Finite Group Databasebeta

← #34186241

Subgroups — #34186241

Order 20224.

522 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 103 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#23772441C211yesmaximal, characteristic
2#23767717C211yesmaximal, characteristic
3#23763478C212yesmaximal, complemented
4#23761307C212yesmaximal, complemented
5C79×#1465C211yesmaximal, characteristic
6C22×C4⋊Dic79C2211yescharacteristic
7#19098641—24no—
8#19098616—21no—
9#19098527—21no—
10#19098527C2212yes—
11#19098502—21no—
12#19098499—21no—
13#19098461—21no—
14#19098443C412yes—
15#19098435C412yes—
16#19098423—21no—
17#19098423C2211yescharacteristic
18#19098331—24no—
19C2×C4×D4×C79C2211yescharacteristic
20#407×C79C2212yes—
21#324×C79—21no—
22#323×C79—21no—
23C23×Dic79C2×C412yes—
24C2×#11929608—22no—
25C2×#11929608D414yes—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—22no—
30C2×C4⋊Dic79—22no—
31C2×Dic79⋊C4—22no—
32C2×C22⋊Dic79—22no—
33C2×C22⋊Dic79C2×C412yes—
34C4⋊(C79⋊C8)—22no—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38C4×C79⋊C8—22no—
39C79⋊#95—22no—
40C79⋊#95—22no—
41C79⋊#95—21no—
42C22×D4×C79C2×C411yescharacteristic
43C23×C4×C79D411yescharacteristic
44C23×C4×C79C2311yescharacteristic
45C2×M16×C79—22no—
46C4×D4×C79—22no—
47C4×D4×C79—22no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C22⋊C8×C79—22no—
53C22⋊C8×C79—22no—
54C22×Dic79—22no—
55C22×Dic79—22no—
56C22×Dic79—24no—
57C22×Dic79—82no—
58#11929608—44no—
59#11929608—82no—
60C2×C79⋊C8—24no—
61C4⋊Dic79—42no—
62C4⋊Dic79—41no—
63C4⋊Dic79—42no—
64C4⋊Dic79—41no—
65Dic79⋊C4—82no—
66C22⋊Dic79—42no—
67C22⋊Dic79—82no—
68C24×C79C22⋊C411yescharacteristic
69C24×C79C22×C411yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—22no—
72C2×D4×C79—41no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—42no—
76C22×C4×C79—42no—
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C2×D412yes—
81C22×C4×C79C2×D411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79#4112yes—
84M16×C79—44no—
85C2×C8×C79—42no—
86C4⋊C4×C79—22no—
87C22⋊C4×C79—22no—
88C22⋊C4×C79—22no—
89C42×C79—21no—
90C42×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79C2×C22⋊C411yescharacteristic
101C23×C79C4×D412yes—
102D4×C79—44no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—22no—
106C2×C4×C79—24no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—41no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—81no—
115C2×C4×C79#10112yes—
116C2×C4×C79#12011yescharacteristic
117C2×C4×C79#12112yes—
118C2×C4×C79#12311yescharacteristic
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—84no—
126C8×C79—82no—
127C79⋊C8—44no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—22no—
131C22×C79—22no—
132C22×C79—22no—
133C22×C79—42no—
134C22×C79—42no—
135C22×C79—42no—
136C22×C79—41no—
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—22no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79#45812yes—
149Dic79—82no—
150Dic79—82no—
151#9510—791nomaximal, Sylow
152C2×C79—21no—
153C2×C79—21no—
154C2×C79—42no—
155C2×C79—81no—
156C2×C79#189011yescharacteristic
157C2×C79#190111yescharacteristic
158C2×C79#191411yescharacteristic
159#2975—791no—
160#2487—791no—
161#1821—792no—
162#1465D7911yescharacteristic
163#1305—792no—
164C79#951011yescharacteristic, complemented, Sylow
165C2×C4×D4C2×D7911yescharacteristic
166C22×C4⋊C4—791no—
167#436—1584no—
168#431—1581no—
169#407—792no—
170#407—1581no—
171#407C2×D7912yes—
172#400—1581no—
173#399—1581no—
174#389—1581no—
175#386—792no—
176#385—792no—
177#380—791no—
178#380—1581no—
179#343—1584no—
180#324—21no—
181#323—21no—
182C22×D4C4×D7911yescharacteristic
183C23×C4—792no—
184C23×C4D31611yescharacteristic
185C23×C4C22×D7911yescharacteristic
186C2×M16—22no—
187C2×M16—794no—
188C2×M16—1582no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—1582no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1582no—
196C2×C4⋊C4—1582no—
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4—792no—
199C2×C22⋊C4—1582no—
200C2×C22⋊C4C4×D7911yescharacteristic
201C2×C22⋊C4C22⋊D7911yescharacteristic
202C2×C42D31611yescharacteristic
203C4⋊C8—1582no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—1582no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209C4×C8—1582no—
210#95—1582no—
211#95—1582no—
212#95—1581no—
213C24(C2×D79)⋊C411yescharacteristic
214C24C2×C4×D7911yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—6322no—
223C22×C4—1582no—
224C22×C4—1582no—
225C22×C4—1584no—
226C22×C4(C2×D79)⋊C411yescharacteristic
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4D4×D7912yes—
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×D31611yescharacteristic
232C22×C4#1192962312yes—
233M16—44no—
234M16—3164no—
235M16—6322no—
236C2×C8—42no—
237C2×C8—1584no—
238C4⋊C4—22no—
239C4⋊C4—3162no—
240C4⋊C4—3161no—
241C4⋊C4—3162no—
242C4⋊C4—3161no—
243C4⋊C4—6322no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—3162no—
247C22⋊C4—6322no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×(C2×D79)⋊C411yescharacteristic
260C23(C22⋊D79)⋊C412yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—3162no—
275C2×C4—3162no—
276C2×C4—3162no—
277C2×C4—3162no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—6324no—
281C2×C4#1457907912yes—
282C2×C4#1457913111yescharacteristic
283C2×C4#1457913312yes—
284C2×C4#1457914511yescharacteristic
285C8—82no—
286C8—3164no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1909844011yescharacteristic
300C22C2×#1457907911yescharacteristic
301C22#1909854511yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—6322no—
308C4—6322no—
309C4#1909875412yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2376381111yescharacteristic
315C2#2376386611yescharacteristic
316C2#2376394511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1465C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#1465D79
hypocenterC79#9510
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #23763478C2semidirect
4. #23761307C2semidirect
164. C79#9510semidirect

Maximal subgroups

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

subgroupindex
#237724412
#237677172
#237634782
#237613072
C79×#14652
#951079