Finite Group Databasebeta

← #34351250

Subgroups — #34351250

Order 20224.

896 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098863C211yesmaximal, characteristic, complemented
2C2×C4×#11929609C211yesmaximal, characteristic
3#23767272C211yesmaximal, characteristic, complemented
4#23764924C211yesmaximal, characteristic, complemented
5#23764882C211yesmaximal, characteristic, complemented
6#23764785C211yesmaximal, characteristic, complemented
7#23764738C211yesmaximal, characteristic, complemented
8#23764130C211yesmaximal, characteristic, complemented
9#23763959C211yesmaximal, characteristic, complemented
10#23763748C211yesmaximal, characteristic
11#23763748C211yesmaximal, characteristic
12#23763661C211yesmaximal, characteristic, complemented
13#23763660C211yesmaximal, characteristic, complemented
14#23763647C211yesmaximal, characteristic, complemented
15C79×#2349C211yesmaximal, characteristic, complemented
16C22×#11929609C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×C4×D316C2211yescharacteristic
19#19098863—24no—
20C4×#11929609—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579094C2211yescharacteristic
23C2×#14579094C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×#14579082C2211yescharacteristic
30C2×#14579081C2211yescharacteristic
31C2×#14579081C2211yescharacteristic
32C2×#14579081C2211yescharacteristic
33C2×C4×C79⋊C8C2211yescharacteristic
34#19098498—21no—
35#19098487—21no—
36#19098465C2211yescharacteristic
37#19098465C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098434—21no—
41#19098422—21no—
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C2×C4×D4×C79C2211yescharacteristic
51C2×C4×D4×C79C2211yescharacteristic
52C2×C4×D4×C79C2211yescharacteristic
53#389×C79C2211yescharacteristic
54#389×C79C2211yescharacteristic
55#385×C79—21no—
56#385×C79—21no—
57#382×C79C2211yescharacteristic
58#380×C79—21no—
59C4×C4⋊C4×C79C2211yescharacteristic
60C2×#95×C79—21no—
61C22×D316C2311yescharacteristic
62C22×C4×D79—21no—
63C2×#11929609—22no—
64C2×#11929609—22no—
65C2×#11929609Q814yes—
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70#14579153—22no—
71#14579153—41no—
72C4×D316—22no—
73C4×D316—41no—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×(C2×D79)⋊C4—21no—
82C2×(C2×D79)⋊C4—21no—
83C2×(C2×D79)⋊C4—21no—
84C2×(C2×D79)⋊C4—21no—
85C2×(C2×D79)⋊C4—21no—
86#14579094—22no—
87#14579094—22no—
88#14579094—22no—
89#14579092—22no—
90#14579092—41no—
91C4⋊(C79⋊C8)—22no—
92#14579082—22no—
93#14579082—22no—
94#14579082D414yes—
95#14579081—22no—
96#14579081—22no—
97#14579081—41no—
98C4×C79⋊C8—22no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C22×D4×C79C2311yescharacteristic
102C23×C4×C79—21no—
103C23×C4×C79—21no—
104C23×C4×C79—21no—
105C4×D4×C79—22no—
106C4×D4×C79—22no—
107C4×D4×C79—22no—
108C4×D4×C79—22no—
109C4×D4×C79—41no—
110C4×D4×C79—41no—
111C2×C4⋊C4×C79—21no—
112C2×C4⋊C4×C79—21no—
113C2×C4⋊C4×C79D411yescharacteristic
114C2×C4⋊C4×C79D411yescharacteristic
115C2×C4⋊C4×C79C2311yescharacteristic
116C2×C4⋊C4×C79C2311yescharacteristic
117C2×C4⋊C4×C79C2311yescharacteristic
118C2×C22⋊C4×C79—21no—
119C2×C22⋊C4×C79—21no—
120C2×C22⋊C4×C79—21no—
121C2×C42×C79C2311yescharacteristic
122C2×C42×C79C2311yescharacteristic
123C2×C42×C79C2311yescharacteristic
124#95×C79—21no—
125#95×C79—21no—
126#95×C79—21no—
127#95×C79—21no—
128#95×C79D411yescharacteristic
129#95×C79D411yescharacteristic
130C23×D79—21no—
131C2×D316—21no—
132C2×D316—21no—
