Finite Group Databasebeta

← #34254092

Subgroups — #34254092

Order 20224.

1278 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
1C2×#19099042C211yesmaximal, characteristic
2#23772422C212yesmaximal, complemented
3#23761722C212yesmaximal, complemented
4#23761718C211yesmaximal, characteristic, complemented
5#23761715C212yesmaximal, complemented
6#23761711C212yesmaximal, complemented
7C2×#19098288C212yesmaximal, complemented
8#324×D79C212yesmaximal, complemented
9C79×#1459C211yesmaximal, characteristic, complemented
10C22×#11929623C411yescharacteristic
11#19099042—24no—
12#19099042—24no—
13C2×C4×Dic158C411yescharacteristic
14C2×#14579126C2211yescharacteristic
15C2×D4×Dic79C411yescharacteristic
16C2×C4×C22⋊D79C411yescharacteristic
17C2×C4×C22⋊D79C2211yescharacteristic
18C2×(C22⋊D79)⋊C4C411yescharacteristic
19C2×(C22⋊D79)⋊C4C2211yescharacteristic
20C2×#14579118C411yescharacteristic
21C2×#14579117C411yescharacteristic
22C2×#14579117C2211yescharacteristic
23C22×Dic79⋊C4C411yescharacteristic
24C22×Dic79⋊C4C2211yescharacteristic
25C2×C4⋊C4×D79C2211yescharacteristic
26C8×C22⋊D79—24no—
27(C22⋊D79)⋊C8—24no—
28#19098613C2214yes—
29C2×C22⋊(C79⋊C8)C2212yes—
30C22⋊C8×D79C2214yes—
31#19098289C2212yes—
32#19098288C2214yes—
33#19098288C2214yes—
34#19098287C2212yes—
35C2×C4×D4×C79C2211yescharacteristic
36C2×C22⋊C8×C79C2212yes—
37#324×C79C2212yes—
38#323×C79C2212yes—
39C2×#11929623—24no—
40C2×#11929623—24no—
41C2×#11929623C818yescomplemented
42C22×Dic158C2×C411yescharacteristic
43C22×C22⋊D79C2×C411yescharacteristic
44C22×C22⋊D79C2×C411yescharacteristic
45C23×Dic79C2×C411yescharacteristic
46C23×Dic79C2×C411yescharacteristic
47C23×Dic79C2×C411yescharacteristic
48C23×Dic79C2×C411yescharacteristic
49C22×C4×D79C2×C411yescharacteristic
50C22×C4×D79C2×C411yescharacteristic
51C22×C4×D79C2311yescharacteristic
52C22×C79⋊C8—22no—
53C2×C8×D79—24no—
54C4×Dic158—22no—
55C4×Dic158—22no—
56#14579126—24no—
57D4×Dic79—24no—
58C4×C22⋊D79—22no—
59C4×C22⋊D79—22no—
60C4×C22⋊D79D414yes—
61C4×C22⋊D79D414yes—
62(C22⋊D79)⋊C4—24no—
63(C22⋊D79)⋊C4—24no—
64#14579118—22no—
65#14579118—22no—
66#14579117—24no—
67#14579117—24no—
68C2×C4⋊Dic79C2×C411yescharacteristic
69C2×Dic79⋊C4—22no—
70C2×Dic79⋊C4—22no—
71C2×Dic79⋊C4—22no—
72C2×Dic79⋊C4—22no—
73C2×Dic79⋊C4C2×C411yescharacteristic
74C2×Dic79⋊C4C2×C411yescharacteristic
75C2×Dic79⋊C4C2×C411yescharacteristic
76C2×Dic79⋊C4C2×C411yescharacteristic
77C2×Dic79⋊C4C2×C411yescharacteristic
78C2×Dic79⋊C4C2×C411yescharacteristic
79C2×Dic79⋊C4C2311yescharacteristic
80C2×Dic79⋊C4C2311yescharacteristic
81C4⋊C4×D79—24no—
82C2×C22⋊Dic79C2×C411yescharacteristic
83C2×C22⋊Dic79C2×C411yescharacteristic
84C2×(C2×D79)⋊C4C2×C411yescharacteristic
85C2×(C2×D79)⋊C4C2×C411yescharacteristic
