Finite Group Databasebeta

← #34448370

Subgroups — #34448370

Order 20224.

3206 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099044C211yesmaximal, characteristic
2C2×#19099043C211yesmaximal, characteristic
3C2×#19099031C211yesmaximal, characteristic
4#23766314C214yesmaximal, complemented
5#23766301C214yesmaximal, complemented
6#23766298C218yesmaximal, complemented
7#23766025C212yesmaximal, complemented
8#23766023C211yesmaximal, characteristic, complemented
9#23766021C212yesmaximal, complemented
10#23765806C212yesmaximal, complemented
11#23765803C211yesmaximal, characteristic, complemented
12C2×#19098432C211yesmaximal, characteristic, complemented
13#384×D79C212yesmaximal, complemented
14C79×#2317C211yesmaximal, characteristic, complemented
15C22×#11929624C2211yescharacteristic
16#19099044—28no—
17#19099043—28no—
18#19099031C4116yescomplemented
19C2×C4×Dic158C2212yes—
20C2×#14579126C2212yes—
21C2×#14579126C2212yes—
22C2×C4×D316C2212yes—
23C2×#14579123C2212yes—
24C2×#14579123C2212yes—
25C2×C4×C22⋊D79C2214yes—
26C2×(C22⋊D79)⋊C4C2214yes—
27C2×(C22⋊D79)⋊C4C2214yes—
28C2×#14579120C2211yescharacteristic
29C2×#14579119C2212yes—
30C2×#14579119C2212yes—
31C2×#14579118C2212yes—
32C22×C4⋊Dic79C2211yescharacteristic
33C22×Dic79⋊C4C2214yes—
34C2×C4⋊C4×D79C2212yes—
35C2×C4⋊C4×D79C2212yes—
36C2×C4⋊C4×D79C2218yes—
37C22×C4×Dic79C2211yescharacteristic
38C2×C42×D79C2212yes—
39#19098462C2214yes—
40#19098460C2218yes—
41#19098458C2218yes—
42#19098438C2214yes—
43#19098437C2218yes—
44#19098434C2218yes—
45#19098433C2212yes—
46#19098432C2212yes—
47#19098432C2218yes—
48#19098432C2214yes—
49#19098432C2211yescharacteristic
50#19098432C2212yes—
51#19098432C2211yescharacteristic
52#19098429C2212yes—
53#19098428C2211yescharacteristic
54#19098428C2211yescharacteristic
55#19098422C2218yes—
56#19098421C2214yes—
57C4×Dic79⋊C4C2214yes—
58#19098414C2218yes—
59C2×#117×C79C2211yescharacteristic
60C2×#117×C79C2211yescharacteristic
61C22×C4⋊C4×C79C2211yescharacteristic
62#389×C79C2214yes—
63#385×C79C2214yes—
64#384×C79C2212yes—
65#382×C79C2212yes—
66C2×#11929624C2×C418yes—
67C2×#11929624D418yes—
68C2×#11929624Q818yes—
69C22×Dic158C2311yescharacteristic
70C22×D316C2311yescharacteristic
71C22×C22⋊D79C2312yes—
72C23×Dic79C2312yes—
73C23×Dic79C2312yes—
74C22×C4×D79C2314yes—
75C22×C4×D79C2314yes—
76C22×C4×D79C2314yes—
77C22×C4×D79C2312yes—
78C4×Dic158—28no—
79#14579126—28no—
80#14579126C2×C4116yescomplemented
