Finite Group Databasebeta

← #34345791

Subgroups — #34345791

Order 20224.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098870C211yesmaximal, characteristic, complemented
2C2×#19098572C211yesmaximal, characteristic
3#23767113C211yesmaximal, characteristic
4#23764938C211yesmaximal, characteristic, complemented
5#23764781C211yesmaximal, characteristic, complemented
6#23764781C211yesmaximal, characteristic, complemented
7#23764551C211yesmaximal, characteristic, complemented
8#23763668C211yesmaximal, characteristic, complemented
9#23763654C211yesmaximal, characteristic, complemented
10#23763597C211yesmaximal, characteristic
11#23763585C211yesmaximal, characteristic
12#23763431C211yesmaximal, characteristic
13#23763405C211yesmaximal, characteristic, complemented
14#23763335C211yesmaximal, characteristic
15Q8×C22⋊C4×C79C211yesmaximal, characteristic
16C22×#11929608C2211yescharacteristic
17C2×#14579153C2211yescharacteristic
18C2×#14579120C2211yescharacteristic
19#19098870—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#19098499—21no—
37#19098486—21no—
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098460—21no—
41#19098446—21no—
42#19098433C2211yescharacteristic
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
61C2×#11929608—22no—
62C2×#11929608—22no—
63C2×#11929608—41no—
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67C22×C79⋊C8C2311yescharacteristic
68#14579153—22no—
69#14579153—41no—
70#14579120—22no—
71#14579120—41no—
72C2×C4⋊Dic79—21no—
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×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×C22⋊Dic79—21no—
83C2×C22⋊Dic79—21no—
84C2×C4×Dic79—21no—
85#14579094—22no—
86#14579094—41no—
87#14579092—22no—
88#14579092—41no—
89C4⋊(C79⋊C8)—22no—
90C4⋊(C79⋊C8)—41no—
91#14579085—22no—
92#14579085—22no—
93#14579085—22no—
94#14579085—22no—
95#14579085—22no—
96#14579085—41no—
97#14579085—41no—
98#14579085D414yes—
99C22⋊(C79⋊C8)—22no—
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×C79Q814yes—
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
136C22×Dic79—21no—
137C22×Dic79—21no—
138C22×Dic79—21no—
139C22×Dic79—21no—
140#11929608—44no—
141C2×C79⋊C8—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
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⋊C8C2×D412yes—
153C4⋊Dic79—21no—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79—42no—
161C4⋊Dic79—81no—
162C4⋊Dic79C2×D412yes—
163Dic79⋊C4—42no—
164Dic79⋊C4—42no—
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—22no—
244C2×C4×C79—21no—
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—
291C79⋊C8—22no—
292C79⋊C8—41no—
293C79⋊C8—41no—
294C79⋊C8—41no—
295C22×C79—22no—
296C22×C79—22no—
297C22×C79—22no—
298C22×C79—41no—
299C22×C79—41no—
300C22×C79—41no—
301C22×C79—41no—
302C22×C79C2×C22⋊Q811yescharacteristic
303C22×C79D4211yescharacteristic
304C22×C79#54611yescharacteristic
305C22×C79C2×#13711yescharacteristic
306C22×C79#57511yescharacteristic
307C22×C79#57711yescharacteristic
308C22×C79#57811yescharacteristic
309C4×C79—22no—
310C4×C79—24no—
311C4×C79—22no—
312C4×C79—42no—
313C4×C79—41no—
314C4×C79—41no—
315C4×C79—41no—
316C4×C79—81no—
317C4×C79—81no—
318C4×C79C2×C22⋊Q811yescharacteristic
319C4×C79#54311yescharacteristic
320C4×C79#54611yescharacteristic