86C2×C4×Dic79C2×C411yescharacteristic
87C2×C4×Dic79C2×C411yescharacteristic
88C2×C4×Dic79C2×C411yescharacteristic
89C2×C4×Dic79C2×C411yescharacteristic
90C2×C4×Dic79C2311yescharacteristic
91Dic79⋊C8—24no—
92C22⋊(C79⋊C8)—22no—
93C22⋊(C79⋊C8)C2314yes—
94(C2×D79)⋊C8—24no—
95C8×Dic79—24no—
96C22×D4×C79C2×C411yescharacteristic
97C23×C4×C79C2×C411yescharacteristic
98C23×C4×C79C2311yescharacteristic
99C22×C8×C79—22no—
100C4×D4×C79—22no—
101C4×D4×C79—22no—
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C22⋊C4×C79C2×C411yescharacteristic
104C2×C22⋊C4×C79C2311yescharacteristic
105C2×C42×C79C2×C411yescharacteristic
106C22⋊C8×C79—22no—
107C22⋊C8×C79C2314yes—
108C23×D79C22×C411yescharacteristic
109#11929623—232no—
110C2×Dic158—22no—
111C2×Dic158—22no—
112C2×Dic158C2×C814yescomplemented
113C2×C22⋊D79—22no—
114C2×C22⋊D79—22no—
115C2×C22⋊D79C22⋊C414yes—
116C2×C22⋊D79C2×C814yes—
117C2×C22⋊D79C2×C814yescomplemented
118C2×C22⋊D79M1614yes—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—22no—
122C22×Dic79—22no—
123C22×Dic79—22no—
124C22×Dic79—22no—
125C22×Dic79—81no—
126C22×Dic79—81no—
127C22×Dic79C2×C814yescomplemented
128C22×Dic79C2×C814yescomplemented
129C22×Dic79C22×C411yescharacteristic
130C22×Dic79C22×C411yescharacteristic
131C22×Dic79C22×C411yescharacteristic
132C22×Dic79C22×C411yescharacteristic
133C22×Dic79C22×C411yescharacteristic
134C22×Dic79C22×C411yescharacteristic
135C22×Dic79C22×C411yescharacteristic
136C2×C4×D79—22no—
137C2×C4×D79—22no—
138C2×C4×D79—22no—
139C2×C4×D79—81no—
140C2×C4×D79—81no—
141C2×C4×D79C2×C814yescomplemented
142C2×C4×D79C2×D414yes—
143C2×C79⋊C8—24no—
144C2×C79⋊C8—82no—
145C8×D79—84no—
146C4⋊Dic79—22no—
147Dic79⋊C4—22no—
148Dic79⋊C4—22no—
149Dic79⋊C4—22no—
150Dic79⋊C4—21no—
151Dic79⋊C4—21no—
152Dic79⋊C4—21no—
153Dic79⋊C4—21no—
154Dic79⋊C4—22no—
155Dic79⋊C4—22no—
156Dic79⋊C4C22⋊C412yes—
157Dic79⋊C4C22⋊C412yes—
158C22⋊Dic79—22no—
159C22⋊Dic79C22⋊C414yes—
160(C2×D79)⋊C4—22no—
161(C2×D79)⋊C4C22⋊C412yes—
162(C2×D79)⋊C4C22⋊C412yes—
163C4×Dic79—22no—
164C4×Dic79—22no—
165C4×Dic79—22no—
166C4×Dic79—22no—
167C4×Dic79C2×D414yes—
168C24×C79C22×C411yescharacteristic
169C24×C79C22×C411yescharacteristic
170C2×D4×C79—22no—
171C2×D4×C79—22no—
172C2×D4×C79C2×C814yescomplemented
173C22×C4×C79—21no—
174C22×C4×C79—21no—
175C22×C4×C79—81no—
176C22×C4×C79—81no—
177C22×C4×C79C22×C411yescharacteristic
178C22×C4×C79C22×C411yescharacteristic
179C22×C4×C79C22×C411yescharacteristic
180C22×C4×C79C22×C411yescharacteristic
181C22×C4×C79C2×D412yes—
182C22×C4×C79C2×D412yes—
183C22×C4×C79C2411yescharacteristic
184C2×C8×C79—24no—
185C2×C8×C79—82no—
186C4⋊C4×C79—22no—
187C22⋊C4×C79—22no—
188C22⋊C4×C79—22no—
189C42×C79—21no—
190C42×C79—21no—
191C23×C79—21no—
192C23×C79—21no—
193C23×C79—41no—
