Finite Group Databasebeta

← #34209656

Subgroups — #34209656

Order 20224.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579119C211yesmaximal, characteristic
2C22×#14579118C211yesmaximal, characteristic
3C22×C4⋊C4×D79C211yesmaximal, characteristic
4#23761630C211yesmaximal, characteristic, complemented
5#23761629C211yesmaximal, characteristic, complemented
6#23761629C211yesmaximal, characteristic, complemented
7Dic79⋊#95C214yesmaximal, complemented
8(C2×D79)⋊#95C214yesmaximal
9C79×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D79C2212yes—
11C23×C4×D79C2211yescharacteristic
12C2×#14579119—24no—
13C2×#14579119—24no—
14C2×#14579119C418yescomplemented
15C2×#14579118—24no—
16C2×#14579118—24no—
17C2×#14579118C418yescomplemented
18C22×C4⋊Dic79C2211yescharacteristic
19C22×C4⋊Dic79C2211yescharacteristic
20C22×Dic79⋊C4C2212yes—
21C22×Dic79⋊C4C2212yes—
22C2×C4⋊C4×D79—24no—
23C2×C4⋊C4×D79—24no—
24C2×C4⋊C4×D79C418yescomplemented
25C22×(C2×D79)⋊C4C2212yes—
26C22×(C2×D79)⋊C4C2212yes—
27C22×C4×Dic79C2211yescharacteristic
28C22×C4×Dic79C2212yes—
29C2×C79⋊#95C2214yes—
30C2×C79⋊#95C2214yes—
31C2×C79⋊#95C2214yes—
32C22×C4⋊C4×C79C2211yescharacteristic
33C22×C4⋊C4×C79C2211yescharacteristic
34C22×C42×C79C2211yescharacteristic
35C2×#95×C79C2214yes—
36C24×D79C2311yescharacteristic
37C23×Dic79C2312yes—
38C23×Dic79C2312yes—
39C23×Dic79C2312yes—
40C23×Dic79C2311yescharacteristic
41C22×C4×D79—24no—
42C22×C4×D79—24no—
43C22×C4×D79—44no—
44C22×C4×D79—44no—
45C22×C4×D79C2×C418yes—
46C22×C4×D79C2×C414yes—
47C22×C4×D79C2×C414yes—
48C22×C4×D79C2×C414yes—
49C22×C4×D79D414yes—
50C22×C4×D79D414yes—
51C22×C4×D79D414yes—
52C22×C4×D79Q814yes—
53#14579119—232no—
54#14579118—232no—
55C2×C4⋊Dic79—22no—
56C2×C4⋊Dic79—22no—
57C2×C4⋊Dic79—22no—
58C2×C4⋊Dic79—22no—
59C2×C4⋊Dic79C2×C414yes—
60C2×C4⋊Dic79C2×C414yescomplemented
61C2×Dic79⋊C4—24no—
62C2×Dic79⋊C4—24no—
63C2×Dic79⋊C4—24no—
64C2×Dic79⋊C4—24no—
65C2×Dic79⋊C4C2×C418yes—
66C2×Dic79⋊C4C2×C418yescomplemented
67C4⋊C4×D79—232no—
68C2×(C2×D79)⋊C4—24no—
69C2×(C2×D79)⋊C4—24no—
70C2×(C2×D79)⋊C4—24no—
71C2×(C2×D79)⋊C4—24no—
72C2×(C2×D79)⋊C4C2×C418yes—
73C2×(C2×D79)⋊C4C2×C418yes—
74C2×C4×Dic79—22no—
75C2×C4×Dic79—22no—
76C2×C4×Dic79—24no—
77C2×C4×Dic79—24no—
78C2×C4×Dic79C2×C414yescomplemented
79C2×C4×Dic79C2×C418yescomplemented
80C79⋊#95—44no—
81C79⋊#95—44no—
82C79⋊#95—44no—
83C23×C4×C79C2312yes—
84C23×C4×C79C2312yes—
85C23×C4×C79C2312yes—
