Finite Group Databasebeta

← #34310741

Subgroups — #34310741

Order 20224.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098782C211yesmaximal, characteristic
2C2×#19098668C211yesmaximal, characteristic, complemented
3C2×C4×#11929612C211yesmaximal, characteristic
4C4⋊C4×Dic158C211yesmaximal, characteristic, complemented
5#23764217C211yesmaximal, characteristic
6#23764163C211yesmaximal, characteristic, complemented
7#23764100C211yesmaximal, characteristic
8#23764079C211yesmaximal, characteristic, complemented
9#23763758C211yesmaximal, characteristic
10#23763744C211yesmaximal, characteristic, complemented
11#23763668C211yesmaximal, characteristic, complemented
12#23763661C211yesmaximal, characteristic
13#23763275C211yesmaximal, characteristic, complemented
14#23763233C211yesmaximal, characteristic, complemented
15C79×#2367C211yesmaximal, characteristic
16C22×#11929614C2211yescharacteristic
17C22×#11929612C2211yescharacteristic
18C2×C4×Dic158C2211yescharacteristic
19C2×C4×Dic158C2211yescharacteristic
20C2×C4×Dic158C2211yescharacteristic
21#19098782C418yescomplemented
22#19098668—24no—
23C4×#11929612—24no—
24C2×#14579092C2211yescharacteristic
25C2×#14579092C2211yescharacteristic
26C2×C4⋊(C79⋊C8)C2211yescharacteristic
27C2×#14579085C2211yescharacteristic
28C2×#14579085C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579084C2211yescharacteristic
31C2×#14579084C2211yescharacteristic
32C2×#14579082C2211yescharacteristic
33C2×#14579082C2211yescharacteristic
34C2×#14579082C2211yescharacteristic
35C2×#14579072C2211yescharacteristic
36C2×#14579072C2211yescharacteristic
37C2×C4×C79⋊C8C2211yescharacteristic
38#19098461C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098460—21no—
41#19098460—21no—
42#19098433C2211yescharacteristic
43#19098432—21no—
44C4⋊C4×Dic79—21no—
45#19098325C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098318C2211yescharacteristic
50#19098318C2211yescharacteristic
51C2×#124×C79C2211yescharacteristic
52C2×C4×Q8×C79C2211yescharacteristic
53C2×C4×D4×C79C2211yescharacteristic
54#391×C79C2211yescharacteristic
55#390×C79C2211yescharacteristic
56#388×C79—21no—
57#388×C79—21no—
58#385×C79—21no—
59C4×C4⋊C4×C79C2211yescharacteristic
60C4×C4⋊C4×C79C2211yescharacteristic
61C4×C22⋊C4×C79—21no—
62C22×Dic158C2311yescharacteristic
63C2×#11929614—22no—
64C2×#11929614—22no—
65C2×#11929614C2×C414yescomplemented
66C2×#11929612—22no—
67C2×#11929612—22no—
68C2×#11929612C2×C414yes—
69C22×C79⋊C8C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71C22×C79⋊C8C2311yescharacteristic
72C22×C79⋊C8C2311yescharacteristic
73C4×Dic158—22no—
74C4×Dic158—22no—
75C4×Dic158—22no—
76C4×Dic158—41no—
77C4×Dic158—41no—
78C4×Dic158C2×C414yescomplemented
79C2×C4⋊Dic79—21no—
80C2×C4⋊Dic79—21no—
81C2×C4⋊Dic79C2311yescharacteristic
82C2×C4⋊Dic79C2311yescharacteristic
83C2×C4⋊Dic79C2311yescharacteristic
84C2×Dic79⋊C4—21no—
85C2×Dic79⋊C4—21no—
86C2×Dic79⋊C4—21no—