194C23×C79—41no—
195C23×C79—41no—
196C23×C79—41no—
197C23×C79C2×C22⋊C412yes—
198C23×C79C22×C812yes—
199C23×C79C22×C812yes—
200C23×C79C2×M1612yes—
201C23×C79C23×C411yescharacteristic
202C22×D79—41no—
203C22×D79—41no—
204C22×D79C2×C22⋊C412yes—
205C22×D79C22×C812yes—
206C22×D79C2×M1612yes—
207D4×C79—28no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—42no—
213C2×C4×C79—41no—
214C2×C4×C79—42no—
215C2×C4×C79—41no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—81no—
219C2×C4×C79—81no—
220C2×C4×C79C2×C22⋊C411yescharacteristic
221C2×C4×C79C2×C22⋊C411yescharacteristic
222C2×C4×C79C22×C812yes—
223C2×C4×C79C22×D412yes—
224C22⋊D79—28no—
225C22⋊D79C22⋊C8116yes—
226C2×Dic79—21no—
227C2×Dic79—21no—
228C2×Dic79—21no—
229C2×Dic79—21no—
230C2×Dic79—28no—
231C2×Dic79—28no—
232C2×Dic79—21no—
233C2×Dic79—21no—
234C2×Dic79—41no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—82no—
239C2×Dic79—82no—
240C2×Dic79C2×C22⋊C412yes—
241C2×Dic79C2×C22⋊C412yes—
242C2×Dic79C2×C22⋊C412yes—
243C2×Dic79C22×C812yes—
244C2×Dic79C22×C812yes—
245C2×Dic79C22×C812yes—
246C2×Dic79C2×M1612yes—
247C4×D79—28no—
248C4×D79—44no—
249C4×D79—82no—
250Dic158—28no—
251C8×C79—44no—
252C79⋊C8—44no—
253C22×C79—21no—
254C22×C79—22no—
255C22×C79—21no—
256C22×C79—22no—
257C22×C79—22no—
258C22×C79—42no—
259C22×C79—42no—
260C22×C79—42no—
261C22×C79—42no—
262C22×C79—41no—
263C22×C79—41no—
264C22×C79C2×C22⋊C814yes—
265C22×C79C22×C22⋊C411yescharacteristic
266C22×C79C23×C811yescharacteristic
267C22×C79C22×M1611yescharacteristic
268C2×D79—22no—
269C2×D79—22no—
270C2×D79—42no—
271C2×D79—42no—
272C2×D79C2×C22⋊C814yes—
273C4×C79—22no—
274C4×C79—21no—
275C4×C79—21no—
276C4×C79—22no—
277C4×C79—41no—
278C4×C79—41no—
279Dic79—22no—
280Dic79—22no—
281Dic79—22no—
282Dic79—22no—
283Dic79—41no—
284Dic79—41no—
285Dic79C2×C22⋊C814yes—
286#17351—791nomaximal, Sylow
287C2×C79—21no—
288C2×C79—21no—
289C2×C79—24no—
290C2×C79—42no—
291C2×C79C22×C22⋊C811yescharacteristic
292C2×C79#288311yescharacteristic
293C2×C79#288911yescharacteristic
294D79—24no—
295C2×#520—791no—
296#2972—792no—
297#1461—792no—
298#1459—792no—
299#1459—791no—
300#1459D7911yescharacteristic, complemented
301#1458—792no—
302C2×#324—792no—
303C2×#324—792no—
304C79#1735111yescharacteristic, complemented, Sylow
305C22×#41—791no—
306#520—1584no—
307#520—1584no—
308C2×C4×Q8—791no—
309C2×C4×Q8—791no—
310C2×C4×D4—791no—
311C2×C4×D4—791no—
312C2×C4×D4—791no—
313C2×C4×D4—791no—
314C2×C4×D4—791no—
315C2×C4×D4C2×D7911yescharacteristic
316C2×#117—791no—
317C2×#117—791no—
318C2×#117—791no—
319C22×C4⋊C4—791no—
320C22×C4⋊C4—791no—
321C22×C4⋊C4—791no—
322C8×D4—1584no—
323C8×D4—1584no—
324#431—794no—
325C2×C22⋊C8—792no—
326C2×C22⋊C8—794no—
327C2×C22⋊C8C2×D7912yes—
328#324—792no—
