Finite Group Databasebeta

← #34345761

Subgroups — #34345761

Order 20224.

928 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×#19098858C211yesmaximal, characteristic, complemented
2C2×#19098572C211yesmaximal, characteristic
3#23767091C211yesmaximal, characteristic
4#23764948C211yesmaximal, characteristic, complemented
5#23764795C211yesmaximal, characteristic, complemented
6#23764781C211yesmaximal, characteristic, complemented
7#23764541C211yesmaximal, characteristic, complemented
8#23763668C211yesmaximal, characteristic, complemented
9#23763668C211yesmaximal, characteristic, complemented
10#23763597C211yesmaximal, characteristic
11#23763585C211yesmaximal, characteristic
12#23763431C211yesmaximal, characteristic
13#23763415C211yesmaximal, characteristic, complemented
14#23763335C211yesmaximal, characteristic
15Q8×C22⋊C4×C79C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579161C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19#19098858—24no—
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26#19098572—24no—
27C2×#14579085C2211yescharacteristic
28C2×#14579085C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579085C2211yescharacteristic
31C2×#14579085C2211yescharacteristic
32C2×#14579085C2211yescharacteristic
33C2×#14579085C2211yescharacteristic
34C2×#14579085C2211yescharacteristic
35C2×C22⋊(C79⋊C8)C2211yescharacteristic
36#19098486—21no—
37#19098475—21no—
38#19098467—21no—
39#19098461C2211yescharacteristic
40#19098461C2211yescharacteristic
41#19098450—21no—
42#19098429C2211yescharacteristic
43#19098423C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50C23×Q8×C79C2211yescharacteristic
51C2×C4×Q8×C79C2211yescharacteristic
52C2×C4×Q8×C79C2211yescharacteristic
53#391×C79—21no—
54#391×C79—21no—
55#391×C79C2211yescharacteristic
56#391×C79C2211yescharacteristic
57#380×C79—21no—
58#380×C79C2211yescharacteristic
59C4×C22⋊C4×C79—21no—
60C4×C22⋊C4×C79C2211yescharacteristic
61C22×Dic158—21no—
62C2×#11929608—22no—
63C2×#11929608—22no—
64C2×#11929608—41no—
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68C22×C79⋊C8C2311yescharacteristic
69#14579161—22no—
70#14579161—41no—
71#14579120—22no—
72#14579120—41no—
73C2×C4⋊Dic79—21no—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82C2×C4×Dic79—21no—
83#14579094—22no—
84#14579094—22no—
85#14579092—41no—
86#14579092—41no—
87C4⋊(C79⋊C8)—22no—
88C4⋊(C79⋊C8)—41no—
89#14579085—22no—
90#14579085—22no—
91#14579085—22no—
92#14579085—22no—
93#14579085—22no—
94#14579085—41no—
95#14579085—41no—
96#14579085D414yes—
97C22⋊(C79⋊C8)—22no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C22×Q8×C79—21no—
104C22×Q8×C79—22no—
105C22×Q8×C79—22no—
106C22×Q8×C79—22no—
107C22×Q8×C79—41no—
108C22×Q8×C79—41no—
109C22×Q8×C79Q814yescomplemented
110C22×Q8×C79C2311yescharacteristic
111C22×Q8×C79C2311yescharacteristic
112C23×C4×C79—21no—
113C23×C4×C79C2311yescharacteristic
114C4×Q8×C79—22no—
115C4×Q8×C79—22no—
116C4×Q8×C79—41no—
117C4×Q8×C79—41no—
118C2×C4⋊C4×C79—21no—
119C2×C4⋊C4×C79—21no—
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C22⋊C4×C79—21no—
123C2×C22⋊C4×C79D411yescharacteristic
124C2×C22⋊C4×C79D411yescharacteristic
125C2×C42×C79—21no—
126C2×C42×C79—21no—
127C2×C42×C79C2311yescharacteristic
128C2×C42×C79C2311yescharacteristic
129#95×C79—21no—
130#95×C79—21no—
131#95×C79—21no—
132#95×C79—21no—
133#95×C79—21no—
134#95×C79D411yescharacteristic
135#95×C79D411yescharacteristic
136C2×Dic158—42no—
