Finite Group Databasebeta

← #34185811

Subgroups — #34185811

Order 20224.

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

normal subgroups only

This group has 422 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#23773780C211yesmaximal, characteristic
2#23762126C211yesmaximal, characteristic
3#23761950C212yesmaximal
4#23761264C212yesmaximal
5C79×#2381C211yesmaximal, characteristic
6#19099120C2211yescharacteristic
7#19098931—22no—
8#19098913C414yescomplemented
9#19098620C2212yes—
10#19098572—22no—
11C4×#11929608C2211yescharacteristic
12#19098312—22no—
13#19098299—24no—
14C2×C4⋊Q8×C79C2211yescharacteristic
15#391×C79—22no—
16#387×C79—24no—
17#383×C79C2212yes—
18C2×#11929608—21no—
19#14579161C2×C412yes—
20#14579153—21no—
21#14579140—22no—
22C4×Dic158—22no—
23#14579120—21no—
24C4⋊(C79⋊C8)—22no—
25#14579085—28no—
26C22⋊(C79⋊C8)—22no—
27#14579072—21no—
28#14579072—22no—
29C4×C79⋊C8C2×C412yes—
30C22×Q8×C79D412yes—
31C4⋊Q8×C79—22no—
32C4⋊Q8×C79C2×C412yes—
33C4⋊Q8×C79D412yes—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—22no—
36C2×C42×C79—22no—
37C2×C42×C79C2311yescharacteristic
38#95×C79—24no—
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—
51C2×Q8×C79—22no—
52C2×Q8×C79—42no—
53C2×Q8×C79—42no—
54C2×Q8×C79C22⋊C414yes—
55C22×C4×C79—22no—
56C22×C4×C79—24no—
57C22×C4×C79C2×D411yescharacteristic
58C22×C4×C79C2×D412yes—
59C4⋊C4×C79—24no—
60C4⋊C4×C79—24no—
61C4⋊C4×C79—82no—
62C42×C79—82no—
63C42×C79C22×C411yescharacteristic
64C42×C79C2×D411yescharacteristic
65C42×C79#4112yes—
66C23×C79#12011yescharacteristic
67Q8×C79—44no—
68Q8×C79—82no—
69C2×C4×C79—22no—
70C2×C4×C79—24no—
71C2×C4×C79—42no—
72C2×C4×C79—42no—
73C2×C4×C79—42no—
74C2×C4×C79—48no—
75C2×C4×C79—44no—
76C2×C4×C79C2×C22⋊C411yescharacteristic
77C2×C4×C79C4×D412yes—
78C2×C4×C79#12112yes—
79C2×C4×C79#12311yescharacteristic
80C2×Dic79—42no—
81Dic158—82no—
82C79⋊C8—42no—
83C79⋊C8—82no—
84C22×C79—22no—
85C22×C79#38611yescharacteristic
86C22×C79#45512yes—
87C4×C79—21no—
88C4×C79—21no—
89C4×C79—22no—
90C4×C79—22no—
91C4×C79—44no—
92C4×C79—84no—
93Dic79—82no—
94#9426—791nomaximal, Sylow
95C2×C79—22no—
96C2×C79#188511yescharacteristic
97C2×C79#190512yes—
98#3139—791no—
99#2381D7911yescharacteristic
100#1558—791no—
101#1515—792no—
102#1288—792no—
103C79#942611yescharacteristic, complemented, Sylow
104#533—791no—
105C2×C4⋊Q8C2×D7911yescharacteristic
106#491—1582no—
107#487—794no—
108#432—792no—
109#419—1582no—
110C4×M16—791no—
111#391—22no—
112#387—24no—
113#383C2×D7912yes—
114#333—1582no—
115#328—1584no—
116C22×Q8C22⋊D7912yes—
117C2×M16—1581no—
118C4⋊Q8—22no—
119C4⋊Q8—792no—
120C4⋊Q8C4×D7912yes—
121C4⋊Q8D31612yes—
122#125—1581no—
123C22⋊Q8—1582no—
124C4×Q8—1582no—
125#117—1581no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C42—22no—
129C2×C42C22×D7911yescharacteristic
130C4⋊C8—1582no—
131#103—1588no—
132C22⋊C8—1582no—
133#97—1581no—
134#97—1582no—
135C4×C8—792no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95C22⋊D7914yes—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—1582no—
144C2×Q8(C2×D79)⋊C414yes—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4C2×C22⋊D7911yescharacteristic
148C22×C4C2×C22⋊D7912yes—
149M16—3162no—
150C2×C8—3162no—
151C2×C8—1582no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—82no—
155C4⋊C4—3162no—
156C4⋊C4—1582no—
157C4⋊C4—1582no—
158C22⋊C4—3161no—
159C42—82no—
160C42—3161no—
161C42C2×C4×D7911yescharacteristic
162C42C2×D31611yescharacteristic
163C42#1192962412yes—
164C23#1457913011yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—6322no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—3162no—
176C2×C4C2×(C2×D79)⋊C411yescharacteristic
177C2×C4C4×C22⋊D7912yes—
178C2×C4#1457913512yes—
179C2×C4#1457914411yescharacteristic
180C8—3162no—
181C8—6322no—
182C22—22no—
183C22#1909844211yescharacteristic
184C22#1909874112yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—84no—
191C4—6322no—
192C2—22no—
193C2#2376378811yescharacteristic
194C2#2376389412yes—
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2381C2
socleC22×C79#386
prime radicalC23×C79#120
cosocleC2×C42×C79C23
hypercenter#2381D79
hypocenterC79#9426
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
8. #19098913C4semidirect
103. C79#9426semidirect

Maximal subgroups

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

subgroupindex
#237737802
#237621262
#237619502
#237612642
C79×#23812
#942679