87C2×Dic79⋊C4—21no—
88C2×Dic79⋊C4—21no—
89C2×C4×Dic79—21no—
90C2×C4×Dic79—21no—
91C2×C4×Dic79—21no—
92#14579092—22no—
93#14579092—22no—
94C4⋊(C79⋊C8)C2×C414yescomplemented
95#14579085—22no—
96#14579085—41no—
97#14579085C2×C414yescomplemented
98#14579084—22no—
99#14579084D414yes—
100#14579082—22no—
101#14579082—41no—
102#14579082C2×C414yes—
103#14579072—22no—
104#14579072—22no—
105C4×C79⋊C8—22no—
106C79⋊#95—21no—
107C79⋊#95—21no—
108C22×Q8×C79C2311yescharacteristic
109C22×D4×C79C2311yescharacteristic
110C23×C4×C79—21no—
111#124×C79—22no—
112#124×C79C2×C414yes—
113C4×Q8×C79—22no—
114C4×Q8×C79—41no—
115C4×D4×C79—22no—
116C4×D4×C79—41no—
117C2×C4⋊C4×C79—21no—
118C2×C4⋊C4×C79D411yescharacteristic
119C2×C4⋊C4×C79D411yescharacteristic
120C2×C4⋊C4×C79C2311yescharacteristic
121C2×C4⋊C4×C79C2311yescharacteristic
122C2×C22⋊C4×C79—21no—
123C2×C22⋊C4×C79—21no—
124C2×C22⋊C4×C79—21no—
125C2×C22⋊C4×C79—21no—
126C2×C22⋊C4×C79—21no—
127C2×C42×C79—21no—
128C2×C42×C79—21no—
129C2×C42×C79C2311yescharacteristic
130C2×C42×C79C2311yescharacteristic
131C2×C42×C79C2311yescharacteristic
132#95×C79—21no—
133#95×C79—21no—
134#95×C79—21no—
135#95×C79D411yescharacteristic
136#95×C79D411yescharacteristic
137C2×Dic158—21no—
138C2×Dic158—21no—
139C2×Dic158—21no—
140C2×Dic158C22×C412yes—
141C2×Dic158C2×D412yes—
142C2×Dic158#4112yes—
143C22×Dic79—21no—
144C22×Dic79—21no—
145C22×Dic79—21no—
146C22×Dic79—21no—
147#11929614—28no—
148#11929612—28no—
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⋊C8—21no—
156C2×C79⋊C8—21no—
157C2×C79⋊C8—21no—
158C2×C79⋊C8C22×C412yes—
159C2×C79⋊C8C22×C412yes—
160C2×C79⋊C8C2×D412yes—
161C4⋊Dic79—21no—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—42no—
167C4⋊Dic79—81no—
168C4⋊Dic79C22×C412yes—
169Dic79⋊C4—24no—
170Dic79⋊C4—42no—
171Dic79⋊C4—42no—
172Dic79⋊C4—81no—
173Dic79⋊C4—81no—
174C4×Dic79—24no—
175C4×Dic79—42no—
176C4×Dic79—42no—
177C24×C79—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—21no—
181C2×Q8×C79—21no—
182C2×Q8×C79—21no—
183C2×Q8×C79C22×C412yes—
184C2×D4×C79—21no—
185C2×D4×C79—21no—
186C2×D4×C79—21no—
187C2×D4×C79—21no—
188C2×D4×C79—21no—
189C2×D4×C79C22×C412yes—
190C22×C4×C79—21no—
191C22×C4×C79—21no—
192C22×C4×C79—21no—
193C22×C4×C79—42no—
194C22×C4×C79—42no—
195C22×C4×C79—81no—
196C22×C4×C79C2×D411yescharacteristic
197C22×C4×C79C2×D411yescharacteristic
198C22×C4×C79C2×D411yescharacteristic
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2×D411yescharacteristic
201C22×C4×C79C2×D411yescharacteristic
202C22×C4×C79C2411yescharacteristic
203C4⋊C4×C79—22no—
204C4⋊C4×C79—21no—
205C4⋊C4×C79—21no—
206C4⋊C4×C79—21no—
207C4⋊C4×C79—41no—
208C4⋊C4×C79—42no—
209C4⋊C4×C79C2×D412yes—
210C22⋊C4×C79—24no—
211C22⋊C4×C79—24no—
212C22⋊C4×C79—42no—
213C22⋊C4×C79—42no—
214C22⋊C4×C79—81no—
215C42×C79—21no—
216C42×C79—21no—
217C42×C79—21no—
218C42×C79—21no—
219C42×C79—42no—
220C42×C79—81no—
221C42×C79C22×C412yes—
222C42×C79#4112yes—
223C23×C79—22no—