133C2×D316—21no—
134C2×D316—21no—
135C2×D316—21no—
136C2×D316C2×Q812yes—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140C2×C4×D79—42no—
141C2×C4×D79—42no—
142C2×C4×D79—81no—
143#11929609—28no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8—21no—
153C2×C79⋊C8—21no—
154C2×C79⋊C8C2×D412yes—
155C2×C79⋊C8C2×Q812yes—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79C2×D412yes—
162Dic79⋊C4—42no—
163Dic79⋊C4—42no—
164Dic79⋊C4—81no—
165Dic79⋊C4—81no—
166(C2×D79)⋊C4—42no—
167(C2×D79)⋊C4—42no—
168(C2×D79)⋊C4—81no—
169(C2×D79)⋊C4—81no—
170(C2×D79)⋊C4—81no—
171C24×C79—21no—
172C2×D4×C79—21no—
173C2×D4×C79—21no—
174C2×D4×C79—21no—
175C2×D4×C79C2×D412yes—
176C2×D4×C79C2×Q812yes—
177C2×D4×C79#4112yes—
178C22×C4×C79—21no—
179C22×C4×C79—24no—
180C22×C4×C79—21no—
181C22×C4×C79—21no—
182C22×C4×C79—42no—
183C22×C4×C79—42no—
184C22×C4×C79—42no—
185C22×C4×C79—42no—
186C22×C4×C79—81no—
187C22×C4×C79—81no—
188C22×C4×C79—81no—
189C22×C4×C79—81no—
190C22×C4×C79C2×D411yescharacteristic
191C22×C4×C79C2×D411yescharacteristic
192C22×C4×C79C2×D411yescharacteristic
193C22×C4×C79C2×D411yescharacteristic
194C22×C4×C79C2×D411yescharacteristic
195C22×C4×C79C2×D411yescharacteristic
196C22×C4×C79C2411yescharacteristic
197C4⋊C4×C79—22no—
198C4⋊C4×C79—21no—
199C4⋊C4×C79—21no—
200C4⋊C4×C79—21no—
201C4⋊C4×C79—21no—
202C4⋊C4×C79—21no—
203C4⋊C4×C79—41no—
204C4⋊C4×C79—81no—
205C4⋊C4×C79—81no—
206C4⋊C4×C79#4112yes—
207C22⋊C4×C79—24no—
208C22⋊C4×C79—42no—
209C22⋊C4×C79—42no—
210C42×C79—21no—
211C42×C79—21no—
212C42×C79—21no—
213C42×C79—21no—
214C42×C79—21no—
215C42×C79—21no—
216C23×C79—22no—
217C23×C79—22no—
218C23×C79—22no—
219C23×C79—41no—
220C23×C79—41no—
221C23×C79—41no—
222C23×C79—41no—
223C23×C79C22×D411yescharacteristic
224C22×D79—22no—
225C22×D79—41no—
226C22×D79—41no—
227C22×D79—41no—
228C22×D79—41no—
229C22×D79—41no—
230C22×D79—41no—
231D4×C79—22no—
232D4×C79—22no—
233D4×C79—22no—
234D4×C79C22⋊Q814yes—
235C2×C4×C79—21no—
236C2×C4×C79—21no—
237C2×C4×C79—21no—
238C2×C4×C79—21no—
239C2×C4×C79—22no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
249C2×C4×C79—21no—
250C2×C4×C79—21no—
251C2×C4×C79—21no—
252C2×C4×C79—21no—
253C2×C4×C79—41no—
254C2×C4×C79—41no—
255C2×C4×C79—41no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—41no—
261C2×C4×C79—84no—
262C2×C4×C79—84no—
263C2×C4×C79—84no—
264C2×C4×C79#13512yes—
265C2×C4×C79C22×D411yescharacteristic
266C2×C4×C79C22×Q811yescharacteristic
267C2×C4×C79C2×#4111yescharacteristic
268C2×C4×C79C2×#4111yescharacteristic
269C2×C4×C79H(5,𝔽2)11yescharacteristic
270C2×C4×C792−1+411yescharacteristic
271C2×Dic79—22no—
272C2×Dic79—41no—
273C2×Dic79—41no—
274C2×Dic79—41no—
275C2×Dic79—41no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279C2×Dic79—41no—
280D316—22no—
281D316—22no—
282D316—41no—
283D316—41no—
284C4×D79—84no—
285C79⋊C8—22no—
286C79⋊C8—22no—
287C79⋊C8—41no—
288C79⋊C8—41no—
289C22×C79—24no—
290C22×C79—42no—
291C22×C79—42no—
292C22×C79—42no—
293C22×C79—42no—
294C22×C79—42no—
295C22×C79—42no—
296C22×C79#52511yescharacteristic
297C22×C79D4211yescharacteristic
298C22×C79D4×Q811yescharacteristic
299C22×C79C2×#13511yescharacteristic
300C22×C79#57511yescharacteristic
301C22×C79#57611yescharacteristic
302C22×C79#57611yescharacteristic
303C2×D79—42no—
304C2×D79—42no—
305C2×D79—42no—
306C2×D79—81no—
307C2×D79—81no—
308C2×D79—81no—
309C2×D79—81no—
310C4×C79—22no—
311C4×C79—41no—
312C4×C79—41no—
313C4×C79—42no—
314C4×C79—41no—
