Finite Group Databasebeta

← #34264389

Subgroups — #34264389

Order 20224.

982 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 982 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773774C211yesmaximal, characteristic
2#23773530C212yesmaximal, complemented
3#23762586C218yesmaximal, complemented
4#23762114C211yesmaximal, characteristic
5#23761847C212yesmaximal, complemented
6C2×#559×C79C211yesmaximal, characteristic
7#19099120C2211yescharacteristic
8#19098927—22no—
9#19098909C2214yes—
10#19098893—22no—
11#19098848C2214yescomplemented
12#19098773C2214yescomplemented
13#19098719—22no—
14#19098668—24no—
15#19098620C2212yes—
16#19098568—22no—
17C4×#11929608C2211yescharacteristic
18#19098307C2214yes—
19Q8⋊(C79⋊C8)C2218yescomplemented
20#559×C79—22no—
21#559×C79—24no—
22#559×C79C2214yes—
23C2×#127×C79C2211yescharacteristic
24C2×#124×C79—22no—
25C2×#123×C79—22no—
26C2×C4×Q8×C79C2212yes—
27C2×C4×D4×C79—22no—
28C2×#11929614—24no—
29C2×#11929608—21no—
30#14579161C2312yes—
31#14579153—21no—
32#14579140—22no—
33C4×Dic158—22no—
34#14579120—21no—
35#14579092—24no—
36C4⋊(C79⋊C8)C2314yes—
37#14579085—24no—
38#14579084—24no—
39#14579082—28no—
40C22⋊(C79⋊C8)—22no—
41#14579072—22no—
42#14579072—21no—
43C4×C79⋊C8C2312yes—
44C22×Q8×C79D412yes—
45C22×D4×C79—22no—
46C22×D4×C79D412yes—
47C23×C4×C79—22no—
48#127×C79—22no—
49#127×C79D412yes—
50#127×C79C2312yes—
51#124×C79—28no—
52#124×C79—28no—
53#123×C79—28no—
54#123×C79—28no—
55C4×Q8×C79—22no—
56C4×Q8×C79—22no—
57C4×Q8×C79D414yes—
58C4×Q8×C79C2314yes—
59C4×D4×C79—28no—
60C4×D4×C79—28no—
61C2×C4⋊C4×C79—22no—
62C2×C4⋊C4×C79D412yes—
63C2×C22⋊C4×C79—24no—
64C2×C22⋊C4×C79—22no—
65C2×C42×C79—22no—
66C2×C42×C79C2311yescharacteristic
67C2×Dic158—22no—
68#11929614—48no—
69#11929608—42no—
70C2×C79⋊C8—22no—
71C2×C79⋊C8—24no—
72C4⋊Dic79—22no—
73C4⋊Dic79—22no—
74Dic79⋊C4—42no—
75C22⋊Dic79—41no—
76C4×Dic79—41no—
77C24×C79—22no—
78C2×Q8×C79—22no—
79C2×Q8×C79—44no—
80C2×Q8×C79C2×D414yes—
81C2×D4×C79—24no—
82C2×D4×C79—22no—
83C2×D4×C79—24no—
84C2×D4×C79—48no—
85C2×D4×C79—44no—
86C2×D4×C79C2×D414yes—
87C22×C4×C79—22no—
88C22×C4×C79—22no—
89C22×C4×C79—28no—
90C22×C4×C79—84no—
91C22×C4×C79C2×D412yes—
92C22×C4×C79C2×D411yescharacteristic
93C4⋊C4×C79—24no—
94C4⋊C4×C79—22no—
95C4⋊C4×C79—24no—
96C4⋊C4×C79C2×D414yes—
97C22⋊C4×C79—216no—
98C22⋊C4×C79—28no—
99C42×C79—24no—
100C42×C79—24no—
101C42×C79C2×D411yescharacteristic
102C42×C79C2×D412yes—
103C42×C79C2411yescharacteristic
104C23×C79—24no—
105C23×C79—48no—
106C23×C79—44no—
107C23×C79#12011yescharacteristic
108Q8×C79—24no—
109Q8×C79—24no—
110Q8×C79—44no—
111D4×C79—44no—
112D4×C79—48no—
113D4×C79—48no—
114D4×C79—82no—
115C2×C4×C79—22no—
116C2×C4×C79—24no—
117C2×C4×C79—24no—
118C2×C4×C79—24no—
119C2×C4×C79—44no—
120C2×C4×C79—44no—
121C2×C4×C79—88no—
122C2×C4×C79#12011yescharacteristic
123C2×C4×C79#12012yes—
124C2×C4×C79C22×D411yescharacteristic
125C2×C4×C79C22×D412yes—
126C2×Dic79—42no—
127Dic158—44no—
128C79⋊C8—42no—
129C79⋊C8—44no—
130C22×C79—22no—
131C22×C79—48no—
132C22×C79—44no—
133C22×C79—88no—
134C22×C79#45512yes—
135C22×C79C2×#12011yescharacteristic
136C4×C79—22no—
137C4×C79—21no—
138C4×C79—21no—
139C4×C79—44no—
140C4×C79—44no—