81C4×D316—28no—
82#14579123—28no—
83#14579123C2×C4116yescomplemented
84C4×C22⋊D79—216no—
85(C22⋊D79)⋊C4—216no—
86(C22⋊D79)⋊C4C2×C4132yescomplemented
87#14579120—24no—
88#14579119—28no—
89#14579119—28no—
90#14579118C2×C4116yescomplemented
91C2×C4⋊Dic79—24no—
92C2×C4⋊Dic79C2311yescharacteristic
93C2×C4⋊Dic79C2311yescharacteristic
94C2×C4⋊Dic79C2312yes—
95C2×C4⋊Dic79C2312yes—
96C2×C4⋊Dic79C2318yes—
97C2×Dic79⋊C4—48no—
98C2×Dic79⋊C4C2314yes—
99C2×Dic79⋊C4C2318yes—
100C2×Dic79⋊C4C2314yes—
101C2×Dic79⋊C4C2314yes—
102C2×Dic79⋊C4C2314yes—
103C2×Dic79⋊C4C2318yes—
104C2×Dic79⋊C4C2314yes—
105C4⋊C4×D79—28no—
106C4⋊C4×D79—28no—
107C4⋊C4×D79—416no—
108C2×C22⋊Dic79C2312yes—
109C2×(C2×D79)⋊C4C2314yes—
110C2×(C2×D79)⋊C4C2314yes—
111C2×(C2×D79)⋊C4C2314yes—
112C2×C4×Dic79C2×C418yescomplemented
113C2×C4×Dic79C2314yes—
114C2×C4×Dic79C2314yes—
115C2×C4×Dic79C2312yes—
116C2×C4×Dic79C2314yes—
117C2×C4×Dic79C2312yes—
118C2×C4×Dic79C2311yescharacteristic
119C2×C4×Dic79C2311yescharacteristic
120C42×D79C2×C4116yescomplemented
121C79⋊#95C2318yes—
122C79⋊#95C2314yes—
123C79⋊#95C2314yes—
124C23×C4×C79C2312yes—
125C23×C4×C79C2311yescharacteristic
126#117×C79—24no—
127#117×C79C2×C418yescomplemented
128C2×C4⋊C4×C79—24no—
129C2×C4⋊C4×C79C2311yescharacteristic
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C4⋊C4×C79C2312yes—
132C2×C4⋊C4×C79C2312yes—
133C2×C4⋊C4×C79C2318yes—
134C2×C4⋊C4×C79C2312yes—
135C2×C22⋊C4×C79C2312yes—
136C2×C22⋊C4×C79C2312yes—
137C2×C42×C79C2312yes—
138C2×C42×C79C2312yes—
139#95×C79C2318yes—
140C23×D79C2412yes—
141#11929624C4⋊C4164yescomplemented
142C2×Dic158C22×C414yes—
143C2×Dic158C2×D414yes—
144C2×Dic158C2×Q814yes—
145C2×D316C22×C414yes—
146C2×D316C2×D414yes—
147C2×D316C2×Q814yes—
148C2×C22⋊D79C22×C418yes—
149C2×C22⋊D79C2×D418yes—
150C2×C22⋊D79C2×Q818yes—
151C22×Dic79—24no—
152C22×Dic79—24no—
153C22×Dic79—24no—
154C22×Dic79—44no—
155C22×Dic79C22×C418yes—
156C22×Dic79C2412yes—
157C22×Dic79C2414yes—
158C22×Dic79C2412yes—
159C22×Dic79C2412yes—
160C22×Dic79C2412yes—
161C22×Dic79C2412yes—
162C2×C4×D79—28no—
163C2×C4×D79—28no—
164C2×C4×D79—28no—
165C2×C4×D79—28no—
166C2×C4×D79—48no—
167C2×C4×D79—48no—
168C2×C4×D79C22×C4116yes—
169C2×C4×D79C22×C418yes—
170C2×C4×D79C2×D418yes—
171C2×C4×D79C2×Q818yes—
172C4⋊Dic79—22no—
173C4⋊Dic79—24no—
174C4⋊Dic79—24no—
175C4⋊Dic79—22no—
176C4⋊Dic79—48no—
177Dic79⋊C4—28no—
178Dic79⋊C4—28no—
179Dic79⋊C4—28no—
180Dic79⋊C4—216no—
