Finite Group Databasebeta

← #34185807

Subgroups — #34185807

Order 20224.

534 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 103 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#23766577C211yesmaximal, characteristic
2#23762114C211yesmaximal, characteristic
3#23761951C212yesmaximal, complemented
4#23761265C212yesmaximal, complemented
5C79×#3138C211yesmaximal, characteristic
6#19098620C2212yes—
7#19098443—24no—
8#19098443—22no—
9#19098431C2212yes—
10#19098307—22no—
11#19098300—24no—
12#533×C79C2211yescharacteristic
13C2×#127×C79C2211yescharacteristic
14#490×C79—22no—
15#486×C79C414yescomplemented
16#418×C79—22no—
17C4×M16×C79C2211yescharacteristic
18C23×Dic79—22no—
19C2×C22⋊Dic79—24no—
20C2×C22⋊Dic79—22no—
21C4⋊(C79⋊C8)—22no—
22C22⋊(C79⋊C8)—22no—
23#14579072—22no—
24C79⋊#95—22no—
25C79⋊#95—22no—
26C79⋊#95D414yes—
27C22×D4×C79—22no—
28C22×D4×C79D412yes—
29C2×M16×C79—21no—
30C4⋊Q8×C79C2×C412yescomplemented
31#127×C79—22no—
32#127×C79C2×C412yes—
33#127×C79D412yes—
34#125×C79—21no—
35C22⋊Q8×C79—22no—
36C4×Q8×C79—22no—
37#117×C79—21no—
38C2×C42×C79C2311yescharacteristic
39#102×C79—28no—
40#97×C79—21no—
41C4×C8×C79C2×C412yescomplemented
42C22×Dic79—24no—
43C22×Dic79—84no—
44C22×Dic79—82no—
45C2×C79⋊C8—42no—
46C22⋊Dic79—84no—
47C22⋊Dic79—82no—
48C24×C79—22no—
49C2×Q8×C79—22no—
50C2×D4×C79—24no—
51C2×D4×C79—24no—
52C2×D4×C79—22no—
53C2×D4×C79—42no—
54C2×D4×C79—42no—
55C2×D4×C79—82no—
56C2×D4×C79—82no—
57C2×D4×C79C22⋊C414yes—
58C22×C4×C79C2×D411yescharacteristic
59C22×C4×C79C2×D412yes—
60M16×C79—42no—
61C2×C8×C79—22no—
62C4⋊C4×C79—22no—
63C4⋊C4×C79—22no—
64C4⋊C4×C79—42no—
65C22⋊C4×C79—41no—
66C42×C79—41no—
67C42×C79C22×C411yescharacteristic
68C42×C79C2×D411yescharacteristic
69C42×C79#4112yes—
70C23×C79—24no—
71C23×C79—44no—
72C23×C79—44no—
73C23×C79—42no—
74C23×C79—42no—
75C23×C79#12011yescharacteristic
76Q8×C79—82no—
77D4×C79—44no—
78D4×C79—82no—
79D4×C79—84no—
80D4×C79—84no—
81C2×C4×C79—22no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79C2×C22⋊C411yescharacteristic
85C2×C4×C79C4×D412yes—
86C2×C4×C79#12112yes—
87C2×C4×C79#12311yescharacteristic
88C2×Dic79—48no—
89C2×Dic79—44no—
90C2×Dic79—88no—
91C8×C79—42no—
92C79⋊C8—82no—
93C22×C79—22no—
94C22×C79—44no—
95C22×C79—48no—
96C22×C79—82no—
97C22×C79—84no—
98C22×C79—82no—
99C22×C79#38611yescharacteristic
100C22×C79#45512yes—
101C4×C79—21no—
102C4×C79—21no—
103C4×C79—22no—
104C4×C79—22no—
105C4×C79—82no—
106Dic79—84no—
107#9425—791nomaximal, Sylow
108C2×C79—22no—
109C2×C79—84no—
110C2×C79#188511yescharacteristic
111C2×C79#190412yes—
112#3138D7911yescharacteristic
113#2380—791no—
114#1555—791no—
115#1515—792no—
116#1288—792no—
117C79#942511yescharacteristic, complemented, Sylow
118#533C2×D7911yescharacteristic
119C2×#127C2×D7911yescharacteristic
120#490—22no—
121#486Dic7914yescomplemented
122#432—792no—
123#418—22no—
124C4×M16C2×D7911yescharacteristic
125#386—1584no—
126#386—1582no—
127#383—792no—
128#331—1582no—
129#328—1584no—
130C22×D4—22no—
131C22×D4C22⋊D7912yes—
132C23×C4—1582no—
133C2×M16—21no—
134C4⋊Q8C2×Dic7912yescomplemented
135#127—22no—
136#127C2×Dic7912yes—
137#127C22⋊D7912yes—
138#125—21no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117—21no—
142C2×C22⋊C4—1584no—
143C2×C22⋊C4—1582no—
144C2×C42C22×D7911yescharacteristic
145C4⋊C8—1582no—
146#102—28no—
147C22⋊C8—1582no—
148#97—21no—
149#97—1582no—
150C4×C8C2×Dic7912yescomplemented
151#95—794no—
152#95—1582no—
153#95—1582no—
154C24—22no—
155C2×Q8—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—82no—
162C2×D4—82no—
163C2×D4C22⋊Dic7914yes—
164C22×C4—6324no—
165C22×C4—6322no—
166C22×C4—1584no—
167C22×C4D4×D7912yes—
168C22×C4C2×C22⋊D7911yescharacteristic
169M16—42no—
170C2×C8—22no—
171C2×C8—3162no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C22⋊C4—41no—
176C22⋊C4—6324no—
177C22⋊C4—6322no—
178C42—41no—
179C42C22×Dic7911yescharacteristic
180C42C2×C22⋊D7911yescharacteristic
181C42#1192962312yes—
182C23—24no—
183C23—44no—
184C23—44no—
185C23—42no—
186C23—42no—
187C23#1457912911yescharacteristic
188Q8—82no—
189D4—44no—
190D4—82no—
191D4—84no—
192D4—84no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—3168no—
197C2×C4—3164no—
198C2×C4—6328no—
199C2×C4C2×C22⋊Dic7911yescharacteristic
200C2×C4D4×Dic7912yes—
201C2×C4#1457913312yes—
202C2×C4#1457914211yescharacteristic
203C8—42no—
204C8—6322no—
205C22—22no—
206C22—44no—
207C22—48no—
208C22—82no—
209C22—82no—
210C22—84no—
211C22#1909844311yescharacteristic
212C22#1909874012yes—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—82no—
218C4—6324no—
219C2—22no—
220C2—84no—
221C2#2376378911yescharacteristic
222C2#2376388912yes—
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3138C2
socleC22×C79#386
prime radicalC2×#127×C79C22
cosocleC2×C42×C79C23
hypercenter#3138D79
hypocenterC79#9425
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
3. #23761951C2semidirect
4. #23761265C2semidirect
15. #486×C79C4semidirect
30. C4⋊Q8×C79C2×C4semidirect
41. C4×C8×C79C2×C4semidirect
117. C79#9425semidirect
121. #486Dic79semidirect
134. C4⋊Q8C2×Dic79semidirect
150. C4×C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237665772
#237621142
#237619512
#237612652
C79×#31382
#942579