Finite Group Databasebeta

← #34329000

Subgroups — #34329000

Order 20224.

814 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×#19098770C211yesmaximal, characteristic, complemented
2C2×#19098624C211yesmaximal, characteristic
3#23766082C211yesmaximal, characteristic
4#23764600C211yesmaximal, characteristic
5#23764593C211yesmaximal, characteristic
6#23764408C211yesmaximal, characteristic
7#23764408C211yesmaximal, characteristic
8#23764324C211yesmaximal, characteristic, complemented
9#23764270C211yesmaximal, characteristic, complemented
10#23764260C211yesmaximal, characteristic, complemented
11#23764166C211yesmaximal, characteristic, complemented
12#23764093C211yesmaximal, characteristic, complemented
13#23764079C211yesmaximal, characteristic, complemented
14#23763412C211yesmaximal, characteristic, complemented
15C79×#2621C211yesmaximal, characteristic
16C22×#11929614C2211yescharacteristic
17C22×#11929614C2211yescharacteristic
18C2×#14579140C2211yescharacteristic
19C2×C4×Dic158C2211yescharacteristic
20#19098770—24no—
21#19098624—24no—
22C2×#14579094C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×#14579085C2211yescharacteristic
28C2×#14579085C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579085C2211yescharacteristic
31C2×#14579085C2211yescharacteristic
32C2×#14579084C2211yescharacteristic
33C2×#14579084C2211yescharacteristic
34C2×C22⋊(C79⋊C8)C2211yescharacteristic
35C2×C22⋊(C79⋊C8)C2211yescharacteristic
36C2×#14579072C2211yescharacteristic
37C2×C4×C79⋊C8C2211yescharacteristic
38#19098469C2211yescharacteristic
39#19098461C2211yescharacteristic
40#19098458—21no—
41#19098457—21no—
42#19098451—21no—
43#19098444—21no—
44#19098423C2211yescharacteristic
45C4×C4⋊Dic79C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C2×C4⋊Q8×C79C2211yescharacteristic
51C2×C22⋊Q8×C79C2211yescharacteristic
52C2×C22⋊Q8×C79C2211yescharacteristic
53C2×#117×C79C2211yescharacteristic
54#398×C79—21no—
55#398×C79—21no—
56#397×C79—21no—
57#397×C79—21no—
58#391×C79C2211yescharacteristic
59#391×C79C2211yescharacteristic
60#382×C79C2211yescharacteristic
61C22×Dic158C2311yescharacteristic
62C2×#11929614—22no—
63C2×#11929614—22no—
64C2×#11929614—22no—
65C2×#11929614—22no—
66C2×#11929614—41no—
67C2×#11929614—41no—
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70C22×C79⋊C8C2311yescharacteristic
71C22×C79⋊C8C2311yescharacteristic
72#14579140—22no—
73#14579140—22no—
74C4×Dic158—22no—
75C4×Dic158—22no—
76C2×C4⋊Dic79C2311yescharacteristic
77C2×C4⋊Dic79C2311yescharacteristic
78C2×C4⋊Dic79C2311yescharacteristic
79C2×Dic79⋊C4—21no—
80C2×Dic79⋊C4—21no—
81C2×Dic79⋊C4—21no—
82C2×C22⋊Dic79—21no—
83C2×C22⋊Dic79—21no—
84C2×C4×Dic79—21no—
85C2×C4×Dic79—21no—
86#14579094—41no—
87#14579092—41no—
88C4⋊(C79⋊C8)—22no—
89C4⋊(C79⋊C8)D414yes—
90#14579085—22no—
91#14579085—22no—
92#14579085—22no—
93#14579085—22no—
94#14579085—41no—
95#14579085—41no—
96#14579084—41no—
97#14579084—41no—
98C22⋊(C79⋊C8)—22no—
99C22⋊(C79⋊C8)—22no—
100#14579072—22no—
101C4×C79⋊C8—22no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C79⋊#95—21no—
106C79⋊#95—21no—
107C22×Q8×C79C2311yescharacteristic
108C22×Q8×C79C2311yescharacteristic
109C23×C4×C79C2311yescharacteristic
110C4⋊Q8×C79—22no—
111C4⋊Q8×C79—22no—
112C22⋊Q8×C79—41no—