181Dic79⋊C4—28no—
182Dic79⋊C4—48no—
183Dic79⋊C4C22×C4116yes—
184C22⋊Dic79—24no—
185(C2×D79)⋊C4—28no—
186(C2×D79)⋊C4—28no—
187(C2×D79)⋊C4C22×C4116yes—
188C4×Dic79—28no—
189C4×Dic79—28no—
190C4×Dic79—24no—
191C4×Dic79C22×C414yes—
192C4×Dic79C22×C4116yes—
193C4×Dic79C22×C414yes—
194C4×Dic79C22×C418yes—
195C24×C79C2411yescharacteristic
196C22×C4×C79—24no—
197C22×C4×C79—44no—
198C22×C4×C79C22×C414yes—
199C22×C4×C79C2×D414yes—
200C22×C4×C79C2×Q814yes—
201C22×C4×C79C2412yes—
202C22×C4×C79C2414yes—
203C22×C4×C79C2412yes—
204C22×C4×C79C2414yes—
205C22×C4×C79C2411yescharacteristic
206C22×C4×C79C2411yescharacteristic
207C4⋊C4×C79—24no—
208C4⋊C4×C79—24no—
209C4⋊C4×C79—22no—
210C4⋊C4×C79—22no—
211C4⋊C4×C79—48no—
212C4⋊C4×C79C22×C418yes—
213C22⋊C4×C79—24no—
214C22⋊C4×C79C22×C418yes—
215C42×C79—24no—
216C42×C79C22×C418yes—
217C23×C79—22no—
218C23×C79—22no—
219C23×C79C23×C412yes—
220C23×C79C22×D412yes—
221C23×C79C22×Q812yes—
222C23×C79C2511yescharacteristic
223C22×D79—24no—
224C22×D79—24no—
225C22×D79C23×C414yes—
226C22×D79C22×D414yes—
227C22×D79C22×Q814yes—
228C2×C4×C79—24no—
229C2×C4×C79—24no—
230C2×C4×C79—22no—
231C2×C4×C79—24no—
232C2×C4×C79—22no—
233C2×C4×C79—28no—
234C2×C4×C79—24no—
235C2×C4×C79—24no—
236C2×C4×C79—84no—
237C2×C4×C79C2×C4⋊C4116yescomplemented
238C2×C4×C79C23×C418yes—
239C2×C4×C79C23×C412yes—
240C2×C4×C79C23×C412yes—
241C2×C4×C79C22×D412yes—
242C2×C4×C79C22×D412yes—
243C2×C4×C79C22×Q812yes—
244C2×C4×C79C22×Q812yes—
245C22⋊D79C2×C4⋊C4132yescomplemented
246C2×Dic79—22no—
247C2×Dic79—22no—
248C2×Dic79—216no—
249C2×Dic79—22no—
250C2×Dic79—24no—
251C2×Dic79—22no—
252C2×Dic79—28no—
253C2×Dic79—24no—
254C2×Dic79—24no—
255C2×Dic79—84no—
256C2×Dic79C23×C414yes—
257C2×Dic79C23×C414yes—
258C2×Dic79C23×C414yes—
259C2×Dic79C22×D414yes—
260C2×Dic79C22×Q814yes—
261C2×Dic79H(5,𝔽2)14yes—
262C2×Dic792−1+414yes—
263D316C2×C4⋊C4116yescomplemented
264C4×D79—232no—
265C4×D79—88no—
266C4×D79—88no—
267C4×D79C2×C4⋊C4132yescomplemented
268Dic158C2×C4⋊C4116yescomplemented
269C22×C79—24no—
270C22×C79—22no—
271C22×C79—22no—
272C22×C79—24no—
273C22×C79C22×C4⋊C414yes—
274C22×C79C24×C411yescharacteristic
275C22×C79C23×D411yescharacteristic
276C22×C79C23×Q811yescharacteristic
277C22×C79C2×H(5,𝔽2)12yes—
278C22×C79C2×2−1+412yes—
279C2×D79—28no—
280C2×D79—24no—
281C2×D79—24no—
282C2×D79—28no—
283C2×D79C22×C4⋊C418yes—
284C4×C79—28no—
285C4×C79—44no—
286C4×C79—42no—
287C4×C79—42no—
288C4×C79C22×C4⋊C414yes—