86C23×C4×C79C2311yescharacteristic
87C2×C4⋊C4×C79—22no—
88C2×C4⋊C4×C79—22no—
89C2×C4⋊C4×C79—22no—
90C2×C4⋊C4×C79—22no—
91C2×C4⋊C4×C79C2×C414yes—
92C2×C4⋊C4×C79C2×C414yescomplemented
93C2×C42×C79—22no—
94C2×C42×C79—22no—
95C2×C42×C79C2×C414yescomplemented
96#95×C79—44no—
97C23×D79—22no—
98C23×D79—22no—
99C23×D79—22no—
100C23×D79—22no—
101C23×D79C22×C412yes—
102C23×D79C22×C412yes—
103C23×D79C22×C412yes—
104C23×D79C2×D412yes—
105C23×D79C2×D412yes—
106C23×D79C2×D412yes—
107C23×D79C2×Q812yes—
108C22×Dic79—22no—
109C22×Dic79—22no—
110C22×Dic79—22no—
111C22×Dic79—22no—
112C22×Dic79—22no—
113C22×Dic79—22no—
114C22×Dic79—22no—
115C22×Dic79—22no—
116C22×Dic79—24no—
117C22×Dic79—24no—
118C22×Dic79—24no—
119C22×Dic79—22no—
120C22×Dic79—22no—
121C22×Dic79—24no—
122C22×Dic79C22×C414yes—
123C22×Dic79C22×C414yes—
124C22×Dic79C22×C414yes—
125C22×Dic79C22×C412yes—
126C22×Dic79C22×C412yes—
127C22×Dic79C22×C412yes—
128C22×Dic79C2×D412yes—
129C22×Dic79C2×D412yes—
130C22×Dic79C2×D412yes—
131C22×Dic79C2×Q812yes—
132C2×C4×D79—216no—
133C2×C4×D79—416no—
134C2×C4×D79—48no—
135C2×C4×D79—416no—
136C2×C4×D79—48no—
137C2×C4×D79C42116yes—
138C2×C4×D79C22⋊C4116yes—
139C2×C4×D79C22⋊C4116yes—
140C2×C4×D79C22⋊C4116yes—
141C2×C4×D79C4⋊C4116yes—
142C2×C4×D79C4⋊C4116yes—
143C2×C4×D79C4⋊C4116yes—
144C4⋊Dic79—28no—
145C4⋊Dic79—28no—
146Dic79⋊C4—216no—
147Dic79⋊C4—216no—
148(C2×D79)⋊C4—216no—
149(C2×D79)⋊C4—216no—
150C4×Dic79—216no—
151C4×Dic79—28no—
152C24×C79C2411yescharacteristic
153C22×C4×C79—22no—
154C22×C4×C79—22no—
155C22×C4×C79—22no—
156C22×C4×C79—22no—
157C22×C4×C79—22no—
158C22×C4×C79—22no—
159C22×C4×C79—22no—
160C22×C4×C79—24no—
161C22×C4×C79—24no—
162C22×C4×C79—22no—
163C22×C4×C79—24no—
164C22×C4×C79—22no—
165C22×C4×C79—24no—
166C22×C4×C79—22no—
167C22×C4×C79C22×C414yes—
168C22×C4×C79C22×C414yes—
169C22×C4×C79C22×C414yes—
170C22×C4×C79C22×C412yes—
171C22×C4×C79C22×C412yes—
172C22×C4×C79C22×C412yes—
173C22×C4×C79C2×D412yes—
174C22×C4×C79C2×D412yes—
175C22×C4×C79C2×D412yes—
176C22×C4×C79C2×Q812yes—
177C4⋊C4×C79—28no—
178C4⋊C4×C79—28no—
179C42×C79—28no—
180C23×C79C23×C411yescharacteristic
181C23×C79C23×C411yescharacteristic
182C23×C79C23×C411yescharacteristic
183C23×C79C22×D411yescharacteristic
184C23×C79C22×D411yescharacteristic
185C23×C79C22×D411yescharacteristic
186C23×C79C22×Q811yescharacteristic
187C23×C79H(5,𝔽2)12yes—
188C23×C79H(5,𝔽2)12yes—
189C23×C792−1+412yes—
190C23×C792−1+412yes—
191C22×D79—24no—
