Finite Group Databasebeta

← #34185806

Subgroups — #34185806

Order 20224.

534 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 99 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#23762114C211yesmaximal, characteristic
3#23761950C212yesmaximal, complemented
4#23761264C212yesmaximal, complemented
5C79×#2380C211yesmaximal, characteristic
6#19099120C2211yescharacteristic
7#19098927—22no—
8#19098909C414yescomplemented
9#19098620C2212yes—
10#19098568—22no—
11C4×#11929608C2211yescharacteristic
12#19098307—22no—
13#19098299—24no—
14C2×#127×C79C2211yescharacteristic
15#386×C79—22no—
16#386×C79—24no—
17#383×C79C2212yes—
18C2×#11929608—21no—
19#14579161C2×C412yescomplemented
20#14579153—21no—
21#14579140—22no—
22C4×Dic158—22no—
23#14579120—21no—
24C4⋊(C79⋊C8)—22no—
25#14579082—28no—
26C22⋊(C79⋊C8)—22no—
27#14579072—21no—
28#14579072—22no—
29C4×C79⋊C8C2×C412yescomplemented
30C22×D4×C79—22no—
31C22×D4×C79D412yes—
32C23×C4×C79—22no—
33#127×C79—22no—
34#127×C79C2×C412yes—
35#127×C79D412yes—
36C2×C22⋊C4×C79—24no—
37C2×C22⋊C4×C79—22no—
38C2×C42×C79C2311yescharacteristic
39#95×C79—22no—
40#95×C79—22no—
41#95×C79D414yes—
42C2×Dic158—22no—
43#11929608—42no—
44C2×C79⋊C8—22no—
45C2×C79⋊C8—42no—
46C4⋊Dic79—22no—
47C4⋊Dic79—22no—
48Dic79⋊C4—42no—
49C22⋊Dic79—41no—
50C4×Dic79—41no—
51C24×C79—22no—
52C2×D4×C79—24no—
53C2×D4×C79—24no—
54C2×D4×C79—22no—
55C2×D4×C79—42no—
56C2×D4×C79—42no—
57C2×D4×C79—82no—
58C2×D4×C79—82no—
59C2×D4×C79C22⋊C414yes—
60C22×C4×C79—24no—
61C22×C4×C79—84no—
62C22×C4×C79—82no—
63C22×C4×C79C2×D411yescharacteristic
64C22×C4×C79C2×D412yes—
65C22⋊C4×C79—84no—
66C22⋊C4×C79—82no—
67C42×C79C22×C411yescharacteristic
68C42×C79C2×D411yescharacteristic
69C42×C79#4112yes—
70C23×C79—24no—
71C23×C79—44no—
72C23×C79—44no—
73C23×C79—42no—
74C23×C79—42no—
75C23×C79#12011yescharacteristic
76D4×C79—44no—
77D4×C79—82no—
78D4×C79—84no—
79D4×C79—84no—
80C2×C4×C79—22no—
81C2×C4×C79—42no—
82C2×C4×C79—48no—
83C2×C4×C79—44no—
84C2×C4×C79—88no—
85C2×C4×C79C2×C22⋊C411yescharacteristic
86C2×C4×C79C4×D412yes—
87C2×C4×C79#12112yes—
88C2×C4×C79#12311yescharacteristic
89C2×Dic79—42no—
90Dic158—82no—
91C79⋊C8—42no—
92C79⋊C8—82no—
93C22×C79—22no—
94C22×C79—44no—
95C22×C79—48no—
96C22×C79—82no—
97C22×C79—82no—
98C22×C79—84no—
99C22×C79#38611yescharacteristic
100C22×C79#45512yes—
101C4×C79—21no—
102C4×C79—21no—
103C4×C79—22no—
104C4×C79—22no—
105C4×C79—84no—
106Dic79—82no—
107#9425—791nomaximal, Sylow
108C2×C79—22no—
109C2×C79—84no—
110C2×C79#188511yescharacteristic
111C2×C79#190412yes—
112#3138—791no—
113#2380D7911yescharacteristic
114#1555—791no—
115#1515—792no—
116#1288—792no—
117C79#942511yescharacteristic, complemented, Sylow
118#533—791no—
119C2×#127C2×D7911yescharacteristic
120#490—1582no—
121#486—794no—
122#432—792no—
123#418—1582no—
124C4×M16—791no—
125#386—22no—
126#386—24no—
127#383C2×D7912yes—
128#331—1582no—
129#328—1584no—
130C22×D4—22no—
131C22×D4C22⋊D7912yes—
132C23×C4—22no—
133C2×M16—1581no—
134C4⋊Q8—792no—
135#127—22no—
136#127C4×D7912yes—
137#127D31612yes—
138#125—1581no—
139C22⋊Q8—1582no—
140C4×Q8—1582no—
141#117—1581no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—22no—
144C2×C42C22×D7911yescharacteristic
145C4⋊C8—1582no—
146#102—1588no—
147C22⋊C8—1582no—
148#97—1581no—
149#97—1582no—
150C4×C8—792no—
151#95—22no—
152#95—22no—
153#95C22⋊D7914yes—
154C24—22no—
155C2×Q8—1582no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4(C2×D79)⋊C414yes—
164C22×C4—24no—
165C22×C4—84no—
166C22×C4—82no—
167C22×C4C2×C22⋊D7911yescharacteristic
168C22×C4C2×C22⋊D7912yes—
169M16—3162no—
170C2×C8—3162no—
171C2×C8—1582no—
172C4⋊C4—3162no—
173C4⋊C4—1582no—
174C4⋊C4—1582no—
175C22⋊C4—84no—
176C22⋊C4—82no—
177C22⋊C4—3161no—
178C42—3161no—
179C42C2×C4×D7911yescharacteristic
180C42C2×D31611yescharacteristic
181C42#1192962412yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1457913011yescharacteristic
188Q8—6322no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—48no—
196C2×C4—44no—
197C2×C4—88no—
198C2×C4—3162no—
199C2×C4C2×(C2×D79)⋊C411yescharacteristic
200C2×C4C4×C22⋊D7912yes—
201C2×C4#1457913512yes—
202C2×C4#1457914411yescharacteristic
203C8—3162no—
204C8—6322no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1909844211yescharacteristic
212C22#1909874112yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—84no—
218C4—6322no—
219C2—22no—
220C2—84no—
221C2#2376378811yescharacteristic
222C2#2376388712yes—
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2380C2
socleC22×C79#386
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenter#2380D79
hypocenterC79#9425
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2380

Complements

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

NH ≅ G/Ntype
3. #23761950C2semidirect
4. #23761264C2semidirect
8. #19098909C4semidirect
19. #14579161C2×C4semidirect
29. C4×C79⋊C8C2×C4semidirect
117. C79#9425semidirect

Maximal subgroups

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

subgroupindex
#237737742
#237621142
#237619502
#237612642
C79×#23802
#942579