Finite Group Databasebeta

← #34200474

Subgroups — #34200474

Order 20224.

470 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 59 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#23769065C211yesmaximal, characteristic, complemented
2#23762352C212yesmaximal
3#23761420C212yesmaximal, complemented
4#23761330C211yesmaximal, characteristic, complemented
5C79×#2022C211yesmaximal, characteristic
6#19098701—22no—
7#19098499—21no—
8#19098475C2212yes—
9#19098467—22no—
10#19098450—22no—
11C4×C22⋊Dic79—21no—
12#19098297—22no—
13#19098287C2212yes—
14#534×C79C2211yescharacteristic
15C2×#123×C79—21no—
16C2×#99×C79—21no—
17#381×C79C2211yescharacteristic
18#342×C79C2211yescharacteristic
19C22×Dic158D412yes—
20C2×#11929612—42no—
21C2×C4⋊Dic79—21no—
22C2×Dic79⋊C4—22no—
23C2×C22⋊Dic79—22no—
24C2×C4×Dic79—21no—
25C2×C4×Dic79—21no—
26#14579084—42no—
27C22⋊(C79⋊C8)—22no—
28C79⋊#95—22no—
29C79⋊#95—21no—
30C79⋊#95—22no—
31C79⋊#95D412yes—
32C22×D4×C79—21no—
33C23×C4×C79D411yescharacteristic
34#123×C79—42no—
35#123×C79—41no—
36#123×C79—42no—
37#121×C79—22no—
38#120×C79—41no—
39C2×C4⋊C4×C79—41no—
40C2×C22⋊C4×C79—21no—
41C2×C22⋊C4×C79—41no—
42C2×C22⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79C2311yescharacteristic
44#99×C79—42no—
45#95×C79—22no—
46C2×Dic158—42no—
47C2×Dic158—44no—
48C22×Dic79—22no—
49C22×Dic79—22no—
50#11929612—84no—
51C2×C79⋊C8—42no—
52C4⋊Dic79—81no—
53Dic79⋊C4—44no—
54C22⋊Dic79—44no—
55C4×Dic79—81no—
56C4×Dic79—81no—
57C24×C79—21no—
58C24×C79C2×D411yescharacteristic
59C2×D4×C79—22no—
60C2×D4×C79—41no—
61C2×D4×C79—41no—
62C2×D4×C79—41no—
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79—41no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—41no—
71C22×C4×C79—41no—
72C22×C4×C79—41no—
73C22×C4×C79SD1612yescomplemented
74C22×C4×C79C2×D412yes—
75C4⋊C4×C79—22no—
76C4⋊C4×C79—44no—
77C22⋊C4×C79—22no—
78C22⋊C4×C79—41no—
79C22⋊C4×C79—42no—
80C22⋊C4×C79—44no—
81C22⋊C4×C79—41no—
82C22⋊C4×C79—41no—
83C23×C79—21no—
84C23×C79—21no—
85C23×C79—42no—
86C23×C79—41no—
87C23×C79—41no—
88C23×C79—41no—
89C23×C79—82no—
90C23×C79#12011yescharacteristic
91C23×C79C2×SD1611yescharacteristic
92C23×C79#13711yescharacteristic
93D4×C79—82no—
94D4×C79—81no—
95D4×C79—81no—
96D4×C79—81no—
97D4×C79—81no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—42no—
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—41no—
110C2×C4×C79—81no—
111C2×C4×C79—82no—
112C2×Dic79—24no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118Dic158—44no—
119Dic158—82no—
120C79⋊C8—82no—
121C22×C79—22no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—41no—
125C22×C79—41no—
126C22×C79—42no—
127C22×C79—81no—
128C22×C79—82no—
129C22×C79—81no—
130C22×C79—82no—
131C22×C79—161no—
132C22×C79#44811yescharacteristic
133C22×C79#45311yescharacteristic
134C22×C79B(4,𝔽2)11yescharacteristic
135C4×C79—42no—
136C4×C79—42no—
137C4×C79—81no—
138C4×C79—82no—
139C4×C79—81no—
140Dic79—44no—
141Dic79—82no—
142#12121—791nomaximal, Sylow
143C2×C79—21no—
144C2×C79—21no—
145C2×C79—41no—
146C2×C79—41no—
147C2×C79—161no—
148C2×C79#159611yescharacteristic
149C2×C79#219712yes—
150#2607—791no—
151#2022D7911yescharacteristic
152#1601—792no—
153#1351—792no—
154#1317—791no—
155C79#1212111yescharacteristic, complemented, Sylow
156#534C2×D7911yescharacteristic
157C2×#123—21no—
158#448—1582no—
159C2×#99—21no—
160#399—1581no—
161#393—792no—
162#391—1582no—
163#387—1582no—
164#381C2×D7911yescharacteristic
165C4×C22⋊C4—1581no—
166#342C2×D7911yescharacteristic
167#327—1582no—
168#323—792no—
169C22×Q8—792no—
170C22×D4—21no—
171C23×C4C22⋊D7911yescharacteristic
172C2×SD16—3162no—
173#123—42no—
174#123—41no—
175#123—42no—
176#121—22no—
177#120—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—1582no—
180C2×C4⋊C4—1581no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—1582no—
184C2×C22⋊C4C22⋊D7911yescharacteristic
185C2×C22⋊C4C22×D7911yescharacteristic
186C2×C42—1581no—
187C2×C42—1581no—
188#103—3162no—
189#99—42no—
190C22⋊C8—1582no—
191#95—22no—
192#95—792no—
193#95—1582no—
194#95—1581no—
195#95—1582no—
196C24—21no—
197C24C2×C22⋊D7911yescharacteristic
198C2×Q8—3162no—
199C2×Q8—3164no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—1582no—
215C22×C4—1582no—
216C22×C4#1192961212yescomplemented
217C22×C4D4×D7912yes—
218SD16—6324no—
219C2×C8—3162no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—3164no—
223C4⋊C4—6321no—
224C22⋊C4—22no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—3164no—
231C42—6321no—
232C42—6321no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—82no—
240C23#1457912911yescharacteristic
241C23C2×#1192961211yescharacteristic
242C23#1457919111yescharacteristic
243Q8—3164no—
244Q8—6322no—
245D4—82no—
246D4—81no—
247D4—81no—
248D4—81no—
249D4—81no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—3162no—
265C2×C4—3162no—
266C2×C4—3162no—
267C2×C4—3162no—
268C2×C4—3162no—
269C2×C4—1584no—
270C8—6322no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—82no—
279C22—82no—
280C22—81no—
281C22—161no—
282C22#1909870111yescharacteristic
283C22#1909873011yescharacteristic
284C22#1909874911yescharacteristic
285C4—42no—
286C4—42no—
287C4—81no—
288C4—82no—
289C4—81no—
290C4—3164no—
291C4—6322no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—161no—
297C2#2376232211yescharacteristic
298C2#2376538712yes—
subgroupG/H
centerC22#19098749
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2022C2
socleC22×C79B(4,𝔽2)
prime radical#534×C79C22
hypercenter#2022D79
hypocenterC79#12121
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. #23769065C2semidirect
3. #23761420C2semidirect
4. #23761330C2semidirect
73. C22×C4×C79SD16semidirect
155. C79#12121semidirect
216. C22×C4#11929612semidirect

Maximal subgroups

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

subgroupindex
#237690652
#237623522
#237614202
#237613302
C79×#20222
#1212179