Finite Group Databasebeta

← #34188064

Subgroups — #34188064

Order 20224.

610 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764362C211yesmaximal, characteristic, complemented
2#23764362C211yesmaximal, characteristic, complemented
3#23764207C211yesmaximal, characteristic
4#23764207C211yesmaximal, characteristic
5#23763528C211yesmaximal, characteristic, complemented
6C2×#19098285C211yesmaximal, characteristic, complemented
7C2×#492×C79C211yesmaximal, characteristic
8C2×#14579082—22no—
9C2×#14579082—22no—
10C2×C4×C79⋊C8C2211yescharacteristic
11C2×C4×C79⋊C8C2211yescharacteristic
12#19098433C2211yescharacteristic
13#19098433C2211yescharacteristic
14#19098326—22no—
15#19098325—22no—
16#19098325—22no—
17#19098285—22no—
18C22×SD16×C79—22no—
19C2×C4⋊Q8×C79C2211yescharacteristic
20C2×#127×C79C2211yescharacteristic
21#492×C79—22no—
22#492×C79C414yescomplemented
23C2×C4×C8×C79C2211yescharacteristic
24C22×C79⋊C8D412yes—
25C22×C79⋊C8D412yes—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32#14579082—44no—
33#14579082—44no—
34C4×C79⋊C8—21no—
35C4×C79⋊C8—21no—
36C4×C79⋊C8—21no—
37C4×C79⋊C8—21no—
38C22×Q8×C79—21no—
39C22×D4×C79—21no—
40C22×D4×C79—21no—
41C22×D4×C79—21no—
42C2×SD16×C79—28no—
43C2×SD16×C79—44no—
44C2×SD16×C79—44no—
45C22×C8×C79D412yes—
46C4⋊Q8×C79—21no—
47C4⋊Q8×C79—21no—
48C4⋊Q8×C79—21no—
49C4⋊Q8×C79C2×C412yes—
50#127×C79—21no—
51#127×C79—21no—
52#127×C79—21no—
53#127×C79C2×C412yes—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—21no—
56C2×C42×C79C2311yescharacteristic
57C4×C8×C79C2×C412yes—
58C4×C8×C79D412yes—
59C22×Dic79—41no—
60C22×Dic79—41no—
61C2×C79⋊C8—22no—
62C2×C79⋊C8—22no—
63C2×C79⋊C8—22no—
64C2×C79⋊C8—22no—
65C2×C79⋊C8—22no—
66C2×C79⋊C8SD1614yescomplemented
67C4⋊Dic79—42no—
68C4⋊Dic79—42no—
69C4⋊Dic79—42no—
70C4⋊Dic79—42no—
71C4⋊Dic79—42no—
72C4⋊Dic79—42no—
73C24×C79—41no—
74C2×Q8×C79—22no—
75C2×Q8×C79—41no—
76C2×Q8×C79—42no—
77C2×Q8×C79—42no—
78C2×D4×C79—22no—
79C2×D4×C79—22no—
80C2×D4×C79—22no—
81C2×D4×C79—41no—
82C2×D4×C79—41no—
83C2×D4×C79—41no—
84C2×D4×C79—42no—
85C2×D4×C79—42no—
86C2×D4×C79—42no—
87C2×D4×C79—42no—
88C2×D4×C79—42no—
89C2×D4×C79—42no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79C2×D411yescharacteristic
94SD16×C79—416no—
95C2×C8×C79—22no—
96C2×C8×C79—22no—
97C2×C8×C79C22⋊C414yes—
98C4⋊C4×C79—22no—
99C4⋊C4×C79—22no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C42×C79C22×C411yescharacteristic
103C42×C79C2×D411yescharacteristic
104C42×C79#4111yescharacteristic
105C42×C79#4111yescharacteristic
106C23×C79—42no—
107C23×C79—81no—
108C23×C79—81no—
109C23×C79—81no—
110C23×C79—81no—
111C23×C79—81no—
112C23×C79—81no—
113C23×C79#12711yescharacteristic
114Q8×C79—44no—
115Q8×C79—82no—
116D4×C79—44no—
117D4×C79—44no—
118D4×C79—44no—
119D4×C79—82no—
120D4×C79—82no—
121D4×C79—82no—
122C2×C4×C79—42no—
123C2×C4×C79—81no—
124C2×C4×C79—81no—
125C2×C4×C79C2×C22⋊C411yescharacteristic
126C2×C4×C79C4×D411yescharacteristic
127C2×C4×C79C4×D411yescharacteristic
128C2×C4×C79#12111yescharacteristic
129C2×C4×C79#12111yescharacteristic
130C2×C4×C79#12411yescharacteristic
131C2×C4×C79C2×SD1612yes—
132C2×C4×C79#13512yes—
133C2×C4×C79AGL(1,ℤ/8)12yes—
134C2×C4×C79AGL(1,ℤ/8)12yes—
135C2×C4×C79AGL(1,ℤ/8)12yes—
136C2×C4×C79#13712yes—
137C2×Dic79—81no—
138C2×Dic79—81no—
139C2×Dic79—81no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142C2×Dic79—81no—
143C8×C79—24no—
144C79⋊C8—24no—
145C79⋊C8—42no—
146C22×C79—82no—
147C22×C79—82no—
148C22×C79—82no—
149C22×C79—161no—
150C22×C79—161no—
151C22×C79—161no—
152C22×C79—161no—
153C22×C79#39011yescharacteristic
154C22×C79#49211yescharacteristic
155C22×C79#49511yescharacteristic
156C22×C79#49611yescharacteristic
157C22×C79#49611yescharacteristic
158C22×C79#49611yescharacteristic
159C22×C79#49711yescharacteristic
160C4×C79—82no—
161C4×C79#41812yes—
162C4×C79C4×SD1612yes—
163C4×C79#44012yes—
164C4×C79#46512yes—
165C4×C79#46812yes—
166C4×C79#49012yes—
167Dic79—161no—
168Dic79—161no—
169#9829—791nomaximal, Sylow
170C2×C79—162no—
171C2×C79#171411yescharacteristic
172C2×C79#171511yescharacteristic
