Finite Group Databasebeta

← #34186123

Subgroups — #34186123

Order 20224.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#23768305C211yesmaximal, characteristic
2#23763396C211yesmaximal, characteristic, complemented
3#23761826C212yesmaximal
4#23761259C212yesmaximal, complemented
5C79×#2982C211yesmaximal, characteristic, complemented
6C2×#14579137—21no—
7C2×C4×D316C2211yescharacteristic
8#19098587C2212yes—
9C2×#14579080—21no—
10#19098493C412yes—
11#19098480C412yes—
12#19098454—22no—
13#19098429C2211yescharacteristic
14#19098420—21no—
15#19098332—21no—
16#19098292—24no—
17C2×C4×Q8×C79C2211yescharacteristic
18C8×Q8×C79—24no—
19#431×C79—22no—
20#431×C79C2212yes—
21C22×D316C2×C411yescharacteristic
22C22×C4×D79—21no—
23C2×#11929608—22no—
24#14579137—42no—
25#14579137—42no—
26C4×D316—22no—
27C4×D316—22no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—22no—
31C2×(C2×D79)⋊C4—22no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—21no—
34C4⋊(C79⋊C8)—22no—
35C4⋊(C79⋊C8)—22no—
36#14579080—42no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95C2×C412yes—
40C22×Q8×C79D411yescharacteristic
41C4×Q8×C79—22no—
42C4×Q8×C79—22no—
43C2×C4⋊C4×C79C2×C412yes—
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C42×C79D412yes—
46C2×C42×C79C2311yescharacteristic
47C4⋊C8×C79—24no—
48C4⋊C8×C79—22no—
49C22⋊C8×C79—22no—
50C4×C8×C79—24no—
51C4×C8×C79D414yes—
52C23×D79—21no—
53C2×D316—22no—
54C2×D316—42no—
55C22×Dic79—22no—
56C22×Dic79—21no—
57C2×C4×D79—42no—
58C2×C4×D79—41no—
59C2×C4×D79—41no—
60C2×C4×D79—42no—
61#11929608—42no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—22no—
65C4⋊Dic79—41no—
66C4⋊Dic79—41no—
67Dic79⋊C4—44no—
68(C2×D79)⋊C4—44no—
69(C2×D79)⋊C4—44no—
70(C2×D79)⋊C4—41no—
71(C2×D79)⋊C4—41no—
72C2×Q8×C79—22no—
73C2×Q8×C79—22no—
74C2×Q8×C79D812yes—
75C2×Q8×C79SD1612yes—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C2×C8×C79—24no—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—22no—
84C42×C79—22no—
85C42×C79—22no—
86C42×C79C2×D412yes—
87C42×C79#4112yes—
88C23×C79C2×C22⋊C411yescharacteristic
89C22×D79—22no—
90C22×D79—41no—
91C22×D79—41no—
92C22×D79—41no—
93C22×D79—41no—
94C22×D79—41no—
95C22×D79—41no—
96Q8×C79—24no—
97Q8×C79—24no—
98C2×C4×C79—22no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—41no—
102C2×C4×C79—41no—
103C2×C4×C79—42no—
104C2×C4×C79#10214yes—
105C2×C4×C79C4×D412yes—
106C2×C4×C79#12011yescharacteristic
107C2×C4×C79#12112yes—
108C2×C4×C79#12311yescharacteristic
109C2×C4×C79C2×D811yescharacteristic
110C2×C4×C79C2×SD1611yescharacteristic
111C2×C4×C79#13512yes—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—41no—
115C2×Dic79—42no—
116C2×Dic79—41no—
117C2×Dic79—41no—
118D316—24no—
119D316—42no—
120C4×D79—82no—
121C4×D79—82no—
122C8×C79—44no—
123C79⋊C8—82no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79#38611yescharacteristic
127C22×C79C2×#10211yescharacteristic
128C22×C79#41611yescharacteristic
129C2×D79—42no—
130C2×D79—42no—
131C2×D79—42no—
132C2×D79—81no—
133C2×D79—81no—
134C2×D79—81no—
135C2×D79—81no—
136C4×C79—22no—
137C4×C79—24no—
138C4×C79—21no—
139C4×C79—21no—
140C4×C79—42no—
141C4×C79#46512yes—
142C4×C79#46612yes—
143Dic79—82no—
144Dic79—81no—
145#9486—791nomaximal, Sylow
146C2×C79—21no—
147C2×C79—21no—
148C2×C79#189111yescharacteristic
149C2×C79#189311yescharacteristic
150C2×C79#191211yescharacteristic
151D79—82no—
