Finite Group Databasebeta

← #34184599

Subgroups — #34184599

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772058C211yesmaximal, characteristic, complemented
2#23763499C211yesmaximal, characteristic, complemented
3#23761916C211yesmaximal, characteristic
4#23761760C211yesmaximal, characteristic
5#23761277C211yesmaximal, characteristic, complemented
6#23761247C211yesmaximal, characteristic, complemented
7C2×#482×C79C211yesmaximal, characteristic
8#19098624—22no—
9#19098620—21no—
10#19098616—21no—
11#19098590—21no—
12#19098527C2211yescharacteristic
13#19098527C2211yescharacteristic
14C2×C22⋊(C79⋊C8)C2211yescharacteristic
15#19098519—21no—
16C4×C22⋊Dic79C2211yescharacteristic
17#19098322—21no—
18#19098322—21no—
19#19098297—22no—
20#19098287—21no—
21#19098287—21no—
22C2×#121×C79C2211yescharacteristic
23C2×#117×C79C2211yescharacteristic
24#482×C79—21no—
25#482×C79—21no—
26#482×C79—21no—
27#482×C79—21no—
28C2×#107×C79C411yescharacteristic, complemented
29C2×#106×C79C411yescharacteristic, complemented
30C2×#102×C79C411yescharacteristic, complemented
31C2×#102×C79C411yescharacteristic, complemented
32C2×C22⋊C8×C79C2211yescharacteristic
33C2×#11929608—21no—
34C2×#11929608—21no—
35C22×C79⋊C8—21no—
36C2×C22⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—22no—
39C4⋊(C79⋊C8)—22no—
40C22⋊(C79⋊C8)—21no—
41C22⋊(C79⋊C8)—21no—
42C22⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)D412yes—
45C22⋊(C79⋊C8)D411yescharacteristic
46C22⋊(C79⋊C8)D411yescharacteristic
47#14579072—22no—
48C4×C79⋊C8—22no—
49C79⋊#95—21no—
50C22×D4×C79—21no—
51C22×D4×C79C2×C411yescharacteristic
52C23×C4×C79C2311yescharacteristic
53C22×C8×C79C2×C411yescharacteristic
54C22×C8×C79C2×C411yescharacteristic
55#121×C79—21no—
56#121×C79—21no—
57#121×C79—21no—
58#121×C79—21no—
59#117×C79—21no—
60#117×C79—21no—
61#117×C79D411yescharacteristic
62#117×C79D411yescharacteristic
63#117×C79D411yescharacteristic
64#117×C79D411yescharacteristic
65C2×C4⋊C4×C79C2×C411yescharacteristic
66C2×C4⋊C4×C79C2×C411yescharacteristic
67C2×C4⋊C4×C79C2×C411yescharacteristic
68C2×C22⋊C4×C79—21no—
69C2×C22⋊C4×C79—21no—
70C2×C42×C79—21no—
71#107×C79—21no—
72#107×C79—21no—
73#106×C79—21no—
74#106×C79—21no—
75#102×C79—21no—
76#102×C79—21no—
77#102×C79—21no—
78#102×C79—21no—
79C22⋊C8×C79—21no—
80C22⋊C8×C79—21no—
81C22×Dic79—41no—
82#11929608—41no—
83#11929608—41no—
84#11929608—41no—
85#11929608—41no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C22⋊Dic79—22no—
93C22⋊Dic79—41no—
94C4×Dic79—41no—
95C4×Dic79—41no—
96C24×C79—21no—
97C24×C79#4111yescharacteristic
98C2×D4×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C2×D4×C79—41no—
103C2×D4×C79—41no—
104C2×D4×C79—82no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79—41no—
109C22×C4×C79—41no—
110C22×C4×C79C22×C411yescharacteristic
111C22×C4×C79C2×D411yescharacteristic
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79#4111yescharacteristic
116C2×C8×C79—21no—
117C2×C8×C79—21no—
118C2×C8×C79—41no—
119C2×C8×C79—41no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79C22⋊C411yescharacteristic
125C4⋊C4×C79C22⋊C411yescharacteristic
126C4⋊C4×C79C22⋊C411yescharacteristic
127C4⋊C4×C79C22⋊C411yescharacteristic
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—21no—
131C22⋊C4×C79—21no—
132C22⋊C4×C79—41no—
133C22⋊C4×C79—41no—
134C42×C79—21no—
135C42×C79—21no—
136C42×C79—21no—
137C42×C79—21no—
138C23×C79—21no—
139C23×C79—21no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—41no—
143C23×C79—41no—
144C23×C79—41no—
145C23×C79—81no—
146C23×C79—81no—
147C23×C79C4×D411yescharacteristic
148C23×C79#12111yescharacteristic
149C23×C79#12311yescharacteristic
150D4×C79—41no—
151D4×C79—41no—
152D4×C79—81no—
153D4×C79—82no—
154D4×C79—82no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—41no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—81no—
172C2×C4×C79—81no—
173C2×C4×C79C2×C22⋊C411yescharacteristic
174C2×C4×C79C4×D411yescharacteristic
175C2×C4×C79#12011yescharacteristic
176C2×C4×C79#12111yescharacteristic
177C2×Dic79—42no—
178C2×Dic79—81no—
179C2×Dic79—81no—
180C8×C79—41no—
181C8×C79—41no—
182C79⋊C8—41no—
183C79⋊C8—41no—
184C79⋊C8—42no—
185C79⋊C8—81no—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—81no—
195C22×C79—81no—
196C22×C79—81no—
197C22×C79—81no—
198C22×C79—81no—
199C22×C79—161no—
200C22×C79—161no—
201C22×C79#38611yescharacteristic
202C22×C79#44311yescharacteristic
203C22×C79#44411yescharacteristic
204C4×C79—21no—
205C4×C79—21no—
206C4×C79—21no—
207C4×C79—41no—
208C4×C79—41no—
209C4×C79—41no—
210C4×C79—41no—
211C4×C79—81no—
212C4×C79#41011yescharacteristic
213C4×C79#41011yescharacteristic
214Dic79—82no—
215#9242—791nomaximal, Sylow