329#324—794no—
330#324—794no—
331#324C2×D7912yes—
332#323—792no—
333#323C2×D7912yes—
334C2×#41—798no—
335C2×#41—1584no—
336C2×#41—1584no—
337C22×Q8—791no—
338C22×D4—791no—
339C22×D4—791no—
340C22×D4C4×D7911yescharacteristic
341C23×C4—791no—
342C23×C4—791no—
343C23×C4—791no—
344C23×C4—791no—
345C23×C4—791no—
346C23×C4—791no—
347C23×C4—791no—
348C23×C4C4×D7911yescharacteristic
349C23×C4C22×D7911yescharacteristic
350C22×C8—22no—
351C22×C8—1584no—
352C22×C8—1582no—
353C4×Q8—1582no—
354C4×Q8—1582no—
355C4×Q8—1584no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—794no—
359C4×D4—794no—
360C4×D4—1584no—
361C4×D4—1584no—
362C4×D4—1582no—
363C4×D4—1582no—
364C4×D4—1584no—
365#117—1582no—
366#117—1582no—
367#117—1584no—
368#117—1584no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—791no—
371C2×C4⋊C4—791no—
372C2×C4⋊C4—791no—
373C2×C4⋊C4—791no—
374C2×C4⋊C4—791no—
375C2×C4⋊C4—791no—
376C2×C4⋊C4—791no—
377C2×C4⋊C4—791no—
378C2×C4⋊C4—1584no—
379C2×C4⋊C4—1582no—
380C2×C4⋊C4—1582no—
381C2×C4⋊C4—1582no—
382C2×C4⋊C4—1582no—
383C2×C4⋊C4C22×D7911yescharacteristic
384C2×C22⋊C4—791no—
385C2×C22⋊C4—791no—
386C2×C22⋊C4—791no—
387C2×C22⋊C4—791no—
388C2×C22⋊C4C4×D7911yescharacteristic
389C2×C22⋊C4C22×D7911yescharacteristic
390C2×C42—791no—
391C2×C42—791no—
392C2×C42—791no—
393C2×C42—791no—
394C2×C42—791no—
395C2×C42C4×D7911yescharacteristic
396C4⋊C8—1584no—
397C22⋊C8—22no—
398C22⋊C8—794no—
399C22⋊C8—1584no—
400C22⋊C8—1582no—
401C22⋊C8C22×D7914yes—
402C4×C8—1584no—
403C24—791no—
404C24C2×C4×D7911yescharacteristic
405C24C2×C4×D7911yescharacteristic
406#41—15832no—
407C2×Q8—794no—
408C2×Q8—1582no—
409C2×Q8—1582no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—794no—
413C2×D4—794no—
414C2×D4—794no—
415C2×D4—794no—
416C2×D4—1582no—
417C2×D4—1582no—
418C2×D4C8×D7914yescomplemented
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—791no—
424C22×C4—791no—
425C22×C4—794no—
426C22×C4—791no—
427C22×C4—791no—
428C22×C4—791no—
429C22×C4—794no—
430C22×C4—794no—
431C22×C4—791no—
432C22×C4—791no—
433C22×C4—794no—
434C22×C4—6321no—
435C22×C4—6321no—
436C22×C4—6321no—
437C22×C4—6321no—
438C22×C4—1582no—
439C22×C4—1582no—
440C22×C4—1582no—
441C22×C4—1582no—
442C22×C4—1582no—
443C22×C4—1582no—
444C22×C4—1582no—
445C22×C4—1582no—
446C22×C4—1582no—
447C22×C4C2×C4×D7911yescharacteristic
448C22×C4C2×C4×D7911yescharacteristic
449C22×C4C2×C4×D7911yescharacteristic
450C22×C4C2×C4×D7911yescharacteristic
451C22×C4D4×D7912yes—
452C22×C4D4×D7912yes—
453C22×C4C23×D7911yescharacteristic
454C2×C8—24no—
455C2×C8—82no—
456C2×C8—6324no—
457C2×C8—6322no—
458C2×C8—1584no—
459C4⋊C4—22no—
460C4⋊C4—792no—
461C4⋊C4—792no—
462C4⋊C4—1582no—
463C4⋊C4—1582no—
464C4⋊C4—1582no—
465C4⋊C4—1582no—
466C4⋊C4—1581no—
467C4⋊C4—1581no—
