Finite Group Databasebeta

← #34186083

Subgroups — #34186083

Order 20224.

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

normal subgroups only

This group has 610 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#23767687C211yesmaximal, characteristic
2#23763500C212yesmaximal
3#23761730C211yesmaximal, characteristic
4#23761253C212yesmaximal
5C79×#2274C211yesmaximal, characteristic
6C2×#14579120C2211yescharacteristic
7#19098527C2212yes—
8#19098502C412yescomplemented
9#19098493C412yescomplemented
10#19098444—22no—
11#19098436—21no—
12#19098426—21no—
13#19098426C2211yescharacteristic
14#19098323—24no—
15#19098289—22no—
16C2×#117×C79—21no—
17C22×C4⋊C4×C79C2211yescharacteristic
18C4×C4⋊C4×C79—24no—
19C4×C22⋊C4×C79—21no—
20C4×C22⋊C4×C79C2212yes—
21C2×#11929608—22no—
22#14579120—22no—
23#14579120—22no—
24C2×C4⋊Dic79C2×C411yescharacteristic
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×Dic79⋊C4—22no—
27C2×C22⋊Dic79—21no—
28C2×C22⋊Dic79—21no—
29C2×C4×Dic79—21no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C79⋊#95—22no—
33C79⋊#95—22no—
34C79⋊#95—22no—
35C79⋊#95—21no—
36C79⋊#95C2×C412yes—
37C23×C4×C79D411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40#117×C79—28no—
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79C2×C412yes—
45C2×C4⋊C4×C79D412yes—
46C2×C22⋊C4×C79—22no—
47C2×C42×C79—22no—
48C2×C42×C79—22no—
49C2×C42×C79D414yes—
50#95×C79—22no—
51#95×C79—22no—
52C22×Dic79—22no—
53C22×Dic79—21no—
54C22×Dic79—21no—
55#11929608—44no—
56C2×C79⋊C8—42no—
57C4⋊Dic79—22no—
58C4⋊Dic79—22no—
59Dic79⋊C4—44no—
60C22⋊Dic79—42no—
61C22⋊Dic79—41no—
62C22⋊Dic79—41no—
63C4×Dic79—42no—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—24no—
68C22×C4×C79—24no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79C22⋊C412yes—
74C22×C4×C79C22⋊C412yes—
75C22×C4×C79C22×C411yescharacteristic
76C22×C4×C79C2×D412yes—
77C22×C4×C79C2×D411yescharacteristic
78C22×C4×C79#4112yes—
79C4⋊C4×C79—28no—
80C4⋊C4×C79—24no—
81C4⋊C4×C79—24no—
82C22⋊C4×C79—28no—
83C42×C79—28no—
84C42×C79—44no—
85C42×C79—82no—
86C23×C79—22no—
87C23×C79—22no—
88C23×C79—41no—
89C23×C79C2×C22⋊C411yescharacteristic
90C23×C79#12011yescharacteristic
91C23×C79#12311yescharacteristic
92C2×C4×C79—22no—
93C2×C4×C79—22no—
94C2×C4×C79—24no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—24no—
98C2×C4×C79—28no—
99C2×C4×C79—22no—
100C2×C4×C79—41no—
101C2×C4×C79—41no—
102C2×C4×C79—42no—
103C2×C4×C79—44no—
104C2×C4×C79—44no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—81no—
108C2×C4×C79#10414yes—
109C2×C4×C79C4×D412yes—
110C2×C4×C79#12112yes—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—41no—
115C2×Dic79—41no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C79⋊C8—82no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79—21no—
122C22×C79—21no—
123C22×C79—21no—
124C22×C79—21no—
125C22×C79—21no—
126C22×C79—44no—
127C22×C79#38611yescharacteristic
128C22×C79C2×#10412yes—
129C22×C79#45812yes—
130C4×C79—22no—
131C4×C79—22no—
132C4×C79—24no—
133C4×C79—48no—
134C4×C79—42no—
135Dic79—82no—
136Dic79—81no—
137Dic79—81no—
138#9478—791nomaximal, Sylow
139C2×C79—22no—
140C2×C79—22no—
141C2×C79—41no—
142C2×C79#189011yescharacteristic
143C2×C79#189611yescharacteristic
144C2×C79#191211yescharacteristic
145#2484—791no—
146#2274D7911yescharacteristic
147#1826—792no—
148#1463—791no—
149#1284—792no—
150C79#947811yescharacteristic, complemented, Sylow
151C2×#117—21no—
152C2×#117—791no—
153C22×C4⋊C4C2×D7911yescharacteristic
154#407—792no—
155#400—792no—
156#398—792no—
157#387—1582no—
158#385—1581no—
159#381—791no—
160#381—1581no—
161C4×C4⋊C4—24no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4C2×D7912yes—
164#339—1584no—
165#324—1582no—
166C23×C4C22⋊D7911yescharacteristic
167C23×C4C22⋊D7911yescharacteristic
168C23×C4C22×D7911yescharacteristic
169C2×M16—1582no—
170#117—28no—
171#117—1582no—
172#117—1582no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—791no—
177C2×C4⋊C4—791no—
178C2×C4⋊C4—1582no—
179C2×C4⋊C4C4×D7912yes—
180C2×C4⋊C4D31612yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1581no—
183C2×C22⋊C4—1581no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—1581no—
187C2×C42C22⋊D7914yes—
188C22⋊C8—1582no—
189C22⋊C8—1582no—
190#95—22no—
191#95—22no—
192#95—792no—
193#95—1582no—
194#95—1582no—
195#95—1582no—
196#95—1581no—
197C24C2×C22⋊D7911yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1582no—
207C22×C4—1581no—
208C22×C4—1581no—
209C22×C4(C2×D79)⋊C412yes—
210C22×C4(C2×D79)⋊C412yes—
211C22×C4C2×C4×D7911yescharacteristic
212C22×C4C2×C22⋊D7912yes—
213C22×C4C2×D31611yescharacteristic
214C22×C4#1192962412yes—
215M16—3164no—
216C2×C8—3162no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—3164no—
221C4⋊C4—1582no—
222C4⋊C4—1582no—
223C22⋊C4—28no—
224C22⋊C4—3162no—
225C22⋊C4—3161no—
226C22⋊C4—3161no—
227C42—28no—
228C42—44no—
229C42—82no—
230C42—3162no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×(C2×D79)⋊C411yescharacteristic
235C23#1457913011yescharacteristic
236C23#1457914411yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4—3162no—
256C2×C4—3161no—
257C2×C4—3161no—
258C2×C4—3162no—
259C2×C4—3162no—
260C2×C4#1457908914yes—
261C2×C4C4×C22⋊D7912yes—
262C2×C4#1457913512yes—
263C8—6322no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1909844211yescharacteristic
273C22C2×#1457908912yes—
274C22#1909875312yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—6322no—
281C4—6321no—
282C4—6321no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2376381311yescharacteristic
287C2#2376384311yescharacteristic
288C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2274C2
socleC22×C79#386
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#2274D79
hypocenterC79#9478
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #19098502C4semidirect
9. #19098493C4semidirect
150. C79#9478semidirect

Maximal subgroups

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

subgroupindex
#237676872
#237635002
#237617302
#237612532
C79×#22742
#947879