Finite Group Databasebeta

← #34201025

Subgroups — #34201025

Order 20224.

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

normal subgroups only

This group has 404 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#23764556C211yesmaximal, characteristic
2#23762407C212yesmaximal, complemented
3#23761406C212yesmaximal, complemented
4#23761332C211yesmaximal, characteristic
5C79×#2724C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098893—22no—
8#19098568—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×#123×C79—21no—
14#398×C79C2212yes—
15#397×C79—22no—
16#390×C79C2211yescharacteristic
17#389×C79—21no—
18#385×C79—22no—
19C2×#11929608—21no—
20#14579160—41no—
21#14579152—41no—
22#14579140—22no—
23#14579120—21no—
24#14579092—42no—
25#14579084—42no—
26#14579082—41no—
27#14579082—41no—
28C22⋊(C79⋊C8)—22no—
29C22×D4×C79D411yescharacteristic
30C23×C4×C79—21no—
31#123×C79—42no—
32#123×C79—42no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79D412yes—
37C2×C4⋊C4×C79C2311yescharacteristic
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—21no—
40C2×C22⋊C4×C79—22no—
41C2×C42×C79D411yescharacteristic
42#95×C79—22no—
43#95×C79—22no—
44#95×C79D412yes—
45C2×Dic158—22no—
46#11929608—41no—
47#11929608—41no—
48C2×C79⋊C8—42no—
49C2×C79⋊C8—41no—
50C4⋊Dic79—21no—
51C4⋊Dic79—21no—
52C4⋊Dic79—22no—
53Dic79⋊C4—42no—
54C22⋊Dic79—41no—
55C4×Dic79—41no—
56C24×C79—21no—
57C2×D4×C79—21no—
58C2×D4×C79—41no—
59C2×D4×C79—41no—
60C2×D4×C79SD1612yes—
61C22×C4×C79—21no—
62C22×C4×C79—22no—
63C22×C4×C79—22no—
64C22×C4×C79—42no—
65C22×C4×C79—81no—
66C22×C4×C79—81no—
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D412yes—
69C4⋊C4×C79—41no—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—42no—
72C4⋊C4×C79—42no—
73C4⋊C4×C79—42no—
74C4⋊C4×C79—81no—
75C22⋊C4×C79—44no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—44no—
78C42×C79—41no—
79C23×C79—22no—
80C23×C79—42no—
81C23×C79—42no—
82C23×C79—41no—
83C23×C79—41no—
84C23×C79#12011yescharacteristic
85D4×C79—42no—
86D4×C79—81no—
87C2×C4×C79—21no—
88C2×C4×C79—21no—
89C2×C4×C79—22no—
90C2×C4×C79—22no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—42no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—84no—
101C2×C4×C79C2×SD1611yescharacteristic
102C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
103C2×Dic79—41no—
104C2×Dic79—41no—
105Dic158—82no—
106C79⋊C8—81no—
107C79⋊C8—82no—
108C22×C79—21no—
109C22×C79—21no—
110C22×C79—44no—
111C22×C79—42no—
112C22×C79—82no—
113C22×C79—81no—
114C22×C79—81no—
115C22×C79#45011yescharacteristic
116C22×C79#45511yescharacteristic
117C22×C79B(4,𝔽2)11yescharacteristic
118C4×C79—21no—
119C4×C79—21no—
120C4×C79—42no—
121C4×C79—42no—
122C4×C79—82no—
123C4×C79—82no—
124Dic79—81no—
125Dic79—81no—
126#12209—791nomaximal, Sylow
127C2×C79—21no—
128C2×C79—21no—
129C2×C79—82no—
130C2×C79#160111yescharacteristic
131C2×C79#219811yescharacteristic
132C2×C79#220311yescharacteristic
133#2724D7911yescharacteristic
134#2027—791no—
135#1611—792no—
136#1346—792no—
137#1318—791no—
138C79#1220911yescharacteristic, complemented, Sylow
139#544—791no—
140C2×#123—21no—
141#483—1582no—
142#418—1581no—
143#398C2×D7912yes—
144#397—22no—
145#390C2×D7911yescharacteristic
146#389—21no—
147#385—22no—
148#343—791no—
149#339—1581no—
150#328—1582no—
151#324—792no—
152C22×D4C22⋊D7911yescharacteristic
153C23×C4—21no—
154C2×M16—1581no—
155C4⋊Q8—3161no—
156#125—3161no—
157#123—42no—
158#123—42no—
159C22⋊Q8—1582no—
160#117—1581no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22⋊D7912yes—
165C2×C4⋊C4C22×D7911yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42C22⋊D7911yescharacteristic
170#106—3162no—
171#103—3162no—
172#102—3161no—
173#102—3161no—
174C22⋊C8—1582no—
175#95—22no—
176#95—22no—
177#95C22⋊D7912yes—
178C24—21no—
179C2×Q8—1582no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4#1192961212yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×C22⋊D7911yescharacteristic
191C22×C4C2×C22⋊D7912yes—
192M16—3161no—
193M16—3161no—
194C2×C8—3162no—
195C2×C8—3161no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—3162no—
203C4⋊C4—1581no—
204C4⋊C4—1581no—
205C4⋊C4—1582no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—3161no—
210C42—41no—
211C42—3161no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#1457913011yescharacteristic
218Q8—6322no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—3161no—
236C2×C4—3161no—
237C2×C4C2×#1192961211yescharacteristic
238C2×C4#1457918711yescharacteristic
239C8—6321no—
240C8—6322no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#1909871411yescharacteristic
249C22#1909874111yescharacteristic
250C22#1909875011yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—82no—
257C4—6321no—
258C4—6321no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2376235211yescharacteristic
263C2#2376539611yescharacteristic
264C2#2376542111yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2724C2
socleC22×C79B(4,𝔽2)
prime radicalC22×D4×C79D4
hypercenter#2724D79
hypocenterC79#12209
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2724

Complements

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

NH ≅ G/Ntype
2. #23762407C2semidirect
3. #23761406C2semidirect
138. C79#12209semidirect

Maximal subgroups

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

subgroupindex
#237645562
#237624072
#237614062
#237613322
C79×#27242
#1220979