113C22⋊Q8×C79—41no—
114C22⋊Q8×C79—41no—
115C22⋊Q8×C79—41no—
116#117×C79—22no—
117#117×C79—22no—
118C2×C4⋊C4×C79—21no—
119C2×C4⋊C4×C79—21no—
120C2×C4⋊C4×C79—21no—
121C2×C4⋊C4×C79—21no—
122C2×C4⋊C4×C79D411yescharacteristic
123C2×C4⋊C4×C79D411yescharacteristic
124C2×C4⋊C4×C79C2311yescharacteristic
125C2×C4⋊C4×C79C2311yescharacteristic
126C2×C22⋊C4×C79—21no—
127C2×C22⋊C4×C79—21no—
128C2×C22⋊C4×C79D411yescharacteristic
129C2×C22⋊C4×C79D411yescharacteristic
130C2×C42×C79C2311yescharacteristic
131C2×C42×C79C2311yescharacteristic
132#95×C79—21no—
133#95×C79—21no—
134#95×C79—21no—
135#95×C79—21no—
136#95×C79—21no—
137#95×C79—21no—
138C2×Dic158—21no—
139C2×Dic158—21no—
140C2×Dic158—21no—
141C2×Dic158—21no—
142C2×Dic158—21no—
143C2×Dic158#4112yes—
144C22×Dic79—21no—
145C22×Dic79—21no—
146C22×Dic79—21no—
147C22×Dic79—21no—
148#11929614—44no—
149#11929614—44no—
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⋊C8—21no—
159C2×C79⋊C8—21no—
160C2×C79⋊C8C2×D412yes—
161C2×C79⋊C8C2×D412yes—
162C4⋊Dic79—21no—
163C4⋊Dic79—21no—
164C4⋊Dic79—21no—
165C4⋊Dic79—21no—
166C4⋊Dic79—21no—
167C4⋊Dic79#4112yes—
168Dic79⋊C4—24no—
169Dic79⋊C4—42no—
170Dic79⋊C4—81no—
171C22⋊Dic79—24no—
172C22⋊Dic79—81no—
173C4×Dic79—42no—
174C4×Dic79—81no—
175C24×C79C2×D411yescharacteristic
176C2×Q8×C79—21no—
177C2×Q8×C79—21no—
178C2×Q8×C79—21no—
179C2×Q8×C79—21no—
180C2×Q8×C79—21no—
181C2×Q8×C79—21no—
182C2×Q8×C79—21no—
183C2×Q8×C79—21no—
184C2×Q8×C79—21no—
185C2×Q8×C79—21no—
186C2×Q8×C79—21no—
187C2×Q8×C79—21no—
188C22×C4×C79—21no—
189C22×C4×C79—21no—
190C22×C4×C79—21no—
191C22×C4×C79—21no—
192C22×C4×C79—21no—
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195C22×C4×C79—21no—
196C22×C4×C79—21no—
197C22×C4×C79—21no—
198C22×C4×C79C2×D411yescharacteristic
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2×D411yescharacteristic
201C22×C4×C79C2×D411yescharacteristic
202C22×C4×C79C2×D411yescharacteristic
203C22×C4×C79C2411yescharacteristic
204C4⋊C4×C79—22no—
205C4⋊C4×C79—22no—
206C4⋊C4×C79—21no—
207C4⋊C4×C79—21no—
208C4⋊C4×C79—21no—
209C4⋊C4×C79—21no—
210C4⋊C4×C79—42no—
211C4⋊C4×C79—42no—
212C4⋊C4×C79—42no—
213C4⋊C4×C79—42no—
214C22⋊C4×C79—22no—
215C22⋊C4×C79—22no—
216C22⋊C4×C79—42no—
217C22⋊C4×C79—42no—
218C42×C79—21no—
219C42×C79—21no—
220C42×C79C2×D412yes—
221C42×C79#4112yes—
222C23×C79—21no—
223C23×C79—21no—
224C23×C79—21no—
225C23×C79—21no—
226C23×C79—21no—
227C23×C79—21no—
228C23×C79—21no—
229C23×C79C22×D411yescharacteristic
230Q8×C79—41no—
231Q8×C79—41no—
232Q8×C79—41no—
233Q8×C79—41no—
234Q8×C79—41no—
235Q8×C79—41no—
236Q8×C79—41no—
237Q8×C79—41no—
238C2×C4×C79—21no—
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—21no—
246C2×C4×C79—21no—
247C2×C4×C79—21no—
248C2×C4×C79—21no—
249C2×C4×C79—21no—
250C2×C4×C79—21no—
251C2×C4×C79—41no—
252C2×C4×C79—41no—
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—41no—
262C2×C4×C79—41no—
263C2×C4×C79—41no—
264C2×C4×C79—41no—
265C2×C4×C79—41no—
266C2×C4×C79—41no—
267C2×C4×C79#12112yes—
268C2×C4×C79#12112yes—
269C2×C4×C79C22×D411yescharacteristic
270C2×C4×C79C2×#4111yescharacteristic
271C2×C4×C79C2×#4111yescharacteristic
272C2×C4×C79C2×#4111yescharacteristic
273C2×C4×C79H(5,𝔽2)11yescharacteristic
274C2×C4×C79H(5,𝔽2)11yescharacteristic
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—