315C4×C79—41no—
316C4×C79—41no—
317C4×C79—81no—
318C4×C79—81no—
319C4×C79C2×C22⋊Q811yescharacteristic
320C4×C79#53911yescharacteristic
321C4×C79#54711yescharacteristic
322C4×C79#55411yescharacteristic
323Dic79—42no—
324Dic79—81no—
325Dic79—81no—
326#26370—791nomaximal, Sylow
327C2×C79—44no—
328C2×C79#244911yescharacteristic
329C2×C79#310211yescharacteristic
330C2×C79#312211yescharacteristic
331C2×C79#326811yescharacteristic
332C2×C79#328011yescharacteristic
333C2×C79#338011yescharacteristic
334C2×C79#338911yescharacteristic
335D79—82no—
336C2×#478—791no—
337C2×C4×D8—791no—
338#2454—791no—
339#2349D7911yescharacteristic, complemented
340#2092—791no—
341#2085—791no—
342#2067—791no—
343#2059—791no—
344#1942—791no—
345#1916—791no—
346#1879—791no—
347#1879—791no—
348#1864—791no—
349#1864—791no—
350#1862—791no—
351C79#2637011yescharacteristic, complemented, Sylow
352C22×D8—791no—
353C2×#125—791no—
354C2×C4×D4—791no—
355C2×C4×D4C2×D7911yescharacteristic
356C2×C4×D4C2×D7911yescharacteristic
357C2×C4×D4C2×D7911yescharacteristic
358#478—1584no—
359C4×D8—1584no—
360C2×#107—791no—
361C2×#107—791no—
362C2×#107—791no—
363C2×#106—791no—
364C2×#106—791no—
365C2×C4⋊C8—791no—
366C2×#102—791no—
367C2×#102—791no—
368C2×#102—791no—
369C2×#102—791no—
370C2×#102—791no—
371C2×#102—791no—
372C2×C4×C8—791no—
373#399—1581no—
374#396—1581no—
375#390—791no—
376#390—791no—
377#389—791no—
378#389—791no—
379#389C2×D7911yescharacteristic
380#389C2×D7911yescharacteristic
381#385—21no—
382#385—21no—
383#385—1581no—
384#382C2×D7911yescharacteristic
385#380—21no—
386#380—1581no—
387C4×C4⋊C4C2×D7911yescharacteristic
388C2×#95—21no—
389#340—791no—
390#340—791no—
391#340—791no—
392#340—791no—
393#336—791no—
394#336—791no—
395#336—791no—
396#336—791no—
397C22×D4—791no—
398C22×D4C22×D7911yescharacteristic
399C23×C4—21no—
400C23×C4—21no—
401C23×C4—21no—
402C23×C4—1581no—
403C2×D8—794no—
404C2×D8—1582no—
405C2×D8—1582no—
406C22×C8—791no—
407C22×C8—791no—
408C22×C8—791no—
409C22×C8—791no—
410#125—3161no—
411#125—1582no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—41no—
418C4×D4—3161no—
419C4×D4—1582no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—791no—
423C2×C4⋊C4—791no—
424C2×C4⋊C4—791no—
425C2×C4⋊C4—1581no—
426C2×C4⋊C4—1581no—
427C2×C4⋊C4—1581no—
428C2×C4⋊C4—1581no—
429C2×C4⋊C4C22⋊D7911yescharacteristic
430C2×C4⋊C4C22⋊D7911yescharacteristic
431C2×C4⋊C4C22×D7911yescharacteristic
432C2×C4⋊C4C22×D7911yescharacteristic
433C2×C4⋊C4C22×D7911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1581no—
438C2×C22⋊C4—1581no—
439C2×C22⋊C4—1581no—
440C2×C22⋊C4—1581no—
441C2×C22⋊C4—1581no—
442C2×C42C22×D7911yescharacteristic
443C2×C42C22×D7911yescharacteristic
444C2×C42C22×D7911yescharacteristic
445#107—1582no—
446#107—1582no—
447#107—1582no—
448#106—3161no—
449#106—1582no—
450C4⋊C8—1582no—
451#102—3161no—
452#102—794no—
453#102—1582no—
454#102—1582no—
455#102—1582no—
456#102—1582no—
457C4×C8—1582no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1581no—
463#95—1581no—
464#95C22⋊D7911yescharacteristic
465#95C22⋊D7911yescharacteristic
466C24—21no—
467C24—1581no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—792no—
472C2×D4—1581no—
473C2×D4—1581no—
474C2×D4—1581no—
475C2×D4—1581no—
476C2×D4—1581no—
477C2×D4D4×D7912yes—
478C2×D4Q8×D7912yes—
479C2×D4#1192962412yes—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—3162no—
493C22×C4—3162no—
494C22×C4—6321no—
495C22×C4—1581no—
