Finite Group Databasebeta

← #34200475

Subgroups — #34200475

Order 20224.

470 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764531C211yesmaximal, characteristic
2#23762351C212yesmaximal
3#23761421C212yesmaximal, complemented
4#23761330C211yesmaximal, characteristic, complemented
5C79×#2607C211yesmaximal, characteristic, complemented
6#19099125C2211yescharacteristic
7C2×#14579144—21no—
8#19098702—22no—
9C2×#14579075—21no—
10#19098330C2211yescharacteristic
11#19098296—22no—
12#19098287C2212yes—
13#399×C79—21no—
14#393×C79C2212yes—
15#391×C79—22no—
16#387×C79—22no—
17#381×C79C2211yescharacteristic
18C4×C22⋊C4×C79—21no—
19C22×C22⋊D79—21no—
20C2×#11929611—42no—
21#14579145—41no—
22#14579144—42no—
23#14579144—42no—
24#14579135—22no—
25#14579131—41no—
26C2×Dic79⋊C4—41no—
27C2×C22⋊Dic79—21no—
28C2×(C2×D79)⋊C4—41no—
29#14579085—42no—
30#14579075—42no—
31C22⋊(C79⋊C8)—22no—
32C22×Q8×C79D412yes—
33C23×C4×C79D411yescharacteristic
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—21no—
36C2×C22⋊C4×C79—22no—
37C2×C22⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39C2×C42×C79—21no—
40C2×C42×C79—21no—
41#95×C79—22no—
42#95×C79—21no—
43#95×C79—22no—
44#95×C79—22no—
45#95×C79D412yes—
46C23×D79—21no—
47C2×D316—22no—
48C2×C22⋊D79—41no—
49C2×C22⋊D79—41no—
50C2×C22⋊D79—41no—
51C2×C22⋊D79—41no—
52C2×C22⋊D79—41no—
53C2×C22⋊D79—41no—
54C22×Dic79—21no—
55C22×Dic79—41no—
56#11929611—84no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59Dic79⋊C4—44no—
60C22⋊Dic79—22no—
61C22⋊Dic79—41no—
62(C2×D79)⋊C4—42no—
63(C2×D79)⋊C4—44no—
64C24×C79C2×D411yescharacteristic
65C2×Q8×C79—44no—
66C2×Q8×C79—42no—
67C22×C4×C79—21no—
68C22×C4×C79—22no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—41no—
73C22×C4×C79—41no—
74C22×C4×C79SD1612yescomplemented
75C22×C4×C79C2×D412yes—
76C4⋊C4×C79—44no—
77C4⋊C4×C79—81no—
78C22⋊C4×C79—41no—
79C22⋊C4×C79—44no—
80C22⋊C4×C79—41no—
81C42×C79—81no—
82C42×C79—81no—
83C23×C79—21no—
84C23×C79—21no—
85C23×C79—42no—
86C23×C79#12011yescharacteristic
87C23×C79C2×SD1611yescharacteristic
88C23×C79#13711yescharacteristic
89C22×D79—41no—
90C22×D79—41no—
91C22×D79—41no—
92C22×D79—82no—
93Q8×C79—44no—
94Q8×C79—82no—
95C2×C4×C79—24no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—42no—
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—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—41no—
109C2×C4×C79—81no—
110C22⋊D79—81no—
111C22⋊D79—81no—
112C22⋊D79—81no—
113C22⋊D79—81no—
114C2×Dic79—42no—
115C2×Dic79—41no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118C2×Dic79—82no—
119D316—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#44811yescharacteristic
129C22×C79#45311yescharacteristic
130C22×C79B(4,𝔽2)11yescharacteristic
131C2×D79—82no—
132C2×D79—81no—
133C2×D79—82no—
134C2×D79—161no—
135C4×C79—42no—
136C4×C79—44no—
137C4×C79—42no—
138C4×C79—82no—
139C4×C79—81no—
140Dic79—82no—
141Dic79—81no—
142#12121—791nomaximal, Sylow
143C2×C79—21no—
144C2×C79—21no—
145C2×C79—41no—
146C2×C79—41no—
147C2×C79#159611yescharacteristic