224C23×C79—41no—
225C23×C79—41no—
226C23×C79—41no—
227C23×C79—41no—
228C23×C79—41no—
229C23×C79—41no—
230C23×C79C22×D411yescharacteristic
231Q8×C79—22no—
232Q8×C79—22no—
233Q8×C79—41no—
234Q8×C79—41no—
235D4×C79—22no—
236D4×C79—22no—
237D4×C79—41no—
238D4×C79—41no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—22no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
249C2×C4×C79—22no—
250C2×C4×C79—21no—
251C2×C4×C79—21no—
252C2×C4×C79—21no—
253C2×C4×C79—21no—
254C2×C4×C79—21no—
255C2×C4×C79—21no—
256C2×C4×C79—41no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—41no—
261C2×C4×C79—41no—
262C2×C4×C79—41no—
263C2×C4×C79—84no—
264C2×C4×C79C4×D412yes—
265C2×C4×C79C4×D412yes—
266C2×C4×C79#13712yes—
267C2×C4×C79C23×C411yescharacteristic
268C2×C4×C79C22×D411yescharacteristic
269C2×C4×C79C2×#4111yescharacteristic
270C2×C4×C79C2×#4111yescharacteristic
271C2×C4×C79H(5,𝔽2)11yescharacteristic
272C2×C4×C792−1+411yescharacteristic
273C2×Dic79—22no—
274C2×Dic79—22no—
275C2×Dic79—22no—
276C2×Dic79—22no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279C2×Dic79—41no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79—41no—
284C2×Dic79—41no—
285Dic158—22no—
286Dic158—22no—
287Dic158—22no—
288Dic158C4×D414yes—
289C79⋊C8—22no—
290C79⋊C8—22no—
291C79⋊C8—22no—
292C79⋊C8—41no—
293C22×C79—42no—
294C22×C79—42no—
295C22×C79—42no—
296C22×C79—81no—
297C22×C79—81no—
298C22×C79—81no—
299C22×C79—81no—
300C22×C79C2×C4×D411yescharacteristic
301C22×C79D4211yescharacteristic
302C22×C79#54811yescharacteristic
303C22×C79C2×#13711yescharacteristic
304C22×C79#57511yescharacteristic
305C22×C79#57711yescharacteristic
306C22×C79#57711yescharacteristic
307C4×C79—22no—
308C4×C79—22no—
309C4×C79—22no—
310C4×C79—42no—
311C4×C79—41no—
312C4×C79—42no—
313C4×C79—41no—
314C4×C79—41no—
315C4×C79—81no—
316C4×C79C2×C4×D411yescharacteristic
317C4×C79#52011yescharacteristic
318C4×C79#54711yescharacteristic
319C4×C79#55311yescharacteristic
320Dic79—24no—
321Dic79—42no—
322Dic79—81no—
323Dic79—81no—
324#23013—791nomaximal, Sylow
325C2×C79—82no—
326C2×C79#236111yescharacteristic
327C2×C79C4×#13711yescharacteristic
328C2×C79#295211yescharacteristic
329C2×C79#326611yescharacteristic
330C2×C79#327111yescharacteristic
331C2×C79#336911yescharacteristic
332C2×C79#337111yescharacteristic
333C2×#464—791no—
334C2×#441—791no—
335C2×C4×SD16—791no—
336#2367D7911yescharacteristic
337Q8×C4⋊C4—791no—
338#1959—791no—
339#1948—791no—
340#1936—791no—
341#1933—791no—
342#1880—791no—
343#1878—791no—
344#1865—791no—
345#1864—791no—
346#1771—791no—
347#1763—791no—
348C79#2301311yescharacteristic, complemented, Sylow
349C22×Q16—791no—
350C22×SD16—791no—
351C2×#124C2×D7911yescharacteristic
352C2×C4×Q8—791no—
353C2×C4×Q8—791no—
354C2×C4×Q8—791no—
355C2×C4×Q8C2×D7911yescharacteristic
356C2×C4×D4C2×D7911yescharacteristic
357#464—798no—
358#441—1584no—
359C4×SD16—1584no—
360C2×#106—791no—
361C2×#106—791no—
