Finite Group Databasebeta

← #34185812

Subgroups — #34185812

Order 20224.

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

normal subgroups only

This group has 422 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#23766581C211yesmaximal, characteristic
2#23762126C211yesmaximal, characteristic
3#23761951C212yesmaximal
4#23761265C212yesmaximal
5C79×#3139C211yesmaximal, characteristic
6#19098620C2212yes—
7#19098470—22no—
8#19098449—24no—
9#19098431C2212yes—
10#19098312—22no—
11#19098300—24no—
12#533×C79C2211yescharacteristic
13C2×C4⋊Q8×C79C2211yescharacteristic
14#491×C79—22no—
15#487×C79C414yescomplemented
16#419×C79—22no—
17C4×M16×C79C2211yescharacteristic
18C2×C4⋊Dic79—22no—
19C2×C4×Dic79—22no—
20C4⋊(C79⋊C8)—22no—
21C22⋊(C79⋊C8)—22no—
22#14579072—22no—
23C79⋊#95—24no—
24C79⋊#95—22no—
25C79⋊#95—22no—
26C79⋊#95D414yes—
27C22×Q8×C79D412yes—
28C2×M16×C79—21no—
29C4⋊Q8×C79—22no—
30C4⋊Q8×C79C2×C412yes—
31C4⋊Q8×C79C2×C412yes—
32C4⋊Q8×C79D412yes—
33#125×C79—21no—
34C22⋊Q8×C79—22no—
35C4×Q8×C79—22no—
36#117×C79—21no—
37C2×C4⋊C4×C79—22no—
38C2×C42×C79C2311yescharacteristic
39#103×C79—28no—
40#97×C79—21no—
41C4×C8×C79C2×C412yes—
42C22×Dic79—24no—
43C2×C79⋊C8—42no—
44C4⋊Dic79—82no—
45C4×Dic79—82no—
46C2×Q8×C79—22no—
47C2×Q8×C79—22no—
48C2×Q8×C79—42no—
49C2×Q8×C79—42no—
50C2×Q8×C79C22⋊C414yes—
51C22×C4×C79—22no—
52C22×C4×C79C2×D411yescharacteristic
53C22×C4×C79C2×D412yes—
54M16×C79—42no—
55C2×C8×C79—22no—
56C4⋊C4×C79—24no—
57C4⋊C4×C79—24no—
58C4⋊C4×C79—22no—
59C4⋊C4×C79—22no—
60C4⋊C4×C79—42no—
61C22⋊C4×C79—41no—
62C42×C79—41no—
63C42×C79C22×C411yescharacteristic
64C42×C79C2×D411yescharacteristic
65C42×C79#4112yes—
66C23×C79#12011yescharacteristic
67Q8×C79—44no—
68Q8×C79—82no—
69Q8×C79—82no—
70C2×C4×C79—22no—
71C2×C4×C79—24no—
72C2×C4×C79—42no—
73C2×C4×C79—42no—
74C2×C4×C79—42no—
75C2×C4×C79—42no—
76C2×C4×C79C2×C22⋊C411yescharacteristic
77C2×C4×C79C4×D412yes—
78C2×C4×C79#12112yes—
79C2×C4×C79#12311yescharacteristic
80C2×Dic79—48no—
81C2×Dic79—44no—
82C8×C79—42no—
83C79⋊C8—82no—
84C22×C79—22no—
85C22×C79#38611yescharacteristic
86C22×C79#45512yes—
87C4×C79—21no—
88C4×C79—21no—
89C4×C79—22no—
90C4×C79—22no—
91C4×C79—44no—
92C4×C79—82no—
93Dic79—84no—
94#9426—791nomaximal, Sylow
95C2×C79—22no—
96C2×C79#188511yescharacteristic
97C2×C79#190512yes—
98#3139D7911yescharacteristic
99#2381—791no—
100#1558—791no—
101#1515—792no—
102#1288—792no—
103C79#942611yescharacteristic, complemented, Sylow
104#533C2×D7911yescharacteristic
105C2×C4⋊Q8C2×D7911yescharacteristic
106#491—22no—
107#487Dic7914yescomplemented
108#432—792no—
109#419—22no—
110C4×M16C2×D7911yescharacteristic
111#391—1582no—
112#387—1584no—
113#383—792no—
114#333—1582no—
115#328—1584no—
116C22×Q8C22⋊D7912yes—
117C2×M16—21no—
118C4⋊Q8—22no—
119C4⋊Q8C2×Dic7912yes—
120C4⋊Q8C2×Dic7912yes—
121C4⋊Q8C22⋊D7912yes—
122#125—21no—
123C22⋊Q8—22no—
124C4×Q8—22no—
125#117—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—1582no—
128C2×C42—1582no—
129C2×C42C22×D7911yescharacteristic
130C4⋊C8—1582no—
131#103—28no—
132C22⋊C8—1582no—
133#97—21no—
134#97—1582no—
135C4×C8C2×Dic7912yes—
136#95—794no—
137#95—1584no—
138#95—1582no—
139#95—1582no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C22⋊Dic7914yes—
145C22×C4—22no—
146C22×C4—1584no—
147C22×C4D4×D7912yes—
148C22×C4C2×C22⋊D7911yescharacteristic
149M16—42no—
150C2×C8—22no—
151C2×C8—3162no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—6322no—
158C22⋊C4—41no—
159C42—41no—
160C42—6322no—
161C42C22×Dic7911yescharacteristic
162C42C2×C22⋊D7911yescharacteristic
163C42#1192962312yes—
164C23#1457912911yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—82no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—3168no—
175C2×C4—3164no—
176C2×C4C2×C22⋊Dic7911yescharacteristic
177C2×C4D4×Dic7912yes—
178C2×C4#1457913312yes—
179C2×C4#1457914211yescharacteristic
180C8—42no—
181C8—6322no—
182C22—22no—
183C22#1909844311yescharacteristic
184C22#1909874012yes—
185C4—21no—
186C4—21no—
187C4—22no—
188C4—22no—
189C4—44no—
190C4—82no—
191C4—6324no—
192C2—22no—
193C2#2376378911yescharacteristic
194C2#2376389612yes—
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3139C2
socleC22×C79#386
prime radicalC23×C79#120
cosocleC2×C42×C79C23
hypercenter#3139D79
hypocenterC79#9426
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
15. #487×C79C4semidirect
103. C79#9426semidirect
107. #487Dic79semidirect

Maximal subgroups

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

subgroupindex
#237665812
#237621262
#237619512
#237612652
C79×#31392
#942679