137C2×Dic158—42no—
138C2×Dic158—81no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—21no—
142C22×Dic79—21no—
143#11929608—44no—
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⋊C8—21no—
155C2×C79⋊C8C2×D412yes—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—21no—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—42no—
164C4⋊Dic79C2×D412yes—
165Dic79⋊C4—42no—
166Dic79⋊C4—81no—
167C22⋊Dic79—42no—
168C22⋊Dic79—81no—
169C4×Dic79—42no—
170C24×C79C2×D411yescharacteristic
171C2×Q8×C79—24no—
172C2×Q8×C79—21no—
173C2×Q8×C79—21no—
174C2×Q8×C79—21no—
175C2×Q8×C79—21no—
176C2×Q8×C79—28no—
177C2×Q8×C79—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—21no—
181C2×Q8×C79—42no—
182C2×Q8×C79—42no—
183C2×Q8×C79—44no—
184C2×Q8×C79—44no—
185C2×Q8×C79—44no—
186C2×Q8×C79C2×D412yes—
187C2×Q8×C79C2×Q812yes—
188C2×Q8×C79C2×Q812yes—
189C2×Q8×C79#4112yes—
190C22×C4×C79—21no—
191C22×C4×C79—21no—
192C22×C4×C79—21no—
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195C22×C4×C79—24no—
196C22×C4×C79—21no—
197C22×C4×C79—21no—
198C22×C4×C79—21no—
199C22×C4×C79—42no—
200C22×C4×C79—42no—
201C22×C4×C79C2×D411yescharacteristic
202C22×C4×C79C2×D411yescharacteristic
203C22×C4×C79C2×D411yescharacteristic
204C22×C4×C79C2×D411yescharacteristic
205C22×C4×C79C2×D411yescharacteristic
206C22×C4×C79C2×Q812yes—
207C22×C4×C79#4112yes—
208C22×C4×C79C2411yescharacteristic
209C4⋊C4×C79—21no—
210C4⋊C4×C79—21no—
211C4⋊C4×C79—21no—
212C4⋊C4×C79—21no—
213C4⋊C4×C79—42no—
214C4⋊C4×C79—42no—
215C22⋊C4×C79—22no—
216C22⋊C4×C79—41no—
217C22⋊C4×C79—42no—
218C42×C79—21no—
219C42×C79—21no—
220C42×C79—21no—
221C42×C79—21no—
222C42×C79—42no—
223C42×C79—42no—
224C23×C79—21no—
225C23×C79—21no—
226C23×C79—21no—
227C23×C79—21no—
228C23×C79—21no—
229C23×C79—21no—
230C23×C79C22⋊Q812yes—
231C23×C79C22×D411yescharacteristic
232Q8×C79—22no—
233Q8×C79—22no—
234Q8×C79—22no—
235Q8×C79—22no—
236Q8×C79—22no—
237Q8×C79—48no—
238Q8×C79—41no—
239Q8×C79—41no—
240Q8×C79C22⋊Q814yes—
241C2×C4×C79—22no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—22no—
245C2×C4×C79—22no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—22no—
249C2×C4×C79—22no—
250C2×C4×C79—22no—
251C2×C4×C79—22no—
252C2×C4×C79—22no—
253C2×C4×C79—21no—
254C2×C4×C79—21no—
255C2×C4×C79—21no—
256C2×C4×C79—21no—
257C2×C4×C79—21no—
258C2×C4×C79—21no—
259C2×C4×C79—21no—
260C2×C4×C79—21no—
261C2×C4×C79—21no—
262C2×C4×C79—48no—
263C2×C4×C79—41no—
264C2×C4×C79—41no—
265C2×C4×C79—41no—
266C2×C4×C79—41no—
267C2×C4×C79—41no—
268C2×C4×C79—41no—
269C2×C4×C79—41no—
270C2×C4×C79—41no—
271C2×C4×C79C22⋊Q812yes—
272C2×C4×C79#13712yes—
273C2×C4×C79C22×D411yescharacteristic
274C2×C4×C79C22×Q811yescharacteristic
275C2×C4×C79C2×#4111yescharacteristic
276C2×C4×C79C2×#4111yescharacteristic
277C2×C4×C79H(5,𝔽2)11yescharacteristic
278C2×C4×C79H(5,𝔽2)11yescharacteristic
279C2×Dic79—22no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79—41no—
284C2×Dic79—41no—
285C2×Dic79—41no—
286C2×Dic79—41no—
287C2×Dic79—41no—
288C2×Dic79—41no—
289C2×Dic79—41no—
290C2×Dic79—41no—
291Dic158—84no—
292C79⋊C8—22no—
293C79⋊C8—41no—
294C79⋊C8—41no—
295C79⋊C8—41no—
296C22×C79—22no—
297C22×C79—22no—
298C22×C79—22no—
299C22×C79—41no—
300C22×C79—41no—
301C22×C79—41no—
302C22×C79—41no—
303C22×C79C2×C22⋊Q811yescharacteristic
304C22×C79D4211yescharacteristic
305C22×C79#54611yescharacteristic
306C22×C79C2×#13711yescharacteristic
307C22×C79#57511yescharacteristic
308C22×C79#57711yescharacteristic
