Finite Group Databasebeta

← #34197677

Subgroups — #34197677

Order 20224.

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

normal subgroups only

This group has 434 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#23761730C212yesmaximal
2#23761346C212yesmaximal, complemented
3#23761344C212yesmaximal, complemented
4C79×#2317C211yesmaximal, characteristic
5#19098527—22no—
6#19098289—22no—
7#19098289C2214yes—
8C2×#117×C79C411yescharacteristic
9C2×#117×C79C411yescharacteristic
10C22×C4⋊C4×C79C2211yescharacteristic
11#389×C79—22no—
12#385×C79—22no—
13#384×C79C411yescharacteristic
14#384×C79C2211yescharacteristic
15#382×C79C411yescharacteristic
16#382×C79C2211yescharacteristic
17C2×#11929608—42no—
18C22⋊(C79⋊C8)—24no—
19C22⋊(C79⋊C8)—42no—
20C23×C4×C79C2×C411yescharacteristic
21C23×C4×C79D412yes—
22#117×C79—22no—
23#117×C79—22no—
24#117×C79—42no—
25C2×C4⋊C4×C79—21no—
26C2×C4⋊C4×C79—22no—
27C2×C4⋊C4×C79—22no—
28C2×C4⋊C4×C79—42no—
29C2×C4⋊C4×C79C2×C411yescharacteristic
30C2×C4⋊C4×C79C2×C411yescharacteristic
31C2×C4⋊C4×C79C2×C411yescharacteristic
32C2×C4⋊C4×C79D412yes—
33C2×C4⋊C4×C79C2311yescharacteristic
34C2×C22⋊C4×C79—21no—
35C2×C22⋊C4×C79—21no—
36C2×C42×C79—21no—
37C2×C42×C79C2×C411yescharacteristic
38C2×C42×C79C2×C411yescharacteristic
39#95×C79—22no—
40#95×C79—22no—
41#11929608—84no—
42C2×C79⋊C8—44no—
43C24×C79C22⋊C411yescharacteristic
44C22×C4×C79—22no—
45C22×C4×C79—22no—
46C22×C4×C79—21no—
47C22×C4×C79—21no—
48C22×C4×C79—22no—
49C22×C4×C79—21no—
50C22×C4×C79—21no—
51C22×C4×C79—22no—
52C22×C4×C79—21no—
53C22×C4×C79—21no—
54C22×C4×C79—82no—
55C22×C4×C79C22⋊C411yescharacteristic
56C22×C4×C79C22⋊C412yes—
57C22×C4×C79C22×C411yescharacteristic
58C22×C4×C79C2×D412yes—
59C4⋊C4×C79—24no—
60C4⋊C4×C79—41no—
61C4⋊C4×C79—42no—
62C4⋊C4×C79—41no—
63C4⋊C4×C79—41no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—82no—
66C4⋊C4×C79—82no—
67C22⋊C4×C79—24no—
68C22⋊C4×C79—42no—
69C42×C79—21no—
70C42×C79—21no—
71C42×C79—42no—
72C42×C79M1612yescomplemented
73C42×C79M1612yescomplemented
74C23×C79—21no—
75C23×C79—21no—
76C23×C79—41no—
77C23×C79—41no—
78C23×C79#9912yes—
79C23×C79C2×C22⋊C411yescharacteristic
80C2×C4×C79—24no—
81C2×C4×C79—24no—
82C2×C4×C79—21no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—44no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—41no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—82no—
102C2×C4×C79#10112yes—
103C2×C4×C79C2×M1611yescharacteristic
104C2×C4×C79C2×M1611yescharacteristic
105C79⋊C8—84no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79—22no—
109C22×C79—22no—
110C22×C79—22no—
111C22×C79—42no—
112C22×C79—42no—
113C22×C79#40711yescharacteristic
114C22×C79C2×#9911yescharacteristic
115C22×C79C2×#10111yescharacteristic
116C4×C79—22no—
117C4×C79—21no—
118C4×C79—21no—
119C4×C79—42no—
120C4×C79—42no—
121C4×C79—42no—
122C4×C79—42no—
123C4×C79—81no—
124C4×C79—81no—
125#11514—791nomaximal, Sylow
126C2×C79—21no—
127C2×C79—21no—
128C2×C79—42no—
129C2×C79#146311yescharacteristic
130C2×C79#212711yescharacteristic
131C2×C79#213311yescharacteristic
132#2317D7911yescharacteristic
133#1463—792no—
134#1325—792no—
135#1324—792no—
136C79#1151411yescharacteristic, complemented, Sylow
137C2×#117Dic7911yescharacteristic
138C2×#117Dic7911yescharacteristic
139C22×C4⋊C4C2×D7911yescharacteristic
140#407—1582no—
141#389—22no—
142#385—22no—
143#384Dic7911yescharacteristic
144#384C2×D7911yescharacteristic
145#382Dic7911yescharacteristic
146#382C2×D7911yescharacteristic
147#324—794no—
148#324—1582no—
149C23×C4C2×Dic7911yescharacteristic
150C23×C4C22⋊D7912yes—
151C2×M16—3162no—
152#117—22no—
153#117—22no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C2×Dic7911yescharacteristic
160C2×C4⋊C4C2×Dic7911yescharacteristic
161C2×C4⋊C4C2×Dic7911yescharacteristic
162C2×C4⋊C4C22⋊D7912yes—
163C2×C4⋊C4C22×D7911yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42C2×Dic7911yescharacteristic
168C2×C42C2×Dic7911yescharacteristic
169C22⋊C8—3162no—
170C22⋊C8—1584no—
171#95—22no—
172#95—22no—
173C24C22⋊Dic7911yescharacteristic
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—82no—
185C22×C4C22⋊Dic7911yescharacteristic
186C22×C4C22⋊Dic7912yes—
187C22×C4C22×Dic7911yescharacteristic
188C22×C4C2×C22⋊D7912yes—
189M16—6324no—
190C2×C8—3164no—
191C4⋊C4—24no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C42—21no—
202C42—21no—
203C42—42no—
204C42#1192960812yescomplemented
205C42#1192960812yescomplemented
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23#1457907612yes—
211C23C2×C22⋊Dic7911yescharacteristic
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4#1457908012yes—
235C2×C4C2×#1192960811yescharacteristic
236C2×C4C2×#1192960811yescharacteristic
237C8—6324no—
238C22—21no—
239C22—21no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22#1909852711yescharacteristic
246C22C2×#1457907611yescharacteristic
247C22C2×#1457908011yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—81no—
256C4—81no—
257C2—21no—
258C2—21no—
259C2—42no—
260C2#2376173011yescharacteristic
261C2#2376506911yescharacteristic
262C2#2376508911yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2317C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2317D79
hypocenterC79#11514
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #23761346C2semidirect
3. #23761344C2semidirect
72. C42×C79M16semidirect
73. C42×C79M16semidirect
136. C79#11514semidirect
204. C42#11929608semidirect
205. C42#11929608semidirect

Maximal subgroups

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

subgroupindex
#237617302
#237613462
#237613442
C79×#23172
#1151479