
126C2×C4×D79—41no—
127C2×C4×D79—48no—
128C2×C4×D79—48no—
129C2×C4×D79—42no—
130C2×C4×D79—42no—
131C2×C4×D79—42no—
132C2×C4×D79—42no—
133C2×C4×D79—42no—
134C2×C4×D79—42no—
135C2×C4×D79C4⋊C414yes—
136C2×C4×D79C4⋊C414yes—
137C2×C4×D79C22×C412yes—
138C2×C4×D79C22×C412yes—
139C2×C4×D79C2×Q812yes—
140C2×C4×D79C2×Q812yes—
141C2×C79⋊C8—22no—
142C2×C79⋊C8—22no—
143C2×C79⋊C8C22×C414yes—
144C8×D79—48no—
145C4⋊Dic79—22no—
146C4⋊Dic79—22no—
147C4⋊Dic79—22no—
148C4⋊Dic79—22no—
149C4⋊Dic79—42no—
150Dic79⋊C4—42no—
151Dic79⋊C4—42no—
152C22⋊Dic79—42no—
153(C2×D79)⋊C4—42no—
154(C2×D79)⋊C4—42no—
155(C2×D79)⋊C4—44no—
156(C2×D79)⋊C4—44no—
157C24×C79C2×D411yescharacteristic
158C22×C4×C79—22no—
159C22×C4×C79—22no—
160C22×C4×C79—21no—
161C22×C4×C79—21no—
162C22×C4×C79—22no—
163C22×C4×C79—42no—
164C22×C4×C79—42no—
165C22×C4×C79—42no—
166C22×C4×C79C22×C412yes—
167C22×C4×C79C2×D411yescharacteristic
168C22×C4×C79C2×Q812yes—
169C22×C4×C79C2411yescharacteristic
170C2×C8×C79—22no—
171C2×C8×C79—22no—
172C2×C8×C79C22×C414yes—
173C4⋊C4×C79—22no—
174C4⋊C4×C79—22no—
175C4⋊C4×C79—22no—
176C4⋊C4×C79—22no—
177C4⋊C4×C79—42no—
178C22⋊C4×C79—42no—
179C23×C79—21no—
180C23×C79—21no—
181C23×C79—21no—
182C23×C79—21no—
183C23×C79C2×C4⋊C412yes—
184C23×C79C2×D812yes—
185C23×C79C2×Q1612yes—
186C23×C79C22×D411yescharacteristic
187C22×D79—21no—
188C22×D79—21no—
189C22×D79—21no—
190C22×D79—21no—
191C22×D79—21no—
192C22×D79—21no—
193C22×D79—21no—
194C22×D79—21no—
195C22×D79—44no—
196C22×D79—44no—
197C22×D79—44no—
198C22×D79—44no—
199C22×D79—44no—
200C22×D79—44no—
201C22×D79#107116yescomplemented
202C22×D79C2×C4⋊C412yes—
203C22×D79C2×C4⋊C412yes—
204C22×D79C2×D812yes—
205C22×D79C2×D812yes—
206C22×D79C2×Q1612yes—
207C22×D79C2×Q1612yes—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—24no—
211C2×C4×C79—24no—
212C2×C4×C79—41no—
213C2×C4×C79—41no—
214C2×C4×C79—42no—
215C2×C4×C79—42no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—42no—
221C2×C4×C79—84no—
222C2×C4×C79C2×C4⋊C412yes—
223C2×C4×C79C23×C411yescharacteristic
224C2×C4×C79C22×Q811yescharacteristic
225C2×C4×C79H(5,𝔽2)12yes—
226C2×C4×C79H(5,𝔽2)12yes—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—21no—
231C2×Dic79—44no—
232C2×Dic79—41no—
233C2×Dic79—41no—
234C2×Dic79—42no—
235C2×Dic79—42no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—42no—
241C2×Dic79—84no—
242C2×Dic79C2×C4⋊C412yes—
243C2×Dic79C2×C4⋊C412yes—
244C4×D79—24no—
245C4×D79—24no—
246C4×D79—24no—
247C4×D79—24no—
248C4×D79—44no—
249C4×D79—44no—
250C4×D79—84no—
251C4×D79—84no—
252C8×C79—24no—
253C79⋊C8—24no—
254C22×C79—22no—
255C22×C79—22no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—22no—
259C22×C79—22no—
260C22×C79C2×#10714yes—
261C22×C79C22×C4⋊C411yescharacteristic
262C22×C79#53411yescharacteristic
263C22×C79#53411yescharacteristic
264C22×C79C22×D811yescharacteristic
265C22×C79C22×Q1611yescharacteristic
266C22×C79#57511yescharacteristic
267C22×C79#57511yescharacteristic
268C2×D79—22no—
269C2×D79—22no—
270C2×D79—22no—
271C2×D79—22no—
272C2×D79—22no—
273C2×D79—22no—
274C2×D79—22no—
275C2×D79—22no—
276C2×D79—22no—
277C2×D79—22no—
278C2×D79—22no—
279C2×D79—22no—
280C2×D79—416no—
281C2×D79C2×#10714yes—
282C2×D79C2×#10714yes—
283C4×C79—22no—
284C4×C79—82no—
285C4×C79—81no—
286C4×C79—81no—
287C4×C79#51812yes—
288C4×C79#54312yes—
289Dic79—22no—
290Dic79—22no—
291Dic79—81no—
292Dic79—81no—
293Dic79—82no—
294#21940—791nomaximal, Sylow
295C2×C79—24no—
