Finite Group Databasebeta

← #34318750

Subgroups — #34318750

Order 20224.

704 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23766044C211yesmaximal, characteristic
2#23764962C211yesmaximal, characteristic, complemented
3#23764929C211yesmaximal, characteristic
4#23764896C211yesmaximal, characteristic, complemented
5#23764891C211yesmaximal, characteristic, complemented
6#23764785C211yesmaximal, characteristic
7#23764771C211yesmaximal, characteristic
8#23764735C211yesmaximal, characteristic, complemented
9#23764543C211yesmaximal, characteristic
10#23763298C211yesmaximal, characteristic, complemented
11C4×#14579094C211yesmaximal, characteristic, complemented
12#23763257C211yesmaximal, characteristic, complemented
13#23763228C211yesmaximal, characteristic
14C4×#14579081C211yesmaximal, characteristic
15C79×#2330C211yesmaximal, characteristic, complemented
16C2×C4×D316C2211yescharacteristic
17C2×#14579094C2211yescharacteristic
18C2×#14579094C2211yescharacteristic
19C2×#14579094C2211yescharacteristic
20C2×#14579092C2211yescharacteristic
21C2×C4⋊(C79⋊C8)C2211yescharacteristic
22C2×C4⋊(C79⋊C8)C2211yescharacteristic
23C2×#14579081C2211yescharacteristic
24C2×#14579081C2211yescharacteristic
25C2×#14579081C2211yescharacteristic
26C2×#14579081C2211yescharacteristic
27C2×#14579072C2211yescharacteristic
28C2×C4×C79⋊C8C2211yescharacteristic
29#19098465C2211yescharacteristic
30#19098465C2211yescharacteristic
31#19098461C2211yescharacteristic
32#19098461C2211yescharacteristic
33#19098452—21no—
34#19098452—21no—
35#19098425—21no—
36#19098422—21no—
37C4×C4⋊Dic79C2211yescharacteristic
38#19098325C2211yescharacteristic
39#19098325C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48C2×#125×C79C2211yescharacteristic
49#389×C79—21no—
50#389×C79—21no—
51#389×C79C2211yescharacteristic
52#389×C79C2211yescharacteristic
53#387×C79—21no—
54#387×C79—21no—
55#384×C79C2211yescharacteristic
56C4×C4⋊C4×C79C2211yescharacteristic
57C4×C4⋊C4×C79C2211yescharacteristic
58#376×C79C2211yescharacteristic
59C22×D316C2311yescharacteristic
60C22×C4×D79—21no—
61C22×C79⋊C8C2311yescharacteristic
62C22×C79⋊C8C2311yescharacteristic
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C4×D316—22no—
66C4×D316Q814yes—
67C2×C4⋊Dic79C2311yescharacteristic
68C2×C4⋊Dic79C2311yescharacteristic
69C2×C4⋊Dic79C2311yescharacteristic
70C2×Dic79⋊C4—21no—
71C2×Dic79⋊C4—21no—
72C2×(C2×D79)⋊C4—21no—
73C2×(C2×D79)⋊C4—21no—
74C2×(C2×D79)⋊C4—21no—
75C2×(C2×D79)⋊C4—21no—
76C2×(C2×D79)⋊C4—21no—
77C2×C4×Dic79—21no—
78#14579094—22no—
79#14579094—22no—
80#14579094—22no—
81#14579092—22no—
82C4⋊(C79⋊C8)—22no—
83C4⋊(C79⋊C8)—22no—
84#14579081—22no—
85#14579081—22no—
86#14579081—22no—
87#14579081—22no—
88#14579072—22no—
89C4×C79⋊C8—22no—
90C79⋊#95—21no—
91C79⋊#95—21no—
92C79⋊#95—21no—
93#125×C79—22no—
94#125×C79—22no—
95C2×C4⋊C4×C79—21no—
96C2×C4⋊C4×C79—21no—
97C2×C4⋊C4×C79—21no—
98C2×C4⋊C4×C79—21no—
99C2×C4⋊C4×C79—21no—
100C2×C4⋊C4×C79—21no—
101C2×C4⋊C4×C79C2311yescharacteristic
102C2×C4⋊C4×C79C2311yescharacteristic
103C2×C4⋊C4×C79C2311yescharacteristic
104C2×C4⋊C4×C79C2311yescharacteristic
105C2×C42×C79—21no—
106C2×C42×C79—21no—
107C2×C42×C79C2311yescharacteristic
108C2×C42×C79C2311yescharacteristic
109C2×C42×C79C2311yescharacteristic
110#95×C79—21no—
111#95×C79—21no—
