Finite Group Databasebeta

← #34175762

Subgroups — #34175762

Order 20224.

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

normal subgroups only

This group has 268 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#23764467C211yesmaximal, characteristic
2#23761287C211yesmaximal, characteristic, complemented
3C79×#1320C211yesmaximal, characteristic, complemented
4#19099363C411yescharacteristic
5#19099074—21no—
6#19098579—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098320C411yescharacteristic
9#407×C79C2211yescharacteristic
10#349×C79—21no—
11C2×#11929624—21no—
12#14579194—41no—
13#14579189—41no—
14#14579183—41no—
15C22×C79⋊C8—41no—
16#14579144—41no—
17#14579140—41no—
18#14579135—41no—
19#14579130—41no—
20C4×C22⋊D79—42no—
21#14579120—21no—
22#14579087—42no—
23C22⋊(C79⋊C8)—22no—
24C22⋊(C79⋊C8)—41no—
25C23×C4×C79D411yescharacteristic
26C2×M16×C79C2×C411yescharacteristic
27C2×M16×C79D411yescharacteristic
28M32×C79—41no—
29C22⋊C8×C79—41no—
30#11929624—41no—
31#11929624—41no—
32#11929624—41no—
33#11929624—41no—
34C2×Dic158—21no—
35C2×D316—21no—
36C2×C22⋊D79—42no—
37C2×C22⋊D79—41no—
38C2×C4×D79—41no—
39Dic316—41no—
40#11929613—42no—
41D632—41no—
42C2×C79⋊C8—42no—
43C2×C79⋊C8—81no—
44C2×C79⋊C8—81no—
45C4⋊Dic79—41no—
46Dic79⋊C4—42no—
47C22⋊Dic79—42no—
48C22⋊Dic79—41no—
49(C2×D79)⋊C4—42no—
50C4×Dic79—22no—
51C24×C79SD1611yescharacteristic
52C22×C4×C79—21no—
53C22×C4×C79—21no—
54C22×C4×C79—41no—
55C22×C4×C79—41no—
56C22×C4×C79C22⋊C411yescharacteristic
57C22×C4×C79D811yescharacteristic
58M16×C79—41no—
59M16×C79—41no—
60C2×C8×C79—21no—
61C2×C8×C79—21no—
62C16×C79—81no—
63C23×C79—21no—
64C23×C79—41no—
65C23×C79—42no—
66C23×C79#10211yescharacteristic
67C22×D79—41no—
68C2×C4×C79—21no—
69C2×C4×C79—21no—
70C2×C4×C79—21no—
71C2×C4×C79—41no—
72C2×C4×C79—41no—
73C2×C4×C79—41no—
74C2×C4×C79—81no—
75C2×C4×C79#9911yescharacteristic
76C2×C4×C79#10411yescharacteristic
77C22⋊D79—81no—
78C22⋊D79—84no—
79C22⋊D79—81no—
80C2×Dic79—42no—
81C2×Dic79—41no—
82D316—41no—
83D316—41no—
84C4×D79—81no—
85C4×D79—81no—
86Dic158—41no—
87Dic158—41no—
88C8×C79—41no—
89C8×C79—41no—
90C79⋊C8—82no—
91C22×C79—22no—
92C22×C79—21no—
93C22×C79—41no—
94C22×C79—41no—
95C22×C79—42no—
96C22×C79—41no—
97C22×C79—81no—
98C22×C79#32711yescharacteristic
99C2×D79—82no—
100C2×D79—81no—
101C4×C79—21no—
102C4×C79—41no—
103C4×C79AΣL(1,𝔽16)11yescharacteristic
104C4×C79#35311yescharacteristic
105Dic79—82no—
106Dic79—81no—
107#7397—791nomaximal, Sylow
108C2×C79—21no—
109C2×C79—41no—
110C2×C79—42no—
111C2×C79#134311yescharacteristic
112D79—161no—
113#2010—791no—
114#1320D7911yescharacteristic, complemented
115#1298—791no—
116C79#739711yescharacteristic, complemented, Sylow
117#575—791no—
118#525—1581no—
119#421—1581no—
120#407C2×D7911yescharacteristic
121C2×C22⋊C8—1581no—
122#349—21no—
123H(3,ℤ/4)—791no—
124C2×#41—1581no—
125C23×C4D31611yescharacteristic
126#137—3161no—
127AGL(1,ℤ/8)—3161no—
128#135—3161no—
129C2×M16C4×D7911yescharacteristic
130C2×M16C22⋊D7911yescharacteristic
131C22×C8—3161no—
132#123—3161no—
133C22⋊Q8—3161no—
134#121—3161no—
135#120—3161no—
136C4×D4—3162no—
137#117—1581no—
138M32—41no—
139#104—3162no—
140C22⋊C8—41no—
141C22⋊C8—3161no—
142C22⋊C8—1582no—
143C24#1192961311yescharacteristic
144#41—3161no—
145#41—3161no—
146#41—3161no—
147#41—3161no—
148C2×Q8—1581no—
149C2×D4—3162no—
150C2×D4—3161no—
151C2×D4—1581no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—3161no—
157C22×C4(C2×D79)⋊C411yescharacteristic
158C22×C4D63211yescharacteristic
159Q16—3161no—
160SD16—3162no—
161D8—3161no—
162M16—41no—
163M16—41no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—3162no—
167C2×C8—6321no—
168C2×C8—6321no—
169C4⋊C4—3161no—
170C4⋊C4—3162no—
171C22⋊C4—3162no—
172C22⋊C4—3162no—
173C22⋊C4—3161no—
174C42—1582no—
175C16—81no—
176C23—21no—
177C23—41no—
178C23—42no—
179C23—3161no—
180C23#1457908311yescharacteristic
181Q8—3161no—
182Q8—3161no—
183D4—3161no—
184D4—3161no—
185D4—6321no—
186D4—6324no—
187D4—6321no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—3162no—
196C2×C4—3161no—
197C2×C4—6321no—
198C2×C4—6321no—
199C2×C4#1457907511yescharacteristic
200C2×C4#1457908711yescharacteristic
201C8—41no—
202C8—41no—
203C8—6322no—
204C22—22no—
205C22—21no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—41no—
210C22—81no—
211C22—6322no—
212C22—6321no—
213C22#1909829811yescharacteristic
214C4—21no—
215C4—41no—
216C4—6322no—
217C4—6321no—
218C4#1909835111yescharacteristic
219C4#1909835511yescharacteristic
220C2—21no—
221C2—41no—
222C2—42no—
223C2—12641no—
224C2#2376139811yescharacteristic
subgroupG/H
centerC4#19098355
commutatorC2×M16×C79C2×C4
Frattini#407C2×D79
FittingC79×#1320C2
socleC2×C79#1343
prime radical#23764467C2
cosocle#407×C79C22
hypercenter#1320D79
hypocenterC79#7397
LCSG ▷ C2×M16×C79 ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #23761287C2semidirect
3. C79×#1320C2semidirect
114. #1320D79semidirect
116. C79#7397semidirect

Maximal subgroups

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

subgroupindex
#237644672
#237612872
C79×#13202
#739779