Finite Group Databasebeta

← #34186239

Subgroups — #34186239

Order 20224.

524 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 111 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#23772429C211yesmaximal, characteristic
2#23763480C212yesmaximal, complemented
3#23761738C211yesmaximal, characteristic
4#23761306C212yesmaximal, complemented
5C79×#2487C211yesmaximal, characteristic
6#19098635—24no—
7#19098620—21no—
8#19098527C2212yes—
9#19098527C2212yes—
10C2×C22⋊(C79⋊C8)—21no—
11#19098332—24no—
12#19098289—21no—
13#19098287—21no—
14C2×C4×D4×C79C2211yescharacteristic
15C22×C4⋊C4×C79C2211yescharacteristic
16#400×C79—21no—
17#399×C79—21no—
18#389×C79—21no—
19#386×C79C412yes—
20#385×C79C412yes—
21#380×C79—21no—
22#380×C79C2211yescharacteristic
23C2×#11929608—22no—
24C2×#11929608D414yes—
25C22×C79⋊C8—22no—
26C4⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32#14579072—22no—
33C22×D4×C79C2×C411yescharacteristic
34C23×C4×C79C2×C412yes—
35C23×C4×C79D411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37C4×D4×C79—22no—
38C4×D4×C79—22no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79—22no—
47C2×C22⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79C2×C412yes—
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C42×C79D411yescharacteristic
51#95×C79—22no—
52#95×C79—22no—
53#95×C79—21no—
54#11929608—44no—
55#11929608—44no—
56C2×C79⋊C8—24no—
57C2×C79⋊C8—42no—
58C2×C79⋊C8—82no—
59C24×C79C22⋊C411yescharacteristic
60C24×C79C22×C411yescharacteristic
61C2×D4×C79—22no—
62C2×D4×C79—22no—
63C2×D4×C79—41no—
64C22×C4×C79—21no—
65C22×C4×C79—21no—
66C22×C4×C79—22no—
67C22×C4×C79—22no—
68C22×C4×C79—24no—
69C22×C4×C79—42no—
70C22×C4×C79—42no—
71C22×C4×C79—82no—
72C22×C4×C79C22⋊C411yescharacteristic
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C2×D412yes—
76C22×C4×C79C2×D411yescharacteristic
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79#4112yes—
79C4⋊C4×C79—22no—
80C4⋊C4×C79—42no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—42no—
83C4⋊C4×C79—41no—
84C4⋊C4×C79—82no—
85C22⋊C4×C79—22no—
86C22⋊C4×C79—22no—
87C22⋊C4×C79—42no—
88C22⋊C4×C79—82no—
89C42×C79—21no—
90C42×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79#10112yes—
100C23×C79C2×C22⋊C411yescharacteristic
101C23×C79C4×D412yes—
102D4×C79—44no—
103C2×C4×C79—24no—
104C2×C4×C79—22no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—41no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—81no—
122C2×C4×C79—84no—
123C2×C4×C79#12011yescharacteristic
124C2×C4×C79#12112yes—
125C2×C4×C79#12311yescharacteristic
126C79⋊C8—44no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—22no—
131C22×C79—22no—
132C22×C79—22no—
133C22×C79—22no—
134C22×C79—42no—
135C22×C79—42no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—81no—
140C22×C79#38611yescharacteristic
141C22×C79C2×#10111yescharacteristic
142C22×C79#41311yescharacteristic
143C4×C79—21no—
144C4×C79—22no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—82no—
149C4×C79—82no—
150C4×C79#45812yes—
151#9509—791nomaximal, Sylow
152C2×C79—21no—
153C2×C79—21no—
154C2×C79—42no—
155C2×C79—42no—
156C2×C79#189011yescharacteristic
157C2×C79#190111yescharacteristic
158C2×C79#191411yescharacteristic
159#2973—791no—
160#2487D7911yescharacteristic
161#1821—792no—
162#1465—791no—
163#1305—792no—
164C79#950911yescharacteristic, complemented, Sylow
165C2×C4×D4C2×D7911yescharacteristic
166C22×C4⋊C4C2×D7911yescharacteristic
167#435—1584no—
168#432—1581no—
169#407—792no—
170#407—792no—
171C2×C22⋊C8—1581no—
172#400—21no—
173#399—21no—
174#389—21no—
175#386Dic7912yes—
176#385Dic7912yes—
177#380—21no—
178#380C2×D7911yescharacteristic
179#343—1584no—
180#324—1581no—
181#323—1581no—
182C22×D4C2×Dic7911yescharacteristic
183C23×C4C2×Dic7912yes—
184C23×C4C22⋊D7911yescharacteristic
185C23×C4C22×D7911yescharacteristic
186C2×M16—794no—
187C2×M16—1582no—
188C22×C8—1582no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic7911yescharacteristic
200C2×C22⋊C4C2×Dic7912yes—
201C2×C22⋊C4C22⋊D7911yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—1582no—
204C22⋊C8—1582no—
205C22⋊C8—1582no—
206C22⋊C8—1582no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209#97—1582no—
210#95—22no—
211#95—22no—
212#95—21no—
213C24C22⋊Dic7911yescharacteristic
214C24C22×Dic7911yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4C22⋊Dic7911yescharacteristic
227C22×C4C22⋊Dic7911yescharacteristic
228C22×C4C22⋊Dic7911yescharacteristic
229C22×C4D4×D7912yes—
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4#1192962312yes—
233M16—3164no—
234M16—3164no—
235C2×C8—3162no—
236C2×C8—6322no—
237C2×C8—1584no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#1457908012yes—
259C23C2×C22⋊Dic7911yescharacteristic
260C23D4×Dic7912yes—
261D4—44no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4#1457912911yescharacteristic
283C2×C4#1457913312yes—
284C2×C4#1457914211yescharacteristic
285C8—3164no—
286C8—6322no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1909844311yescharacteristic
300C22C2×#1457908011yescharacteristic
301C22#1909854711yescharacteristic
302C4—22no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—82no—
308C4—82no—
309C4#1909875212yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#2376381411yescharacteristic
315C2#2376386911yescharacteristic
316C2#2376394811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2487C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2487D79
hypocenterC79#9509
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #23763480C2semidirect
4. #23761306C2semidirect
164. C79#9509semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237634802
#237617382
#237613062
C79×#24872
#950979