Finite Group Databasebeta

← #34180869

Subgroups — #34180869

Order 20224.

718 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 131 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#23763483C211yesmaximal, characteristic, complemented
2#23763471C211yesmaximal, characteristic, complemented
3#23761748C214yesmaximal, complemented
4C79×#3491C211yesmaximal, characteristic
5#19098527C2212yes—
6C2×C22⋊(C79⋊C8)C2212yes—
7#19098332—22no—
8#19098318—22no—
9#19098289—24no—
10#19098287—24no—
11#552×C79—24no—
12#551×C79—24no—
13#543×C79C411yescharacteristic
14#539×C79—22no—
15#534×C79C411yescharacteristic
16#529×C79—22no—
17C2×#123×C79C2212yes—
18#518×C79C412yes—
19C2×#117×C79C2211yescharacteristic
20C2×#11929608—22no—
21C22×C79⋊C8—22no—
22C22⋊(C79⋊C8)—24no—
23C22⋊(C79⋊C8)—24no—
24C22×D4×C79C2×C412yes—
25C23×C4×C79C2311yescharacteristic
26#126×C79—24no—
27#126×C79—24no—
28#125×C79—22no—
29#125×C79—24no—
30#124×C79—22no—
31#123×C79—24no—
32#123×C79—22no—
33#123×C79—22no—
34#123×C79—22no—
35#123×C79—24no—
36C22⋊Q8×C79—24no—
37C22⋊Q8×C79—22no—
38C22⋊Q8×C79—24no—
39#121×C79—22no—
40#120×C79—22no—
41C4×Q8×C79—22no—
42C4×D4×C79—22no—
43C4×D4×C79—24no—
44C4×D4×C79—24no—
45#117×C79—24no—
46#117×C79—22no—
47#117×C79—24no—
48C2×C4⋊C4×C79D412yes—
49C2×C4⋊C4×C79D414yes—
50C2×C22⋊C4×C79—22no—
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79C2×C412yes—
54C2×C22⋊C4×C79D414yes—
55C2×C42×C79D412yes—
56#11929608—44no—
57C2×C79⋊C8—44no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—42no—
60C24×C79C22⋊C412yes—
61C24×C79C22×C411yescharacteristic
62C2×Q8×C79—22no—
63C2×D4×C79—22no—
64C2×D4×C79—22no—
65C2×D4×C79—22no—
66C2×D4×C79—44no—
67C22×C4×C79—24no—
68C22×C4×C79—42no—
69C22×C4×C79C22⋊C412yes—
70C22×C4×C79C22⋊C412yes—
71C22×C4×C79C22⋊C414yes—
72C22×C4×C79C22⋊C412yes—
73C22×C4×C79C2×D411yescharacteristic
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79C2×D414yes—
76C4⋊C4×C79—24no—
77C4⋊C4×C79—24no—
78C4⋊C4×C79—22no—
79C4⋊C4×C79—22no—
80C4⋊C4×C79—24no—
81C4⋊C4×C79—24no—
82C4⋊C4×C79—24no—
83C22⋊C4×C79—24no—
84C22⋊C4×C79—24no—
85C22⋊C4×C79—22no—
86C22⋊C4×C79—22no—
87C22⋊C4×C79—22no—
88C22⋊C4×C79—22no—
89C22⋊C4×C79—24no—
90C22⋊C4×C79—24no—
91C42×C79—24no—
92C42×C79—22no—
93C42×C79—22no—
94C23×C79—22no—
95C23×C79—22no—
96C23×C79—42no—
97C23×C79—42no—
98C23×C79—44no—
99C23×C79C2×C22⋊C412yes—
100C23×C79C2×C22⋊C411yescharacteristic
101Q8×C79—82no—
102D4×C79—44no—
103D4×C79—44no—
104D4×C79—82no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—24no—
111C2×C4×C79—44no—
112C2×C4×C79—42no—
113C2×C4×C79—44no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—84no—
117C2×C4×C79#12012yes—
118C2×C4×C79#12012yes—
119C79⋊C8—82no—
120C79⋊C8—82no—
121C22×C79—22no—
122C22×C79—22no—
123C22×C79—24no—
124C22×C79—42no—
125C22×C79—44no—
126C22×C79—82no—
127C22×C79—162no—