296C2×C79#231511yescharacteristic
297C2×C79C22×#10711yescharacteristic
298C2×C79#291111yescharacteristic
299C2×C79#318611yescharacteristic
300C2×C79#318611yescharacteristic
301C2×C79#318911yescharacteristic
302C2×C79#318911yescharacteristic
303D79—24no—
304D79—24no—
305C22×C22⋊C8—791no—
306C2×#380—791no—
307C2×#380—791no—
308#1852—792no—
309#1852—791no—
310#1852—791no—
311#1852—792no—
312#1852—791no—
313#1852D7911yescharacteristic, complemented
314#1808—792no—
315#1808—792no—
316C79#2194011yescharacteristic, complemented, Sylow
317C24×C4—791no—
318C23×C8—792no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4—791no—
322C22×C4⋊C4C2×D7911yescharacteristic
323C22×C22⋊C4—1581no—
324C22×C22⋊C4—1581no—
325C2×#107—792no—
326C2×#107—792no—
327C2×#107—792no—
328C2×#107C2×D7912yescomplemented
329C2×C22⋊C8—791no—
330C2×C22⋊C8—792no—
331C2×C22⋊C8—798no—
332C2×C22⋊C8C2×D7911yescharacteristic
333#380—791no—
334#380—791no—
335#380—791no—
336#380—792no—
337#380—792no—
338#380—1581no—
339#380—1581no—
340#380—1581no—
341#380—1581no—
342#380C2×D7911yescharacteristic
343C2×#95—1581no—
344C2×#95—1581no—
345#340—792no—
346#340—792no—
347#340—792no—
348#340C2×D7912yescomplemented
349C25—791no—
350C23×C4—21no—
351C23×C4—3162no—
352C23×C4—3162no—
353C23×C4—791no—
354C23×C4—794no—
355C23×C4—791no—
356C23×C4—791no—
357C23×C4—794no—
358C23×C4—791no—
359C23×C4—791no—
360C23×C4—1581no—
361C23×C4—1581no—
362C23×C4—1581no—
363C23×C4C22×D7911yescharacteristic
364C22×C8—3162no—
365C22×C8—3162no—
366C22×C8—798no—
367C22×C8—792no—
368C22×C8C22×D7912yes—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—42no—
371C2×C4⋊C4—3162no—
372C2×C4⋊C4—3162no—
373C2×C4⋊C4—3162no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—792no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—1581no—
378C2×C4⋊C4—1581no—
379C2×C4⋊C4—1581no—
380C2×C4⋊C4C22×D7911yescharacteristic
381C2×C4⋊C4C22×D7911yescharacteristic
382C2×C4⋊C4C22×D7912yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—3164no—
385C2×C22⋊C4—3164no—
386C2×C22⋊C4—1581no—
387C2×C22⋊C4—1581no—
388C2×C22⋊C4—1582no—
389C2×C22⋊C4—1582no—
390C2×C22⋊C4—1581no—
391#107—42no—
392#107—3162no—
393#107—3162no—
394#107—3162no—
395C22⋊C8—794no—
396C22⋊C8—798no—
397C22⋊C8C4×D7914yescomplemented
398#95—21no—
399#95—1581no—
400#95—1581no—
401#95—1582no—
402#95—1582no—
403#95—1581no—
404C24—3161no—
405C24—3161no—
406C24—3161no—
407C24—3161no—
408C24—794no—
409C24—794no—
410C24—794no—
411C24—791no—
412C24—791no—
413C24D4×D7911yescharacteristic
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—3161no—
423C22×C4—3161no—
424C22×C4—3161no—
425C22×C4—3161no—
426C22×C4—3168no—
427C22×C4—3168no—
428C22×C4—3162no—
429C22×C4—3162no—
430C22×C4—3162no—
431C22×C4—3162no—
432C22×C4—3162no—
433C22×C4—3162no—
434C22×C4—3161no—
435C22×C4—3161no—
436C22×C4—3162no—
437C22×C4—3162no—
438C22×C4—3162no—
439C22×C4—794no—
440C22×C4—794no—
441C22×C4—794no—
442C22×C4—791no—
443C22×C4—791no—
444C22×C4—792no—
445C22×C4—792no—
446C22×C4—792no—
447C22×C4—792no—
448C22×C4—1581no—
449C22×C4—1582no—
450C22×C4—1581no—
451C22×C4—1582no—
452C22×C4—1582no—
453C22×C4—1582no—
454C22×C4—1582no—
455C22×C4C2×C4×D7912yes—
456C22×C4D4×D7911yescharacteristic
457C22×C4Q8×D7912yes—
458C22×C4C23×D7911yescharacteristic
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—3168no—
462C2×C8—794no—
463C2×C8—1582no—
464C2×C8—1582no—
465C2×C8C2×C4×D7914yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—3162no—
