Finite Group Databasebeta

← #34183387

Subgroups — #34183387

Order 20224.

468 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772448C211yesmaximal, characteristic
2#23763499C211yesmaximal, characteristic, complemented
3#23763471C211yesmaximal, characteristic, complemented
4#23761891C211yesmaximal, characteristic
5#23761288C211yesmaximal, characteristic, complemented
6#23761246C211yesmaximal, characteristic, complemented
7C79×#3091C211yesmaximal, characteristic
8#19098641—21no—
9#19098635—21no—
10#19098635—21no—
11D4×C79⋊C8—21no—
12#19098624—21no—
13#19098530—21no—
14#19098527C2211yescharacteristic
15C2×C22⋊(C79⋊C8)C2211yescharacteristic
16C2×C22⋊(C79⋊C8)C2211yescharacteristic
17C4×C22⋊Dic79C2211yescharacteristic
18#19098322—21no—
19#19098322—21no—
20#19098318—21no—
21#19098318—21no—
22#19098300—21no—
23#19098297—21no—
24#525×C79C2211yescharacteristic
25C2×#117×C79C2211yescharacteristic
26#485×C79—21no—
27#484×C79C411yescharacteristic, complemented
28#483×C79C411yescharacteristic, complemented
29#482×C79—21no—
30#481×C79C411yescharacteristic, complemented
31#480×C79C411yescharacteristic, complemented
32#427×C79—21no—
33#416×C79—21no—
34C2×C22⋊C8×C79C2211yescharacteristic
35C2×#11929608—21no—
36C2×#11929608D411yescharacteristic
37C2×#11929608D411yescharacteristic
38C22×C79⋊C8—21no—
39C22×C79⋊C8—21no—
40C22×C79⋊C8D411yescharacteristic
41C22×C79⋊C8D411yescharacteristic
42C2×C22⋊Dic79—21no—
43C2×C4×Dic79—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C22⋊(C79⋊C8)—21no—
47C22⋊(C79⋊C8)—21no—
48C22⋊(C79⋊C8)—21no—
49C22⋊(C79⋊C8)—21no—
50C22⋊(C79⋊C8)—21no—
51C22⋊(C79⋊C8)—21no—
52C22⋊(C79⋊C8)—21no—
53C22⋊(C79⋊C8)—21no—
54#14579072—21no—
55C4×C79⋊C8—21no—
56C79⋊#95—21no—
57C2×#41×C79D411yescharacteristic
58C23×C4×C79C2311yescharacteristic
59C22×C8×C79—21no—
60#123×C79—21no—
61C22⋊Q8×C79C2×C411yescharacteristic
62C22⋊Q8×C79D411yescharacteristic
63#121×C79C2×C411yescharacteristic
64#121×C79D411yescharacteristic
65#120×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68#117×C79—21no—
69#117×C79—21no—
70#117×C79—21no—
71#117×C79—21no—
72#117×C79D411yescharacteristic
73C2×C4⋊C4×C79C2×C411yescharacteristic
74C2×C4⋊C4×C79C2×C411yescharacteristic
75C2×C22⋊C4×C79—21no—
76C2×C42×C79—21no—
77#107×C79—21no—
78#107×C79—21no—
79#106×C79—21no—
80#106×C79—21no—
81#103×C79—21no—
82#103×C79—21no—
83#102×C79—21no—
84#102×C79—21no—
85C22⋊C8×C79—21no—
86C22⋊C8×C79C2×C411yescharacteristic
87C22⋊C8×C79C2×C411yescharacteristic
88C22×Dic79—41no—
89#11929608—41no—
90#11929608—41no—
91#11929608—81no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—21no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—41no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8—81no—
102C22⋊Dic79—22no—
103C22⋊Dic79—41no—
104C4×Dic79—41no—
105C4×Dic79—41no—
106C24×C79#4111yescharacteristic
107#41×C79—41no—
108#41×C79—41no—
109C2×Q8×C79C22⋊C411yescharacteristic
110C2×D4×C79—21no—
111C2×D4×C79—21no—
112C2×D4×C79—21no—
113C2×D4×C79C22⋊C411yescharacteristic
114C22×C4×C79—21no—
115C22×C4×C79—21no—
116C22×C4×C79—21no—
117C22×C4×C79—41no—
118C22×C4×C79—41no—
119C22×C4×C79C22×C411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79C2×D411yescharacteristic
124C22×C4×C79#4111yescharacteristic
125C2×C8×C79—21no—
126C2×C8×C79—21no—
