Finite Group Databasebeta

← #34197673

Subgroups — #34197673

Order 20224.

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

normal subgroups only

This group has 446 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#23761734C212yesmaximal
2#23761346C212yesmaximal, complemented
3#23761344C212yesmaximal, complemented
4C79×#2285C211yesmaximal, characteristic
5#19098620—22no—
6#19098289C2214yes—
7#19098287—22no—
8C2×#117×C79C411yescharacteristic
9C2×#117×C79C2211yescharacteristic
10C22×C42×C79C411yescharacteristic
11#384×C79C411yescharacteristic
12#384×C79C2211yescharacteristic
13#382×C79C411yescharacteristic
14#382×C79C2211yescharacteristic
15#380×C79—22no—
16C4×C4⋊C4×C79—22no—
17C4⋊(C79⋊C8)—42no—
18C22⋊(C79⋊C8)—24no—
19#14579072—42no—
20C23×C4×C79—21no—
21C23×C4×C79C2×C411yescharacteristic
22#117×C79—41no—
23#117×C79—41no—
24#117×C79—41no—
25#117×C79—41no—
26C2×C4⋊C4×C79—22no—
27C2×C4⋊C4×C79—22no—
28C2×C4⋊C4×C79C2×C411yescharacteristic
29C2×C4⋊C4×C79C2×C411yescharacteristic
30C2×C4⋊C4×C79C2×C411yescharacteristic
31C2×C4⋊C4×C79C2311yescharacteristic
32C2×C22⋊C4×C79—21no—
33C2×C22⋊C4×C79D412yes—
34C2×C42×C79—21no—
35C2×C42×C79—22no—
36C2×C42×C79—22no—
37C2×C42×C79—21no—
38C2×C42×C79C2×C411yescharacteristic
39C2×C42×C79C2×C411yescharacteristic
40C2×C42×C79D412yes—
41#95×C79—22no—
42#95×C79—22no—
43C2×C79⋊C8—44no—
44C24×C79C22⋊C411yescharacteristic
45C22×C4×C79—22no—
46C22×C4×C79—21no—
47C22×C4×C79—21no—
48C22×C4×C79—21no—
49C22×C4×C79—21no—
50C22×C4×C79—21no—
51C22×C4×C79—24no—
52C22×C4×C79—21no—
53C22×C4×C79—21no—
54C22×C4×C79—21no—
55C22×C4×C79—42no—
56C22×C4×C79—42no—
57C22×C4×C79C22⋊C411yescharacteristic
58C22×C4×C79C22⋊C412yes—
59C22×C4×C79C22×C411yescharacteristic
60C22×C4×C79C2×D412yes—
61C4⋊C4×C79—41no—
62C4⋊C4×C79—41no—
63C4⋊C4×C79—44no—
64C4⋊C4×C79—44no—
65C4⋊C4×C79—41no—
66C4⋊C4×C79—41no—
67C22⋊C4×C79—42no—
68C22⋊C4×C79—42no—
69C42×C79—24no—
70C42×C79—22no—
71C42×C79—22no—
72C42×C79—21no—
73C42×C79—21no—
74C42×C79—41no—
75C42×C79—41no—
76C42×C79M1612yes—
77C42×C79M1612yes—
78C23×C79—21no—
79C23×C79—21no—
80C23×C79—41no—
81C23×C79—41no—
82C23×C79#10112yes—
83C23×C79C2×C22⋊C411yescharacteristic
84C2×C4×C79—22no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—22no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—48no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79#9911yescharacteristic
108C2×C4×C79#9911yescharacteristic
109C2×C4×C79C2×M1611yescharacteristic
110C2×C4×C79C2×M1611yescharacteristic
111C79⋊C8—84no—
112C22×C79—21no—
113C22×C79—21no—
114C22×C79—22no—
115C22×C79—42no—
116C22×C79—42no—
117C22×C79—41no—
118C22×C79—41no—
119C22×C79#40711yescharacteristic
120C22×C79C2×#9911yescharacteristic
121C22×C79C2×#10111yescharacteristic
122C4×C79—24no—
123C4×C79—21no—
124C4×C79—21no—
125C4×C79—21no—
126C4×C79—21no—
127C4×C79—42no—
128C4×C79—42no—
129C4×C79—42no—
130C4×C79—81no—
131C4×C79—81no—
132#11513—791nomaximal, Sylow
133C2×C79—21no—
134C2×C79—21no—
135C2×C79—42no—
136C2×C79#146311yescharacteristic
137C2×C79#212611yescharacteristic
138C2×C79#213211yescharacteristic
139#2285D7911yescharacteristic
140#1464—792no—
141#1325—792no—
142#1324—792no—
143C79#1151311yescharacteristic, complemented, Sylow
144C2×#117Dic7911yescharacteristic
145C2×#117C2×D7911yescharacteristic
146C22×C42Dic7911yescharacteristic
147#432—1582no—
148#384Dic7911yescharacteristic
149#384C2×D7911yescharacteristic
150#382Dic7911yescharacteristic
151#382C2×D7911yescharacteristic
152#380—22no—
153C4×C4⋊C4—22no—
154#324—794no—
155#323—1582no—
156C23×C4—21no—
157C23×C4C2×Dic7911yescharacteristic
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic7911yescharacteristic
165C2×C4⋊C4C2×Dic7911yescharacteristic
166C2×C4⋊C4C2×Dic7911yescharacteristic
167C2×C4⋊C4C22×D7911yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4C22⋊D7912yes—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42C2×Dic7911yescharacteristic
175C2×C42C2×Dic7911yescharacteristic
176C2×C42C22⋊D7912yes—
177C4⋊C8—3162no—
178C22⋊C8—1584no—
179#97—3162no—
180#95—22no—
181#95—22no—
182C24C22⋊Dic7911yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C22⋊Dic7912yes—
196C22×C4C22⋊Dic7911yescharacteristic
197C22×C4C22×Dic7911yescharacteristic
198C22×C4C2×C22⋊D7912yes—
199C2×C8—3164no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42—41no—
214C42—41no—
215C42#1192960812yes—
216C42#1192960812yes—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23#1457908012yes—
222C23C2×C22⋊Dic7911yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4#1457907611yescharacteristic
247C2×C4#1457907611yescharacteristic
248C2×C4C2×#1192960811yescharacteristic
249C2×C4C2×#1192960811yescharacteristic
250C8—6324no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22#1909852711yescharacteristic
259C22C2×#1457907611yescharacteristic
260C22C2×#1457908011yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—21no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—81no—
270C4—81no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2#2376173011yescharacteristic
275C2#2376506511yescharacteristic
276C2#2376508511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2285C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#2285D79
hypocenterC79#11513
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #23761346C2semidirect
3. #23761344C2semidirect
143. C79#11513semidirect

Maximal subgroups

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

subgroupindex
#237617342
#237613462
#237613442
C79×#22852
#1151379