Finite Group Databasebeta

← #34180772

Subgroups — #34180772

Order 20224.

818 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#19099168C211yesmaximal, characteristic
2#23763477C211yesmaximal, characteristic, complemented
3#23761930C214yesmaximal, complemented
4C79×#1822C211yesmaximal, characteristic, complemented
5#19099168—28no—
6C2×#14579158C411yescharacteristic, complemented
7C2×#14579153C411yescharacteristic, complemented
8C2×#14579146—22no—
9C2×#14579135C2212yes—
10C2×#14579134—22no—
11C2×C4×D316C412yescomplemented
12#19098426C2212yes—
13#19098330—22no—
14#19098298—28no—
15C2×#117×C79C2211yescharacteristic
16#407×C79C2212yes—
17#343×C79—22no—
18C22×D316—22no—
19C22×D316C2×C412yes—
20C22×C22⋊D79—22no—
21C22×C4×D79—22no—
22#14579158—24no—
23#14579153—24no—
24#14579146—216no—
25#14579135—24no—
26#14579135D418yes—
27#14579134—216no—
28C4×D316—24no—
29C4×D316—24no—
30C2×C4⋊Dic79C2×C412yes—
31C2×Dic79⋊C4—22no—
32C2×C22⋊Dic79—22no—
33C2×(C2×D79)⋊C4—22no—
34C2×(C2×D79)⋊C4—22no—
35C79⋊#95—24no—
36C23×C4×C79C2311yescharacteristic
37C2×M16×C79—22no—
38#117×C79—24no—
39C2×C4⋊C4×C79D412yes—
40C2×C22⋊C4×C79D412yes—
41C2×C42×C79C2×C411yescharacteristic
42C2×C42×C79C2×C411yescharacteristic
43C22⋊C8×C79—24no—
44C23×D79—22no—
45C2×D316—28no—
46C2×D316—22no—
47C2×D316—44no—
48C2×D316—84no—
49C2×D316C22⋊C414yes—
50C2×C22⋊D79—28no—
51C2×C22⋊D79—84no—
52C22×Dic79—22no—
53C22×Dic79—42no—
54C2×C4×D79—28no—
55C2×C4×D79—84no—
56C4⋊Dic79—22no—
57C4⋊Dic79C22⋊C414yes—
58Dic79⋊C4—28no—
59C22⋊Dic79—44no—
60(C2×D79)⋊C4—28no—
61(C2×D79)⋊C4—28no—
62C24×C79C2×D411yescharacteristic
63C22×C4×C79—42no—
64C22×C4×C79C22⋊C412yes—
65C22×C4×C79C22⋊C412yes—
66C22×C4×C79C22×C411yescharacteristic
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D414yes—
69M16×C79—44no—
70C2×C8×C79—42no—
71C4⋊C4×C79—24no—
72C22⋊C4×C79—24no—
73C42×C79—22no—
74C42×C79—22no—
75C23×C79—22no—
76C23×C79—42no—
77C23×C79C2×C22⋊C411yescharacteristic
78C23×C79#12012yes—
79C22×D79—24no—
80C22×D79—44no—
81C22×D79—44no—
82C22×D79—44no—
83C2×C4×C79—22no—
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—44no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79C2×C22⋊C412yes—
91C2×C4×C79#12012yes—
92C22⋊D79—88no—
93C2×Dic79—24no—
94C2×Dic79—44no—
95C2×Dic79—82no—
96C2×Dic79—84no—
97D316—44no—
98D316—44no—
99D316—88no—
100C4×D79—88no—
101C8×C79—82no—
102C22×C79—22no—
103C22×C79—24no—
104C22×C79—22no—
105C22×C79—82no—
106C22×C79#37911yescharacteristic
107C22×C79#41012yes—
108C2×D79—44no—
109C2×D79—44no—
110C2×D79—44no—
111C2×D79—88no—
112C4×C79—22no—
113C4×C79—22no—
114C4×C79—42no—
115C4×C79—42no—
116Dic79—44no—
117Dic79—162no—
118#8518—791nomaximal, Sylow
119C2×C79—22no—
120C2×C79—42no—
121C2×C79#178211yescharacteristic
122C2×C79#179512yes—
123D79—84no—
124C2×#540—791no—
125#1822D7911yescharacteristic, complemented
