Finite Group Databasebeta

← #34200869

Subgroups — #34200869

Order 20224.

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

normal subgroups only

This group has 360 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#23764629C211yesmaximal, characteristic
2#23762436C212yesmaximal
3#23761427C212yesmaximal, complemented
4#23761344C211yesmaximal, characteristic, complemented
5C79×#2844C211yesmaximal, characteristic, complemented
6#19099144C2211yescharacteristic
7#19098881—22no—
8C2×#14579089—21no—
9#19098562—21no—
10#19098332C2211yescharacteristic
11#19098296—22no—
12#19098289C2212yes—
13#400×C79—22no—
14#400×C79C2212yes—
15#395×C79—22no—
16#395×C79—21no—
17#382×C79—21no—
18#382×C79C2211yescharacteristic
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—
30C22×Q8×C79—21no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—22no—
34C2×C4⋊C4×C79D412yes—
35C2×C4⋊C4×C79D411yescharacteristic
36C2×C4⋊C4×C79C2311yescharacteristic
37C2×C42×C79D411yescharacteristic
38#95×C79—22no—
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—
59C2×Q8×C79—82no—
60C2×Q8×C79—81no—
61C22×C4×C79—21no—
62C22×C4×C79—22no—
63C22×C4×C79—22no—
64C22×C4×C79—21no—
65C22×C4×C79C2×D411yescharacteristic
66C22×C4×C79C2×D412yes—
67C4⋊C4×C79—41no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—42no—
70C4⋊C4×C79—81no—
71C4⋊C4×C79—81no—
72C4⋊C4×C79—82no—
73C42×C79—21no—
74C42×C79SD1612yes—
75C23×C79#12011yescharacteristic
76C22×D79—41no—
77Q8×C79—84no—
78C2×C4×C79—21no—
79C2×C4×C79—21no—
80C2×C4×C79—22no—
81C2×C4×C79—22no—
82C2×C4×C79—41no—
83C2×C4×C79—41no—
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—42no—
91C2×C4×C79—41no—
92C2×C4×C79—42no—
93C2×C4×C79C2×SD1611yescharacteristic
94C2×C4×C79#13711yescharacteristic
95C22⋊D79—81no—
96C22⋊D79—81no—
97C2×Dic79—41no—
98D316—41no—
99D316—41no—
100D316—82no—
101C4×D79—81no—
102C4×D79—81no—
103Dic158—41no—
104Dic158—41no—
105C79⋊C8—81no—
106C79⋊C8—82no—
107C22×C79—21no—
108C22×C79—21no—
109C22×C79#44811yescharacteristic
110C22×C79#45411yescharacteristic
111C22×C79B(4,𝔽2)11yescharacteristic
112C2×D79—81no—
113C2×D79—81no—
114C2×D79—81no—
115C4×C79—21no—
116C4×C79—21no—
117C4×C79—42no—
118C4×C79—42no—
119C4×C79—82no—
120C4×C79—81no—
121C4×C79—82no—
122Dic79—81no—
123#12185—791nomaximal, Sylow
124C2×C79—21no—
125C2×C79—21no—
126C2×C79#159911yescharacteristic
127C2×C79#220111yescharacteristic
128C2×C79#220211yescharacteristic
129D79—161no—
130#2844D7911yescharacteristic, complemented
131#2040—791no—
132#1616—792no—
133#1353—792no—
134#1324—791no—
135C79#1218511yescharacteristic, complemented, Sylow
136#537—791no—
137#481—1582no—
138C2×#104—1581no—
139#417—1581no—
140#400—22no—
141#400C2×D7912yes—
142#395—22no—
143#395—21no—
144#382—21no—
145#382C2×D7911yescharacteristic
146#343—791no—
147#327—1582no—
148#324—792no—
149C2×#41—1581no—
150C22×Q8—21no—
151C2×M16—1581no—
152C22⋊Q8—3161no—
153#121—3161no—
154#121—1582no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C22⋊D7911yescharacteristic
159C2×C4⋊C4C22⋊D7912yes—
160C2×C4⋊C4C22×D7911yescharacteristic
161C2×C42C22⋊D7911yescharacteristic
162#106—3162no—
163#104—3162no—
164#103—3161no—
165#102—3161no—
166#102—3162no—
167C22⋊C8—1582no—
168#95—22no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95C22⋊D7912yes—
174#41—3161no—
175#41—3161no—
176#41—3161no—
177#41—3161no—
178C2×Q8—82no—
179C2×Q8—81no—
180C2×Q8—1581no—
181C2×D4—3161no—
182C2×D4—1581no—
183C2×D4—1582no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—3161no—
189C22×C4C2×C22⋊D7911yescharacteristic
190C22×C4C2×C22⋊D7912yes—
191M16—3161no—
192M16—3161no—
193C2×C8—3162no—
194C2×C8—3161no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—82no—
201C4⋊C4—1582no—
202C22⋊C4—3162no—
203C42—21no—
204C42#1192961112yes—
205C23—3161no—
206C23#1457913011yescharacteristic
207Q8—84no—
208Q8—3161no—
209Q8—3161no—
210D4—3161no—
211D4—3161no—
212D4—6322no—
213D4—6321no—
214D4—6321no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—3161no—
231C2×C4—6321no—
232C2×C4—6321no—
233C2×C4C2×#1192961111yescharacteristic
234C2×C4#1457919211yescharacteristic
235C8—6321no—
236C8—6322no—
237C22—21no—
238C22—21no—
239C22—6321no—
240C22—6321no—
241C22—6321no—
242C22#1909870211yescharacteristic
243C22#1909873611yescharacteristic
244C22#1909875011yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4—6321no—
253C2—21no—
254C2—21no—
255C2—12641no—
256C2#2376234011yescharacteristic
257C2#2376541311yescharacteristic
258C2#2376541711yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2844C2
socleC22×C79B(4,𝔽2)
prime radical#19099144C22
hypercenter#2844D79
hypocenterC79#12185
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
3. #23761427C2semidirect
4. #23761344C2semidirect
5. C79×#2844C2semidirect
130. #2844D79semidirect
135. C79#12185semidirect

Maximal subgroups

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

subgroupindex
#237646292
#237624362
#237614272
#237613442
C79×#28442
#1218579