309C22×C79#57811yescharacteristic
310C4×C79—22no—
311C4×C79—24no—
312C4×C79—22no—
313C4×C79—41no—
314C4×C79—42no—
315C4×C79—41no—
316C4×C79—41no—
317C4×C79—81no—
318C4×C79—81no—
319C4×C79C2×C22⋊Q811yescharacteristic
320C4×C79#54311yescharacteristic
321C4×C79#54611yescharacteristic
322C4×C79#55111yescharacteristic
323Dic79—42no—
324Dic79—81no—
325Dic79—81no—
326Dic79—81no—
327#25979—791nomaximal, Sylow
328C2×C79—42no—
329C2×C79#243711yescharacteristic
330C2×C79#307811yescharacteristic
331C2×C79#308211yescharacteristic
332C2×C79#327011yescharacteristic
333C2×C79#327511yescharacteristic
334C2×C79#331311yescharacteristic
335C2×C79#332911yescharacteristic
336C2×#477—791no—
337C2×#419—791no—
338#2439—791no—
339Q8×C22⋊C4D7911yescharacteristic
340#2097—791no—
341#2068—791no—
342#2066—791no—
343#2024—791no—
344#1865—791no—
345#1865—791no—
346#1854—791no—
347#1851—791no—
348#1809—791no—
349#1805—791no—
350#1787—791no—
351C79#2597911yescharacteristic, complemented, Sylow
352C23×Q8C2×D7911yescharacteristic
353C22×M16—791no—
354C2×C4⋊Q8—791no—
355C2×C4×Q8C2×D7911yescharacteristic
356C2×C4×Q8C2×D7911yescharacteristic
357C2×#117—791no—
358#477—1584no—
359C2×#107—791no—
360C2×#107—791no—
361C2×#106—791no—
362C2×#106—791no—
363C2×C4⋊C8—791no—
364C2×C4⋊C8—791no—
365#419—1584no—
366C2×#103—791no—
367C2×#103—791no—
368C2×#103—791no—
369C2×#103—791no—
370C2×#103—791no—
371C2×#103—791no—
372C2×#103—791no—
373C2×#103—791no—
374C2×C22⋊C8—791no—
375#396—1581no—
376#393—1581no—
377#391—21no—
378#391—21no—
379#391—1581no—
380#391C2×D7911yescharacteristic
381#391C2×D7911yescharacteristic
382#389—791no—
383#389—791no—
384#387—1581no—
385#382—791no—
386#380—21no—
387#380—791no—
388#380C2×D7911yescharacteristic
389C4×C22⋊C4—21no—
390C4×C22⋊C4C2×D7911yescharacteristic
391#340—791no—
392#340—791no—
393#340—791no—
394#340—791no—
395#336—791no—
396#336—791no—
397C22×Q8—21no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—41no—
402C22×Q8—41no—
403C22×Q8—1581no—
404C22×Q8Dic15814yescomplemented
405C22×Q8C22×D7911yescharacteristic
406C22×Q8C22×D7911yescharacteristic
407C23×C4—21no—
408C23×C4C22×D7911yescharacteristic
409C2×M16—3161no—
410C2×M16—1582no—
411C2×M16—1582no—
412C22×C8—791no—
413C22×C8—791no—
414C22×C8—791no—
415C22×C8—791no—
416C4⋊Q8—3161no—
417C4⋊Q8—1582no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—41no—
421C4×Q8—41no—
422#117—3161no—
423#117—1582no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—791no—
428C2×C4⋊C4—791no—
429C2×C4⋊C4—791no—
430C2×C4⋊C4—1581no—
431C2×C4⋊C4—1581no—
432C2×C4⋊C4—1581no—
433C2×C4⋊C4C22×D7911yescharacteristic
434C2×C4⋊C4C22×D7911yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1581no—
437C2×C22⋊C4—1581no—
438C2×C22⋊C4C22⋊D7911yescharacteristic
439C2×C22⋊C4C22⋊D7911yescharacteristic
440C2×C42—21no—
441C2×C42—21no—
442C2×C42—1581no—
443C2×C42C22×D7911yescharacteristic
444C2×C42C22×D7911yescharacteristic
445#107—1582no—
446#107—1582no—
447#106—3161no—
448#106—3161no—
449C4⋊C8—3161no—
450C4⋊C8—1582no—
451#103—3161no—
452#103—3161no—
453#103—794no—
454#103—1582no—
455#103—1582no—
456#103—1582no—
457#103—1582no—
458#103—1582no—
459C22⋊C8—1582no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1581no—
466#95—1581no—
467#95—1581no—
468#95—1581no—
469#95—1581no—
470#95C22⋊D7911yescharacteristic
471#95C22⋊D7911yescharacteristic
472C24C2×C22⋊D7911yescharacteristic
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—24no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8—3162no—
