Finite Group Databasebeta

← #34180783

Subgroups — #34180783

Order 20224.

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

normal subgroups only

This group has 690 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×#19099175C211yesmaximal, characteristic
2#23763483C211yesmaximal, characteristic
3#23761932C214yesmaximal
4C79×#1820C211yesmaximal, characteristic
5#19099175—28no—
6C2×#14579161C411yescharacteristic, complemented
7C2×#14579154—22no—
8C2×#14579153C411yescharacteristic, complemented
9C2×#14579151—22no—
10C2×#14579140C2212yes—
11C2×C4×Dic158C412yescomplemented
12#19098527C2212yes—
13#19098332—22no—
14#19098299—28no—
15C2×#117×C79C2211yescharacteristic
16#381×C79C2212yes—
17#342×C79—22no—
18C22×Dic158C2×C412yes—
19C2×#11929608—22no—
20#14579161—24no—
21#14579154—216no—
22#14579153—24no—
23#14579151—216no—
24#14579140—24no—
25#14579140D418yes—
26C4×Dic158—24no—
27C4×Dic158—24no—
28C2×C4⋊Dic79—22no—
29C2×C4⋊Dic79C2×C412yes—
30C2×Dic79⋊C4—22no—
31C2×Dic79⋊C4—22no—
32C2×Dic79⋊C4—22no—
33C2×C22⋊Dic79—22no—
34C2×C4×Dic79—22no—
35C22⋊(C79⋊C8)—24no—
36C23×C4×C79C2311yescharacteristic
37#117×C79—24no—
38C2×C4⋊C4×C79D412yes—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79D412yes—
41C2×C42×C79C2×C411yescharacteristic
42C2×C42×C79C2×C411yescharacteristic
43#95×C79—24no—
44C2×Dic158—22no—
45C2×Dic158—44no—
46C2×Dic158C22⋊C414yes—
47C22×Dic79—22no—
48C22×Dic79—22no—
49#11929608—44no—
50C2×C79⋊C8—42no—
51C4⋊Dic79—28no—
52C4⋊Dic79—22no—
53C4⋊Dic79C22⋊C414yes—
54Dic79⋊C4—28no—
55Dic79⋊C4—28no—
56Dic79⋊C4—28no—
57C22⋊Dic79—28no—
58C4×Dic79—28no—
59C24×C79C2×D411yescharacteristic
60C22×C4×C79—42no—
61C22×C4×C79—42no—
62C22×C4×C79C22⋊C412yes—
63C22×C4×C79C22⋊C412yes—
64C22×C4×C79C22×C411yescharacteristic
65C22×C4×C79C2×D411yescharacteristic
66C22×C4×C79C2×D414yes—
67C4⋊C4×C79—24no—
68C22⋊C4×C79—24no—
69C22⋊C4×C79—44no—
70C42×C79—22no—
71C42×C79—22no—
72C23×C79—22no—
73C23×C79—42no—
74C23×C79C2×C22⋊C411yescharacteristic
75C23×C79#12012yes—
76C2×C4×C79—22no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—44no—
81C2×C4×C79—42no—
82C2×C4×C79—42no—
83C2×C4×C79—82no—
84C2×C4×C79—84no—
85C2×C4×C79C2×C22⋊C412yes—
86C2×C4×C79#12012yes—
87C2×Dic79—24no—
88C2×Dic79—24no—
89C2×Dic79—44no—
90C2×Dic79—44no—
91Dic158—44no—
92Dic158—44no—
93C79⋊C8—82no—
94C22×C79—22no—
95C22×C79—24no—
96C22×C79—22no—
97C22×C79—82no—
98C22×C79#37911yescharacteristic
99C22×C79#41012yes—
100C4×C79—22no—
101C4×C79—22no—
102C4×C79—42no—
103C4×C79—42no—
104C4×C79—162no—
105Dic79—44no—
106Dic79—44no—
107#8521—791nomaximal, Sylow
108C2×C79—22no—
109C2×C79—42no—
110C2×C79#178211yescharacteristic
111C2×C79#179612yes—
112C2×#541—791no—
113#1822—791no—
114#1820D7911yescharacteristic
115#1511—794no—
116C79#852111yescharacteristic, complemented, Sylow
117#541—1588no—
118C2×C4⋊Q8—791no—
119C2×#126—1582no—
120C2×#125—791no—
121C2×#125—1582no—
122C2×C22⋊Q8—792no—
123C2×C4×Q8—792no—
124C2×#117C2×D7911yescharacteristic
125#407—792no—
126#381C2×D7912yes—
127#343—1582no—
128#342—22no—
129#328—1588no—
130C22×Q8—792no—
131C23×C4C22×D7911yescharacteristic
132C2×M16—1582no—
133C4⋊Q8—1584no—
134#126—15816no—
135#125—1584no—
136#125—15816no—
137C22⋊Q8—798no—
138C22⋊Q8—1584no—
139C4×Q8—1584no—
140C4×Q8—1584no—
141#117—24no—
142C2×C4⋊C4—792no—
143C2×C4⋊C4—1582no—
144C2×C4⋊C4—1582no—
145C2×C4⋊C4—1582no—
146C2×C4⋊C4—1582no—
147C2×C4⋊C4D31612yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1582no—
150C2×C22⋊C4C22⋊D7912yes—
151C2×C42—1582no—
152C2×C42C4×D7911yescharacteristic
153C2×C42C4×D7911yescharacteristic
154C22⋊C8—1584no—
155#95—24no—
156C24C2×C22⋊D7911yescharacteristic
157C2×Q8—3164no—
158C2×Q8—794no—
159C2×Q8—1582no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—1582no—
163C22×C4—1582no—
164C22×C4(C2×D79)⋊C412yes—
165C22×C4(C2×D79)⋊C412yes—
166C22×C4C2×C4×D7911yescharacteristic
167C22×C4D4×D7914yes—
168C22×C4C2×D31611yescharacteristic
169M16—3164no—
170C2×C8—3162no—
171C4⋊C4—24no—
172C4⋊C4—794no—
173C4⋊C4—1588no—
174C4⋊C4—1588no—
175C4⋊C4—1588no—
176C4⋊C4—1582no—
177C4⋊C4—1588no—
178C22⋊C4—24no—
179C22⋊C4—44no—
180C22⋊C4—1588no—
181C42—22no—
182C42—22no—
183C42—1588no—
184C23—22no—
185C23—42no—
186C23C2×(C2×D79)⋊C411yescharacteristic
187C23#1457912912yes—
188Q8—3164no—
189Q8—3164no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—82no—
198C2×C4—84no—
199C2×C4—3164no—
200C2×C4—3164no—
201C2×C4—1584no—
202C2×C4—1584no—
203C2×C4C22⋊C4×D7912yes—
204C2×C4#1457913112yes—
205C8—6322no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1909841811yescharacteristic
211C22#1909853612yes—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—162no—
217C4—3164no—
218C4—3164no—
219C2—22no—
220C2—42no—
221C2#2376331711yescharacteristic
222C2#2376337312yes—
subgroupG/H
centerC22#19098418
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1820C2
socleC22×C79#379
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#1820D79
hypocenterC79#8521
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
6. C2×#14579161C4semidirect
8. C2×#14579153C4semidirect
11. C2×C4×Dic158C4semidirect
116. C79#8521semidirect

Maximal subgroups

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

subgroupindex
C2×#190991752
#237634832
#237619322
C79×#18202
#852179