126#1820—791no—
127#1510—794no—
128C79#851811yescharacteristic, complemented, Sylow
129#540—1588no—
130C2×#127—791no—
131C2×#125—791no—
132C2×#123—1582no—
133C2×#121—792no—
134C2×#121—1582no—
135C2×C4×D4—792no—
136C2×#117C2×D7911yescharacteristic
137#407C2×D7912yes—
138#381—792no—
139#343—22no—
140#342—1582no—
141#327—1588no—
142C22×D4—792no—
143C22×D4—1582no—
144C22×D4—1582no—
145C23×C4—1582no—
146C23×C4C22×D7911yescharacteristic
147C2×M16—22no—
148#127—1584no—
149#125—1584no—
150#123—15816no—
151#121—798no—
152#121—15816no—
153#121—1584no—
154C4×D4—1584no—
155C4×D4—1584no—
156#117—24no—
157C2×C4⋊C4—792no—
158C2×C4⋊C4—1582no—
159C2×C4⋊C4C22⋊D7912yes—
160C2×C22⋊C4—1582no—
161C2×C22⋊C4—1582no—
162C2×C22⋊C4—1582no—
163C2×C22⋊C4D31612yes—
164C2×C42C4×D7911yescharacteristic
165C2×C42C4×D7911yescharacteristic
166C22⋊C8—24no—
167#95—1584no—
168C24—1582no—
169C24C2×D31611yescharacteristic
170C2×D4—3164no—
171C2×D4—794no—
172C2×D4—6324no—
173C2×D4—6324no—
174C2×D4—1588no—
175C2×D4—1588no—
176C2×D4—1582no—
177C22×C4—42no—
178C22×C4—3162no—
179C22×C4—6324no—
180C22×C4—1588no—
181C22×C4—1582no—
182C22×C4(C2×D79)⋊C412yes—
183C22×C4(C2×D79)⋊C412yes—
184C22×C4C2×C4×D7911yescharacteristic
185C22×C4D4×D7914yes—
186C22×C4C2×C22⋊D7911yescharacteristic
187M16—44no—
188C2×C8—42no—
189C4⋊C4—24no—
190C4⋊C4—794no—
191C4⋊C4—1588no—
192C4⋊C4—1582no—
193C22⋊C4—24no—
194C22⋊C4—3164no—
195C22⋊C4—1588no—
196C22⋊C4—1588no—
197C42—22no—
198C42—22no—
199C23—22no—
200C23—42no—
201C23—3164no—
202C23—3164no—
203C23—3164no—
204C23—1584no—
205C23C2×(C2×D79)⋊C411yescharacteristic
206C23#1457913112yes—
207D4—3164no—
208D4—3164no—
209D4—6328no—
210D4—6328no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—3164no—
219C2×C4—6328no—
220C2×C4—6322no—
221C2×C4—6324no—
222C2×C4—1584no—
223C2×C4C22⋊C4×D7912yes—
224C2×C4#1457912912yes—
225C8—82no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—82no—
230C22—3164no—
231C22—3164no—
232C22—3164no—
233C22—6328no—
234C22#1909841811yescharacteristic
235C22#1909853612yes—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—3164no—
241C4—12642no—
242C2—22no—
243C2—42no—
244C2—6324no—
245C2#2376331711yescharacteristic
246C2#2376337112yes—
subgroupG/H
centerC22#19098418
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1822C2
socleC22×C79#379
prime radicalC2×#19099168C2
cosocleC23×C4×C79C23
hypercenter#1822D79
hypocenterC79#8518
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #23763477C2semidirect
3. #23761930C2semidirect
4. C79×#1822C2semidirect
6. C2×#14579158C4semidirect
7. C2×#14579153C4semidirect
11. C2×C4×D316C4semidirect
125. #1822D79semidirect
128. C79#8518semidirect

Maximal subgroups

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

subgroupindex
C2×#190991682
#237634772
#237619302
C79×#18222
#851879