127C2×C8×C79—41no—
128C2×C8×C79—41no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79—21no—
131C4⋊C4×C79—21no—
132C4⋊C4×C79—21no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79—21no—
135C4⋊C4×C79C22⋊C411yescharacteristic
136C4⋊C4×C79C22⋊C411yescharacteristic
137C22⋊C4×C79—21no—
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C22⋊C4×C79—21no—
141C22⋊C4×C79—21no—
142C22⋊C4×C79—41no—
143C22⋊C4×C79—41no—
144C42×C79—21no—
145C42×C79—41no—
146C42×C79—41no—
147C23×C79—21no—
148C23×C79—21no—
149C23×C79—21no—
150C23×C79—41no—
151C23×C79—41no—
152C23×C79C4×D411yescharacteristic
153C23×C79#12111yescharacteristic
154C23×C79#12311yescharacteristic
155Q8×C79—41no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—81no—
160D4×C79—81no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—21no—
167C2×C4×C79—21no—
168C2×C4×C79—21no—
169C2×C4×C79—21no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—42no—
176C2×C4×C79—41no—
177C2×C4×C79—41no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79C2×C22⋊C411yescharacteristic
181C2×C4×C79C4×D411yescharacteristic
182C2×C4×C79#12011yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×Dic79—42no—
185C2×Dic79—81no—
186C2×Dic79—81no—
187C8×C79—41no—
188C8×C79—41no—
189C79⋊C8—41no—
190C79⋊C8—41no—
191C79⋊C8—41no—
192C79⋊C8—41no—
193C79⋊C8—81no—
194C22×C79—21no—
195C22×C79—21no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—21no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—41no—
205C22×C79—81no—
206C22×C79—81no—
207C22×C79#38611yescharacteristic
208C22×C79#44311yescharacteristic
209C22×C79#44411yescharacteristic
210C4×C79—21no—
211C4×C79—21no—
212C4×C79—21no—
213C4×C79—41no—
214C4×C79—41no—
215C4×C79—41no—
216C4×C79—81no—
217C4×C79—81no—
218C4×C79#45811yescharacteristic
219C4×C79#45811yescharacteristic
220Dic79—82no—
221#9038—791nomaximal, Sylow
222C2×C79—21no—
223C2×C79—21no—
224C2×C79—41no—
225C2×C79—41no—
226C2×C79—81no—
227C2×C79#187611yescharacteristic
228C2×C79#188111yescharacteristic
229C2×C79#189011yescharacteristic
230#3091D7911yescharacteristic
231#2976—791no—
232#1826—791no—
233#1818—791no—
234#1501—791no—
235#1298—791no—
236#1282—791no—
237C79#903811yescharacteristic, complemented, Sylow
238#525C2×D7911yescharacteristic
239C2×#117C2×D7911yescharacteristic
240#485—21no—
241#484Dic7911yescharacteristic, complemented
242#483Dic7911yescharacteristic, complemented
243#482—21no—
244#481Dic7911yescharacteristic, complemented
245#480Dic7911yescharacteristic, complemented
246#436—1581no—
247#435—1581no—
248#435—1581no—
249C8×D4—1581no—
250#433—1581no—
251#427—21no—
252#416—21no—
253#408—1581no—
254#407—791no—
255C2×C22⋊C8—791no—
256C2×C22⋊C8—791no—
257C2×C22⋊C8C2×D7911yescharacteristic
258C4×C22⋊C4—791no—
259#339—1581no—
260#339—1581no—
261#336—1581no—
262#336—1581no—
263#328—1581no—
264#327—1581no—
265C2×#41C22⋊D7911yescharacteristic
266C23×C4C22×D7911yescharacteristic
267C2×M16—791no—
268C2×M16—791no—
269C2×M16—1581no—
270C22×C8—21no—
271C22×C8—791no—
272C22×C8—791no—
273C22×C8—1581no—
274C22×C8—1581no—
275#123—21no—
276C22⋊Q8C2×Dic7911yescharacteristic
277C22⋊Q8C22⋊D7911yescharacteristic
278#121C2×Dic7911yescharacteristic
279#121C22⋊D7911yescharacteristic
280#120—21no—
281C4×D4—21no—