496C22×C4—1581no—
497C22×C4—1581no—
498C22×C4C2×C22⋊D7911yescharacteristic
499C22×C4C2×C22⋊D7911yescharacteristic
500C22×C4C2×C22⋊D7911yescharacteristic
501C22×C4C2×C22⋊D7911yescharacteristic
502C22×C4C2×C22⋊D7911yescharacteristic
503C22×C4C2×C22⋊D7911yescharacteristic
504C22×C4C23×D7911yescharacteristic
505D8—1588no—
506C2×C8—792no—
507C2×C8—792no—
508C2×C8—1581no—
509C2×C8—1581no—
510C2×C8—1581no—
511C2×C8—1581no—
512C2×C8—1581no—
513C2×C8—1581no—
514C2×C8—1581no—
515C2×C8—1581no—
516C2×C8—1581no—
517C2×C8—1581no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—3162no—
528C4⋊C4—3162no—
529C4⋊C4—792no—
530C4⋊C4—6321no—
531C4⋊C4—6321no—
532C4⋊C4—1581no—
533C4⋊C4—1581no—
534C4⋊C4—1581no—
535C4⋊C4—1581no—
536C4⋊C4—1581no—
537C4⋊C4#1192962412yes—
538C22⋊C4—24no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—3162no—
542C22⋊C4—3162no—
543C22⋊C4—6321no—
544C22⋊C4—6321no—
545C22⋊C4—6321no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—3161no—
560C23—3161no—
561C23—3161no—
562C23—3161no—
563C23—3161no—
564C23—3161no—
565C23—1582no—
566C23C22×C22⋊D7911yescharacteristic
567D4—22no—
568D4—22no—
569D4—22no—
570D4—3161no—
571D4—3161no—
572D4—1582no—
573D4—1582no—
574D4#1457913914yes—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—3161no—
605C2×C4—3161no—
606C2×C4—3161no—
607C2×C4—3161no—
608C2×C4—3161no—
609C2×C4—3161no—
610C2×C4—3161no—
611C2×C4—3161no—
612C2×C4—6324no—
613C2×C4—1582no—
614C2×C4#1457918112yes—
615C2×C4C2×D4×D7911yescharacteristic
616C2×C4C2×Q8×D7911yescharacteristic
617C2×C4C2×#1192962411yescharacteristic
618C2×C4C2×#1192962411yescharacteristic
619C2×C4#1457920611yescharacteristic
620C2×C4#1457920911yescharacteristic
621C8—3161no—
622C8—3161no—
623C8—1582no—
624C8—1582no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—3162no—
633C22—3162no—
634C22—3162no—
635C22—6321no—
636C22—6321no—
637C22—6321no—
638C22—6321no—
639C22#1909907411yescharacteristic
640C22D4×C22⋊D7911yescharacteristic
641C22Q8×C22⋊D7911yescharacteristic
642C22C2×#1457918111yescharacteristic
643C22#1909936111yescharacteristic
644C22#1909936611yescharacteristic
645C22#1909936611yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—3162no—
656C4—6321no—
657C4—6321no—
658C4C2×#1457913911yescharacteristic
659C4#1909916011yescharacteristic
660C4#1909921411yescharacteristic
661C4#1909926811yescharacteristic
662C2—44no—
663C2—6322no—
664C2#2376720911yescharacteristic
665C2#2377341811yescharacteristic
666C2#2377364311yescharacteristic
667C2#2377493611yescharacteristic
668C2#2377508111yescharacteristic
669C2#2377626311yescharacteristic
670C2#2377635411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2349C2
socleC23×C79C22×D4
prime radicalC22×#11929609C22
hypercenter#2349D79
hypocenterC79#26370
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C2×#19098863C2semidirect
3. #23767272C2semidirect
4. #23764924C2semidirect
5. #23764882C2semidirect
6. #23764785C2semidirect
7. #23764738C2semidirect
8. #23764130C2semidirect
9. #23763959C2semidirect
12. #23763661C2semidirect
13. #23763660C2semidirect
14. #23763647C2semidirect
15. C79×#2349C2semidirect
339. #2349D79semidirect
351. C79#26370semidirect

Maximal subgroups

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

subgroupindex
C2×#190988632
C2×C4×#119296092
#237672722
#237649242
#237648822
#237647852
#237647382
#237641302
#237639592
#237637482
#237636612
#237636602
#237636472
C79×#23492
#2637079