216C2×C79—21no—
217C2×C79—21no—
218C2×C79—41no—
219C2×C79—41no—
220C2×C79—161no—
221C2×C79#187611yescharacteristic
222C2×C79#189911yescharacteristic
223C2×C79#190311yescharacteristic
224C2×#482D7911yescharacteristic
225#2924—791no—
226#1826—791no—
227#1507—791no—
228#1471—791no—
229#1293—791no—
230#1282—791no—
231C79#924211yescharacteristic, complemented, Sylow
232C2×#121C2×D7911yescharacteristic
233C2×#117C2×D7911yescharacteristic
234#482—21no—
235#482—21no—
236#482—21no—
237#482—21no—
238#433—1582no—
239#432—1581no—
240#431—1581no—
241C2×#107Dic7911yescharacteristic, complemented
242C2×#106Dic7911yescharacteristic, complemented
243#424—1581no—
244C2×#102Dic7911yescharacteristic, complemented
245C2×#102Dic7911yescharacteristic, complemented
246#407—791no—
247#407—791no—
248C2×C22⋊C8—791no—
249C2×C22⋊C8C2×D7911yescharacteristic
250#405—1581no—
251C4×C22⋊C4—791no—
252#339—1581no—
253#339—1581no—
254#327—1582no—
255#323—1581no—
256#323—1581no—
257C22×D4—21no—
258C22×D4C2×Dic7911yescharacteristic
259C23×C4C22×D7911yescharacteristic
260C2×M16—1581no—
261C2×M16—1581no—
262C22×C8—1581no—
263C22×C8C2×Dic7911yescharacteristic
264C22×C8C2×Dic7911yescharacteristic
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#117—21no—
270#117—21no—
271#117C22⋊D7911yescharacteristic
272#117C22⋊D7911yescharacteristic
273#117C22⋊D7911yescharacteristic
274#117C22⋊D7911yescharacteristic
275C2×C4⋊C4C2×Dic7911yescharacteristic
276C2×C4⋊C4C2×Dic7911yescharacteristic
277C2×C4⋊C4C2×Dic7911yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—1581no—
281C2×C42—21no—
282C2×C42—1581no—
283#107—21no—
284#107—21no—
285#106—21no—
286#106—21no—
287C4⋊C8—1582no—
288C4⋊C8—1582no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—791no—
296C22⋊C8—791no—
297C22⋊C8—792no—
298C22⋊C8—1581no—
299C22⋊C8—1581no—
300C22⋊C8—1581no—
301C22⋊C8—1581no—
302#97—1582no—
303C4×C8—1582no—
304#95—1581no—
305C24—21no—
306C24#1192962311yescharacteristic
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—82no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—3161no—
320C22×C4C22×Dic7911yescharacteristic
321C22×C4D4×D7911yescharacteristic
322C22×C4D4×D7911yescharacteristic
323C22×C4C2×C22⋊D7911yescharacteristic
324C22×C4C2×C22⋊D7911yescharacteristic
325C22×C4#1192962311yescharacteristic
326M16—3161no—
327M16—3161no—
328M16—3161no—
329M16—3161no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—3161no—
335C2×C8—3161no—
336C2×C8—3161no—
337C2×C8—1581no—
338C2×C8—1582no—
339C2×C8—1581no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4C22⋊Dic7911yescharacteristic
345C4⋊C4C22⋊Dic7911yescharacteristic
346C4⋊C4C22⋊Dic7911yescharacteristic
347C4⋊C4C22⋊Dic7911yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—3161no—
355C22⋊C4—1582no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—3161no—
361C42—3161no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—81no—
370C23—81no—
371C23D4×Dic7911yescharacteristic
372C23#1457913311yescharacteristic
373C23#1457914211yescharacteristic
374D4—41no—
375D4—41no—
376D4—81no—
377D4—82no—
378D4—82no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—3162no—
398C2×C4—6321no—
399C2×C4—6321no—
400C2×C4C2×C22⋊Dic7911yescharacteristic
401C2×C4D4×Dic7911yescharacteristic
402C2×C4#1457912911yescharacteristic
403C2×C4#1457913311yescharacteristic
404C8—41no—
405C8—41no—
406C8—3161no—
407C8—3161no—
408C8—3162no—
409C8—6321no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—161no—
424C22—161no—
425C22#1909844311yescharacteristic
426C22#1909868011yescharacteristic
427C22#1909868511yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—6322no—
437C4#1909853711yescharacteristic
438C4#1909853711yescharacteristic
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—161no—
444C2#2376372911yescharacteristic
445C2#2376385911yescharacteristic
446C2#2376388211yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#482×C79C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenterC2×#482D79
hypocenterC79#9242
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#482

Complements

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

NH ≅ G/Ntype
1. #23772058C2semidirect
2. #23763499C2semidirect
5. #23761277C2semidirect
6. #23761247C2semidirect
28. C2×#107×C79C4semidirect
29. C2×#106×C79C4semidirect
30. C2×#102×C79C4semidirect
31. C2×#102×C79C4semidirect
231. C79#9242semidirect
241. C2×#107Dic79semidirect
242. C2×#106Dic79semidirect
244. C2×#102Dic79semidirect
245. C2×#102Dic79semidirect

Maximal subgroups

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

subgroupindex
#237720582
#237634992
#237619162
#237617602
#237612772
#237612472
C2×#482×C792
#924279