112#95×C79—21no—
113#95×C79—21no—
114#95×C79D411yescharacteristic
115#95×C79D411yescharacteristic
116#95×C79D411yescharacteristic
117#95×C79D411yescharacteristic
118C23×D79—21no—
119C2×D316—21no—
120C2×D316—21no—
121C2×D316—21no—
122C2×D316—21no—
123C2×D316—21no—
124C2×D316C2×Q812yes—
125C22×Dic79—21no—
126C22×Dic79—21no—
127C22×Dic79—21no—
128C2×C4×D79—24no—
129C2×C4×D79—81no—
130C2×C4×D79—81no—
131C2×C79⋊C8—21no—
132C2×C79⋊C8—21no—
133C2×C79⋊C8—21no—
134C2×C79⋊C8—21no—
135C2×C79⋊C8—21no—
136C2×C79⋊C8—21no—
137C2×C79⋊C8—21no—
138C2×C79⋊C8—21no—
139C2×C79⋊C8—21no—
140C2×C79⋊C8—21no—
141C2×C79⋊C8#4112yes—
142C2×C79⋊C8#4112yes—
143C4⋊Dic79—21no—
144C4⋊Dic79—21no—
145C4⋊Dic79—21no—
146C4⋊Dic79—21no—
147C4⋊Dic79—21no—
148C4⋊Dic79C2×Q812yes—
149Dic79⋊C4—81no—
150Dic79⋊C4—81no—
151(C2×D79)⋊C4—24no—
152(C2×D79)⋊C4—81no—
153(C2×D79)⋊C4—81no—
154(C2×D79)⋊C4—81no—
155(C2×D79)⋊C4—81no—
156C4×Dic79—81no—
157C22×C4×C79—21no—
158C22×C4×C79—21no—
159C22×C4×C79—21no—
160C22×C4×C79—21no—
161C22×C4×C79C2×D411yescharacteristic
162C22×C4×C79C2×D411yescharacteristic
163C22×C4×C79C2×D411yescharacteristic
164C22×C4×C79C2×D411yescharacteristic
165C22×C4×C79C2×D411yescharacteristic
166C22×C4×C79C2×D411yescharacteristic
167C22×C4×C79C2411yescharacteristic
168C4⋊C4×C79—24no—
169C4⋊C4×C79—24no—
170C4⋊C4×C79—21no—
171C4⋊C4×C79—21no—
172C4⋊C4×C79—21no—
173C4⋊C4×C79—21no—
174C4⋊C4×C79—21no—
175C4⋊C4×C79—21no—
176C4⋊C4×C79—81no—
177C4⋊C4×C79—81no—
178C4⋊C4×C79—81no—
179C4⋊C4×C79—81no—
180C4⋊C4×C79#4112yes—
181C4⋊C4×C79#4112yes—
182C42×C79—21no—
183C42×C79—21no—
184C42×C79—21no—
185C42×C79—21no—
186C42×C79—81no—
187C42×C79—81no—
188C42×C79C2×Q812yes—
189C42×C79#4112yes—
190C23×C79C22×D411yescharacteristic
191C22×D79—22no—
192C22×D79—41no—
193C22×D79—41no—
194C22×D79—41no—
195C22×D79—41no—
196C22×D79—41no—
197C22×D79—41no—
198C2×C4×C79—21no—
199C2×C4×C79—21no—
200C2×C4×C79—21no—
201C2×C4×C79—21no—
202C2×C4×C79—21no—
203C2×C4×C79—21no—
204C2×C4×C79—22no—
205C2×C4×C79—22no—
206C2×C4×C79—21no—
207C2×C4×C79—21no—
208C2×C4×C79—21no—
209C2×C4×C79—21no—
210C2×C4×C79—21no—
211C2×C4×C79—21no—
212C2×C4×C79—21no—
213C2×C4×C79—21no—
214C2×C4×C79—21no—
215C2×C4×C79—21no—
216C2×C4×C79—41no—
217C2×C4×C79—41no—
218C2×C4×C79—41no—
219C2×C4×C79—41no—
220C2×C4×C79—41no—
221C2×C4×C79—41no—
222C2×C4×C79—41no—
223C2×C4×C79—41no—
224C2×C4×C79—41no—
225C2×C4×C79—41no—
226C2×C4×C79C22⋊Q812yes—
227C2×C4×C79C22⋊Q812yes—
228C2×C4×C79C22×Q811yescharacteristic
229C2×C4×C79C2×#4111yescharacteristic
230C2×C4×C79C2×#4111yescharacteristic
231C2×C4×C79C2×#4111yescharacteristic
232C2×C4×C79C2×#4111yescharacteristic
233C2×C4×C79C2×#4111yescharacteristic
234C2×Dic79—22no—
235C2×Dic79—41no—
236C2×Dic79—41no—
237C2×Dic79—41no—
238C2×Dic79—41no—
239C2×Dic79—41no—
240C2×Dic79—41no—
241C2×Dic79—41no—
242C2×Dic79—41no—
243D316—22no—
244D316—22no—
245D316—41no—
246D316—41no—
247C4×D79—84no—
248C79⋊C8—22no—
249C79⋊C8—22no—
250C79⋊C8—41no—
251C79⋊C8—41no—
252C22×C79C2×C22⋊Q811yescharacteristic
253C22×C79#52511yescharacteristic
254C22×C79#53211yescharacteristic
255C22×C79#57611yescharacteristic
256C22×C79#57611yescharacteristic
257C22×C79#57711yescharacteristic
258C22×C79#57811yescharacteristic
259C2×D79—42no—
260C2×D79—42no—
261C2×D79—42no—
262C2×D79—81no—
263C2×D79—81no—