289C4×C79C22×C4⋊C414yes—
290Dic79—24no—
291Dic79—42no—
292Dic79—44no—
293Dic79—42no—
294Dic79C22×C4⋊C418yes—
295Dic79#52018yescomplemented
296#34183—791nomaximal, Sylow
297C2×C79—24no—
298C2×C79C23×C4⋊C411yescharacteristic
299C2×C79C2×#52012yes—
300C2×C79#349811yescharacteristic
301C2×C79#349911yescharacteristic
302C2×C79#350312yes—
303D79—28no—
304C2×#520—791no—
305C2×#520—791no—
306C2×C4×#41—791no—
307Q8×C4⋊C4—794no—
308D4×C4⋊C4—794no—
309D4×C4⋊C4—798no—
310#2317—792no—
311#2317—792no—
312#2317—791no—
313#2317D7911yescharacteristic, complemented
314#2285—792no—
315#2285—791no—
316C2×#384—792no—
317C2×#384—791no—
318C79#3418311yescharacteristic, complemented, Sylow
319C22×#41—791no—
320#520—1588no—
321#520—1588no—
322C4×#41—7916no—
323C2×C4×Q8—792no—
324C2×C4×Q8—792no—
325C2×C4×Q8—792no—
326C2×C4×D4—794no—
327C2×C4×D4—792no—
328C2×C4×D4—792no—
329C2×C4×D4—794no—
330C2×C4×D4—794no—
331C2×C4×D4—792no—
332C2×#117—791no—
333C2×#117—792no—
334C2×#117—792no—
335C2×#117—792no—
336C2×#117C2×D7911yescharacteristic
337C2×#117C2×D7911yescharacteristic
338C22×C4⋊C4—794no—
339C22×C4⋊C4—798no—
340C22×C4⋊C4—791no—
341C22×C4⋊C4—792no—
342C22×C4⋊C4—792no—
343C22×C4⋊C4C2×D7911yescharacteristic
344C22×C42—791no—
345C22×C42—792no—
346#389—798no—
347#389—794no—
348#389—798no—
349#389C2×D7914yes—
350#385—798no—
351#385—794no—
352#385—798no—
353#385C2×D7914yes—
354#384—792no—
355#384—794no—
356#384—798no—
357#384—791no—
358#384—792no—
359#384—791no—
360#384—792no—
361#384C2×D7912yes—
362#382—791no—
363#382—791no—
364#382—792no—
365#382C2×D7912yes—
366#380—794no—
367#380—798no—
368C4×C4⋊C4—794no—
369C4×C4⋊C4—798no—
370C2×#41—798no—
371C2×#41—798no—
372C2×#41—798no—
373C22×Q8—791no—
374C22×D4—792no—
375C22×D4—791no—
376C23×C4—792no—
377C23×C4—794no—
378C23×C4—794no—
379C23×C4—792no—
380C23×C4—792no—
381C23×C4—794no—
382C23×C4C22×D7912yes—
383C23×C4C22×D7911yescharacteristic
384C4×Q8—7916no—
385C4×Q8—1588no—
386C4×Q8—1588no—
387C4×D4—7916no—
388C4×D4—7932no—
389C4×D4—1588no—
390C4×D4—15816no—
391C4×D4—1588no—
392C4×D4—15816no—
393#117—24no—
394#117—7916no—
395#117—1588no—
396#117—1588no—
397#117—1584no—
398#117C4×D7918yescomplemented
399C2×C4⋊C4—24no—
400C2×C4⋊C4—31616no—
401C2×C4⋊C4—3168no—
402C2×C4⋊C4—798no—
403C2×C4⋊C4—798no—
404C2×C4⋊C4—794no—
405C2×C4⋊C4—792no—
406C2×C4⋊C4—791no—
407C2×C4⋊C4—794no—
408C2×C4⋊C4—798no—
409C2×C4⋊C4—794no—
410C2×C4⋊C4—794no—
411C2×C4⋊C4—792no—
412C2×C4⋊C4—791no—