285C2×Dic79—41no—
286C2×Dic79—41no—
287Dic158—22no—
288Dic158—22no—
289Dic158—41no—
290Dic158—41no—
291C79⋊C8—22no—
292C79⋊C8—22no—
293C79⋊C8—41no—
294C79⋊C8—41no—
295C22×C79—41no—
296C22×C79—41no—
297C22×C79—41no—
298C22×C79—41no—
299C22×C79—41no—
300C22×C79—41no—
301C22×C79—41no—
302C22×C79C2×#12111yescharacteristic
303C22×C79#52511yescharacteristic
304C22×C79#53511yescharacteristic
305C22×C79#57711yescharacteristic
306C22×C79#57811yescharacteristic
307C22×C79#57811yescharacteristic
308C22×C79#57811yescharacteristic
309C4×C79—22no—
310C4×C79—22no—
311C4×C79—41no—
312C4×C79—41no—
313C4×C79—41no—
314C4×C79—81no—
315C4×C79—81no—
316C4×C79—81no—
317C4×C79—81no—
318C4×C79#54611yescharacteristic
319C4×C79#55011yescharacteristic
320C4×C79#55311yescharacteristic
321C4×C79#55511yescharacteristic
322Dic79—42no—
323Dic79—42no—
324Dic79—81no—
325Dic79—81no—
326#24664—791nomaximal, Sylow
327C2×C79—81no—
328C2×C79#241411yescharacteristic
329C2×C79#305911yescharacteristic
330C2×C79#306311yescharacteristic
331C2×C79#312011yescharacteristic
332C2×C79#312711yescharacteristic
333C2×C79#322311yescharacteristic
334C2×C79#322511yescharacteristic
335C2×#462—791no—
336C2×#433—791no—
337#2621D7911yescharacteristic
338#2323—791no—
339#2036—791no—
340#2035—791no—
341#2001—791no—
342#2001—791no—
343#1981—791no—
344#1971—791no—
345#1969—791no—
346#1949—791no—
347#1935—791no—
348#1933—791no—
349#1805—791no—
350C79#2466411yescharacteristic, complemented, Sylow
351C22×Q16—791no—
352C22×Q16—791no—
353C2×C4⋊Q8C2×D7911yescharacteristic
354C2×C22⋊Q8—791no—
355C2×C22⋊Q8C2×D7911yescharacteristic
356C2×C22⋊Q8C2×D7911yescharacteristic
357C2×C4×Q8—791no—
358C2×#117C2×D7911yescharacteristic
359#462—1584no—
360#433—1584no—
361C2×#107—791no—
362C2×#106—791no—
363C2×C4⋊C8—791no—
364C2×C4⋊C8—791no—
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—
374C2×C22⋊C8—791no—
375C2×#97—791no—
376C2×C4×C8—791no—
377#398—21no—
378#398—21no—
379#397—21no—
380#397—21no—
381#391—791no—
382#391C2×D7911yescharacteristic
383#391C2×D7911yescharacteristic
384#389—791no—
385#389—1581no—
386#388—1581no—
387#388—1581no—
388#387—1581no—
389#382C2×D7911yescharacteristic
390#380—791no—
391C4×C4⋊C4—791no—
392#340—791no—
393#340—791no—
394#340—791no—
395#340—791no—
396C22×Q8—791no—
397C22×Q8C22×D7911yescharacteristic
398C22×Q8C22×D7911yescharacteristic
399C23×C4C22×D7911yescharacteristic
400C2×Q16—3161no—
401C2×Q16—3161no—
402C2×Q16—1582no—
403C2×Q16—1582no—
404C2×Q16—1582no—
405C2×Q16—1582no—
406C22×C8—791no—
407C22×C8—791no—
408C22×C8—791no—
409C22×C8—791no—
410C4⋊Q8—22no—
411C4⋊Q8—22no—
412C22⋊Q8—41no—
413C22⋊Q8—41no—
414C22⋊Q8—41no—
415C22⋊Q8—41no—
416C22⋊Q8—1582no—
417C22⋊Q8—1582no—
418C4×Q8—1582no—
419C4×Q8—1582no—
420#117—22no—
421#117—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—791no—
428C2×C4⋊C4—791no—
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—1581no—
439C2×C22⋊C4—1581no—
440C2×C22⋊C4C22⋊D7911yescharacteristic
441C2×C22⋊C4C22⋊D7911yescharacteristic
442C2×C42—1581no—
443C2×C42—1581no—
444C2×C42C22×D7911yescharacteristic
445C2×C42C22×D7911yescharacteristic
446#107—3161no—
447#106—3161no—
448C4⋊C8—794no—
449C4⋊C8—1582no—
450#103—3161no—
451#103—3161no—
452#103—3161no—
453#103—3161no—
454#103—1582no—
