Finite Group Databasebeta

← #34186231

Subgroups — #34186231

Order 20224.

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

normal subgroups only

This group has 542 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#23767708C211yesmaximal, characteristic
3#23763478C212yesmaximal, complemented
4#23761307C212yesmaximal, complemented
5C79×#1465C211yesmaximal, characteristic
6C2×#14579142C412yes—
7C22×C4⋊Dic79C2211yescharacteristic
8#19098641—24no—
9#19098616—21no—
10#19098527—21no—
11#19098527C2212yes—
12#19098493—21no—
13#19098488—21no—
14#19098461—21no—
15#19098424C412yes—
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—
26#14579142—44no—
27C2×C4⋊Dic79—21no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—22no—
31C2×C4⋊Dic79—22no—
32C2×Dic79⋊C4—22no—
33C2×C22⋊Dic79—22no—
34C2×C22⋊Dic79C2×C412yes—
35C4⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38C22⋊(C79⋊C8)—22no—
39C4×C79⋊C8—22no—
40C79⋊#95—22no—
41C79⋊#95—22no—
42C79⋊#95—21no—
43C22×D4×C79C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C23×C4×C79C2311yescharacteristic
46C2×M16×C79—22no—
47C4×D4×C79—22no—
48C4×D4×C79—22no—
49C2×C4⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C22⋊C8×C79—22no—
54C22⋊C8×C79—22no—
55C22×Dic79—22no—
56C22×Dic79—22no—
57C22×Dic79—24no—
58C22×Dic79—42no—
59C22×Dic79—42no—
60#11929608—44no—
61#11929608—82no—
62C2×C79⋊C8—24no—
63C4⋊Dic79—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—42no—
66C4⋊Dic79—41no—
67Dic79⋊C4—44no—
68C22⋊Dic79—42no—
69C22⋊Dic79—44no—
70C24×C79C22⋊C411yescharacteristic
71C24×C79C22×C411yescharacteristic
72C2×D4×C79—22no—
73C2×D4×C79—22no—
74C2×D4×C79—41no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C2×D412yes—
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79#4112yes—
86M16×C79—44no—
87C2×C8×C79—42no—
88C4⋊C4×C79—22no—
89C22⋊C4×C79—22no—
90C22⋊C4×C79—22no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—21no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79C2×C22⋊C411yescharacteristic
103C23×C79C4×D412yes—
104D4×C79—44no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—24no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—41no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—81no—
117C2×C4×C79#10012yes—
118C2×C4×C79#12011yescharacteristic
119C2×C4×C79#12112yes—
120C2×C4×C79#12311yescharacteristic
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126C2×Dic79—42no—
127C2×Dic79—84no—
128C8×C79—82no—
129C79⋊C8—44no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—22no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—42no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—81no—
142C22×C79#38611yescharacteristic
143C22×C79C2×#10011yescharacteristic
144C22×C79#41311yescharacteristic
145C4×C79—21no—
146C4×C79—22no—
147C4×C79—22no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79#45812yes—
151Dic79—82no—
152Dic79—82no—
153#9508—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—42no—
157C2×C79—81no—
158C2×C79#189011yescharacteristic
159C2×C79#190011yescharacteristic
160C2×C79#191511yescharacteristic
161#2975—791no—
162#2486—791no—
163#1821—792no—
164#1465D7911yescharacteristic
165#1305—792no—
166C79#950811yescharacteristic, complemented, Sylow
167C2×#123—792no—
168C2×C4×D4C2×D7911yescharacteristic
169C22×C4⋊C4—791no—
170#436—1584no—
171#431—1581no—
172#407—792no—
173#407—1581no—
174#407C2×D7912yes—
175#398—1581no—
176#397—1581no—
177#389—1581no—
178#381—792no—
179#380—791no—
180#380—1581no—
181#343—1584no—
182#324—21no—
183#323—21no—
184C22×D4C4×D7911yescharacteristic
185C23×C4—792no—
186C23×C4D31611yescharacteristic
187C23×C4C22×D7911yescharacteristic
188C2×M16—22no—
189C2×M16—794no—
190C2×M16—1582no—
191#123—3164no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1582no—
195C2×C4⋊C4—1582no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4—1582no—
200C2×C4⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4—792no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4C4×D7911yescharacteristic
204C2×C22⋊C4C22⋊D7911yescharacteristic
205C2×C42D31611yescharacteristic
206C4⋊C8—1582no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1582no—
210C22⋊C8—1582no—
211C22⋊C8—1582no—
212C4×C8—1582no—
213#95—1582no—
214#95—1582no—
215#95—1581no—
216C24(C2×D79)⋊C411yescharacteristic
217C24C2×C4×D7911yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—3162no—
226C22×C4—3162no—
227C22×C4—1582no—
228C22×C4—1582no—
229C22×C4—1584no—
230C22×C4(C2×D79)⋊C411yescharacteristic
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4D4×D7912yes—
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4C2×D31611yescharacteristic
236C22×C4#1192962312yes—
237M16—44no—
238M16—3164no—
239M16—6322no—
240C2×C8—42no—
241C2×C8—1584no—
242C4⋊C4—22no—
243C4⋊C4—3162no—
244C4⋊C4—3164no—
245C4⋊C4—3161no—
246C4⋊C4—3162no—
247C4⋊C4—3161no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—3164no—
251C22⋊C4—3162no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×(C2×D79)⋊C411yescharacteristic
264C23(C22⋊D79)⋊C412yes—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—3162no—
281C2×C4—3162no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—6324no—
285C2×C4#1457907712yes—
286C2×C4#1457913111yescharacteristic
287C2×C4#1457913312yes—
288C2×C4#1457914511yescharacteristic
289C8—82no—
290C8—3164no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1909844011yescharacteristic
304C22C2×#1457907711yescharacteristic
305C22#1909854511yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—6322no—
312C4—6322no—
313C4#1909875412yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2#2376381111yescharacteristic
319C2#2376386111yescharacteristic
320C2#2376395011yescharacteristic
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#9508
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
166. C79#9508semidirect

Maximal subgroups

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

subgroupindex
#237724412
#237677082
#237634782
#237613072
C79×#14652
#950879