321C4×C79#55111yescharacteristic
322Dic79—42no—
323Dic79—81no—
324Dic79—81no—
325Dic79—81no—
326#25981—791nomaximal, Sylow
327C2×C79—42no—
328C2×C79#243711yescharacteristic
329C2×C79#307811yescharacteristic
330C2×C79#308211yescharacteristic
331C2×C79#327111yescharacteristic
332C2×C79#327611yescharacteristic
333C2×C79#332011yescharacteristic
334C2×C79#332711yescharacteristic
335C2×#479—791no—
336C2×#419—791no—
337#2441—791no—
338Q8×C22⋊C4D7911yescharacteristic
339#2095—791no—
340#2066—791no—
341#2066—791no—
342#2026—791no—
343#1865—791no—
344#1863—791no—
345#1854—791no—
346#1851—791no—
347#1809—791no—
348#1803—791no—
349#1787—791no—
350C79#2598111yescharacteristic, complemented, Sylow
351C23×Q8C2×D7911yescharacteristic
352C22×M16—791no—
353C2×#125—791no—
354C2×C4×Q8C2×D7911yescharacteristic
355C2×C4×Q8C2×D7911yescharacteristic
356C2×#117—791no—
357#479—1584no—
358C2×#107—791no—
359C2×#107—791no—
360C2×#106—791no—
361C2×#106—791no—
362C2×C4⋊C8—791no—
363C2×C4⋊C8—791no—
364#419—1584no—
365C2×#103—791no—
366C2×#103—791no—
367C2×#103—791no—
368C2×#103—791no—
369C2×#103—791no—
370C2×#103—791no—
371C2×#103—791no—
372C2×#103—791no—
373C2×C22⋊C8—791no—
374#399—1581no—
375#396—1581no—
376#391—21no—
377#391—21no—
378#391C2×D7911yescharacteristic
379#391C2×D7911yescharacteristic
380#389—791no—
381#389—791no—
382#389—1581no—
383#387—1581no—
384#384—791no—
385#380—21no—
386#380—791no—
387#380C2×D7911yescharacteristic
388C4×C22⋊C4—21no—
389C4×C22⋊C4C2×D7911yescharacteristic
390#340—791no—
391#340—791no—
392#340—791no—
393#340—791no—
394#336—791no—
395#336—791no—
396C22×Q8—21no—
397C22×Q8—22no—
398C22×Q8—22no—
399C22×Q8—22no—
400C22×Q8—41no—
401C22×Q8—41no—
402C22×Q8Dic15814yes—
403C22×Q8C22×D7911yescharacteristic
404C22×Q8C22×D7911yescharacteristic
405C23×C4—21no—
406C23×C4C22×D7911yescharacteristic
407C2×M16—3161no—
408C2×M16—1582no—
409C2×M16—1582no—
410C22×C8—791no—
411C22×C8—791no—
412C22×C8—791no—
413C22×C8—791no—
414#125—3161no—
415#125—1582no—
416C4×Q8—22no—
417C4×Q8—22no—
418C4×Q8—41no—
419C4×Q8—41no—
420#117—3161no—
421#117—1582no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—791no—
425C2×C4⋊C4—791no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—791no—
428C2×C4⋊C4—1581no—
429C2×C4⋊C4—1581no—
430C2×C4⋊C4—1581no—
431C2×C4⋊C4—1581no—
432C2×C4⋊C4—1581no—
433C2×C4⋊C4—1581no—
434C2×C4⋊C4C22×D7911yescharacteristic
435C2×C4⋊C4C22×D7911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1581no—
438C2×C22⋊C4—1581no—
439C2×C22⋊C4C22⋊D7911yescharacteristic
440C2×C22⋊C4C22⋊D7911yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—1581no—
444C2×C42C22×D7911yescharacteristic
445C2×C42C22×D7911yescharacteristic
446#107—3161no—
447#107—1582no—
448#106—3161no—
449#106—1582no—
450C4⋊C8—3161no—
451C4⋊C8—1582no—
452#103—3161no—
453#103—3161no—
454#103—794no—
455#103—1582no—
456#103—1582no—
457#103—1582no—
458#103—1582no—
459#103—1582no—
460C22⋊C8—1582no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—1581no—
467#95—1581no—
468#95—1581no—
469#95C22⋊D7911yescharacteristic
470#95C22⋊D7911yescharacteristic
471C24C2×C22⋊D7911yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—24no—
477C2×Q8—28no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—42no—
483C2×Q8—42no—