148C2×C79#219712yes—
149D79—161no—
150#2607D7911yescharacteristic, complemented
151#2022—791no—
152#1601—792no—
153#1351—792no—
154#1317—791no—
155C79#1212111yescharacteristic, complemented, Sylow
156#534—791no—
157C2×#123—1581no—
158#448—1582no—
159C2×#99—1581no—
160#399—21no—
161#393C2×D7912yes—
162#391—22no—
163#387—22no—
164#381C2×D7911yescharacteristic
165C4×C22⋊C4—21no—
166#342—791no—
167#327—1582no—
168#323—792no—
169C22×Q8C22⋊D7912yes—
170C22×D4—1581no—
171C23×C4C22⋊D7911yescharacteristic
172C2×SD16—3162no—
173#123—3161no—
174#123—3162no—
175#123—3162no—
176#121—1582no—
177#120—3161no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—3161no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—3161no—
183C2×C22⋊C4—1581no—
184C2×C22⋊C4C22⋊D7911yescharacteristic
185C2×C22⋊C4C22×D7911yescharacteristic
186C2×C42—21no—
187C2×C42—21no—
188#103—3162no—
189#99—3162no—
190C22⋊C8—1582no—
191#95—22no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C22⋊D7912yes—
196C24—1581no—
197C24C2×C22⋊D7911yescharacteristic
198C2×Q8—44no—
199C2×Q8—42no—
200C2×D4—3161no—
201C2×D4—3161no—
202C2×D4—3161no—
203C2×D4—3161no—
204C2×D4—3161no—
205C2×D4—3161no—
206C2×D4—1582no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—3161no—
215C22×C4—1581no—
216C22×C4#1192961112yescomplemented
217C22×C4C2×C22⋊D7912yes—
218SD16—6324no—
219C2×C8—3162no—
220C4⋊C4—44no—
221C4⋊C4—81no—
222C4⋊C4—3164no—
223C4⋊C4—1582no—
224C22⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—41no—
227C22⋊C4—3162no—
228C22⋊C4—3164no—
229C22⋊C4—3161no—
230C22⋊C4—1582no—
231C42—81no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—42no—
236C23—3161no—
237C23—3161no—
238C23—3161no—
239C23—6322no—
240C23#1457913011yescharacteristic
241C23C2×#1192961111yescharacteristic
242C23#1457919211yescharacteristic
243Q8—44no—
244Q8—82no—
245D4—6322no—
246D4—6321no—
247D4—6321no—
248D4—6321no—
249D4—6321no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—81no—
265C2×C4—3162no—
266C2×C4—3161no—
267C2×C4—3161no—
268C2×C4—3161no—
269C2×C4—6322no—
270C8—6322no—
271C22—22no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—81no—
278C22—6322no—
279C22—6322no—
280C22—6321no—
281C22—12641no—
282C22#1909870211yescharacteristic
283C22#1909873111yescharacteristic
284C22#1909875011yescharacteristic
285C4—42no—
286C4—44no—
287C4—42no—
288C4—82no—
289C4—81no—
290C4—6322no—
291C4—6321no—
292C2—21no—
293C2—21no—
294C2—41no—
295C2—41no—
296C2—12641no—
297C2#2376232311yescharacteristic
298C2#2376539012yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2607C2
socleC22×C79B(4,𝔽2)
prime radical#19099125C22
hypercenter#2607D79
hypocenterC79#12121
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
3. #23761421C2semidirect
4. #23761330C2semidirect
5. C79×#2607C2semidirect
74. C22×C4×C79SD16semidirect
150. #2607D79semidirect
155. C79#12121semidirect
216. C22×C4#11929611semidirect

Maximal subgroups

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

subgroupindex
#237645312
#237623512
#237614212
#237613302
C79×#26072
#1212179