192C22×D79—24no—
193C22×D79—24no—
194C22×D79—24no—
195C22×D79—24no—
196C22×D79—24no—
197C22×D79—22no—
198C22×D79—24no—
199C22×D79—22no—
200C22×D79—24no—
201C22×D79—22no—
202C22×D79—22no—
203C22×D79—24no—
204C22×D79—24no—
205C22×D79—24no—
206C22×D79—24no—
207C22×D79C2×C4214yes—
208C22×D79C2×C22⋊C414yes—
209C22×D79C2×C22⋊C414yes—
210C22×D79C2×C22⋊C414yes—
211C22×D79C2×C4⋊C414yes—
212C22×D79C2×C4⋊C414yes—
213C22×D79C2×C4⋊C414yes—
214C2×C4×C79—24no—
215C2×C4×C79—24no—
216C2×C4×C79—24no—
217C2×C4×C79—28no—
218C2×C4×C79—28no—
219C2×C4×C79—24no—
220C2×C4×C79—24no—
221C2×C4×C79—42no—
222C2×C4×C79—44no—
223C2×C4×C79—42no—
224C2×C4×C79—42no—
225C2×C4×C79—42no—
226C2×C4×C79—42no—
227C2×C4×C79—42no—
228C2×C4×C79—42no—
229C2×C4×C79—44no—
230C2×C4×C79—42no—
231C2×C4×C79C2×C4214yes—
232C2×C4×C79C2×C22⋊C414yes—
233C2×C4×C79C2×C22⋊C414yes—
234C2×C4×C79C2×C22⋊C414yes—
235C2×C4×C79C2×C4⋊C414yes—
236C2×C4×C79C2×C4⋊C414yes—
237C2×C4×C79C2×C4⋊C414yes—
238C2×Dic79—24no—
239C2×Dic79—24no—
240C2×Dic79—24no—
241C2×Dic79—28no—
242C2×Dic79—24no—
243C2×Dic79—24no—
244C2×Dic79—28no—
245C2×Dic79—42no—
246C2×Dic79—44no—
247C2×Dic79—42no—
248C2×Dic79—42no—
249C2×Dic79—42no—
250C2×Dic79—42no—
251C2×Dic79—42no—
252C2×Dic79—42no—
253C2×Dic79—42no—
254C2×Dic79—44no—
255C2×Dic79C2×C4214yes—
256C2×Dic79C2×C22⋊C414yes—
257C2×Dic79C2×C22⋊C414yes—
258C2×Dic79C2×C22⋊C414yes—
259C2×Dic79C2×C4⋊C414yes—
260C2×Dic79C2×C4⋊C414yes—
261C2×Dic79C2×C4⋊C414yes—
262C4×D79—432no—
263C4×D79#95164yescomplemented
264C22×C79C22×C4211yescharacteristic
265C22×C79C22×C22⋊C411yescharacteristic
266C22×C79C22×C22⋊C411yescharacteristic
267C22×C79C22×C22⋊C411yescharacteristic
268C22×C79C22×C4⋊C411yescharacteristic
269C22×C79C22×C4⋊C411yescharacteristic
270C22×C79C22×C4⋊C411yescharacteristic
271C22×C79#51812yes—
272C22×C79#51912yes—
273C22×C79#52012yes—
274C22×C79#52012yes—
275C22×C79#52012yes—
276C22×C79#52012yes—
277C22×C79#53511yescharacteristic
278C22×C79#53512yes—
279C22×C79#53511yescharacteristic
280C22×C79#53612yes—
281C22×C79#53611yescharacteristic
282C22×C79#53611yescharacteristic
283C22×C79#53712yes—
284C22×C79#53712yes—
285C22×C79#54412yes—
286C22×C79#54412yes—
287C2×D79—24no—
288C2×D79—24no—
289C2×D79—28no—
290C2×D79—24no—
291C2×D79—24no—
292C2×D79—28no—
293C2×D79—24no—
294C2×D79—28no—
295C2×D79—24no—
296C2×D79—28no—
297C2×D79C2×#9518yes—
298C4×C79—44no—
299C4×C79—44no—
300C4×C79—44no—
301C4×C79C2×#9518yes—
302Dic79—44no—