413C2×C4⋊C4—794no—
414C2×C4⋊C4—1588no—
415C2×C4⋊C4—1588no—
416C2×C4⋊C4—1584no—
417C2×C4⋊C4C22×D7911yescharacteristic
418C2×C4⋊C4C22×D7912yes—
419C2×C4⋊C4C22×D7912yes—
420C2×C4⋊C4C22×D7911yescharacteristic
421C2×C4⋊C4C22×D7918yes—
422C2×C4⋊C4C22×D7912yes—
423C2×C22⋊C4—794no—
424C2×C22⋊C4—792no—
425C2×C22⋊C4—794no—
426C2×C22⋊C4—794no—
427C2×C22⋊C4C22×D7912yes—
428C2×C22⋊C4C22×D7912yes—
429C2×C42—798no—
430C2×C42—791no—
431C2×C42—794no—
432C2×C42—792no—
433C2×C42—791no—
434C2×C42—7916no—
435C2×C42—794no—
436C2×C42—794no—
437C2×C42—792no—
438C2×C42C22×D7912yes—
439C2×C42C22×D7912yes—
440#95—794no—
441#95—794no—
442#95—798no—
443#95C22×D7918yes—
444C24—792no—
445C24C23×D7911yescharacteristic
446#41—7964no—
447C2×Q8—794no—
448C2×Q8—794no—
449C2×Q8—794no—
450C2×D4—798no—
451C2×D4—798no—
452C2×D4—798no—
453C2×D4—794no—
454C2×D4—794no—
455C2×D4—794no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—3168no—
459C22×C4—3168no—
460C22×C4—3164no—
461C22×C4—792no—
462C22×C4—792no—
463C22×C4—798no—
464C22×C4—798no—
465C22×C4—798no—
466C22×C4—792no—
467C22×C4—798no—
468C22×C4—7916no—
469C22×C4—792no—
470C22×C4—794no—
471C22×C4—792no—
472C22×C4—1588no—
473C22×C4—1588no—
474C22×C4—1588no—
475C22×C4—1588no—
476C22×C4—1584no—
477C22×C4—1584no—
478C22×C4—1584no—
479C22×C4C2×C4×D7914yes—
480C22×C4D4×D7914yes—
481C22×C4Q8×D7914yes—
482C22×C4C23×D7912yes—
483C22×C4C23×D7912yes—
484C22×C4C23×D7914yes—
485C22×C4C23×D7914yes—
486C22×C4C23×D7911yescharacteristic
487C22×C4C23×D7911yescharacteristic
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—48no—
493C4⋊C4—3168no—
494C4⋊C4—3168no—
495C4⋊C4—7916no—
496C4⋊C4—1588no—
497C4⋊C4—15816no—
498C4⋊C4—1588no—
499C4⋊C4—1588no—
500C4⋊C4—1588no—
501C4⋊C4—1582no—
502C4⋊C4—1584no—
503C4⋊C4—1584no—
504C4⋊C4—1582no—
505C4⋊C4C2×C4×D7918yes—
506C22⋊C4—24no—
507C22⋊C4—7916no—
508C22⋊C4—1588no—
509C22⋊C4—1588no—
510C22⋊C4—1584no—
511C22⋊C4C2×C4×D7918yes—
512C42—24no—
513C42—794no—
514C42—794no—
515C42—7916no—
516C42—798no—
517C42—1588no—
518C42—1588no—
519C42—1584no—
520C42C2×C4×D7918yes—
521C23—22no—
522C23—22no—
523C23—794no—
524C23—794no—
525C23—794no—
526C23—1584no—
527C23—1584no—
528C23C22×C4×D7912yes—
529C23C2×D4×D7912yes—
530C23C2×Q8×D7912yes—
531C23C24×D7911yescharacteristic
532Q8—7916no—
533D4—7932no—
534D4—7916no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—84no—
544C2×C4—794no—
545C2×C4—794no—
546C2×C4—794no—
547C2×C4—794no—
