Finite Group Databasebeta

← #34327084

Subgroups — #34327084

Order 20224.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098719C211yesmaximal, characteristic
2C2×#19098624C211yesmaximal, characteristic
3#23769197C211yesmaximal, characteristic
4#23764795C211yesmaximal, characteristic, complemented
5#23764781C211yesmaximal, characteristic, complemented
6#23764722C211yesmaximal, characteristic, complemented
7#23764722C211yesmaximal, characteristic, complemented
8#23764270C211yesmaximal, characteristic, complemented
9#23764260C211yesmaximal, characteristic, complemented
10#23764093C211yesmaximal, characteristic, complemented
11#23764079C211yesmaximal, characteristic, complemented
12#23763712C211yesmaximal, characteristic
13#23763599C211yesmaximal, characteristic
14#23763335C211yesmaximal, characteristic
15C79×#2310C211yesmaximal, characteristic
16C22×#11929614C2211yescharacteristic
17C22×#11929614C2211yescharacteristic
18C2×#14579140C2211yescharacteristic
19#19098719—24no—
20#19098624—24no—
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×C4⋊(C79⋊C8)C2211yescharacteristic
24C2×C4⋊(C79⋊C8)C2211yescharacteristic
25C2×#14579085C2211yescharacteristic
26C2×#14579085C2211yescharacteristic
27C2×#14579085C2211yescharacteristic
28C2×#14579085C2211yescharacteristic
29C2×#14579085C2211yescharacteristic
30C2×#14579085C2211yescharacteristic
31C2×#14579084C2211yescharacteristic
32C2×#14579084C2211yescharacteristic
33C2×C22⋊(C79⋊C8)C2211yescharacteristic
34C2×C22⋊(C79⋊C8)C2211yescharacteristic
35C2×#14579072C2211yescharacteristic
36C2×C4×C79⋊C8C2211yescharacteristic
37#19098501—21no—
38#19098494—21no—
39#19098491—21no—
40#19098491—21no—
41#19098469C2211yescharacteristic
42#19098469C2211yescharacteristic
43#19098461C2211yescharacteristic
44#19098461C2211yescharacteristic
45#19098423C2211yescharacteristic
46#19098325C2211yescharacteristic
47#19098325C2211yescharacteristic
48#19098325C2211yescharacteristic
49#19098325C2211yescharacteristic
50C23×Q8×C79C2211yescharacteristic
51C2×#117×C79C2211yescharacteristic
52C2×#117×C79C2211yescharacteristic
53#391×C79—21no—
54#391×C79—21no—
55#391×C79C2211yescharacteristic
56#391×C79C2211yescharacteristic
57#391×C79C2211yescharacteristic
58#391×C79C2211yescharacteristic
59#381×C79—21no—
60#381×C79—21no—
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—
74C2×C4⋊Dic79C2311yescharacteristic
75C2×C4⋊Dic79C2311yescharacteristic
76C2×C4⋊Dic79C2311yescharacteristic
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×Dic79⋊C4—21no—
80C2×C22⋊Dic79—21no—
81C2×C22⋊Dic79—21no—
82#14579094—41no—
83#14579092—41no—
84C4⋊(C79⋊C8)—22no—
85C4⋊(C79⋊C8)—22no—
86#14579085—22no—
87#14579085—22no—
88#14579085—22no—
89#14579085—22no—
90#14579085—41no—
91#14579085—41no—
92#14579084—41no—
93#14579084—41no—
94C22⋊(C79⋊C8)—22no—
95C22⋊(C79⋊C8)D414yes—
96#14579072—22no—
97C4×C79⋊C8—22no—
98C79⋊#95—21no—
99C79⋊#95—21no—
100C79⋊#95—21no—
101C79⋊#95—21no—
102C79⋊#95—21no—
103C79⋊#95—21no—
104C79⋊#95—21no—
105C22×Q8×C79—22no—
106C22×Q8×C79—22no—
107C22×Q8×C79—21no—
108C22×Q8×C79—22no—
109C22×Q8×C79—22no—
110C22×Q8×C79—41no—
111C22×Q8×C79—41no—
112C22×Q8×C79C2311yescharacteristic
113C22×Q8×C79C2311yescharacteristic
114C23×C4×C79—21no—
115C23×C4×C79C2311yescharacteristic
116#117×C79—22no—
117#117×C79—22no—
118#117×C79—41no—
119#117×C79—41no—