303Dic79—44no—
304Dic79—44no—
305Dic79C2×#9518yes—
306#13795—791nomaximal, Sylow
307C2×C79C22×#9511yescharacteristic
308C2×C79#227412yes—
309C2×C79#230911yescharacteristic
310C2×C79#231011yescharacteristic
311C2×C79#231112yes—
312C2×C79#231111yescharacteristic
313C2×C79#231111yescharacteristic
314C2×C79#231612yes—
315C2×C79#231711yescharacteristic
316C2×C79#231712yes—
317C2×C79#231711yescharacteristic
318D79—28no—
319C22×#117—791no—
320C22×#117—791no—
321C23×C4⋊C4—791no—
322C4⋊#95—791no—
323C4⋊#95—794no—
324C4⋊#95—791no—
325C4⋊#95—791no—
326C4⋊#95D7911yescharacteristic, complemented
327C22⋊#95—794no—
328C79#1379511yescharacteristic, complemented, Sylow
329C24×C4—792no—
330C24×C4—791no—
331C2×#117—798no—
332C2×#117—798no—
333C2×#117—1584no—
334C2×#117—1584no—
335C2×#117—1584no—
336C2×#117—1584no—
337C22×C4⋊C4—792no—
338C22×C4⋊C4—791no—
339C22×C4⋊C4—792no—
340C22×C4⋊C4—791no—
341C22×C4⋊C4—798no—
342C22×C4⋊C4—1584no—
343C22×C4⋊C4—1584no—
344C22×C4⋊C4C2×D7911yescharacteristic
345C22×C4⋊C4C2×D7911yescharacteristic
346C22×C22⋊C4—792no—
347C22×C22⋊C4—792no—
348C22×C42—792no—
349C22×C42—791no—
350C22×C42C2×D7911yescharacteristic
351C2×#95—794no—
352C2×#95—794no—
353C2×#95—794no—
354C2×#95C2×D7914yes—
355C25—791no—
356C23×C4—3164no—
357C23×C4—3164no—
358C23×C4—798no—
359C23×C4—791no—
360C23×C4—794no—
361C23×C4—794no—
362C23×C4—794no—
363C23×C4—794no—
364C23×C4—794no—
365C23×C4—794no—
366C23×C4—794no—
367C23×C4—792no—
368C23×C4—792no—
369C23×C4—792no—
370C23×C4—1584no—
371C23×C4—1584no—
372C23×C4C22×D7912yes—
373C23×C4C22×D7912yes—
374C23×C4C22×D7912yes—
375C23×C4C22×D7911yescharacteristic
376#117—15832no—
377#117—15832no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—798no—
383C2×C4⋊C4—794no—
384C2×C4⋊C4—798no—
385C2×C4⋊C4—794no—
386C2×C4⋊C4—15832no—
387C2×C4⋊C4—1584no—
388C2×C4⋊C4—1584no—
389C2×C4⋊C4—1584no—
390C2×C4⋊C4—1584no—
391C2×C4⋊C4—1582no—
392C2×C4⋊C4—1582no—
393C2×C4⋊C4—1582no—
394C2×C4⋊C4—1582no—
395C2×C4⋊C4C4×D7914yes—
396C2×C4⋊C4C4×D7914yescomplemented
397C2×C22⋊C4—798no—
398C2×C22⋊C4—798no—
399C2×C22⋊C4—1584no—
400C2×C22⋊C4—1584no—
401C2×C22⋊C4—1584no—
402C2×C22⋊C4—1584no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—798no—
406C2×C42—794no—
407C2×C42—1584no—
408C2×C42—1584no—
409C2×C42—1582no—
410C2×C42—1582no—
411C2×C42C4×D7914yescomplemented
412#95—44no—
413#95—3164no—
414#95—3164no—
415#95—3164no—
416C24—792no—
417C24—792no—
418C24—792no—
419C24—792no—