468C4⋊C4—1581no—
469C4⋊C4—1581no—
470C4⋊C4—1582no—
471C4⋊C4—1582no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—794no—
475C22⋊C4—792no—
476C22⋊C4—792no—
477C22⋊C4—1582no—
478C22⋊C4—1582no—
479C42—21no—
480C42—21no—
481C42—794no—
482C42—1582no—
483C42—1582no—
484C42—1582no—
485C42—1582no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—3161no—
493C23—3161no—
494C23—792no—
495C23—792no—
496C23—792no—
497C23C22⋊C4×D7912yes—
498C23C2×C8×D7912yes—
499C23C2×C8×D7912yes—
500C23M16×D7912yes—
501C23C22×C4×D7911yescharacteristic
502Q8—1588no—
503D4—28no—
504D4—7916no—
505D4—1588no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—3164no—
519C2×C4—3161no—
520C2×C4—3161no—
521C2×C4—3161no—
522C2×C4—3161no—
523C2×C4—792no—
524C2×C4—792no—
525C2×C4—792no—
526C2×C4—792no—
527C2×C4—792no—
528C2×C4—792no—
529C2×C4—792no—
530C2×C4—6322no—
531C2×C4—6322no—
532C2×C4—6322no—
533C2×C4—1588no—
534C2×C4—1581no—
535C2×C4—1581no—
536C2×C4—1588no—
537C2×C4—1581no—
538C2×C4—1581no—
539C2×C4—1588no—
540C2×C4—1581no—
541C2×C4—1581no—
542C2×C4C22⋊C4×D7911yescharacteristic
543C2×C4C22⋊C4×D7911yescharacteristic
544C2×C4C2×C8×D7912yes—
545C2×C4C2×D4×D7912yes—
546C8—44no—
547C8—3164no—
548C22—21no—
549C22—22no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—41no—
559C22—3162no—
560C22—3162no—
561C22—794no—
562C22—1582no—
563C22—1582no—
564C22C22⋊C8×D7914yes—
565C22C2×C22⋊C4×D7911yescharacteristic
566C22C22×C8×D7911yescharacteristic
567C22C2×M16×D7911yescharacteristic
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—41no—
573C4—41no—
574C4—3161no—
575C4—3161no—
576C4—794no—
577C4—1582no—
578C4—1582no—
579C4—1582no—
580C4—1582no—
581C2—21no—
582C2—21no—
583C2—24no—
584C2—42no—
585C2—1584no—
586C2C2×C22⋊C8×D7911yescharacteristic
587C2#2377185811yescharacteristic
588C2#2377188111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D79
commutatorC22×C79C23×C8
FrattiniC22×C4C23×D79
FittingC79×#1459C2
socleC22×C79C22×C22⋊C4
prime radicalC22×#11929623C4
cosocleC22×C4×C79C24
hypercenter#1459D79
hypocenterC79#17351
LCSG ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #23772422C2semidirect
3. #23761722C2semidirect
4. #23761718C2semidirect
5. #23761715C2semidirect
6. #23761711C2semidirect
7. C2×#19098288C2semidirect
8. #324×D79C2semidirect
9. C79×#1459C2semidirect
41. C2×#11929623C8semidirect
112. C2×Dic158C2×C8semidirect
117. C2×C22⋊D79C2×C8semidirect
127. C22×Dic79C2×C8semidirect
128. C22×Dic79C2×C8semidirect
141. C2×C4×D79C2×C8semidirect
172. C2×D4×C79C2×C8semidirect
300. #1459D79semidirect
304. C79#17351semidirect
418. C2×D4C8×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990422
#237724222
#237617222
#237617182
#237617152
#237617112
C2×#190982882
#324×D792
C79×#14592
#1735179