Finite Group Databasebeta

← #34197434

Subgroups — #34197434

Order 20224.

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

normal subgroups only

This group has 434 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#23761714C212yesmaximal
2#23761344C214yesmaximal, complemented
3C79×#2317C211yesmaximal, characteristic
4#19098616—22no—
5#19098289C2214yes—
6#19098287—22no—
7C2×#117×C79C411yescharacteristic
8C2×#117×C79C2211yescharacteristic
9C22×C4⋊C4×C79C411yescharacteristic
10#389×C79—22no—
11#385×C79—22no—
12#384×C79C412yes—
13#382×C79C2212yes—
14C4⋊(C79⋊C8)—42no—
15C22⋊(C79⋊C8)—24no—
16C4×C79⋊C8—42no—
17C23×C4×C79—21no—
18C23×C4×C79C2×C411yescharacteristic
19#117×C79—22no—
20#117×C79—42no—
21#117×C79C814yescomplemented
22C2×C4⋊C4×C79—21no—
23C2×C4⋊C4×C79—21no—
24C2×C4⋊C4×C79—24no—
25C2×C4⋊C4×C79—42no—
26C2×C4⋊C4×C79C2×C412yes—
27C2×C4⋊C4×C79C2×C411yescharacteristic
28C2×C4⋊C4×C79C2311yescharacteristic
29C2×C22⋊C4×C79—21no—
30C2×C22⋊C4×C79D412yes—
31C2×C42×C79C2×C412yes—
32C2×C42×C79D412yes—
33#95×C79—24no—
34C2×C79⋊C8—44no—
35C24×C79C22⋊C411yescharacteristic
36C22×C4×C79—22no—
37C22×C4×C79—22no—
38C22×C4×C79—22no—
39C22×C4×C79—22no—
40C22×C4×C79—42no—
41C22×C4×C79—82no—
42C22×C4×C79C22⋊C412yes—
43C22×C4×C79C22⋊C411yescharacteristic
44C22×C4×C79C2×C812yes—
45C22×C4×C79M1612yes—
46C22×C4×C79C22×C411yescharacteristic
47C22×C4×C79C2×D412yes—
48C4⋊C4×C79—24no—
49C4⋊C4×C79—41no—
50C4⋊C4×C79—41no—
51C4⋊C4×C79—42no—
52C4⋊C4×C79—42no—
53C4⋊C4×C79—84no—
54C22⋊C4×C79—24no—
55C22⋊C4×C79—42no—
56C42×C79—22no—
57C42×C79—22no—
58C42×C79—22no—
59C42×C79C2×C814yes—
60C23×C79—21no—
61C23×C79—21no—
62C23×C79—41no—
63C23×C79—41no—
64C23×C79C22⋊C812yes—
65C23×C79C2×C22⋊C411yescharacteristic
66C2×C4×C79—21no—
67C2×C4×C79—21no—
68C2×C4×C79—24no—
69C2×C4×C79—24no—
70C2×C4×C79—24no—
71C2×C4×C79—22no—
72C2×C4×C79—22no—
73C2×C4×C79—42no—
74C2×C4×C79—42no—
75C2×C4×C79—42no—
76C2×C4×C79—42no—
77C2×C4×C79—44no—
78C2×C4×C79—42no—
79C2×C4×C79—84no—
80C2×C4×C79C22⋊C812yes—
81C2×C4×C79#9912yes—
82C2×C4×C79#10112yes—
83C2×C4×C79C22×C811yescharacteristic
84C2×C4×C79C2×M1611yescharacteristic
85C79⋊C8—84no—
86C22×C79—21no—
87C22×C79—21no—
88C22×C79—22no—
89C22×C79—42no—
90C22×C79—42no—
91C22×C79—41no—
92C22×C79—41no—
93C22×C79C2×C22⋊C811yescharacteristic
94C22×C79C2×#9911yescharacteristic
95C22×C79C2×#10111yescharacteristic
96C4×C79—21no—
97C4×C79—21no—
98C4×C79—44no—
99C4×C79—42no—
100C4×C79—42no—
101C4×C79—82no—
102C4×C79#32414yes—
103#11449—791nomaximal, Sylow
104C2×C79—21no—
105C2×C79—21no—
106C2×C79—42no—
107C2×C79C2×#32411yescharacteristic
108C2×C79#212911yescharacteristic
109C2×C79#213411yescharacteristic
110#2317D7911yescharacteristic
111#1458—792no—
112#1324—794no—
113C79#1144911yescharacteristic, complemented, Sylow
114C2×#117Dic7911yescharacteristic
115C2×#117C2×D7911yescharacteristic
116C22×C4⋊C4Dic7911yescharacteristic
117#431—1582no—
118#389—22no—
119#385—22no—
120#384Dic7912yes—
121#382C2×D7912yes—
122#324—794no—
123#323—1582no—
124C23×C4—21no—
125C23×C4C2×Dic7911yescharacteristic
126#117—22no—
127#117—42no—
128#117C79⋊C814yescomplemented
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4C2×Dic7912yes—
134C2×C4⋊C4C2×Dic7911yescharacteristic
135C2×C4⋊C4C22×D7911yescharacteristic
136C2×C22⋊C4—21no—
137C2×C22⋊C4C22⋊D7912yes—
138C2×C42C2×Dic7912yes—
139C2×C42C22⋊D7912yes—
140C4⋊C8—3162no—
141C22⋊C8—1584no—
142C4×C8—3162no—
143#95—24no—
144C24C22⋊Dic7911yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—82no—
151C22×C4C22⋊Dic7912yes—
152C22×C4C22⋊Dic7911yescharacteristic
153C22×C4C2×C79⋊C812yes—
154C22×C4#1192960812yes—
155C22×C4C22×Dic7911yescharacteristic
156C22×C4C2×C22⋊D7912yes—
157C2×C8—3164no—
158C4⋊C4—24no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—84no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42C2×C79⋊C814yes—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23C22⋊(C79⋊C8)12yes—
175C23C2×C22⋊Dic7911yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4C22⋊(C79⋊C8)12yes—
191C2×C4#1457907612yes—
192C2×C4#1457908012yes—
193C2×C4C22×C79⋊C811yescharacteristic
194C2×C4C2×#1192960811yescharacteristic
195C8—6324no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×C22⋊(C79⋊C8)11yescharacteristic
204C22C2×#1457907611yescharacteristic
205C22C2×#1457908011yescharacteristic
206C4—21no—
207C4—21no—
208C4—44no—
209C4—42no—
210C4—42no—
211C4—82no—
212C4#1909828914yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#1909828911yescharacteristic
217C2#2376507511yescharacteristic
218C2#2376509211yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC2×C4×C79C22×C8
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2317C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2317D79
hypocenterC79#11449
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #23761344C2semidirect
21. #117×C79C8semidirect
113. C79#11449semidirect
128. #117C79⋊C8semidirect

Maximal subgroups

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

subgroupindex
#237617142
#237613442
C79×#23172
#1144979