548C2×C4—794no—
549C2×C4—794no—
550C2×C4—7932no—
551C2×C4—794no—
552C2×C4—6328no—
553C2×C4—6328no—
554C2×C4—6324no—
555C2×C4—15832no—
556C2×C4—1588no—
557C2×C4—1584no—
558C2×C4—1584no—
559C2×C4—1582no—
560C2×C4—1584no—
561C2×C4—1582no—
562C2×C4—15816no—
563C2×C4—1582no—
564C2×C4—1582no—
565C2×C4C4⋊C4×D79116yescomplemented
566C2×C4C22×C4×D7918yes—
567C2×C4C22×C4×D7912yes—
568C2×C4C22×C4×D7912yes—
569C2×C4C2×D4×D7912yes—
570C2×C4C2×D4×D7912yes—
571C2×C4C2×Q8×D7912yes—
572C2×C4C2×Q8×D7912yes—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—798no—
578C22—1588no—
579C22—1584no—
580C22—1584no—
581C22—1588no—
582C22C2×C4⋊C4×D7914yes—
583C22C23×C4×D7911yescharacteristic
584C22C22×D4×D7911yescharacteristic
585C22C22×Q8×D7911yescharacteristic
586C22H(5,𝔽2)×D7912yes—
587C222−1+4×D7912yes—
588C4—28no—
589C4—44no—
590C4—42no—
591C4—42no—
592C4—3162no—
593C4—3164no—
594C4—3162no—
595C4—798no—
596C4—798no—
597C4—1584no—
598C4C2×C4⋊C4×D7914yes—
599C4C2×C4⋊C4×D7914yes—
600C2—24no—
601C2—1588no—
602C2C22×C4⋊C4×D7911yescharacteristic
603C2#520×D7912yes—
604C2#2377717511yescharacteristic
605C2#2377718211yescharacteristic
606C2#2377721512yes—
subgroupG/H
center, FrattiniC23C24×D79
commutatorC22×C79C24×C4
FittingC79×#2317C2
socle, cosocleC23×C79C25
prime radicalC22×#11929624C22
hypercenter#2317D79
hypocenterC79#34183
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #23766314C2semidirect
5. #23766301C2semidirect
6. #23766298C2semidirect
7. #23766025C2semidirect
8. #23766023C2semidirect
9. #23766021C2semidirect
10. #23765806C2semidirect
11. #23765803C2semidirect
12. C2×#19098432C2semidirect
13. #384×D79C2semidirect
14. C79×#2317C2semidirect
18. #19099031C4semidirect
80. #14579126C2×C4semidirect
83. #14579123C2×C4semidirect
86. (C22⋊D79)⋊C4C2×C4semidirect
90. #14579118C2×C4semidirect
112. C2×C4×Dic79C2×C4semidirect
120. C42×D79C2×C4semidirect
127. #117×C79C2×C4semidirect
141. #11929624C4⋊C4semidirect
237. C2×C4×C79C2×C4⋊C4semidirect
245. C22⋊D79C2×C4⋊C4semidirect
263. D316C2×C4⋊C4semidirect
267. C4×D79C2×C4⋊C4semidirect
268. Dic158C2×C4⋊C4semidirect
295. Dic79#520semidirect
313. #2317D79semidirect
318. C79#34183semidirect
398. #117C4×D79semidirect
565. C2×C4C4⋊C4×D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190990442
C2×#190990432
C2×#190990312
#237663142
#237663012
#237662982
#237660252
#237660232
#237660212
#237658062
#237658032
C2×#190984322
#384×D792
C79×#23172
#3418379