362C2×C4⋊C8—791no—
363C2×#103—791no—
364C2×#103—791no—
365C2×#103—791no—
366C2×#103—791no—
367C2×#103—791no—
368C2×#102—791no—
369C2×#102—791no—
370C2×#102—791no—
371C2×#97—791no—
372C2×#97—791no—
373C2×C4×C8—791no—
374#391C2×D7911yescharacteristic
375#390C2×D7911yescharacteristic
376#389—791no—
377#389—791no—
378#389—1581no—
379#389—1581no—
380#388—21no—
381#388—21no—
382#385—21no—
383#384—791no—
384#384—1581no—
385C4×C4⋊C4—1581no—
386C4×C4⋊C4C2×D7911yescharacteristic
387C4×C4⋊C4C2×D7911yescharacteristic
388C4×C22⋊C4—21no—
389#340—791no—
390#340—791no—
391#340—791no—
392#340—791no—
393#336—791no—
394#336—791no—
395C22×Q8—791no—
396C22×Q8C22×D7911yescharacteristic
397C22×D4C22×D7911yescharacteristic
398C23×C4—21no—
399C2×Q16—794no—
400C2×Q16—1582no—
401C2×Q16—1582no—
402C2×SD16—794no—
403C2×SD16—1582no—
404C2×SD16—1582no—
405C22×C8—791no—
406C22×C8—791no—
407C22×C8—791no—
408C22×C8—791no—
409#124—22no—
410#124C4×D7914yes—
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—3161no—
414C4×Q8—3161no—
415C4×Q8—794no—
416C4×Q8—1582no—
417C4×Q8—1582no—
418C4×Q8—1582no—
419C4×D4—22no—
420C4×D4—41no—
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⋊C4—1581no—
430C2×C4⋊C4—1581no—
431C2×C4⋊C4—1581no—
432C2×C4⋊C4C22⋊D7911yescharacteristic
433C2×C4⋊C4C22⋊D7911yescharacteristic
434C2×C4⋊C4C22×D7911yescharacteristic
435C2×C4⋊C4C22×D7911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—1581no—
444C2×C42—1581no—
445C2×C42—1581no—
446C2×C42C22×D7911yescharacteristic
447C2×C42C22×D7911yescharacteristic
448C2×C42C22×D7911yescharacteristic
449#106—1582no—
450#106—1582no—
451C4⋊C8—794no—
452#103—3161no—
453#103—794no—
454#103—794no—
455#103—1582no—
456#103—1582no—
457#102—3161no—
458#102—794no—
459#102—1582no—
460#97—1582no—
461#97—1582no—
462C4×C8—1582no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—1581no—
467#95—1581no—
468#95C22⋊D7911yescharacteristic
469#95C22⋊D7911yescharacteristic
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—792no—
477C2×Q8—792no—
478C2×Q8—792no—
479C2×Q8—1581no—
480C2×Q8—1581no—
481C2×Q8—1581no—
482C2×Q8C2×C4×D7912yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D7912yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—81no—
495C22×C4—1581no—
496C22×C4—1581no—
497C22×C4—1581no—
498C22×C4—1581no—
499C22×C4C2×C22⋊D7911yescharacteristic
500C22×C4C2×C22⋊D7911yescharacteristic
501C22×C4C2×C22⋊D7911yescharacteristic
502C22×C4C2×C22⋊D7911yescharacteristic
503C22×C4C2×C22⋊D7911yescharacteristic
504C22×C4C2×C22⋊D7911yescharacteristic
505C22×C4C23×D7911yescharacteristic
506Q16—1588no—
507SD16—1588no—
508C2×C8—792no—
509C2×C8—792no—
510C2×C8—792no—
511C2×C8—1581no—
512C2×C8—1581no—
513C2×C8—1581no—
514C2×C8—1581no—
515C2×C8—1581no—
516C2×C8—1581no—
517C2×C8—1581no—
518C2×C8—1581no—
519C2×C8—1581no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—42no—
526C4⋊C4—3162no—
527C4⋊C4—3162no—
528C4⋊C4—3162no—
529C4⋊C4—792no—
530C4⋊C4—6321no—
