Finite Group Databasebeta

← #34185903

Subgroups — #34185903

Order 20224.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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#23773803C211yesmaximal, characteristic
2#23762027C211yesmaximal, characteristic, complemented
3#23761907C212yesmaximal
4#23761293C212yesmaximal, complemented
5C79×#2371C211yesmaximal, characteristic, complemented
6#19099113C2211yescharacteristic
7#19098933C412yescomplemented
8#19098929C412yescomplemented
9#19098920—22no—
10#19098620C2212yes—
11#19098562—22no—
12C4×#11929608C2211yescharacteristic
13#19098305—22no—
14#19098296—22no—
15#19098296—22no—
16C2×#125×C79C2211yescharacteristic
17#387×C79—22no—
18#387×C79—22no—
19C4×C4⋊C4×C79—22no—
20#376×C79C2212yes—
21C2×#11929624—21no—
22C2×#11929608—21no—
23#14579161C2×C411yescharacteristic
24#14579158C2×C411yescharacteristic
25#14579149—21no—
26#14579135—22no—
27C4×D316—22no—
28C4⋊(C79⋊C8)—22no—
29#14579084—24no—
30#14579081—24no—
31C22⋊(C79⋊C8)—22no—
32#14579072—22no—
33#14579072C2×C412yes—
34C4×C79⋊C8—21no—
35#125×C79—22no—
36#125×C79C2×C412yes—
37#125×C79D412yes—
38C2×C4⋊C4×C79—22no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79D412yes—
41C2×C42×C79—22no—
42C2×C42×C79—22no—
43C2×C42×C79C2311yescharacteristic
44#95×C79—22no—
45#95×C79—22no—
46#95×C79—22no—
47#95×C79D412yes—
48#95×C79D412yes—
49#11929624—44no—
50C2×Dic158—21no—
51C2×D316—21no—
52C2×D316—22no—
53C2×C22⋊D79—41no—
54C2×C4×D79—41no—
55#11929608—42no—
56C2×C79⋊C8—22no—
57C2×C79⋊C8—42no—
58C4⋊Dic79—22no—
59(C2×D79)⋊C4—42no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—22no—
63C22×C4×C79C2×D411yescharacteristic
64C22×C4×C79C2×D412yes—
65C4⋊C4×C79—24no—
66C4⋊C4×C79—24no—
67C4⋊C4×C79—22no—
68C4⋊C4×C79—44no—
69C4⋊C4×C79C22⋊C414yes—
70C42×C79—44no—
71C42×C79—44no—
72C42×C79C22×C411yescharacteristic
73C42×C79C2×D411yescharacteristic
74C42×C79#4112yes—
75C23×C79#12011yescharacteristic
76C22×D79—41no—
77C2×C4×C79—22no—
78C2×C4×C79—24no—
79C2×C4×C79—24no—
80C2×C4×C79—42no—
81C2×C4×C79—42no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79C2×C22⋊C411yescharacteristic
89C2×C4×C79C4×D412yes—
90C2×C4×C79#12112yes—
91C2×C4×C79#12311yescharacteristic
92C22⋊D79—82no—
93C2×Dic79—41no—
94D316—42no—
95D316—82no—
96C4×D79—82no—
97Dic158—42no—
98C79⋊C8—42no—
99C79⋊C8—82no—
100C22×C79—22no—
101C22×C79#38611yescharacteristic
102C22×C79#45412yes—
103C2×D79—81no—
104C2×D79—82no—
105C4×C79—21no—
106C4×C79—21no—
107C4×C79—22no—
108C4×C79—22no—
109C4×C79—44no—
110C4×C79—44no—
111C4×C79—82no—
112Dic79—81no—
113#9442—791nomaximal, Sylow
114C2×C79—22no—
115C2×C79#188811yescharacteristic
116C2×C79#190612yes—
117D79—161no—
118#3142—791no—
119#2371D7911yescharacteristic, complemented
120#1534—791no—
121#1505—792no—
122#1300—792no—
123C79#944211yescharacteristic, complemented, Sylow
124#532—791no—
125C2×#125C2×D7911yescharacteristic
126#491—792no—
127#490—792no—
128#488—1582no—
129#432—792no—
130#417—1582no—
131C4×M16—791no—
132#387—22no—
133#387—22no—
134C4×C4⋊C4—22no—
135#376C2×D7912yes—
136#330—1582no—
137#327—1582no—
138#327—1582no—
139C2×#41—1581no—
140C2×M16—1581no—
141C4⋊Q8—791no—
142#127—791no—
143#125—22no—
144#125C4×D7912yes—
145#125D31612yes—
146#124—1581no—
147#121—1582no—
148C4×D4—1582no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊D7912yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×D7911yescharacteristic
155C4⋊C8—1582no—
156#103—1584no—
157#102—1584no—
158C22⋊C8—1582no—
159#97—792no—
160#97—1582no—
161C4×C8—1581no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95C22⋊D7912yes—
166#95C22⋊D7912yes—
167#41—3164no—
168C2×Q8—1581no—
169C2×D4—3161no—
170C2×D4—1581no—
171C2×D4—1582no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—3161no—
176C22×C4C2×C22⋊D7911yescharacteristic
177C22×C4C2×C22⋊D7912yes—
178M16—3162no—
179C2×C8—3162no—
180C2×C8—1582no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—1582no—
186C4⋊C4(C2×D79)⋊C414yes—
187C22⋊C4—3162no—
188C42—44no—
189C42—44no—
190C42C2×C4×D7911yescharacteristic
191C42C2×D31611yescharacteristic
192C42#1192962412yes—
193C23—3161no—
194C23#1457913011yescharacteristic
195Q8—3162no—
196D4—3162no—
197D4—6322no—
198D4—6322no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—3161no—
211C2×C4—6322no—
212C2×C4C2×(C2×D79)⋊C411yescharacteristic
213C2×C4C4×C22⋊D7912yes—
214C2×C4#1457913512yes—
215C2×C4#1457914411yescharacteristic
216C8—3162no—
217C8—6322no—
218C22—22no—
219C22—6321no—
220C22—6322no—
221C22#1909844211yescharacteristic
222C22#1909873612yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—6321no—
231C2—22no—
232C2—12641no—
233C2#2376380311yescharacteristic
234C2#2376390112yes—
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2371C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2371D79
hypocenterC79#9442
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #23762027C2semidirect
4. #23761293C2semidirect
5. C79×#2371C2semidirect
7. #19098933C4semidirect
8. #19098929C4semidirect
119. #2371D79semidirect
123. C79#9442semidirect

Maximal subgroups

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

subgroupindex
#237738032
#237620272
#237619072
#237612932
C79×#23712
#944279