120C2×C4⋊C4×C79D411yescharacteristic
121C2×C4⋊C4×C79D411yescharacteristic
122C2×C4⋊C4×C79C2311yescharacteristic
123C2×C4⋊C4×C79C2311yescharacteristic
124C2×C22⋊C4×C79—21no—
125C2×C22⋊C4×C79D411yescharacteristic
126C2×C22⋊C4×C79D411yescharacteristic
127C2×C42×C79—21no—
128C2×C42×C79C2311yescharacteristic
129C2×C42×C79C2311yescharacteristic
130#95×C79—21no—
131#95×C79—21no—
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—21no—
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—21no—
168Dic79⋊C4—42no—
169Dic79⋊C4—81no—
170Dic79⋊C4—81no—
171C22⋊Dic79—42no—
172C22⋊Dic79—81no—
173C24×C79C2×D411yescharacteristic
174C2×Q8×C79—24no—
175C2×Q8×C79—21no—
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—44no—
186C2×Q8×C79—44no—
187C2×Q8×C79—44no—
188C2×Q8×C79—44no—
189C2×Q8×C79—42no—
190C2×Q8×C79—42no—
191C2×Q8×C79#4112yes—
192C2×Q8×C79#4112yes—
193C22×C4×C79—21no—
194C22×C4×C79—21no—
195C22×C4×C79—21no—
196C22×C4×C79—21no—
197C22×C4×C79—24no—
198C22×C4×C79—21no—
199C22×C4×C79—21no—
200C22×C4×C79—21no—
201C22×C4×C79—21no—
202C22×C4×C79—42no—
203C22×C4×C79—42no—
204C22×C4×C79C2×D411yescharacteristic
205C22×C4×C79C2×D411yescharacteristic
206C22×C4×C79C2×D411yescharacteristic
207C22×C4×C79C2×D411yescharacteristic
208C22×C4×C79C2×D411yescharacteristic
209C22×C4×C79C2×D412yes—
210C22×C4×C79#4112yes—
211C22×C4×C79C2411yescharacteristic
212C4⋊C4×C79—22no—
213C4⋊C4×C79—22no—
214C4⋊C4×C79—21no—
215C4⋊C4×C79—21no—
216C4⋊C4×C79—21no—
217C4⋊C4×C79—21no—
218C22⋊C4×C79—22no—
219C22⋊C4×C79—22no—
220C22⋊C4×C79—42no—
221C42×C79—21no—
222C42×C79—21no—
223C42×C79—21no—
224C42×C79—21no—
225C42×C79—42no—
226C23×C79—21no—
227C23×C79—21no—
228C23×C79—21no—
229C23×C79—21no—
230C23×C79—21no—
231C23×C79—21no—
232C23×C79#12112yes—
233C23×C79C22×D411yescharacteristic
234Q8×C79—22no—
235Q8×C79—22no—
236Q8×C79—22no—
237Q8×C79—22no—
238Q8×C79—48no—
239Q8×C79—41no—
240Q8×C79—41no—
241Q8×C79—41no—
242Q8×C79—41no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—22no—
246C2×C4×C79—22no—
247C2×C4×C79—22no—
248C2×C4×C79—22no—
249C2×C4×C79—21no—
250C2×C4×C79—21no—
251C2×C4×C79—21no—
252C2×C4×C79—21no—
253C2×C4×C79—22no—
254C2×C4×C79—22no—
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—21no—
263C2×C4×C79—48no—
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×C79—41no—
272C2×C4×C79—41no—
273C2×C4×C79—41no—
274C2×C4×C79#12112yes—
275C2×C4×C79C22×D411yescharacteristic
276C2×C4×C79C2×#4111yescharacteristic
277C2×C4×C79C2×#4111yescharacteristic
278C2×C4×C79C2×#4111yescharacteristic
279C2×C4×C79H(5,𝔽2)11yescharacteristic
280C2×C4×C79H(5,𝔽2)11yescharacteristic
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—
291C2×Dic79—41no—
292C2×Dic79—41no—
293Dic158—41no—
294Dic158—41no—
295Dic158—41no—
296Dic158—41no—
297C79⋊C8—22no—
298C79⋊C8—22no—
299C79⋊C8—41no—
300C79⋊C8—41no—
301C22×C79—22no—
302C22×C79—22no—
303C22×C79—22no—
304C22×C79—41no—
305C22×C79—41no—
306C22×C79—41no—
307C22×C79—41no—
308C22×C79C2×#12111yescharacteristic
309C22×C79#52511yescharacteristic
310C22×C79#53511yescharacteristic
311C22×C79#57711yescharacteristic
312C22×C79#57811yescharacteristic
313C22×C79#57811yescharacteristic
314C22×C79#57811yescharacteristic
315C4×C79—22no—
316C4×C79—42no—
317C4×C79—42no—
318C4×C79—41no—
319C4×C79—41no—
320C4×C79—41no—
321C4×C79—41no—
322C4×C79—81no—