152#2982D7911yescharacteristic, complemented
153#2536—791no—
154#1801—791no—
155#1486—792no—
156#1286—792no—
157C79#948611yescharacteristic, complemented, Sylow
158C2×C22⋊Q8—1581no—
159C2×C4×Q8C2×D7911yescharacteristic
160C2×C4×D4—791no—
161C8×Q8—24no—
162#431—22no—
163#431C2×D7912yes—
164#423—792no—
165C2×#101—1581no—
166#398—792no—
167#394—792no—
168#388—1582no—
169#382—791no—
170#380—1581no—
171#343—1581no—
172D4⋊C8—1584no—
173C22×Q8D31611yescharacteristic
174C22×D4—791no—
175C23×C4—1581no—
176C2×M16—1582no—
177C22⋊Q8—3162no—
178C22⋊Q8—3162no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1582no—
182C4×D4—1582no—
183C2×C4⋊C4—791no—
184C2×C4⋊C4—1582no—
185C2×C4⋊C4—1581no—
186C2×C4⋊C4C4×D7912yes—
187C2×C4⋊C4D31611yescharacteristic
188C2×C22⋊C4—1582no—
189C2×C22⋊C4—1582no—
190C2×C22⋊C4—1581no—
191C2×C42C22⋊D7912yes—
192C2×C42C22×D7911yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—1582no—
196C4⋊C8—1582no—
197#101—3162no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8C22⋊D7914yes—
201#95—792no—
202#95—1582no—
203#95—1581no—
204C24—1581no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8D63212yes—
208C2×Q8#1192961312yes—
209C2×D4—3162no—
210C2×D4—1582no—
211C22×C4—3162no—
212C22×C4—3161no—
213C22×C4—3161no—
214C22×C4—3162no—
215C22×C4—1582no—
216C22×C4—1581no—
217C22×C4(C2×D79)⋊C412yes—
218C22×C4(C2×D79)⋊C412yes—
219C22×C4C2×C4×D7911yescharacteristic
220C22×C4C2×C22⋊D7911yescharacteristic
221C22×C4C2×D31611yescharacteristic
222M16—3162no—
223M16—3162no—
224C2×C8—24no—
225C2×C8—3162no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—3164no—
229C4⋊C4—3161no—
230C4⋊C4—3161no—
231C4⋊C4—1582no—
232C22⋊C4—3164no—
233C22⋊C4—3164no—
234C22⋊C4—3161no—
235C22⋊C4—3161no—
236C42—22no—
237C42—22no—
238C42C2×C22⋊D7912yes—
239C42#1192962412yes—
240C23—3161no—
241C23—3161no—
242C23—3161no—
243C23—3161no—
244C23—3161no—
245C23—3161no—
246C23—1582no—
247C23C2×(C2×D79)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—3162no—
251D4—1584no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4—3161no—
261C2×C4—3162no—
262C2×C4—3161no—
263C2×C4—3161no—
264C2×C4—6322no—
265C2×C4—6322no—
266C2×C4#1457908314yes—
267C2×C4C4×C22⋊D7912yes—
268C2×C4#1457913011yescharacteristic
269C2×C4#1457913512yes—
270C2×C4#1457914411yescharacteristic
271C2×C4C2×D63211yescharacteristic
272C2×C4C2×#1192961311yescharacteristic
273C2×C4#1457918312yes—
274C8—44no—
275C8—6322no—
276C22—21no—
277C22—21no—
278C22—3162no—
279C22—3162no—
280C22—3162no—
281C22—6321no—
282C22—6321no—
283C22—6321no—
284C22—6321no—
285C22#1909844211yescharacteristic
286C22C2×#1457908311yescharacteristic
287C22#1909856011yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—6322no—
294C4—6321no—
295C4#1909879212yes—
296C4#1909879912yes—
297C2—21no—
298C2—21no—
299C2—6322no—
300C2#2376381811yescharacteristic
301C2#2376382811yescharacteristic
302C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2982C2
socleC22×C79#386
prime radicalC22×D316C2×C4
cosocleC2×C42×C79C23
hypercenter#2982D79
hypocenterC79#9486
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #23763396C2semidirect
4. #23761259C2semidirect
5. C79×#2982C2semidirect
152. #2982D79semidirect
157. C79#9486semidirect

Maximal subgroups

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

subgroupindex
#237683052
#237633962
#237618262
#237612592
C79×#29822
#948679