Finite Group Databasebeta

← #34219323

Subgroups — #34219323

Order 20224.

1530 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 371 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
1C22×#14579155C211yesmaximal, characteristic
2C2×#19098431C211yesmaximal, characteristic, complemented
3#23761650C216yesmaximal, complemented
4#23761642C216yesmaximal
5C2×#384×C79C211yesmaximal, characteristic, complemented
6C2×#14579155—24no—
7C2×#14579155—24no—
8C2×#14579155C418yescomplemented
9C22×Dic79⋊C4C2213yes—
10C22×(C2×D79)⋊C4C2213yes—
11C22×(C2×D79)⋊C4C2214yes—
12#19098431—24no—
13C2×C79⋊#95C2216yes—
14C2×C79⋊#95C22112yes—
15C22×C4⋊C4×C79C2216yes—
16C22×C42×C79C2211yescharacteristic
17#384×C79—24no—
18C24×D79C2311yescharacteristic
19C23×Dic79C2314yes—
20C23×Dic79C2313yes—
21#14579155—232no—
22C2×Dic79⋊C4—26no—
23C2×Dic79⋊C4—26no—
24C2×Dic79⋊C4C2×C4112yes—
25C2×(C2×D79)⋊C4—26no—
26C2×(C2×D79)⋊C4—26no—
27C2×(C2×D79)⋊C4—412no—
28C2×(C2×D79)⋊C4C2×C4112yes—
29C79⋊#95—212no—
30C79⋊#95—412no—
31C23×C4×C79C2314yes—
32C23×C4×C79C2313yes—
33C2×C4⋊C4×C79—212no—
34C2×C4⋊C4×C79—412no—
35C2×C42×C79C2×C414yes—
36C2×C42×C79D414yes—
37C2×C42×C79D414yes—
38C23×D79—21no—
39C23×D79—21no—
40C23×D79—26no—
41C23×D79—26no—
42C23×D79C22×C412yes—
43C22×Dic79—23no—
44C22×Dic79—23no—
45C22×Dic79—24no—
46C22×Dic79—26no—
47C22×Dic79—212no—
48C22×Dic79—26no—
49C22×Dic79—212no—
50C22×Dic79C22×C416yes—
51Dic79⋊C4—224no—
52(C2×D79)⋊C4—224no—
53(C2×D79)⋊C4—416no—
54C24×C79C2411yescharacteristic
55C22×C4×C79—26no—
56C22×C4×C79—26no—
57C22×C4×C79—24no—
58C22×C4×C79—212no—
59C22×C4×C79—212no—
60C22×C4×C79C22×C416yes—
61C22×C4×C79C2×D416yes—
62C22×C4×C79C2×D416yes—
63C4⋊C4×C79—424no—
64C42×C79C22⋊C4116yescomplemented
65C23×C79C23×C411yescharacteristic
66C23×C79C22×D411yescharacteristic
67C23×C79C22×D411yescharacteristic
68C23×C79H(5,𝔽2)16yes—
69C23×C792−1+416yes—
70C22×D79—22no—
71C22×D79—212no—
72C22×D79—43no—
73C22×D79—43no—
74C22×D79—46no—
75C22×D79—44no—
76C22×D79—412no—
77C2×C4×C79—212no—
78C2×C4×C79—26no—
79C2×C4×C79—26no—
80C2×C4×C79—212no—
81C2×C4×C79—412no—
82C2×C4×C79—44no—
83C2×C4×C79—412no—
84C2×C4×C79C2×C22⋊C4112yes—
85C2×Dic79—26no—
86C2×Dic79—212no—
87C2×Dic79—43no—
88C2×Dic79—43no—
89C2×Dic79—46no—
90C2×Dic79—412no—
91C2×Dic79—44no—
92C2×Dic79—412no—
93C22×C79C22×C22⋊C411yescharacteristic
94C22×C79#52016yes—
95C22×C79#53513yes—
96C22×C79#53613yes—
97C22×C79#53716yes—
98C22×C79#56014yes—
99C22×C79#563112yes—
100C2×D79—46no—
101C2×D79—48no—
102C2×D79—81no—
103C2×D79—83no—
104C2×D79—84no—
105C4×C79—212no—
106C4×C79—84no—
107Dic79—46no—
108Dic79—84no—
109#14665—791nomaximal, Sylow
110C2×C79#231113yes—
111C2×C79#238314yes—
112C2×C79#281811yescharacteristic
113C2×C79#282113yes—
114C2×C79#282314yes—
115D79—82no—
116C22×#126—791no—
117C2×#384D7911yescharacteristic, complemented
