Finite Group Databasebeta

← #34185826

Subgroups — #34185826

Order 20224.

508 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 95 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#23773857C211yesmaximal, characteristic
2#23766581C211yesmaximal, characteristic, complemented
3#23761948C212yesmaximal
4#23761266C212yesmaximal, complemented
5C79×#1558C211yesmaximal, characteristic, complemented
6C2×#14579192—22no—
7#19099113C2211yescharacteristic
8#19098955—22no—
9#19098942C412yescomplemented
10#19098937C412yescomplemented
11C4×#11929608C2211yescharacteristic
12#19098470—22no—
13#19098449—24no—
14#19098431C2212yes—
15#19098298—24no—
16C2×C4⋊Q8×C79C2211yescharacteristic
17#432×C79C2212yes—
18#333×C79—22no—
19C2×#11929624—21no—
20#14579192—44no—
21#14579192—44no—
22C2×#11929614—24no—
23C2×#11929611—24no—
24C2×#11929608—21no—
25#14579161C2×C411yescharacteristic
26#14579158C2×C411yescharacteristic
27#14579149—21no—
28#14579135—22no—
29C4×D316—22no—
30C2×C4⋊Dic79—22no—
31C2×C4×Dic79—22no—
32#14579072—21no—
33C4×C79⋊C8C2×C412yes—
34C79⋊#95—24no—
35C79⋊#95—22no—
36C79⋊#95—22no—
37C79⋊#95D414yes—
38C22×Q8×C79D412yes—
39C4⋊Q8×C79—22no—
40C4⋊Q8×C79C2×C412yes—
41C4⋊Q8×C79D412yes—
42C2×C4⋊C4×C79—22no—
43C2×C42×C79C2311yescharacteristic
44C4⋊C8×C79—22no—
45C22⋊C8×C79—22no—
46#97×C79—22no—
47#11929624—42no—
48#11929624—42no—
49C2×Dic158—21no—
50C2×D316—21no—
51C2×D316—22no—
52C2×C22⋊D79—41no—
53C22×Dic79—24no—
54C2×C4×D79—41no—
55#11929614—48no—
56#11929611—48no—
57#11929608—24no—
58C2×C79⋊C8—22no—
59C4⋊Dic79—22no—
60C4⋊Dic79—82no—
61(C2×D79)⋊C4—42no—
62C4×Dic79—82no—
63C2×Q8×C79—22no—
64C2×Q8×C79—42no—
65C2×Q8×C79—42no—
66C2×Q8×C79C22⋊C414yes—
67C22×C4×C79—22no—
68C22×C4×C79C2×D411yescharacteristic
69C22×C4×C79C2×D412yes—
70C2×C8×C79—42no—
71C4⋊C4×C79—24no—
72C4⋊C4×C79—24no—
73C42×C79C22×C411yescharacteristic
74C42×C79C2×D411yescharacteristic
75C42×C79#4112yes—
76C23×C79#12011yescharacteristic
77C22×D79—41no—
78Q8×C79—44no—
79Q8×C79—82no—
80C2×C4×C79—22no—
81C2×C4×C79—24no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79C2×C22⋊C411yescharacteristic
86C2×C4×C79C4×D412yes—
87C2×C4×C79#12112yes—
88C2×C4×C79#12311yescharacteristic
89C22⋊D79—82no—
90C2×Dic79—48no—
91C2×Dic79—41no—
92C2×Dic79—44no—
93D316—42no—
94D316—82no—
95C4×D79—81no—
96C4×D79—81no—
97Dic158—42no—
98C8×C79—82no—
99C79⋊C8—24no—
100C22×C79—22no—
101C22×C79#38611yescharacteristic
102C22×C79#45312yes—
103C2×D79—81no—
104C2×D79—82no—
105C4×C79—21no—
106C4×C79—21no—
107C4×C79—22no—
108C4×C79—22no—
109C4×C79—44no—
110Dic79—84no—
111Dic79—81no—
112#9429—791nomaximal, Sylow
113C2×C79—22no—
114C2×C79#188711yescharacteristic
115C2×C79#190512yes—
116D79—161no—
117#3152—791no—
118#2381—791no—
119#1558D7911yescharacteristic, complemented
120#1514—792no—
121#1288—792no—
122C79#942911yescharacteristic, complemented, Sylow
123C2×#137—1582no—
124#532—791no—
125C2×C4⋊Q8C2×D7911yescharacteristic
126#497—1582no—
127#494—792no—
128#492—792no—
129#432C2×D7912yes—
130C4×M16—791no—
131#391—1582no—
132#387—1584no—
133#383—792no—
134#333—22no—
135#327—1584no—
136C2×#41—1581no—
137C22×Q8D31612yes—
138#137—3164no—
139#137—3164no—
140C2×Q16—1584no—
141C2×SD16—1584no—
142C2×M16—1581no—
143C4⋊Q8—22no—
144C4⋊Q8—791no—
145C4⋊Q8C4×D7912yes—
146C4⋊Q8C22⋊D7912yes—
147#127—791no—
148#124—1581no—
149#121—1582no—
150C4×D4—1582no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—1582no—
153C2×C42—1582no—
154C2×C42C22×D7911yescharacteristic
155C4⋊C8—22no—
156C22⋊C8—22no—
157#97—22no—
158#97—1581no—
159C4×C8—792no—
160#95—794no—
161#95—1584no—
162#95—1582no—
163#95—1582no—
164#41—3162no—
165#41—3162no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—1581no—
170C2×Q8(C2×D79)⋊C414yes—
171C2×D4—3161no—
172C2×D4—1581no—
173C2×D4—1582no—
174C22×C4—22no—
175C22×C4—3161no—
176C22×C4—1584no—
177C22×C4D4×D7912yes—
178C22×C4C2×D31611yescharacteristic
179Q16—3168no—
180SD16—3168no—
181M16—1584no—
182C2×C8—42no—
183C2×C8—1582no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—6322no—
187C4⋊C4—1582no—
188C22⋊C4—3162no—
189C42—6322no—
190C42C2×C4×D7911yescharacteristic
191C42C2×C22⋊D7911yescharacteristic
192C42#1192962312yes—
193C23—3161no—
194C23#1457913111yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—3162no—
198D4—3162no—
199D4—6322no—
200D4—6322no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—3168no—
207C2×C4—3161no—
208C2×C4—3164no—
209C2×C4—6321no—
210C2×C4—6321no—
211C2×C4C2×(C2×D79)⋊C411yescharacteristic
212C2×C4(C22⋊D79)⋊C412yes—
213C2×C4#1457913312yes—
214C2×C4#1457914511yescharacteristic
215C8—82no—
216C8—1584no—
217C22—22no—
218C22—6321no—
219C22—6322no—
220C22#1909844011yescharacteristic
221C22#1909873312yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—6324no—
228C4—6321no—
229C2—22no—
230C2—12641no—
231C2#2376379611yescharacteristic
232C2#2376389212yes—
subgroupG/H
centerC22#19098440
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1558C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1558D79
hypocenterC79#9429
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #23766581C2semidirect
4. #23761266C2semidirect
5. C79×#1558C2semidirect
9. #19098942C4semidirect
10. #19098937C4semidirect
119. #1558D79semidirect
122. C79#9429semidirect

Maximal subgroups

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

subgroupindex
#237738572
#237665812
#237619482
#237612662
C79×#15582
#942979