323C4×C79—81no—
324C4×C79#54611yescharacteristic
325C4×C79#54611yescharacteristic
326C4×C79#55111yescharacteristic
327C4×C79#55111yescharacteristic
328Dic79—81no—
329Dic79—81no—
330Dic79—81no—
331Dic79—81no—
332#24514—791nomaximal, Sylow
333C2×C79—42no—
334C2×C79#241111yescharacteristic
335C2×C79#305911yescharacteristic
336C2×C79#306311yescharacteristic
337C2×C79#312111yescharacteristic
338C2×C79#312711yescharacteristic
339C2×C79#322311yescharacteristic
340C2×C79#322511yescharacteristic
341C2×#451—791no—
342C2×#433—791no—
343#2625—791no—
344#2310D7911yescharacteristic
345#2068—791no—
346#2066—791no—
347#2056—791no—
348#2056—791no—
349#1971—791no—
350#1969—791no—
351#1935—791no—
352#1933—791no—
353#1873—791no—
354#1854—791no—
355#1787—791no—
356C79#2451411yescharacteristic, complemented, Sylow
357C23×Q8C2×D7911yescharacteristic
358C22×Q16—791no—
359C22×Q16—791no—
360C2×C22⋊Q8—791no—
361C2×#117C2×D7911yescharacteristic
362C2×#117C2×D7911yescharacteristic
363#451—1584no—
364#433—1584no—
365C2×#107—791no—
366C2×#106—791no—
367C2×C4⋊C8—791no—
368C2×C4⋊C8—791no—
369C2×#103—791no—
370C2×#103—791no—
371C2×#103—791no—
372C2×#103—791no—
373C2×#103—791no—
374C2×#103—791no—
375C2×#103—791no—
376C2×#103—791no—
377C2×C22⋊C8—791no—
378C2×C22⋊C8—791no—
379C2×#97—791no—
380C2×C4×C8—791no—
381#400—1581no—
382#398—1581no—
383#397—1581no—
384#397—1581no—
385#391—21no—
386#391—21no—
387#391—791no—
388#391—791no—
389#391C2×D7911yescharacteristic
390#391C2×D7911yescharacteristic
391#391C2×D7911yescharacteristic
392#391C2×D7911yescharacteristic
393#389—791no—
394#389—791no—
395#381—21no—
396#381—21no—
397#380—791no—
398#340—791no—
399#340—791no—
400#340—791no—
401#340—791no—
402C22×Q8—22no—
403C22×Q8—22no—
404C22×Q8—21no—
405C22×Q8—22no—
406C22×Q8—22no—
407C22×Q8—41no—
408C22×Q8—41no—
409C22×Q8—791no—
410C22×Q8C22×D7911yescharacteristic
411C22×Q8C22×D7911yescharacteristic
412C23×C4—21no—
413C23×C4C22×D7911yescharacteristic
414C2×Q16—3161no—
415C2×Q16—3161no—
416C2×Q16—1582no—
417C2×Q16—1582no—
418C2×Q16—1582no—
419C2×Q16—1582no—
420C22×C8—791no—
421C22×C8—791no—
422C22×C8—791no—
423C22×C8—791no—
424C22⋊Q8—1582no—
425C22⋊Q8—1582no—
426#117—22no—
427#117—22no—
428#117—41no—
429#117—41no—
430C2×C4⋊C4—791no—
431C2×C4⋊C4—791no—
432C2×C4⋊C4—791no—
433C2×C4⋊C4—1581no—
434C2×C4⋊C4—1581no—
435C2×C4⋊C4—1581no—
436C2×C4⋊C4C22⋊D7911yescharacteristic
437C2×C4⋊C4C22⋊D7911yescharacteristic
438C2×C4⋊C4C22×D7911yescharacteristic
439C2×C4⋊C4C22×D7911yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—1581no—
442C2×C22⋊C4—1581no—
443C2×C22⋊C4C22⋊D7911yescharacteristic
444C2×C22⋊C4C22⋊D7911yescharacteristic
445C2×C42—21no—
446C2×C42C22×D7911yescharacteristic
447C2×C42C22×D7911yescharacteristic
448#107—3161no—
449#106—3161no—
450C4⋊C8—1582no—
451C4⋊C8—1582no—
452#103—3161no—
453#103—3161no—
454#103—3161no—
455#103—3161no—
456#103—1582no—
457#103—1582no—
458#103—1582no—
459#103—1582no—
460C22⋊C8—794no—
461C22⋊C8—1582no—
462#97—1582no—
463C4×C8—1582no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—1581no—
473#95—1581no—
474#95—1581no—
475#95—1581no—
476#95—1581no—
477#95—1581no—
478#95—1581no—
479C24C2×C22⋊D7911yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—1581no—
498C2×Q8—1581no—
499C2×Q8—1581no—
500C2×Q8—1581no—
501C2×Q8—1581no—
502C2×Q8—1581no—
503C2×Q8#1192962412yes—
