Finite Group Databasebeta

← #34181802

Subgroups — #34181802

Order 20224.

546 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 107 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#23761727C212yesmaximal
2#23761306C214yesmaximal, complemented
3C79×#2315C211yesmaximal, characteristic
4#19098527C2214yes—
5#19098287—22no—
6#19098287—22no—
7C22×C22⋊C4×C79C411yescharacteristic
8C22×C22⋊C4×C79C411yescharacteristic
9C22×C22⋊C4×C79C2211yescharacteristic
10#385×C79—22no—
11#385×C79—22no—
12#380×C79C412yes—
13#380×C79C2212yes—
14C2×#11929608—24no—
15C22⋊(C79⋊C8)—24no—
16C22⋊(C79⋊C8)—24no—
17C25×C79C2×C411yescharacteristic
18C23×C4×C79—21no—
19C23×C4×C79—21no—
20C23×C4×C79C2×C411yescharacteristic
21C23×C4×C79C2311yescharacteristic
22C2×C4⋊C4×C79—22no—
23C2×C4⋊C4×C79—22no—
24C2×C22⋊C4×C79—24no—
25C2×C22⋊C4×C79—21no—
26C2×C22⋊C4×C79—21no—
27C2×C22⋊C4×C79—42no—
28C2×C22⋊C4×C79—42no—
29C2×C22⋊C4×C79C2×C412yes—
30C2×C22⋊C4×C79C2×C412yes—
31C2×C22⋊C4×C79D412yes—
32C2×C22⋊C4×C79D412yes—
33#95×C79—22no—
34#95×C79—22no—
35#11929608—84no—
36C2×C79⋊C8—44no—
37C24×C79—22no—
38C24×C79—22no—
39C24×C79—22no—
40C24×C79—81no—
41C24×C79—81no—
42C24×C79C22⋊C411yescharacteristic
43C24×C79C22⋊C411yescharacteristic
44C24×C79C22×C411yescharacteristic
45C22×C4×C79—22no—
46C22×C4×C79—22no—
47C22×C4×C79—22no—
48C22×C4×C79—42no—
49C22×C4×C79—42no—
50C22×C4×C79—81no—
51C22×C4×C79—81no—
52C22×C4×C79—81no—
53C22×C4×C79—81no—
54C22×C4×C79—81no—
55C22×C4×C79—81no—
56C22×C4×C79C22⋊C412yes—
57C22×C4×C79C2×D412yes—
58C22×C4×C79#4114yes—
59C4⋊C4×C79—82no—
60C4⋊C4×C79—82no—
61C22⋊C4×C79—24no—
62C22⋊C4×C79—24no—
63C22⋊C4×C79—42no—
64C22⋊C4×C79—42no—
65C22⋊C4×C79—42no—
66C22⋊C4×C79—42no—
67C23×C79—28no—
68C23×C79—21no—
69C23×C79—21no—
70C23×C79—21no—
71C23×C79—21no—
72C23×C79—21no—
73C23×C79—21no—
74C23×C79—41no—
75C23×C79—41no—
76C23×C79—41no—
77C23×C79—41no—
78C23×C79—41no—
79C23×C79—41no—
80C23×C79—82no—
81C23×C79—82no—
82C23×C79—84no—
83C23×C79—84no—
84C23×C79#10012yes—
85C23×C79#10112yes—
86C23×C79C2×C22⋊C411yescharacteristic
87C23×C79#11712yes—
88C2×C4×C79—22no—
89C2×C4×C79—44no—
90C2×C4×C79—42no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79—42no—
94C2×C4×C79—42no—
95C2×C4×C79—42no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79—82no—
99C2×C4×C79—84no—
100C2×C4×C79—84no—
101C2×C4×C79#12314yes—
102C79⋊C8—84no—
103C22×C79—21no—
104C22×C79—21no—
105C22×C79—22no—
106C22×C79—22no—
107C22×C79—22no—
108C22×C79—22no—
109C22×C79—22no—
110C22×C79—42no—
111C22×C79—42no—
112C22×C79—42no—
113C22×C79—42no—
114C22×C79—42no—
115C22×C79—42no—
116C22×C79—41no—
117C22×C79—41no—
118C22×C79—41no—
119C22×C79—41no—
120C22×C79—88no—
121C22×C79#38111yescharacteristic
122C22×C79C2×#10011yescharacteristic
123C22×C79C2×#10111yescharacteristic
124C4×C79—24no—
125C4×C79—42no—
126C4×C79—82no—
127C4×C79—82no—
128#8756—791nomaximal, Sylow
129C2×C79—21no—
130C2×C79—21no—
131C2×C79—42no—
132C2×C79—42no—
133C2×C79—42no—
134C2×C79#182111yescharacteristic
135C2×C79#182811yescharacteristic
136C2×C79#182911yescharacteristic
137#2315D7911yescharacteristic
138#1462—792no—
139#1305—794no—
140C79#875611yescharacteristic, complemented, Sylow
141C22×C22⋊C4Dic7911yescharacteristic
142C22×C22⋊C4Dic7911yescharacteristic
143C22×C22⋊C4C2×D7911yescharacteristic
144#407—794no—
145#385—22no—
146#385—22no—
147#380Dic7912yes—
148#380C2×D7912yes—
149#323—1582no—
150#323—1582no—
151C25C2×Dic7911yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×Dic7911yescharacteristic
155C23×C4C22×D7911yescharacteristic
156C2×M16—1584no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4C2×Dic7912yes—
165C2×C22⋊C4C2×Dic7912yes—
166C2×C22⋊C4C22⋊D7912yes—
167C2×C22⋊C4C22⋊D7912yes—
168C22⋊C8—1584no—
169C22⋊C8—1584no—
170#95—22no—
171#95—22no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—81no—
176C24—81no—
177C24C22⋊Dic7911yescharacteristic
178C24C22⋊Dic7911yescharacteristic
179C24C22×Dic7911yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C22⋊Dic7912yes—
192C22×C4C2×C22⋊D7912yes—
193C22×C4#1192962414yes—
194M16—6324no—
195C2×C8—3164no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C23—28no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23#1457907812yes—
222C23#1457908012yes—
223C23C2×C22⋊Dic7911yescharacteristic
224C23#1457912012yes—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4#1457914414yes—
239C8—6324no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—88no—
258C22#1909842611yescharacteristic
259C22C2×#1457907811yescharacteristic
260C22C2×#1457908011yescharacteristic
261C4—24no—
262C4—42no—
263C4—82no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2#2376348011yescharacteristic
271C2#2376350811yescharacteristic
272C2#2376351111yescharacteristic
subgroupG/H
centerC22#19098426
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2315C2
socleC22×C79#381
prime radicalC25×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2315D79
hypocenterC79#8756
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #23761306C2semidirect
140. C79#8756semidirect

Maximal subgroups

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

subgroupindex
#237617272
#237613062
C79×#23152
#875679