118C2×#383—791no—
119#1441—796no—
120#1439—796no—
121C79#1466511yescharacteristic, complemented, Sylow
122C2×#126—798no—
123C2×#126—1584no—
124C2×#126—1584no—
125C22×C4⋊C4—793no—
126C22×C4⋊C4C2×D7916yes—
127C22×C22⋊C4—794no—
128C22×C22⋊C4—793no—
129C22×C42C2×D7911yescharacteristic
130#384—24no—
131#383—1584no—
132C2×#95—7912no—
133C2×#95—796no—
134C25—791no—
135C23×C4—793no—
136C23×C4—794no—
137C23×C4C22×D7914yes—
138C23×C4C22×D7913yes—
139#126—15832no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—412no—
142C2×C4⋊C4—7912no—
143C2×C4⋊C4—1586no—
144C2×C4⋊C4—1586no—
145C2×C22⋊C4—31612no—
146C2×C22⋊C4—7912no—
147C2×C22⋊C4—1586no—
148C2×C22⋊C4—1586no—
149C2×C42C4×D7914yes—
150C2×C42D31614yes—
151C2×C42C22⋊D7914yes—
152#95—31612no—
153#95—15812no—
154C24—792no—
155C24—1586no—
156C24—1586no—
157C24—1581no—
158C24—1581no—
159C24C23×D7911yescharacteristic
160C22×C4—26no—
161C22×C4—26no—
162C22×C4—212no—
163C22×C4—212no—
164C22×C4—24no—
165C22×C4—796no—
166C22×C4—15812no—
167C22×C4—15812no—
168C22×C4—1584no—
169C22×C4—1586no—
170C22×C4—1586no—
171C22×C4—1583no—
172C22×C4—1583no—
173C22×C4C2×C4×D7916yes—
174C22×C4C2×C22⋊D7916yes—
175C22×C4C2×D31616yes—
176C4⋊C4—424no—
177C4⋊C4—15824no—
178C22⋊C4—31616no—
179C22⋊C4—15824no—
180C42(C2×D79)⋊C4116yescomplemented
181C23—3163no—
182C23—3166no—
183C23—3164no—
184C23—31612no—
185C23—3163no—
186C23—1582no—
187C23—15812no—
188C23C22×C4×D7911yescharacteristic
189C23C22×C22⋊D7911yescharacteristic
190C23C22×D31611yescharacteristic
191C23#1457920616yes—
192C23#1457920916yes—
193C2×C4—26no—
194C2×C4—212no—
195C2×C4—212no—
196C2×C4—26no—
197C2×C4—44no—
198C2×C4—412no—
199C2×C4—412no—
200C2×C4—3163no—
201C2×C4—3166no—
202C2×C4—3163no—
203C2×C4—31612no—
204C2×C4—3164no—
205C2×C4—31612no—
206C2×C4—1586no—
207C2×C4—15812no—
208C2×C4C2×(C2×D79)⋊C4112yes—
209C22—3168no—
210C22—3166no—
211C22—6321no—
212C22—6323no—
213C22—6324no—
214C22C22×(C2×D79)⋊C411yescharacteristic
215C22#1909904416yes—
216C22#1909913013yes—
217C22#1909913713yes—
218C22#1909914416yes—
219C22#1909930014yes—
220C22#19099312112yes—
221C4—212no—
222C4—84no—
223C4—3166no—
224C4—6324no—
225C2—6322no—
226C2#2376599313yes—
227C2#2376659314yes—
228C2#2377144111yescharacteristic
229C2#2377145913yes—
230C2#2377147414yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#384×C79C2
socle, cosocleC24×C79C24
prime radicalC24×D79C23
hypercenterC2×#384D79
hypocenterC79#14665
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#19098431C2semidirect
3. #23761650C2semidirect
5. C2×#384×C79C2semidirect
8. C2×#14579155C4semidirect
64. C42×C79C22⋊C4semidirect
117. C2×#384D79semidirect
121. C79#14665semidirect
180. C42(C2×D79)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145791552
C2×#190984312
#237616502
#237616422
C2×#384×C792
#1466579