420C24—792no—
421C24—792no—
422C24—792no—
423C24—1582no—
424C24—1582no—
425C24—1582no—
426C24—1582no—
427C24C23×D7911yescharacteristic
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—31616no—
443C22×C4—3168no—
444C22×C4—31616no—
445C22×C4—3168no—
446C22×C4—792no—
447C22×C4—792no—
448C22×C4—792no—
449C22×C4—792no—
450C22×C4—792no—
451C22×C4—792no—
452C22×C4—792no—
453C22×C4—7916no—
454C22×C4—7916no—
455C22×C4—7916no—
456C22×C4—7916no—
457C22×C4—7916no—
458C22×C4—7916no—
459C22×C4—7916no—
460C22×C4—794no—
461C22×C4—794no—
462C22×C4—794no—
463C22×C4—15816no—
464C22×C4—1582no—
465C22×C4—1584no—
466C22×C4—1582no—
467C22×C4—1584no—
468C22×C4—1584no—
469C22×C4—1582no—
470C22×C4—1582no—
471C22×C4—1584no—
472C22×C4—1582no—
473C22×C4—1582no—
474C22×C4—1582no—
475C22×C4—1582no—
476C22×C4—1582no—
477C22×C4—1582no—
478C22×C4C2×C4×D7914yes—
479C22×C4C2×C4×D7914yes—
480C22×C4C2×C4×D7914yes—
481C22×C4C2×C4×D7912yes—
482C22×C4C2×C4×D7912yes—
483C22×C4C2×C4×D7912yes—
484C22×C4D4×D7912yes—
485C22×C4D4×D7912yes—
486C22×C4D4×D7912yes—
487C22×C4Q8×D7912yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—15816no—
491C4⋊C4—15816no—
492C4⋊C4—1588no—
493C4⋊C4—1588no—
494C22⋊C4—15816no—
495C22⋊C4—15816no—
496C42—28no—
497C42—15816no—
498C42—1588no—
499C23—794no—
500C23—794no—
501C23—794no—
502C23—794no—
503C23—794no—
504C23—794no—
505C23—794no—
506C23—1584no—
507C23—1582no—
508C23—1584no—
509C23—1584no—
510C23—1584no—
511C23—1584no—
512C23—1584no—
513C23—1584no—
514C23—1584no—
515C23—1584no—
516C23—1584no—
517C23—1584no—
518C23—1582no—
519C23—1584no—
520C23—1582no—
521C23—1582no—
522C23C22×C4×D7911yescharacteristic
523C23C22×C4×D7911yescharacteristic
524C23C22×C4×D7911yescharacteristic
525C23C2×D4×D7911yescharacteristic
526C23C2×D4×D7911yescharacteristic
527C23C2×D4×D7911yescharacteristic
528C23C2×Q8×D7911yescharacteristic
529C23#1457920612yes—
530C23#1457920712yes—
531C23#1457920812yes—
532C23#1457920912yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—31632no—
551C2×C4—3162no—
552C2×C4—3162no—
553C2×C4—3162no—
554C2×C4—3162no—
555C2×C4—3164no—
556C2×C4—3162no—
557C2×C4—3164no—
558C2×C4—3162no—
559C2×C4—3162no—
560C2×C4—3162no—
561C2×C4—794no—
562C2×C4—794no—
563C2×C4—794no—
564C2×C4—794no—
565C2×C4—794no—
566C2×C4—794no—
567C2×C4—794no—
568C2×C4—7964no—
569C2×C4—1584no—
570C2×C4—1584no—
571C2×C4—1584no—
572C2×C4—1588no—
573C2×C4—1584no—
574C2×C4—1584no—
575C2×C4—1588no—