504C2×Q8#1192962412yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—24no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—1581no—
517C22×C4—1581no—
518C22×C4—1581no—
519C22×C4—1581no—
520C22×C4D4×D7912yes—
521C22×C4C2×C22⋊D7911yescharacteristic
522C22×C4C2×C22⋊D7911yescharacteristic
523C22×C4C2×C22⋊D7911yescharacteristic
524C22×C4C2×C22⋊D7911yescharacteristic
525C22×C4C2×C22⋊D7911yescharacteristic
526C22×C4#1192962312yes—
527C22×C4C23×D7911yescharacteristic
528Q16—3164no—
529Q16—3164no—
530C2×C8—792no—
531C2×C8—792no—
532C2×C8—1581no—
533C2×C8—1581no—
534C2×C8—1581no—
535C2×C8—1581no—
536C2×C8—1581no—
537C2×C8—1581no—
538C2×C8—1581no—
539C2×C8—1581no—
540C2×C8—1581no—
541C2×C8—1581no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—3162no—
549C4⋊C4—6321no—
550C4⋊C4—6321no—
551C4⋊C4—1581no—
552C4⋊C4—1581no—
553C4⋊C4—1581no—
554C4⋊C4—1581no—
555C4⋊C4—1581no—
556C4⋊C4—1581no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—42no—
560C22⋊C4—3162no—
561C22⋊C4—6321no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1457913312yes—
574C23C22×C22⋊D7911yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—22no—
578Q8—22no—
579Q8—48no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—3161no—
585Q8—3161no—
586Q8—3161no—
587Q8—3161no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—48no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—3161no—
620C2×C4—3161no—
621C2×C4—3161no—
622C2×C4—3161no—
623C2×C4—3161no—
624C2×C4—3161no—
625C2×C4—3161no—
626C2×C4—3161no—
627C2×C4—3161no—
628C2×C4—3161no—
629C2×C4—3161no—
630C2×C4—3161no—
631C2×C4#1457913312yes—
632C2×C4C2×D4×D7911yescharacteristic
633C2×C4C2×#1192962311yescharacteristic
634C2×C4C2×#1192962411yescharacteristic
635C2×C4C2×#1192962411yescharacteristic
636C2×C4#1457920611yescharacteristic
637C2×C4#1457920611yescharacteristic
638C8—3161no—
639C8—3161no—
640C8—1582no—
641C8—1582no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1457913311yescharacteristic
650C22#1909907411yescharacteristic
651C22#1909913011yescharacteristic
652C22#1909937111yescharacteristic
653C22#1909937811yescharacteristic
654C22#1909937811yescharacteristic
655C22#1909937811yescharacteristic
656C4—22no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—6321no—
666C4—6321no—
667C4—6321no—
668C4—6321no—
669C4#1909920311yescharacteristic
670C4#1909920311yescharacteristic
671C4#1909923911yescharacteristic
672C4#1909923911yescharacteristic
673C2—42no—
674C2#2376678511yescharacteristic
675C2#2377305211yescharacteristic
676C2#2377308711yescharacteristic
677C2#2377362911yescharacteristic
678C2#2377370011yescharacteristic
679C2#2377448311yescharacteristic
680C2#2377450911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#2310C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
hypercenter#2310D79
hypocenterC79#24514
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
4. #23764795C2semidirect
5. #23764781C2semidirect
6. #23764722C2semidirect
7. #23764722C2semidirect
8. #23764270C2semidirect
9. #23764260C2semidirect
10. #23764093C2semidirect
11. #23764079C2semidirect
356. C79#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#190987192
C2×#190986242
#237691972
#237647952
#237647812
#237647222
#237642702
#237642602
#237640932
#237640792
#237637122
#237635992
#237633352
C79×#23102
#2451479