484C2×Q8—44no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8D4×D7912yes—
488C2×Q8C2×Dic15812yes—
489C2×Q8C2×Dic15812yes—
490C2×Q8#1192962312yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—1581no—
503C22×C4—1581no—
504C22×C4—1581no—
505C22×C4—1581no—
506C22×C4C2×C22⋊D7911yescharacteristic
507C22×C4C2×C22⋊D7911yescharacteristic
508C22×C4C2×C22⋊D7911yescharacteristic
509C22×C4C2×C22⋊D7911yescharacteristic
510C22×C4C2×C22⋊D7911yescharacteristic
511C22×C4C2×Dic15812yes—
512C22×C4#1192962412yes—
513C22×C4C23×D7911yescharacteristic
514M16—3164no—
515C2×C8—792no—
516C2×C8—1581no—
517C2×C8—1581no—
518C2×C8—1581no—
519C2×C8—1581no—
520C2×C8—1581no—
521C2×C8—1581no—
522C2×C8—1581no—
523C2×C8—1581no—
524C2×C8—1581no—
525C2×C8—1581no—
526C2×C8—1581no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—3162no—
534C4⋊C4—3162no—
535C4⋊C4—3162no—
536C4⋊C4—3162no—
537C4⋊C4—792no—
538C4⋊C4—6321no—
539C4⋊C4—6321no—
540C4⋊C4—1581no—
541C4⋊C4—1581no—
542C4⋊C4—1581no—
543C4⋊C4—1581no—
544C4⋊C4—1581no—
545C4⋊C4—1581no—
546C4⋊C4—1581no—
547C22⋊C4—22no—
548C22⋊C4—41no—
549C22⋊C4—42no—
550C22⋊C4—3162no—
551C22⋊C4—6321no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—42no—
557C42—42no—
558C42—3162no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1457914012yes—
566C23C22×C22⋊D7911yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—48no—
573Q8—41no—
574Q8—41no—
575Q8#1457913814yes—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—3161no—
607C2×C4—3161no—
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—1582no—
618C2×C4#1457914012yes—
619C2×C4#1457919212yes—
620C2×C4C2×D4×D7911yescharacteristic
621C2×C4C22×Dic15811yescharacteristic
622C2×C4C2×#1192962311yescharacteristic
623C2×C4C2×#1192962411yescharacteristic
624C2×C4#1457920611yescharacteristic
625C2×C4#1457920611yescharacteristic
626C8—3161no—
627C8—3161no—
628C8—3161no—
629C8—1582no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1457914011yescharacteristic
638C22D4×C22⋊D7911yescharacteristic
639C22#1909920711yescharacteristic
640C22C2×#1457919211yescharacteristic
641C22#1909936111yescharacteristic
642C22#1909937111yescharacteristic
643C22#1909937811yescharacteristic
644C4—22no—
645C4—24no—
646C4—22no—
647C4—41no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—3162no—
654C4—6321no—
655C4—6321no—
656C4—6321no—
657C4C2×#1457913811yescharacteristic
658C4#1909918911yescharacteristic
659C4#1909920311yescharacteristic
660C4#1909923911yescharacteristic
661C2—42no—
662C2#2376706111yescharacteristic
663C2#2377320511yescharacteristic
664C2#2377324211yescharacteristic
665C2#2377498611yescharacteristic
666C2#2377504611yescharacteristic
667C2#2377557111yescharacteristic
668C2#2377566811yescharacteristic
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#25981
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×#19098870C2semidirect
4. #23764938C2semidirect
5. #23764781C2semidirect
6. #23764781C2semidirect
7. #23764551C2semidirect
8. #23763668C2semidirect
9. #23763654C2semidirect
13. #23763405C2semidirect
350. C79#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#190988702
C2×#190985722
#237671132
#237649382
#237647812
#237645512
#237636682
#237636542
#237635972
#237635852
#237634312
#237634052
#237633352
Q8×C22⋊C4×C792
#2598179