Finite Group Databasebeta

← #34200845

Subgroups — #34200845

Order 20224.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764624C211yesmaximal, characteristic
2#23762436C212yesmaximal
3#23761427C212yesmaximal, complemented
4#23761346C211yesmaximal, characteristic, complemented
5C79×#2571C211yesmaximal, characteristic, complemented
6#19099144C2211yescharacteristic
7#19098881—22no—
8C2×#14579089—21no—
9#19098562—21no—
10#19098332C2211yescharacteristic
11#19098296—22no—
12#19098289C2212yes—
13#400×C79—21no—
14#400×C79C2212yes—
15#389×C79—22no—
16#384×C79C2211yescharacteristic
17#382×C79—22no—
18C4×C4⋊C4×C79—21no—
19C2×#11929624—21no—
20C2×#11929608—21no—
21#14579141—41no—
22#14579136—41no—
23#14579135—22no—
24#14579092—42no—
25#14579089—42no—
26#14579084—41no—
27#14579081—42no—
28#14579081—41no—
29C22⋊(C79⋊C8)—22no—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—22no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79D412yes—
35C2×C4⋊C4×C79D411yescharacteristic
36C2×C4⋊C4×C79C2311yescharacteristic
37C2×C42×C79—22no—
38C2×C42×C79D411yescharacteristic
39#95×C79—21no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79—22no—
43#95×C79D412yes—
44#11929624—41no—
45#11929624—41no—
46#11929624—41no—
47#11929624—41no—
48C2×Dic158—21no—
49C2×D316—21no—
50C2×D316—22no—
51C2×C22⋊D79—41no—
52C2×C4×D79—41no—
53#11929608—41no—
54#11929608—41no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—41no—
57C4⋊Dic79—22no—
58(C2×D79)⋊C4—42no—
59C22×C4×C79—21no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—21no—
63C22×C4×C79C2×D411yescharacteristic
64C22×C4×C79C2×D412yes—
65C4⋊C4×C79—41no—
66C4⋊C4×C79—42no—
67C4⋊C4×C79—44no—
68C4⋊C4×C79—42no—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—81no—
72C42×C79—21no—
73C42×C79—44no—
74C42×C79SD1612yes—
75C23×C79#12011yescharacteristic
76C22×D79—41no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—21no—
81C2×C4×C79—21no—
82C2×C4×C79—22no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—41no—
91C2×C4×C79—42no—
92C2×C4×C79C2×SD1611yescharacteristic
93C2×C4×C79#13711yescharacteristic
94C22⋊D79—81no—
95C22⋊D79—81no—
96C2×Dic79—41no—
97D316—41no—
98D316—41no—
99D316—82no—
100C4×D79—81no—
101C4×D79—81no—
102Dic158—41no—
103Dic158—41no—
104C79⋊C8—81no—
105C79⋊C8—82no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79#44811yescharacteristic
109C22×C79#45411yescharacteristic
110C22×C79B(4,𝔽2)11yescharacteristic
111C2×D79—81no—
112C2×D79—81no—
113C2×D79—81no—
114C4×C79—24no—
115C4×C79—21no—
116C4×C79—21no—
117C4×C79—42no—
118C4×C79—82no—
119C4×C79—81no—
120C4×C79—82no—
121Dic79—81no—
122#12181—791nomaximal, Sylow
123C2×C79—21no—
124C2×C79—21no—
125C2×C79#159911yescharacteristic
126C2×C79#219811yescharacteristic
127C2×C79#220411yescharacteristic
128D79—161no—
129#2571D7911yescharacteristic, complemented
130#2039—791no—
131#1616—792no—
132#1353—792no—
133#1325—791no—
134C79#1218111yescharacteristic, complemented, Sylow
135#537—791no—
136#481—1582no—
137C2×#104—1581no—
138#417—1581no—
139#400—21no—
140#400C2×D7912yes—
141#389—22no—
142#384C2×D7911yescharacteristic
143#382—22no—
144C4×C4⋊C4—21no—
145#343—791no—
146#327—1582no—
147#324—792no—
148C2×#41—1581no—
149C2×M16—1581no—
150C22⋊Q8—3161no—
151#121—3161no—
152#121—1582no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D7911yescharacteristic
158C2×C4⋊C4C22⋊D7912yes—
159C2×C4⋊C4C22×D7911yescharacteristic
160C2×C42—22no—
161C2×C42C22⋊D7911yescharacteristic
162#106—3162no—
163#104—3162no—
164#103—3161no—
165#102—3161no—
166#102—3162no—
167C22⋊C8—1582no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95C22⋊D7912yes—
173#41—3161no—
174#41—3161no—
175#41—3161no—
176#41—3161no—
177C2×Q8—1581no—
178C2×D4—3161no—
179C2×D4—1581no—
180C2×D4—1582no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—3161no—
186C22×C4C2×C22⋊D7911yescharacteristic
187C22×C4C2×C22⋊D7912yes—
188M16—3161no—
189M16—3161no—
190C2×C8—3162no—
191C2×C8—3161no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—1582no—
200C22⋊C4—3162no—
201C42—21no—
202C42—44no—
203C42#1192961112yes—
204C23—3161no—
205C23#1457913011yescharacteristic
206Q8—3161no—
207Q8—3161no—
208D4—3161no—
209D4—3161no—
210D4—6322no—
211D4—6321no—
212D4—6321no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—3161no—
229C2×C4—6321no—
230C2×C4—6321no—
231C2×C4C2×#1192961111yescharacteristic
232C2×C4#1457919211yescharacteristic
233C8—6321no—
234C8—6322no—
235C22—21no—
236C22—21no—
237C22—6321no—
238C22—6321no—
239C22—6321no—
240C22#1909870211yescharacteristic
241C22#1909873611yescharacteristic
242C22#1909875011yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—6321no—
251C2—21no—
252C2—21no—
253C2—12641no—
254C2#2376234011yescharacteristic
255C2#2376539611yescharacteristic
256C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2571C2
socleC22×C79B(4,𝔽2)
prime radical#19099144C22
hypercenter#2571D79
hypocenterC79#12181
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
3. #23761427C2semidirect
4. #23761346C2semidirect
5. C79×#2571C2semidirect
129. #2571D79semidirect
134. C79#12181semidirect

Maximal subgroups

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

subgroupindex
#237646242
#237624362
#237614272
#237613462
C79×#25712
#1218179