128C22×C79#37911yescharacteristic
129C22×C79#41012yes—
130C4×C79—22no—
131C4×C79—22no—
132C4×C79—42no—
133C4×C79—42no—
134C4×C79—44no—
135C4×C79—42no—
136#8540—791nomaximal, Sylow
137C2×C79—22no—
138C2×C79—42no—
139C2×C79—82no—
140C2×C79#178211yescharacteristic
141C2×C79#179112yes—
142#3491D7911yescharacteristic
143#1822—791no—
144#1818—791no—
145#1468—794no—
146C79#854011yescharacteristic, complemented, Sylow
147#552—24no—
148#551—24no—
149#543Dic7911yescharacteristic
150#539—22no—
151#534Dic7911yescharacteristic
152#529—22no—
153C2×#123C2×D7912yes—
154#518Dic7912yes—
155C2×#117C2×D7911yescharacteristic
156#407—792no—
157C2×C22⋊C8—792no—
158#343—1582no—
159#336—1582no—
160#324—1584no—
161#323—1584no—
162C22×D4C2×Dic7912yes—
163C23×C4C22×D7911yescharacteristic
164C2×M16—1582no—
165C22×C8—1582no—
166#126—24no—
167#126—24no—
168#125—22no—
169#125—24no—
170#124—22no—
171#123—24no—
172#123—22no—
173#123—22no—
174#123—22no—
175#123—24no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—24no—
179#121—22no—
180#120—22no—
181C4×Q8—22no—
182C4×D4—22no—
183C4×D4—24no—
184C4×D4—24no—
185#117—24no—
186#117—22no—
187#117—24no—
188C2×C4⋊C4C22⋊D7912yes—
189C2×C4⋊C4C22⋊D7914yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×Dic7911yescharacteristic
192C2×C22⋊C4C2×Dic7911yescharacteristic
193C2×C22⋊C4C2×Dic7912yes—
194C2×C22⋊C4C22⋊D7914yes—
195C2×C42C22⋊D7912yes—
196C22⋊C8—1584no—
197C22⋊C8—1584no—
198C24C22⋊Dic7912yes—
199C24C22×Dic7911yescharacteristic
200C2×Q8—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4C22⋊Dic7912yes—
208C22×C4C22⋊Dic7912yes—
209C22×C4C22⋊Dic7914yes—
210C22×C4C22⋊Dic7912yes—
211C22×C4C2×C22⋊D7911yescharacteristic
212C22×C4C2×C22⋊D7911yescharacteristic
213C22×C4C2×C22⋊D7914yes—
214M16—3164no—
215C2×C8—3164no—
216C2×C8—3162no—
217C2×C8—3162no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23C2×C22⋊Dic7912yes—
242C23C2×C22⋊Dic7911yescharacteristic
243Q8—82no—
244D4—44no—
245D4—44no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4#1457913012yes—
260C2×C4#1457913012yes—
261C8—6322no—
262C8—6322no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—44no—
268C22—82no—
269C22—162no—
270C22#1909841911yescharacteristic
271C22#1909853712yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2#2376331811yescharacteristic
282C2#2376335112yes—
subgroupG/H
centerC22#19098419
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#3491C2
socleC22×C79#379
prime radical#534×C79C4
cosocleC23×C4×C79C23
hypercenter#3491D79
hypocenterC79#8540
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #23763483C2semidirect
2. #23763471C2semidirect
3. #23761748C2semidirect
146. C79#8540semidirect

Maximal subgroups

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

subgroupindex
#237634832
#237634712
#237617482
C79×#34912
#854079