264C2×D79—81no—
265C2×D79—81no—
266C4×C79—22no—
267C4×C79—22no—
268C4×C79—41no—
269C4×C79—42no—
270C4×C79—41no—
271C4×C79—42no—
272C4×C79—41no—
273C4×C79—41no—
274C4×C79—81no—
275C4×C79—81no—
276C4×C79#52911yescharacteristic
277C4×C79#52911yescharacteristic
278C4×C79#53311yescharacteristic
279C4×C79#53311yescharacteristic
280Dic79—42no—
281Dic79—81no—
282Dic79—81no—
283#23791—791nomaximal, Sylow
284C2×C79#239611yescharacteristic
285C2×C79#308111yescharacteristic
286C2×C79#308311yescharacteristic
287C2×C79#311911yescharacteristic
288C2×C79#312411yescharacteristic
289C2×C79#317611yescharacteristic
290C2×C79#317811yescharacteristic
291D79—82no—
292#2330D7911yescharacteristic, complemented
293#2319—791no—
294#2100—791no—
295#2093—791no—
296#2087—791no—
297#2086—791no—
298#2067—791no—
299#2065—791no—
300#2059—791no—
301#2025—791no—
302#1776—791no—
303C4×#107—791no—
304#1768—791no—
305#1762—791no—
306C4×#102—791no—
307C79#2379111yescharacteristic, complemented, Sylow
308C2×#125C2×D7911yescharacteristic
309C2×C4×D4—791no—
310C2×#107—791no—
311C2×#107—791no—
312C2×#107—791no—
313C2×#106—791no—
314C2×C4⋊C8—791no—
315C2×C4⋊C8—791no—
316C2×#102—791no—
317C2×#102—791no—
318C2×#102—791no—
319C2×#102—791no—
320C2×#97—791no—
321C2×C4×C8—791no—
322#390—791no—
323#390—791no—
324#389—21no—
325#389—21no—
326#389—791no—
327#389—791no—
328#389C2×D7911yescharacteristic
329#389C2×D7911yescharacteristic
330#388—1581no—
331#388—1581no—
332#387—21no—
333#387—21no—
334#384C2×D7911yescharacteristic
335#381—1581no—
336#380—1581no—
337C4×C4⋊C4—791no—
338C4×C4⋊C4C2×D7911yescharacteristic
339C4×C4⋊C4C2×D7911yescharacteristic
340#376C2×D7911yescharacteristic
341#340—791no—
342#340—791no—
343#340—791no—
344#340—791no—
345#340—791no—
346#340—791no—
347#340—791no—
348#340—791no—
349#336—791no—
350#336—791no—
351C22×D4—791no—
352C23×C4—1581no—
353C22×C8—791no—
354C22×C8—791no—
355C22×C8—791no—
356C22×C8—791no—
357#125—22no—
358#125—22no—
359C4×D4—794no—
360C4×D4—1582no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—791no—
368C2×C4⋊C4—791no—
369C2×C4⋊C4—791no—
370C2×C4⋊C4—1581no—
371C2×C4⋊C4—1581no—
372C2×C4⋊C4C22×D7911yescharacteristic
373C2×C4⋊C4C22×D7911yescharacteristic
374C2×C4⋊C4C22×D7911yescharacteristic
375C2×C4⋊C4C22×D7911yescharacteristic
376C2×C22⋊C4—1581no—
377C2×C22⋊C4—1581no—
378C2×C22⋊C4—1581no—
379C2×C22⋊C4—1581no—
380C2×C22⋊C4—1581no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—1581no—
384C2×C42C22×D7911yescharacteristic
385C2×C42C22×D7911yescharacteristic
386C2×C42C22×D7911yescharacteristic
387#107—1582no—
388#107—1582no—
389#107—1582no—
390#106—1582no—
391C4⋊C8—1582no—
392C4⋊C8—1582no—
393#102—1582no—
394#102—1582no—
395#102—1582no—
396#102—1582no—
397#97—1582no—
398C4×C8—1582no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1581no—
404#95—1581no—
405#95—1581no—
406#95C22⋊D7911yescharacteristic
407#95C22⋊D7911yescharacteristic
408#95C22⋊D7911yescharacteristic
409#95C22⋊D7911yescharacteristic
410C24—1581no—
411C2×D4—792no—
412C2×D4—1581no—
413C2×D4—1581no—
414C2×D4—1581no—
415C2×D4—1581no—
416C2×D4—1581no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—6321no—
422C22×C4—6321no—
423C22×C4—1581no—
424C22×C4—1581no—
425C22×C4—1581no—
426C22×C4—1584no—
427C22×C4C2×C22⋊D7911yescharacteristic
