Finite Group Databasebeta

← #34197633

Subgroups — #34197633

Order 20224.

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

normal subgroups only

This group has 662 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, complemented
2#23761330C212yesmaximal, complemented
3#23761328C212yesmaximal
4C79×#2280C211yesmaximal, characteristic
5#19098620—22no—
6#19098289—22no—
7#19098287C2214yes—
8C24×C4×C79C411yescharacteristic
9C2×#117×C79C411yescharacteristic
10C2×#117×C79C2211yescharacteristic
11#381×C79C411yescharacteristic
12#381×C79C2211yescharacteristic
13#380×C79—22no—
14#379×C79C411yescharacteristic
15#379×C79C2211yescharacteristic
16C4×C22⋊C4×C79—22no—
17C4⋊(C79⋊C8)—42no—
18C22⋊(C79⋊C8)—24no—
19#14579072—42no—
20C25×C79C2×C411yescharacteristic
21C23×C4×C79—22no—
22C23×C4×C79—21no—
23C23×C4×C79—21no—
24C23×C4×C79—22no—
25C23×C4×C79—42no—
26C23×C4×C79—42no—
27C23×C4×C79C2×C411yescharacteristic
28C23×C4×C79C2×C411yescharacteristic
29#117×C79—42no—
30#117×C79—42no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79D412yes—
33C2×C22⋊C4×C79—22no—
34C2×C22⋊C4×C79—22no—
35C2×C22⋊C4×C79C2×C411yescharacteristic
36C2×C22⋊C4×C79C2×C411yescharacteristic
37C2×C22⋊C4×C79C2×C411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39C2×C42×C79—21no—
40C2×C42×C79D412yes—
41#95×C79—22no—
42#95×C79—22no—
43C2×C79⋊C8—44no—
44C24×C79—21no—
45C24×C79—21no—
46C24×C79—41no—
47C24×C79—41no—
48C24×C79—44no—
49C24×C79M1612yes—
50C24×C79C22×C411yescharacteristic
51C22×C4×C79—21no—
52C22×C4×C79—21no—
53C22×C4×C79—22no—
54C22×C4×C79—22no—
55C22×C4×C79—24no—
56C22×C4×C79—24no—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—48no—
62C22×C4×C79—48no—
63C22×C4×C79—42no—
64C22×C4×C79—42no—
65C22×C4×C79—42no—
66C22×C4×C79—42no—
67C22×C4×C79—41no—
68C22×C4×C79—41no—
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C412yes—
72C22×C4×C79M1612yes—
73C22×C4×C79C2×D412yes—
74C4⋊C4×C79—42no—
75C4⋊C4×C79—42no—
76C22⋊C4×C79—41no—
77C22⋊C4×C79—41no—
78C22⋊C4×C79—44no—
79C22⋊C4×C79—44no—
80C22⋊C4×C79—41no—
81C22⋊C4×C79—41no—
82C42×C79—22no—
83C42×C79—22no—
84C42×C79—42no—
85C23×C79—24no—
86C23×C79—22no—
87C23×C79—21no—
88C23×C79—21no—
89C23×C79—42no—
90C23×C79—44no—
91C23×C79—42no—
92C23×C79—42no—
93C23×C79—42no—
94C23×C79—44no—
95C23×C79—44no—
96C23×C79—48no—
97C23×C79—44no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79C2×C22⋊C411yescharacteristic
101C23×C79C2×M1611yescharacteristic
102C23×C79C2×M1611yescharacteristic
103C2×C4×C79—24no—
104C2×C4×C79—24no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—22no—
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—48no—
113C2×C4×C79—48no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79#9912yes—
125C2×C4×C79#10012yes—
126C79⋊C8—84no—
127C22×C79—22no—
128C22×C79—24no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—41no—
132C22×C79—41no—
133C22×C79—44no—
134C22×C79—42no—
135C22×C79—42no—
136C22×C79—44no—
137C22×C79—44no—
138C22×C79—48no—
139C22×C79—44no—
140C22×C79—42no—
141C22×C79—42no—
142C22×C79#40711yescharacteristic
143C22×C79C2×#9911yescharacteristic
144C22×C79C2×#10011yescharacteristic
145C4×C79—22no—
146C4×C79—22no—
147C4×C79—42no—
148C4×C79—42no—
149C4×C79—42no—
150C4×C79—81no—
151C4×C79—81no—
152#11501—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79—44no—
156C2×C79—41no—
157C2×C79—41no—
158C2×C79#146211yescharacteristic
159C2×C79#212012yes—
160#2280D7911yescharacteristic
161#1464—792no—
162#1317—792no—
163#1316—792no—
164C79#1150111yescharacteristic, complemented, Sylow
165C24×C4Dic7911yescharacteristic
166C2×#117Dic7911yescharacteristic
167C2×#117C2×D7911yescharacteristic
168#432—1582no—
169#381Dic7911yescharacteristic
170#381C2×D7911yescharacteristic
171#380—22no—
172#379Dic7911yescharacteristic
173#379C2×D7911yescharacteristic
174C4×C22⋊C4—22no—
175#324—1582no—
176#323—794no—
177C25C2×Dic7911yescharacteristic
178C23×C4—22no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4C2×Dic7911yescharacteristic
185C23×C4C2×Dic7911yescharacteristic
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22⋊D7912yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C2×Dic7911yescharacteristic
193C2×C22⋊C4C2×Dic7911yescharacteristic
194C2×C22⋊C4C2×Dic7911yescharacteristic
195C2×C22⋊C4C22×D7911yescharacteristic
196C2×C42—21no—
197C2×C42C22⋊D7912yes—
198C4⋊C8—3162no—
199C22⋊C8—1584no—
200#97—3162no—
201#95—22no—
202#95—22no—
203C24—21no—
204C24—21no—
205C24—41no—
206C24—41no—
207C24—44no—
208C24#1192960812yes—
209C24C22×Dic7911yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—48no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4C22⋊Dic7911yescharacteristic
229C22×C4C22⋊Dic7912yes—
230C22×C4C22⋊Dic7911yescharacteristic
231C22×C4#1192960812yes—
232C22×C4C2×C22⋊D7912yes—
233C2×C8—3164no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C42—22no—
243C42—22no—
244C42—42no—
245C23—24no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—42no—
250C23—44no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—44no—
255C23—44no—
256C23—48no—
257C23—44no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊Dic7911yescharacteristic
261C23C2×#1192960811yescharacteristic
262C23C2×#1192960811yescharacteristic
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#1457907612yes—
285C2×C4#1457907812yes—
286C8—6324no—
287C22—22no—
288C22—24no—
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—44no—
297C22—44no—
298C22—48no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22#1909852711yescharacteristic
303C22C2×#1457907611yescharacteristic
304C22C2×#1457907811yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—41no—
316C2—41no—
317C2#2376172711yescharacteristic
318C2#2376504112yes—
subgroupG/H
centerC22C2×#14579076
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2280C2
socleC22×C79C2×#99
prime radicalC25×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#2280D79
hypocenterC79#11501
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2280

Complements

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

NH ≅ G/Ntype
1. #23761734C2semidirect
2. #23761330C2semidirect
164. C79#11501semidirect

Maximal subgroups

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

subgroupindex
#237617342
#237613302
#237613282
C79×#22802
#1150179