472C4⋊C4—3162no—
473C4⋊C4—3162no—
474C4⋊C4—1582no—
475C4⋊C4—1582no—
476C4⋊C4—1582no—
477C4⋊C4—1582no—
478C22⋊C4—42no—
479C22⋊C4—3162no—
480C22⋊C4—3162no—
481C22⋊C4—3164no—
482C22⋊C4—3164no—
483C22⋊C4—3162no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—3164no—
489C23—3164no—
490C23—3164no—
491C23—3164no—
492C23—3164no—
493C23—3164no—
494C23—7916no—
495C23—792no—
496C23—792no—
497C23—792no—
498C23—792no—
499C23—792no—
500C23—792no—
501C23—1581no—
502C23—1581no—
503C23—1581no—
504C23—1581no—
505C23—1581no—
506C23—1581no—
507C23—1581no—
508C23—1581no—
509C23C4⋊C4×D7912yes—
510C23D8×D7912yes—
511C23Q16×D7912yes—
512C23C2×D4×D7911yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
527C2×C4—3164no—
528C2×C4—3164no—
529C2×C4—3161no—
530C2×C4—3162no—
531C2×C4—3161no—
532C2×C4—3161no—
533C2×C4—3161no—
534C2×C4—3162no—
535C2×C4—3161no—
536C2×C4—3162no—
537C2×C4—3161no—
538C2×C4—3164no—
539C2×C4—792no—
540C2×C4—792no—
541C2×C4—6324no—
542C2×C4—6324no—
543C2×C4—6324no—
544C2×C4—1581no—
545C2×C4—1581no—
546C2×C4—1581no—
547C2×C4—1581no—
548C2×C4—1584no—
549C2×C4—1584no—
550C2×C4—1584no—
551C2×C4—1584no—
552C2×C4C4⋊C4×D7912yes—
553C2×C4C22×C4×D7911yescharacteristic
554C2×C4C2×Q8×D7911yescharacteristic
555C2×C4#1457920612yes—
556C2×C4#1457920712yes—
557C8—24no—
558C8—1584no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—31616no—
566C22—794no—
567C22—794no—
568C22—1582no—
569C22—1582no—
570C22—1582no—
571C22—1582no—
572C22—1582no—
573C22—1582no—
574C22—1582no—
575C22—1582no—
576C22—1582no—
577C22—1582no—
578C22—1582no—
579C22—1582no—
580C22#107×D7914yes—
581C22C2×C4⋊C4×D7911yescharacteristic
582C22#1909912311yescharacteristic
583C22#1909912411yescharacteristic
584C22C2×D8×D7911yescharacteristic
585C22C2×Q16×D7911yescharacteristic
586C22#1909935911yescharacteristic
587C22#1909936211yescharacteristic
588C4—22no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—6321no—
593C4—6321no—
594C4—6322no—
595C4—1582no—
596C4—1582no—
597C4#1909903512yes—
598C4#1909918812yes—
599C2—24no—
600C2—1584no—
601C2—1584no—
602C2#2376601111yescharacteristic
603C2C2×#107×D7911yescharacteristic
604C2#2377198711yescharacteristic
605C2#2377414911yescharacteristic
606C2#2377415011yescharacteristic
607C2#2377417311yescharacteristic
608C2#2377417411yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#1852C2
socleC23×C79C22×D4
prime radicalC24×D79D4
cosocleC22×C4×C79C24
hypercenter#1852D79
hypocenterC79#21940
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #23763590C2semidirect
5. #23763589C2semidirect
6. #23763588C2semidirect
7. #23763587C2semidirect
8. #23763586C2semidirect
9. #23763428C2semidirect
10. #23763426C2semidirect
11. C79×#1852C2semidirect
19. C2×#14579095C22semidirect
20. C2×#14579094C22semidirect
21. C2×#14579094C22semidirect
24. C22⋊C8×D79C4semidirect
36. #19098326C22semidirect
37. #19098325C22semidirect
38. #19098325C22semidirect
40. C2×#107×C79C22semidirect
43. #340×C79C22semidirect
60. C2×C8×D79C2×C4semidirect
80. C22⋊(C79⋊C8)C2×C4semidirect
81. (C2×D79)⋊C8C2×C4semidirect
97. C22⋊C8×C79C2×C4semidirect
201. C22×D79#107semidirect
313. #1852D79semidirect
316. C79#21940semidirect
328. C2×#107C2×D79semidirect
348. #340C2×D79semidirect
397. C22⋊C8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C8×D792
C2×#190984202
#380×D792
#237635902
#237635892
#237635882
#237635872
#237635862
#237634282
#237634262
C79×#18522
#2194079