428C22×C4C2×C22⋊D7911yescharacteristic
429C22×C4C2×C22⋊D7911yescharacteristic
430C22×C4C2×C22⋊D7911yescharacteristic
431C22×C4C2×C22⋊D7911yescharacteristic
432C22×C4C2×C22⋊D7911yescharacteristic
433C22×C4C23×D7911yescharacteristic
434C2×C8—792no—
435C2×C8—792no—
436C2×C8—1581no—
437C2×C8—1581no—
438C2×C8—1581no—
439C2×C8—1581no—
440C2×C8—1581no—
441C2×C8—1581no—
442C2×C8—1581no—
443C2×C8—1581no—
444C2×C8—1581no—
445C2×C8—1581no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—792no—
459C4⋊C4—6321no—
460C4⋊C4—6321no—
461C4⋊C4—1581no—
462C4⋊C4—1581no—
463C4⋊C4—1581no—
464C4⋊C4—1581no—
465C4⋊C4—1581no—
466C4⋊C4#1192962412yes—
467C4⋊C4#1192962412yes—
468C22⋊C4—6321no—
469C22⋊C4—6321no—
470C22⋊C4—6321no—
471C22⋊C4—6321no—
472C22⋊C4—1584no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—81no—
478C42—81no—
479C42—6321no—
480C42Q8×D7912yes—
481C42#1192962512yes—
482C23—3161no—
483C23—3161no—
484C23—3161no—
485C23—3161no—
486C23—3161no—
487C23—3161no—
488C23—1582no—
489C23C22×C22⋊D7911yescharacteristic
490D4—3161no—
491D4—3161no—
492D4—1582no—
493D4—1582no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—3161no—
523C2×C4—3161no—
524C2×C4—3161no—
525C2×C4—3161no—
526C2×C4—3161no—
527C2×C4—3161no—
528C2×C4—3161no—
529C2×C4—3161no—
530C2×C4—6324no—
531C2×C4—1582no—
532C2×C4#1457913712yes—
533C2×C4#1457913712yes—
534C2×C4C2×Q8×D7911yescharacteristic
535C2×C4#41×D7911yescharacteristic
536C2×C4#41×D7911yescharacteristic
537C2×C4C2×#1192962411yescharacteristic
538C2×C4C2×#1192962411yescharacteristic
539C2×C4C2×#1192962511yescharacteristic
540C8—3161no—
541C8—3161no—
542C8—1582no—
543C8—1582no—
544C22—3162no—
545C22—3162no—
546C22—3162no—
547C22—6321no—
548C22—6321no—
549C22—6321no—
550C22—6321no—
551C22C2×#1457913711yescharacteristic
552C22#1909907411yescharacteristic
553C22#1909910911yescharacteristic
554C22#1909936611yescharacteristic
555C22#1909936611yescharacteristic
556C22#1909937111yescharacteristic
557C22#1909937811yescharacteristic
558C4—22no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—41no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—3162no—
569C4—6321no—
570C4—6321no—
571C4#1909910011yescharacteristic
572C4#1909910011yescharacteristic
573C4#1909911711yescharacteristic
574C4#1909911711yescharacteristic
575C2—6322no—
576C2#2376667011yescharacteristic
577C2#2377322511yescharacteristic
578C2#2377324711yescharacteristic
579C2#2377359911yescharacteristic
580C2#2377366211yescharacteristic
581C2#2377404411yescharacteristic
582C2#2377407011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2330C2
socleC23×C79C22×D4
prime radicalC22×D316C23
hypercenter#2330D79
hypocenterC79#23791
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ #376 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. #23764962C2semidirect
4. #23764896C2semidirect
5. #23764891C2semidirect
8. #23764735C2semidirect
10. #23763298C2semidirect
11. C4×#14579094C2semidirect
12. #23763257C2semidirect
15. C79×#2330C2semidirect
292. #2330D79semidirect
307. C79#23791semidirect

Maximal subgroups

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

subgroupindex
#237660442
#237649622
#237649292
#237648962
#237648912
#237647852
#237647712
#237647352
#237645432
#237632982
C4×#145790942
#237632572
#237632282
C4×#145790812
C79×#23302
#2379179