173C2×C79#171711yescharacteristic
174C2×C79#171911yescharacteristic
175C2×C79#196111yescharacteristic
176C2×C79#196811yescharacteristic
177C2×C79#197311yescharacteristic
178C2×#492D7911yescharacteristic
179#1990—791no—
180#1990—791no—
181#1957—791no—
182#1957—791no—
183#1834—791no—
184C2×#322—791no—
185C79#982911yescharacteristic, complemented, Sylow
186C22×SD16—22no—
187C2×C4⋊Q8C2×D7911yescharacteristic
188C2×#127C2×D7911yescharacteristic
189#492—22no—
190#492Dic7914yescomplemented
191C2×#102—1582no—
192C2×#102—1582no—
193C2×C4×C8—791no—
194C2×C4×C8—791no—
195C2×C4×C8C2×D7911yescharacteristic
196#384—791no—
197#384—791no—
198#340—1582no—
199#340—1582no—
200#340—1582no—
201#322—1582no—
202C22×Q8—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4—21no—
206C2×SD16—28no—
207C2×SD16—44no—
208C2×SD16—44no—
209C22×C8—792no—
210C22×C8—792no—
211C22×C8C22⋊D7912yes—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C2×Dic7912yes—
216#127—21no—
217#127—21no—
218#127—21no—
219#127C2×Dic7912yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4—1581no—
225C2×C4⋊C4—1581no—
226C2×C4⋊C4—1581no—
227C2×C4⋊C4—1581no—
228C2×C42C22×D7911yescharacteristic
229#102—3164no—
230#102—3164no—
231C4×C8—1581no—
232C4×C8—1581no—
233C4×C8—1581no—
234C4×C8—1581no—
235C4×C8C2×Dic7912yes—
236C4×C8C22⋊D7912yes—
237C24—41no—
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C22×C4—41no—
255C22×C4—3161no—
256C22×C4—3161no—
257C22×C4D4×D7911yescharacteristic
258C22×C4D4×D7911yescharacteristic
259C22×C4C2×C22⋊D7911yescharacteristic
260SD16—416no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—794no—
264C2×C8—1582no—
265C2×C8—1582no—
266C2×C8—1582no—
267C2×C8—1582no—
268C2×C8—1582no—
269C2×C8C22⋊Dic7914yes—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—3162no—
275C4⋊C4—3162no—
276C4⋊C4—3162no—
277C4⋊C4—3162no—
278C4⋊C4—3162no—
279C4⋊C4—3162no—
280C42C22×Dic7911yescharacteristic
281C42C2×C22⋊D7911yescharacteristic
282C42#1192962311yescharacteristic
283C42#1192962311yescharacteristic
284C23—42no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23#1457915711yescharacteristic
292Q8—44no—
293Q8—82no—
294D4—44no—
295D4—44no—
296D4—44no—
297D4—82no—
298D4—82no—
299D4—82no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—6321no—
304C2×C4—6321no—
305C2×C4—6321no—
306C2×C4—6321no—
307C2×C4—6321no—
308C2×C4—6321no—
309C2×C4C2×C22⋊Dic7911yescharacteristic
310C2×C4D4×Dic7911yescharacteristic
311C2×C4D4×Dic7911yescharacteristic
312C2×C4#1457913211yescharacteristic
313C2×C4#1457913211yescharacteristic
314C2×C4#1457914811yescharacteristic
315C2×C4SD16×D7912yes—
316C2×C4#1457918212yes—
317C2×C4#1457918712yes—
318C2×C4#1457918712yes—
319C2×C4#1457918812yes—
320C2×C4#1457919012yes—
321C8—24no—
322C8—3162no—
323C8—1584no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22#1909846611yescharacteristic
332C22#1909893411yescharacteristic
333C22#1909894411yescharacteristic
334C22#1909894911yescharacteristic
335C22#1909895011yescharacteristic
336C22#1909895011yescharacteristic
337C22#1909895411yescharacteristic
338C4—82no—
339C4—12641no—
340C4—12641no—
341C4#1909856812yes—
342C4SD16×Dic7912yes—
343C4#1909866412yes—
344C4#1909878912yes—
345C4#1909880612yes—
346C4#1909892712yes—
347C2—162no—
348C2#2376302011yescharacteristic
349C2#2376302711yescharacteristic
350C2#2376303711yescharacteristic
351C2#2376305111yescharacteristic
352C2#2376422511yescharacteristic
353C2#2376425611yescharacteristic
354C2#2376428311yescharacteristic
subgroupG/H
centerC23#14579157
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#492×C79C2
socleC23×C79#127
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenterC2×#492D79
hypocenterC79#9829
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. #23764362C2semidirect
2. #23764362C2semidirect
5. #23763528C2semidirect
6. C2×#19098285C2semidirect
22. #492×C79C4semidirect
66. C2×C79⋊C8SD16semidirect
185. C79#9829semidirect
190. #492Dic79semidirect

Maximal subgroups

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

subgroupindex
#237643622
#237642072
#237635282
C2×#190982852
C2×#492×C792
#982979