282C4×D4—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117—21no—
287#117C22⋊D7911yescharacteristic
288C2×C4⋊C4C2×Dic7911yescharacteristic
289C2×C4⋊C4C2×Dic7911yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—1581no—
292C2×C42—21no—
293C2×C42—1581no—
294#107—21no—
295#107—21no—
296#106—21no—
297#106—21no—
298C4⋊C8—1581no—
299C4⋊C8—1581no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304C22⋊C8—21no—
305C22⋊C8—1581no—
306C22⋊C8—1581no—
307C22⋊C8—1581no—
308C22⋊C8—1581no—
309C22⋊C8—1581no—
310C22⋊C8—1581no—
311C22⋊C8—1581no—
312C22⋊C8—1581no—
313C22⋊C8C2×Dic7911yescharacteristic
314C22⋊C8C2×Dic7911yescharacteristic
315#97—1581no—
316C4×C8—1581no—
317#95—1581no—
318C24#1192962311yescharacteristic
319#41—41no—
320#41—41no—
321C2×Q8C22⋊Dic7911yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4C22⋊Dic7911yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—3161no—
332C22×C4C22×Dic7911yescharacteristic
333C22×C4D4×D7911yescharacteristic
334C22×C4D4×D7911yescharacteristic
335C22×C4C2×C22⋊D7911yescharacteristic
336C22×C4C2×C22⋊D7911yescharacteristic
337C22×C4#1192962311yescharacteristic
338M16—3161no—
339M16—3161no—
340M16—6321no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—3161no—
346C2×C8—3161no—
347C2×C8—3161no—
348C2×C8—3161no—
349C2×C8—3161no—
350C2×C8—6321no—
351C2×C8—1581no—
352C2×C8—1581no—
353C2×C8—1581no—
354C2×C8—1581no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4C22⋊Dic7911yescharacteristic
362C4⋊C4C22⋊Dic7911yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—3161no—
371C22⋊C4—1582no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—3161no—
376C42—3161no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23D4×Dic7911yescharacteristic
383C23#1457913311yescharacteristic
384C23#1457914211yescharacteristic
385Q8—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—3162no—
411C2×C4—6321no—
412C2×C4—6321no—
413C2×C4C2×C22⋊Dic7911yescharacteristic
414C2×C4D4×Dic7911yescharacteristic
415C2×C4#1457912911yescharacteristic
416C2×C4#1457913311yescharacteristic
417C8—41no—
418C8—41no—
419C8—3161no—
420C8—3161no—
421C8—3161no—
422C8—3161no—
423C8—6321no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22#1909844311yescharacteristic
438C22#1909868011yescharacteristic
439C22#1909868511yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—6322no—
449C4#1909875211yescharacteristic
450C4#1909875211yescharacteristic
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#2376372911yescharacteristic
457C2#2376376411yescharacteristic
458C2#2376381411yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#3091C2
socleC22×C79#386
prime radical#525×C79C22
cosocleC23×C4×C79C23
hypercenter#3091D79
hypocenterC79#9038
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3091

Complements

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

NH ≅ G/Ntype
2. #23763499C2semidirect
3. #23763471C2semidirect
5. #23761288C2semidirect
6. #23761246C2semidirect
27. #484×C79C4semidirect
28. #483×C79C4semidirect
30. #481×C79C4semidirect
31. #480×C79C4semidirect
237. C79#9038semidirect
241. #484Dic79semidirect
242. #483Dic79semidirect
244. #481Dic79semidirect
245. #480Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724482
#237634992
#237634712
#237618912
#237612882
#237612462
C79×#30912
#903879