489C2×Q8—3162no—
490C2×Q8—6321no—
491C2×Q8D4×D7912yes—
492C2×Q8C2×Dic15812yes—
493C2×Q8C2×Dic15812yes—
494C2×Q8#1192962312yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—1581no—
507C22×C4—1581no—
508C22×C4—1581no—
509C22×C4—1581no—
510C22×C4C2×C22⋊D7911yescharacteristic
511C22×C4C2×C22⋊D7911yescharacteristic
512C22×C4C2×C22⋊D7911yescharacteristic
513C22×C4C2×C22⋊D7911yescharacteristic
514C22×C4C2×C22⋊D7911yescharacteristic
515C22×C4C2×Dic15812yes—
516C22×C4#1192962412yes—
517C22×C4C23×D7911yescharacteristic
518M16—3164no—
519C2×C8—792no—
520C2×C8—1581no—
521C2×C8—1581no—
522C2×C8—1581no—
523C2×C8—1581no—
524C2×C8—1581no—
525C2×C8—1581no—
526C2×C8—1581no—
527C2×C8—1581no—
528C2×C8—1581no—
529C2×C8—1581no—
530C2×C8—1581no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—3162no—
538C4⋊C4—3162no—
539C4⋊C4—792no—
540C4⋊C4—6321no—
541C4⋊C4—1581no—
542C4⋊C4—1581no—
543C4⋊C4—1581no—
544C4⋊C4—1581no—
545C4⋊C4—1581no—
546C4⋊C4—1581no—
547C4⋊C4—1581no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—42no—
551C22⋊C4—3162no—
552C22⋊C4—6321no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—42no—
559C42—3162no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1457914012yes—
567C23C22×C22⋊D7911yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8—6324no—
577Q8#1457913814yes—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—3161no—
609C2×C4—3161no—
610C2×C4—3161no—
611C2×C4—3161no—
612C2×C4—3161no—
613C2×C4—3161no—
614C2×C4—3161no—
615C2×C4—3161no—
616C2×C4—3161no—
617C2×C4—3161no—
618C2×C4—3161no—
619C2×C4—1582no—
620C2×C4#1457914012yes—
621C2×C4#1457919212yes—
622C2×C4C2×D4×D7911yescharacteristic
623C2×C4C22×Dic15811yescharacteristic
624C2×C4C2×#1192962311yescharacteristic
625C2×C4C2×#1192962411yescharacteristic
626C2×C4#1457920611yescharacteristic
627C2×C4#1457920611yescharacteristic
628C8—3161no—
629C8—3161no—
630C8—3161no—
631C8—1582no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1457914011yescharacteristic
640C22D4×C22⋊D7911yescharacteristic
641C22#1909920711yescharacteristic
642C22C2×#1457919211yescharacteristic
643C22#1909936111yescharacteristic
644C22#1909937111yescharacteristic
645C22#1909937811yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—3162no—
656C4—6321no—
657C4—6321no—
658C4—6321no—
659C4C2×#1457913811yescharacteristic
660C4#1909918911yescharacteristic
661C4#1909920311yescharacteristic
662C4#1909923911yescharacteristic
663C2—42no—
664C2#2376706111yescharacteristic
665C2#2377320511yescharacteristic
666C2#2377324211yescharacteristic
667C2#2377497111yescharacteristic
668C2#2377503211yescharacteristic
669C2#2377546611yescharacteristic
670C2#2377569411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingQ8×C22⋊C4×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenterQ8×C22⋊C4D79
hypocenterC79#25979
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19098858C2semidirect
4. #23764948C2semidirect
5. #23764795C2semidirect
6. #23764781C2semidirect
7. #23764541C2semidirect
8. #23763668C2semidirect
9. #23763668C2semidirect
13. #23763415C2semidirect
109. C22×Q8×C79Q8semidirect
351. C79#25979semidirect
404. C22×Q8Dic158semidirect

Maximal subgroups

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

subgroupindex
C2×#190988582
C2×#190985722
#237670912
#237649482
#237647952
#237647812
#237645412
#237636682
#237635972
#237635852
#237634312
#237634152
#237633352
Q8×C22⋊C4×C792
#2597979