141C4×C79#57514yes—
142Dic79—82no—
143#18531—791nomaximal, Sylow
144C2×C79—22no—
145C2×C79—84no—
146C2×C79#300512yes—
147C2×C79C2×#45511yescharacteristic
148C2×#559D7911yescharacteristic
149#3138—791no—
150#3113—792no—
151#1640—798no—
152#1555—791no—
153#1491—792no—
154C79#1853111yescharacteristic, complemented, Sylow
155#559—22no—
156#559—24no—
157#559C2×D7914yes—
158#533—791no—
159C2×#127C2×D7911yescharacteristic
160C2×#124—22no—
161C2×#123—22no—
162C2×C4×Q8C2×D7912yes—
163C2×C4×D4—22no—
164#490—1582no—
165#486—794no—
166#483—1582no—
167#475—794no—
168#462—794no—
169#451—1582no—
170#441—1584no—
171#432—792no—
172#418—1582no—
173C4×M16—791no—
174#331—794no—
175Q8⋊C8—798no—
176C22×Q8C22⋊D7912yes—
177C22×D4—22no—
178C22×D4C22⋊D7912yes—
179C23×C4—22no—
180C2×Q16—1584no—
181C2×M16—1581no—
182C4⋊Q8—792no—
183#127—22no—
184#127C22⋊D7912yes—
185#127C22×D7912yes—
186#125—1581no—
187#124—28no—
188#124—28no—
189#123—28no—
190#123—28no—
191C22⋊Q8—1582no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—1582no—
195C4×Q8C22⋊D7914yes—
196C4×Q8C22×D7914yes—
197C4×D4—28no—
198C4×D4—28no—
199#117—1581no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D7912yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C22×D7911yescharacteristic
206#106—1584no—
207C4⋊C8—794no—
208#103—1584no—
209#103—1584no—
210#102—1588no—
211C22⋊C8—1582no—
212#97—1581no—
213#97—1582no—
214C4×C8—792no—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—1582no—
219C2×Q8C2×C22⋊D7914yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D7914yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D7912yes—
231C22×C4C2×C22⋊D7911yescharacteristic
232Q16—3168no—
233M16—3162no—
234C2×C8—1582no—
235C2×C8—1584no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—3162no—
240C4⋊C4—1582no—
241C4⋊C4—1582no—
242C4⋊C4C2×C22⋊D7914yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—3161no—
246C42—24no—
247C42—24no—
248C42—3161no—
249C42C2×C22⋊D7911yescharacteristic
250C42C2×C22⋊D7912yes—
251C42C23×D7911yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1457913011yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—3164no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—3162no—
272C2×C4#1457913012yes—
273C2×C4#1457913011yescharacteristic
274C2×C4C22×C22⋊D7911yescharacteristic
275C2×C4C22×C22⋊D7912yes—
276C8—3162no—
277C8—3164no—
278C22—22no—
279C22—44no—
280C22—48no—
281C22—88no—
282C22#1909874112yes—
283C22C2×#1457913011yescharacteristic
284C4—21no—
285C4—21no—
286C4—22no—
287C4—44no—
288C4—44no—
289C4—6322no—
290C4#1909936114yes—
291C2—22no—
292C2—84no—
293C2#2377264212yes—
294C2C2×#1909874111yescharacteristic
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#559×C79C2
socleC22×C79C2×#120
prime radicalC2×#127×C79C22
hypercenterC2×#559D79
hypocenterC79#18531
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #23773530C2semidirect
3. #23762586C2semidirect
5. #23761847C2semidirect
11. #19098848C22semidirect
12. #19098773C22semidirect
19. Q8⋊(C79⋊C8)C22semidirect
154. C79#18531semidirect

Maximal subgroups

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

subgroupindex
#237737742
#237735302
#237625862
#237621142
#237618472
C2×#559×C792
#1853179