455#103—1582no—
456#103—1582no—
457#103—1582no—
458C22⋊C8—1582no—
459C22⋊C8—1582no—
460#97—1582no—
461C4×C8—1582no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—1581no—
469#95—1581no—
470#95—1581no—
471#95—1581no—
472#95—1581no—
473C24C2×C22⋊D7911yescharacteristic
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—792no—
487C2×Q8—1581no—
488C2×Q8—1581no—
489C2×Q8—1581no—
490C2×Q8—1581no—
491C2×Q8—1581no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
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×C4C23×D7911yescharacteristic
512Q16—3164no—
513Q16—3164no—
514C2×C8—792no—
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—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—3162no—
537C4⋊C4—792no—
538C4⋊C4—6321no—
539C4⋊C4—1584no—
540C4⋊C4—1581no—
541C4⋊C4—1581no—
542C4⋊C4—1581no—
543C4⋊C4—1581no—
544C4⋊C4—1581no—
545C22⋊C4—22no—
546C22⋊C4—22no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—6321no—
550C22⋊C4—1584no—
551C42—21no—
552C42—21no—
553C42—3162no—
554C42—6321no—
555C42D4×D7912yes—
556C42#1192962312yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23C22×C22⋊D7911yescharacteristic
565Q8—41no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—3161no—
574Q8—3161no—
575Q8—1582no—
576Q8—1582no—
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—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
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—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—1582no—
617C2×C4—1582no—
618C2×C4#1457913312yes—
619C2×C4#1457913312yes—
620C2×C4C2×D4×D7911yescharacteristic
621C2×C4#41×D7911yescharacteristic
622C2×C4#41×D7911yescharacteristic
623C2×C4C2×#1192962311yescharacteristic
624C2×C4#1457920611yescharacteristic
625C2×C4#1457920611yescharacteristic
626C8—3161no—
627C8—3161no—
628C8—1582no—
629C8—1582no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1457913311yescharacteristic
638C22#1909907011yescharacteristic
639C22#1909913011yescharacteristic
640C22#1909937111yescharacteristic
641C22#1909937811yescharacteristic
642C22#1909937811yescharacteristic
643C22#1909937811yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—3162no—
654C4—3162no—
655C4—6321no—
656C4—6321no—
657C4#1909921011yescharacteristic
658C4#1909923111yescharacteristic
659C4#1909926011yescharacteristic
660C4#1909927711yescharacteristic
661C2—81no—
662C2#2376681911yescharacteristic
663C2#2377305211yescharacteristic
664C2#2377308711yescharacteristic
665C2#2377360811yescharacteristic
666C2#2377369511yescharacteristic
667C2#2377448311yescharacteristic
668C2#2377450911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2621C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2621D79
hypocenterC79#24664
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
1. C2×#19098770C2semidirect
8. #23764324C2semidirect
9. #23764270C2semidirect
10. #23764260C2semidirect
11. #23764166C2semidirect
12. #23764093C2semidirect
13. #23764079C2semidirect
14. #23763412C2semidirect
350. C79#24664semidirect

Maximal subgroups

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

subgroupindex
C2×#190987702
C2×#190986242
#237660822
#237646002
#237645932
#237644082
#237643242
#237642702
#237642602
#237641662
#237640932
#237640792
#237634122
C79×#26212
#2466479