531C4⋊C4—6321no—
532C4⋊C4—6321no—
533C4⋊C4—1581no—
534C4⋊C4—1581no—
535C4⋊C4—1581no—
536C4⋊C4—1581no—
537C4⋊C4—1581no—
538C4⋊C4—1584no—
539C4⋊C4D4×D7912yes—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—81no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—81no—
551C42—3162no—
552C42—3162no—
553C42—1584no—
554C42C2×C4×D7912yes—
555C42#1192962412yes—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C22×C22⋊D7911yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—794no—
569Q8—1582no—
570Q8—1582no—
571Q8—1582no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
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—84no—
601C2×C4—3161no—
602C2×C4—3161no—
603C2×C4—3161no—
604C2×C4—3161no—
605C2×C4—3161no—
606C2×C4—3161no—
607C2×C4—3161no—
608C2×C4—3161no—
609C2×C4—1582no—
610C2×C4—1582no—
611C2×C4—1582no—
612C2×C4—1582no—
613C2×C4C4×C22⋊D7912yes—
614C2×C4C4×C22⋊D7912yes—
615C2×C4#1457919112yes—
616C2×C4C22×C4×D7911yescharacteristic
617C2×C4C2×D4×D7911yescharacteristic
618C2×C4#41×D7911yescharacteristic
619C2×C4C2×#1192962411yescharacteristic
620C2×C4#1457920611yescharacteristic
621C2×C4#1457920811yescharacteristic
622C8—3161no—
623C8—1582no—
624C8—1582no—
625C8—1582no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C2×C4×C22⋊D7911yescharacteristic
634C22D4×C22⋊D7911yescharacteristic
635C22#1909922011yescharacteristic
636C22C2×#1457919111yescharacteristic
637C22#1909936111yescharacteristic
638C22#1909937111yescharacteristic
639C22#1909937411yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—42no—
646C4—41no—
647C4—42no—
648C4—81no—
649C4—3162no—
650C4—6321no—
651C4—6321no—
652C4—1584no—
653C4C4×D4×D7911yescharacteristic
654C4#1909904211yescharacteristic
655C4#1909921811yescharacteristic
656C4#1909926311yescharacteristic
657C2—82no—
658C2#2376641211yescharacteristic
659C2C4×#1457919111yescharacteristic
660C2#2377228111yescharacteristic
661C2#2377490611yescharacteristic
662C2#2377498611yescharacteristic
663C2#2377613611yescharacteristic
664C2#2377616211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC79×#2367C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
cosocleC22×C4×C79C24
hypercenter#2367D79
hypocenterC79#23013
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
2. C2×#19098668C2semidirect
4. C4⋊C4×Dic158C2semidirect
6. #23764163C2semidirect
8. #23764079C2semidirect
10. #23763744C2semidirect
11. #23763668C2semidirect
13. #23763275C2semidirect
14. #23763233C2semidirect
21. #19098782C4semidirect
65. C2×#11929614C2×C4semidirect
78. C4×Dic158C2×C4semidirect
94. C4⋊(C79⋊C8)C2×C4semidirect
97. #14579085C2×C4semidirect
348. C79#23013semidirect

Maximal subgroups

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

subgroupindex
C2×#190987822
C2×#190986682
C2×C4×#119296122
C4⋊C4×Dic1582
#237642172
#237641632
#237641002
#237640792
#237637582
#237637442
#237636682
#237636612
#237632752
#237632332
C79×#23672
#2301379