Finite Group Databasebeta

← #34197868

Subgroups — #34197868

Order 20224.

446 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 95 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#23761741C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3#23761330C214yesmaximal, complemented
4C79×#2486C211yesmaximal, characteristic
5#19098620—21no—
6C2×C22⋊(C79⋊C8)—21no—
7#19098289—21no—
8#19098287—21no—
9#19098287C2212yes—
10#19098287C2212yes—
11C2×#123×C79C412yes—
12C2×C4×D4×C79C2211yescharacteristic
13C22×C4⋊C4×C79C411yescharacteristic
14#398×C79—21no—
15#397×C79—21no—
16#389×C79—21no—
17#381×C79C2212yes—
18#380×C79—21no—
19#380×C79C411yescharacteristic
20C22×C79⋊C8—41no—
21C4⋊(C79⋊C8)—41no—
22C22⋊(C79⋊C8)—22no—
23C22⋊(C79⋊C8)—22no—
24C22⋊(C79⋊C8)—41no—
25#14579072—41no—
26C22×D4×C79C2×C411yescharacteristic
27C23×C4×C79C2×C412yes—
28C23×C4×C79C2×C411yescharacteristic
29C23×C4×C79D411yescharacteristic
30#123×C79—44no—
31C4×D4×C79—41no—
32C4×D4×C79—41no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—41no—
38C2×C4⋊C4×C79—41no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79C2×C412yes—
42C2×C22⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79C2311yescharacteristic
44C2×C42×C79D411yescharacteristic
45#95×C79—21no—
46#95×C79—22no—
47#95×C79—22no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—42no—
50C2×C79⋊C8—82no—
51C24×C79C22⋊C411yescharacteristic
52C24×C79C22×C411yescharacteristic
53C2×D4×C79—22no—
54C2×D4×C79—22no—
55C2×D4×C79—41no—
56C22×C4×C79—21no—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—24no—
63C22×C4×C79—42no—
64C22×C4×C79—42no—
65C22×C4×C79—42no—
66C22×C4×C79—42no—
67C22×C4×C79C22⋊C411yescharacteristic
68C22×C4×C79C22⋊C411yescharacteristic
69C22×C4×C79C22⋊C411yescharacteristic
70C22×C4×C79C2×D411yescharacteristic
71C22×C4×C79C2×D411yescharacteristic
72C4⋊C4×C79—41no—
73C4⋊C4×C79—42no—
74C4⋊C4×C79—42no—
75C4⋊C4×C79—44no—
76C4⋊C4×C79—41no—
77C4⋊C4×C79—41no—
78C22⋊C4×C79—41no—
79C22⋊C4×C79—41no—
80C22⋊C4×C79—44no—
81C22⋊C4×C79—42no—
82C42×C79—21no—
83C42×C79—21no—
84C23×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79—41no—
89C23×C79—41no—
90C23×C79—41no—
91C23×C79—41no—
92C23×C79#10012yes—
93C23×C79C2×C22⋊C411yescharacteristic
94C23×C79#13112yes—
95D4×C79—44no—
96C2×C4×C79—22no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—81no—
116C2×C4×C79—84no—
117C2×C4×C79#9911yescharacteristic
118C2×C4×C79#9911yescharacteristic
119C79⋊C8—82no—
120C79⋊C8—82no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—22no—
124C22×C79—22no—
125C22×C79—42no—
126C22×C79—42no—
127C22×C79—42no—
128C22×C79—41no—
129C22×C79—41no—
130C22×C79—41no—
131C22×C79—41no—
132C22×C79—81no—
133C22×C79#40811yescharacteristic
134C22×C79C2×#9911yescharacteristic
135C22×C79C2×#10011yescharacteristic
136C4×C79—21no—
137C4×C79—41no—
138C4×C79—41no—
139C4×C79—41no—
140C4×C79—41no—
141C4×C79—82no—
142C4×C79—82no—
143C4×C79#35212yes—
144#11558—791nomaximal, Sylow
145C2×C79—21no—
146C2×C79—21no—
147C2×C79—42no—
148C2×C79—42no—
149C2×C79#147111yescharacteristic
150C2×C79C2×#35211yescharacteristic
151C2×C79#212311yescharacteristic
152#2486D7911yescharacteristic
153#1466—791no—
154#1459—791no—
155#1317—794no—
156C79#1155811yescharacteristic, complemented, Sylow
157C2×#123Dic7912yes—
158C2×C4×D4C2×D7911yescharacteristic
159C22×C4⋊C4Dic7911yescharacteristic
160#432—1581no—
161C2×C22⋊C8—1581no—
162#398—21no—
163#397—21no—
164#389—21no—
165#381C2×D7912yes—
166#380—21no—
167#380Dic7911yescharacteristic
168#324—1581no—
169#323—792no—
170#323—792no—
171#323—1581no—
172C22×D4C2×Dic7911yescharacteristic
173C23×C4C2×Dic7912yes—
174C23×C4C2×Dic7911yescharacteristic
175C23×C4C22⋊D7911yescharacteristic
176C22×C8—3161no—
177#123—44no—
178C4×D4—41no—
179C4×D4—41no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C22⋊D7911yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4C2×Dic7912yes—
189C2×C22⋊C4C22⋊D7911yescharacteristic
190C2×C22⋊C4C22×D7911yescharacteristic
191C2×C42C22⋊D7911yescharacteristic
192C4⋊C8—3161no—
193C22⋊C8—3161no—
194C22⋊C8—1582no—
195C22⋊C8—1582no—
196#97—3161no—
197#95—21no—
198#95—22no—
199#95—22no—
200C24C22⋊Dic7911yescharacteristic
201C24C22×Dic7911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4C22⋊Dic7911yescharacteristic
217C22×C4C22⋊Dic7911yescharacteristic
218C22×C4C22⋊Dic7911yescharacteristic
219C22×C4C2×C22⋊D7911yescharacteristic
220C22×C4C2×C22⋊D7911yescharacteristic
221C2×C8—3162no—
222C2×C8—3162no—
223C2×C8—6322no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C42—21no—
235C42—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23#1457907812yes—
245C23C2×C22⋊Dic7911yescharacteristic
246C23#1457916812yes—
247D4—44no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—81no—
268C2×C4—84no—
269C2×C4#1457907611yescharacteristic
270C2×C4#1457907611yescharacteristic
271C8—6322no—
272C8—6322no—
273C22—21no—
274C22—21no—
275C22—22no—
276C22—22no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#1909853011yescharacteristic
286C22C2×#1457907611yescharacteristic
287C22C2×#1457907811yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—82no—
295C4#1909835312yes—
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2#2376176011yescharacteristic
301C2C2×#1909835311yescharacteristic
302C2#2376505411yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2486C2
socleC22×C79C2×#99
prime radicalC22×D4×C79C2×C4
cosocleC2×C22⋊C4×C79C23
hypercenter#2486D79
hypocenterC79#11558
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #23761330C2semidirect
156. C79#11558semidirect

Maximal subgroups

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

subgroupindex
#237617412
#237617182
#237613302
C79×#24862
#1155879