Finite Group Databasebeta

← #34175754

Subgroups — #34175754

Order 20224.

284 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 37 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#23761330C211yesmaximal, characteristic, complemented
3C79×#1306C211yesmaximal, characteristic, complemented
4#19099125C411yescharacteristic
5C2×#14579144—21no—
6C2×#14579075—21no—
7#19098330C411yescharacteristic
8#19098287—21no—
9#407×C79—21no—
10#381×C79C2211yescharacteristic
11C22×C22⋊D79—21no—
12#14579145—41no—
13#14579144—42no—
14#14579144—41no—
15#14579144—41no—
16#14579135—41no—
17#14579131—41no—
18C2×Dic79⋊C4—41no—
19C2×C22⋊Dic79—21no—
20C2×(C2×D79)⋊C4—41no—
21#14579075—42no—
22C22⋊(C79⋊C8)—41no—
23C23×C4×C79D411yescharacteristic
24C2×M16×C79—41no—
25C2×C22⋊C4×C79C2×C411yescharacteristic
26C2×C22⋊C4×C79D411yescharacteristic
27C22⋊C8×C79—22no—
28C22⋊C8×C79—41no—
29#95×C79—41no—
30C23×D79—21no—
31C2×D316—41no—
32C2×C22⋊D79—41no—
33C2×C22⋊D79—41no—
34C2×C22⋊D79—41no—
35C2×C22⋊D79—41no—
36C2×C22⋊D79—41no—
37C2×C22⋊D79—41no—
38C22×Dic79—21no—
39C22×Dic79—41no—
40C2×C79⋊C8—81no—
41C4⋊Dic79—41no—
42Dic79⋊C4—42no—
43Dic79⋊C4—42no—
44C22⋊Dic79—22no—
45C22⋊Dic79—41no—
46(C2×D79)⋊C4—42no—
47(C2×D79)⋊C4—42no—
48(C2×D79)⋊C4—42no—
49C24×C79C22⋊C411yescharacteristic
50C22×C4×C79—21no—
51C22×C4×C79—21no—
52C22×C4×C79—21no—
53C22×C4×C79—21no—
54C22×C4×C79—41no—
55C22×C4×C79—41no—
56C22×C4×C79D811yescharacteristic
57C22×C4×C79SD1611yescharacteristic
58M16×C79—81no—
59M16×C79—81no—
60C2×C8×C79—42no—
61C22⋊C4×C79—41no—
62C22⋊C4×C79—41no—
63C23×C79—21no—
64C23×C79—21no—
65C23×C79—81no—
66C23×C79#9911yescharacteristic
67C23×C79#10211yescharacteristic
68C23×C79#10411yescharacteristic
69C22×D79—41no—
70C22×D79—41no—
71C22×D79—41no—
72C22×D79—82no—
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—41no—
76C2×C4×C79—41no—
77C2×C4×C79—41no—
78C2×C4×C79—41no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—81no—
82C2×C4×C79—81no—
83C2×C4×C79—81no—
84C22⋊D79—81no—
85C22⋊D79—81no—
86C22⋊D79—81no—
87C22⋊D79—81no—
88C2×Dic79—42no—
89C2×Dic79—41no—
90C2×Dic79—41no—
91C2×Dic79—41no—
92C2×Dic79—82no—
93D316—161no—
94C8×C79—82no—
95C79⋊C8—161no—
96C22×C79—21no—
97C22×C79—21no—
98C22×C79—41no—
99C22×C79—41no—
100C22×C79—41no—
101C22×C79—42no—
102C22×C79—81no—
103C22×C79#32711yescharacteristic
104C22×C79AΣL(1,𝔽16)11yescharacteristic
105C22×C79#35311yescharacteristic
106C2×D79—82no—
107C2×D79—82no—
108C2×D79—81no—
109C2×D79—161no—
110C4×C79—41no—
111C4×C79—41no—
112C4×C79—81no—
113C4×C79—81no—
114Dic79—82no—
115Dic79—81no—
116#7395—791nomaximal, Sylow
117C2×C79—21no—
118C2×C79—21no—
119C2×C79—21no—
120C2×C79—41no—
121C2×C79—41no—
122C2×C79#134311yescharacteristic
123D79—161no—
124#2022—791no—
125#1317—791no—
126#1306D7911yescharacteristic, complemented
127C79#739511yescharacteristic, complemented, Sylow
128#534—791no—
129C2×#123—1581no—
130C2×#99—1581no—
131#407—21no—
132#381C2×D7911yescharacteristic
133#342—791no—
134#323—1581no—
135C22×D4—1581no—
136C23×C4C22⋊D7911yescharacteristic
137C2×M16—41no—
138#123—3161no—
139#123—3161no—
140#123—3161no—
141#123—3162no—
142#121—3161no—
143#120—3161no—
144C2×C4⋊C4—3161no—
145C2×C22⋊C4—3161no—
146C2×C22⋊C4—1581no—
147C2×C22⋊C4C4×D7911yescharacteristic
148C2×C22⋊C4D31611yescharacteristic
149#99—3162no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152C22⋊C8—3161no—
153#95—41no—
154C24—1581no—
155C24(C2×D79)⋊C411yescharacteristic
156C2×D4—3161no—
157C2×D4—3161no—
158C2×D4—3161no—
159C2×D4—3161no—
160C2×D4—3161no—
161C2×D4—3161no—
162C2×D4—3161no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—3161no—
170C22×C4—1581no—
171C22×C4#1192960911yescharacteristic
172C22×C4#1192961111yescharacteristic
173M16—81no—
174M16—81no—
175C2×C8—42no—
176C2×C8—6321no—
177C4⋊C4—3161no—
178C4⋊C4—3162no—
179C4⋊C4—3162no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—3162no—
183C22⋊C4—3162no—
184C22⋊C4—3162no—
185C22⋊C4—3161no—
186C22⋊C4—1582no—
187C23—21no—
188C23—21no—
189C23—81no—
190C23—3161no—
191C23—3161no—
192C23—3161no—
193C23—6322no—
194C23#1457907511yescharacteristic
195C23#1457908111yescharacteristic
196C23#1457908911yescharacteristic
197D4—6321no—
198D4—6321no—
199D4—6321no—
200D4—6321no—
201D4—12641no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—3162no—
214C2×C4—3161no—
215C2×C4—3161no—
216C2×C4—3161no—
217C2×C4—6322no—
218C8—82no—
219C8—12641no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22—6322no—
228C22—6322no—
229C22—6321no—
230C22—12641no—
231C22#1909829611yescharacteristic
232C22#1909834911yescharacteristic
233C22#1909835511yescharacteristic
234C4—41no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—6322no—
239C4—6321no—
240C2—21no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2—12641no—
246C2#2376139611yescharacteristic
subgroupG/H
centerC2#23761396
commutatorC2×C22⋊C4×C79C2×C4
Frattini#381C2×D79
FittingC79×#1306C2
socleC2×C79#1343
prime radical#19099125C4
cosocle#381×C79C22
hypercenter#1306D79
hypocenterC79#7395
LCSG ▷ C2×C22⋊C4×C79 ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ #1306

Complements

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

NH ≅ G/Ntype
2. #23761330C2semidirect
3. C79×#1306C2semidirect
126. #1306D79semidirect
127. C79#7395semidirect

Maximal subgroups

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

subgroupindex
#237645312
#237613302
C79×#13062
#739579