576C2×C4C42×D7914yes—
577C2×C4C22⋊C4×D7914yes—
578C2×C4C22⋊C4×D7914yes—
579C2×C4C22⋊C4×D7914yes—
580C2×C4C4⋊C4×D7914yes—
581C2×C4C4⋊C4×D7914yes—
582C2×C4C4⋊C4×D7914yes—
583C22—798no—
584C22—1584no—
585C22—1588no—
586C22—1584no—
587C22—1584no—
588C22—1584no—
589C22—1588no—
590C22—1584no—
591C22—1584no—
592C22—1588no—
593C22—1588no—
594C22C2×C42×D7911yescharacteristic
595C22C2×C22⋊C4×D7911yescharacteristic
596C22C2×C22⋊C4×D7911yescharacteristic
597C22C2×C22⋊C4×D7911yescharacteristic
598C22C2×C4⋊C4×D7911yescharacteristic
599C22C2×C4⋊C4×D7911yescharacteristic
600C22C2×C4⋊C4×D7911yescharacteristic
601C22#1909903512yes—
602C22#1909903912yes—
603C22#1909904212yes—
604C22#1909904312yes—
605C22#1909904412yes—
606C22#1909904512yes—
607C22#1909912611yescharacteristic
608C22#1909912912yes—
609C22#1909913111yescharacteristic
610C22#1909913311yescharacteristic
611C22#1909913612yes—
612C22#1909913811yescharacteristic
613C22#1909914212yes—
614C22#1909914312yes—
615C22#1909919012yes—
616C22#1909919312yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—3164no—
621C4—3164no—
622C4—3164no—
623C4—798no—
624C4#95×D7918yes—
625C2—1588no—
626C2C2×#95×D7911yescharacteristic
627C2#2376575412yes—
628C2#2376597911yescharacteristic
629C2#2376598311yescharacteristic
630C2#2376598912yes—
631C2#2376599011yescharacteristic
632C2#2376599111yescharacteristic
633C2#2376601512yes—
634C2#2376602111yescharacteristic
635C2#2376602212yes—
636C2#2376602511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D79
commutatorC22×C79C22×C42
FittingC79×C4⋊#95C2
socle, cosocleC24×C79C24
prime radicalC24×D79C23
hypercenterC4⋊#95D79
hypocenterC79#13795
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #23761630C2semidirect
5. #23761629C2semidirect
6. #23761629C2semidirect
7. Dic79⋊#95C2semidirect
9. C79×C4⋊#95C2semidirect
14. C2×#14579119C4semidirect
17. C2×#14579118C4semidirect
24. C2×C4⋊C4×D79C4semidirect
60. C2×C4⋊Dic79C2×C4semidirect
66. C2×Dic79⋊C4C2×C4semidirect
78. C2×C4×Dic79C2×C4semidirect
79. C2×C4×Dic79C2×C4semidirect
92. C2×C4⋊C4×C79C2×C4semidirect
95. C2×C42×C79C2×C4semidirect
263. C4×D79#95semidirect
326. C4⋊#95D79semidirect
328. C79#13795semidirect
396. C2×C4⋊C4C4×D79semidirect
411. C2×C42C4×D79semidirect

Maximal subgroups

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

subgroupindex
C22×#145791192
C22×#145791182
C22×C4⋊C4×D792
#237616302
#237616292
Dic79⋊#952
(C2×D79)⋊#952
C79×C4⋊#952
#1379579