Finite Group Databasebeta

← #34175778

Subgroups — #34175778

Order 20224.

282 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 41 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#23764482C211yesmaximal, characteristic
2#23761332C211yesmaximal, characteristic
3C79×#1393C211yesmaximal, characteristic
4C22×#11929612—21no—
5C2×#14579161C411yescharacteristic, complemented
6C2×#14579084—21no—
7#19098318C411yescharacteristic, complemented
8#19098289—21no—
9C2×#99×C79—21no—
10#390×C79C2211yescharacteristic
11C22×Dic158—21no—
12C2×#11929612—42no—
13C2×#11929612—42no—
14C2×#11929612—41no—
15C2×#11929612—41no—
16C22×C79⋊C8—21no—
17#14579161—42no—
18C2×C4⋊Dic79—41no—
19#14579084—42no—
20C22⋊(C79⋊C8)—41no—
21C22×D4×C79D411yescharacteristic
22C2×C4⋊C4×C79D411yescharacteristic
23C2×C22⋊C4×C79—41no—
24C2×C22⋊C4×C79—41no—
25C2×C42×C79C2×C411yescharacteristic
26#99×C79—22no—
27#99×C79—41no—
28C2×Dic158—22no—
29C2×Dic158—41no—
30C2×Dic158—41no—
31C2×Dic158—41no—
32C2×Dic158—41no—
33C2×Dic158—41no—
34C22×Dic79—41no—
35#11929612—44no—
36#11929612—44no—
37C2×C79⋊C8—22no—
38C2×C79⋊C8—41no—
39C2×C79⋊C8—41no—
40C2×C79⋊C8—81no—
41C4⋊Dic79—42no—
42C4⋊Dic79—42no—
43C24×C79—21no—
44C2×D4×C79—21no—
45C2×D4×C79—41no—
46C2×D4×C79—41no—
47C2×D4×C79D811yescharacteristic
48C2×D4×C79SD1611yescharacteristic
49C22×C4×C79—21no—
50C22×C4×C79—21no—
51C22×C4×C79—81no—
52C22×C4×C79C22⋊C411yescharacteristic
53C4⋊C4×C79—41no—
54C22⋊C4×C79—42no—
55C22⋊C4×C79—82no—
56C22⋊C4×C79—81no—
57C42×C79—41no—
58C23×C79—21no—
59C23×C79—21no—
60C23×C79—41no—
61C23×C79—41no—
62C23×C79—81no—
63C23×C79—81no—
64C23×C79#9911yescharacteristic
65D4×C79—41no—
66D4×C79—41no—
67D4×C79—81no—
68C2×C4×C79—21no—
69C2×C4×C79—21no—
70C2×C4×C79—22no—
71C2×C4×C79—41no—
72C2×C4×C79—41no—
73C2×C4×C79—41no—
74C2×C4×C79—81no—
75C2×C4×C79—82no—
76C2×C4×C79—161no—
77C2×C4×C79—161no—
78C2×C4×C79#10211yescharacteristic
79C2×C4×C79#10411yescharacteristic
80C2×Dic79—42no—
81C2×Dic79—81no—
82C2×Dic79—81no—
83Dic158—42no—
84Dic158—42no—
85Dic158—81no—
86Dic158—81no—
87C79⋊C8—42no—
88C79⋊C8—161no—
89C22×C79—21no—
90C22×C79—21no—
91C22×C79—41no—
92C22×C79—41no—
93C22×C79—81no—
94C22×C79—81no—
95C22×C79—81no—
96C22×C79—81no—
97C22×C79—161no—
98C22×C79#32711yescharacteristic
99C22×C79AΣL(1,𝔽16)11yescharacteristic
100C22×C79#35311yescharacteristic
101C4×C79—21no—
102C4×C79—21no—
103C4×C79—42no—
104C4×C79—81no—
105C4×C79—162no—
106Dic79—82no—
107#7403—791nomaximal, Sylow
108C2×C79—21no—
109C2×C79—21no—
110C2×C79—81no—
111C2×C79—81no—
112C2×C79#134311yescharacteristic
113C2×C79#140711yescharacteristic
114C2×C79#140911yescharacteristic
115#2013—791no—
116#1393D7911yescharacteristic
117#1318—791no—
118C79#740311yescharacteristic, complemented, Sylow
119C22×SD16—1581no—
120C2×C4⋊Q8—791no—
121C2×#103—1581no—
122C2×#99—21no—
123#390C2×D7911yescharacteristic
124#336—791no—
125#324—1581no—
126C22×Q8—1581no—
127C22×D4C22⋊D7911yescharacteristic
128C2×SD16—3162no—
129C2×SD16—3161no—
130C2×SD16—3161no—
131C2×SD16—3162no—
132C22×C8—1581no—
133C4⋊Q8—3162no—
134C2×C4⋊C4—3161no—
135C2×C4⋊C4D31611yescharacteristic
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C42C4×D7911yescharacteristic
139#103—3162no—
140#99—22no—
141#99—41no—
142C22⋊C8—3161no—
143C24—21no—
144C2×Q8—3161no—
145C2×Q8—3161no—
146C2×Q8—3161no—
147C2×Q8—3161no—
148C2×Q8—3161no—
149C2×Q8—1582no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4#1192960911yescharacteristic
154C2×D4#1192961111yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—81no—
158C22×C4—3161no—
159C22×C4(C2×D79)⋊C411yescharacteristic
160SD16—3164no—
161SD16—3164no—
162C2×C8—3161no—
163C2×C8—3161no—
164C2×C8—6321no—
165C2×C8—1582no—
166C4⋊C4—41no—
167C4⋊C4—3162no—
168C4⋊C4—3162no—
169C22⋊C4—42no—
170C22⋊C4—82no—
171C22⋊C4—81no—
172C42—41no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23—81no—
178C23—81no—
179C23#1457907511yescharacteristic
180Q8—3162no—
181Q8—3162no—
182Q8—6321no—
183Q8—6321no—
184D4—41no—
185D4—41no—
186D4—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4—161no—
196C2×C4—161no—
197C2×C4—3162no—
198C2×C4—6321no—
199C2×C4—6321no—
200C2×C4#1457908111yescharacteristic
201C2×C4#1457908911yescharacteristic
202C8—3162no—
203C8—12641no—
204C22—21no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—161no—
213C22#1909829611yescharacteristic
214C22#1909834911yescharacteristic
215C22#1909835511yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—81no—
220C4—162no—
221C4—6322no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2376139611yescharacteristic
227C2#2376155911yescharacteristic
228C2#2376156411yescharacteristic
subgroupG/H
centerC22#19098355
commutatorC2×C42×C79C2×C4
Frattini#390C2×D79
FittingC79×#1393C2
socleC22×C79#353
prime radicalC22×D4×C79D4
cosocle#390×C79C22
hypercenter#1393D79
hypocenterC79#7403
LCSG ▷ C2×C42×C79 ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1393

Complements

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

NH ≅ G/Ntype
5. C2×#14579161C4semidirect
7. #19098318C4semidirect
118. C79#7403semidirect

Maximal subgroups

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

subgroupindex
#237644822
#237613322
C79×#13932
#740379