Finite Group Databasebeta

← #34189259

Subgroups — #34189259

Order 20224.

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

normal subgroups only

This group has 528 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#23773924C211yesmaximal, characteristic, complemented
2#23773408C212yesmaximal
3#23761802C212yesmaximal, complemented
4#23761231C211yesmaximal, characteristic, complemented
5C79×#1955C211yesmaximal, characteristic
6#19099120C2212yes—
7#19098972—22no—
8#19098972—22no—
9#19098893—22no—
10#19098853—22no—
11#19098768—22no—
12#19098714—22no—
13C4×#11929612—24no—
14#19098616C2212yes—
15#19098601—22no—
16#19098293—24no—
17C2×C4×D4×C79C2211yescharacteristic
18C2×C4⋊C8×C79C2211yescharacteristic
19C2×C22⋊C8×C79—22no—
20C2×#97×C79C2211yescharacteristic
21C2×#11929612—24no—
22#14579161D412yes—
23#14579161D412yes—
24#14579153—22no—
25#14579140—22no—
26#14579140—22no—
27C4×Dic158—22no—
28C4×Dic158—22no—
29#14579120—22no—
30#14579095—22no—
31#14579095—22no—
32#14579093—24no—
33#14579092—24no—
34C4⋊(C79⋊C8)—22no—
35#14579084—24no—
36#14579082—24no—
37C22⋊(C79⋊C8)—22no—
38C4×C79⋊C8D414yes—
39C22×D4×C79D411yescharacteristic
40C23×C4×C79—21no—
41C22×C8×C79—21no—
42C22×C8×C79D412yes—
43C4×D4×C79—22no—
44C4×D4×C79—22no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C22⋊C4×C79—21no—
47C2×C42×C79C2311yescharacteristic
48C4⋊C8×C79—22no—
49C22⋊C8×C79—44no—
50#97×C79—22no—
51C2×Dic158—22no—
52C2×Dic158—22no—
53#11929612—48no—
54C2×C79⋊C8—24no—
55C4⋊Dic79—22no—
56C4⋊Dic79—22no—
57C4⋊Dic79—22no—
58C4⋊Dic79—22no—
59Dic79⋊C4—42no—
60Dic79⋊C4—42no—
61C22⋊Dic79—42no—
62C4×Dic79—42no—
63C24×C79—21no—
64C2×D4×C79—22no—
65C2×D4×C79—42no—
66C22×C4×C79—21no—
67C22×C4×C79—41no—
68C22×C4×C79—41no—
69C22×C4×C79—42no—
70C22×C4×C79—42no—
71C22×C4×C79C2×D411yescharacteristic
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79#4111yescharacteristic
74C2×C8×C79—22no—
75C2×C8×C79—22no—
76C2×C8×C79—41no—
77C2×C8×C79—42no—
78C2×C8×C79SD1614yescomplemented
79C4⋊C4×C79—22no—
80C22⋊C4×C79—42no—
81C42×C79C2×D412yes—
82C42×C79C2×D412yes—
83C23×C79—22no—
84C23×C79—41no—
85C23×C79—42no—
86C23×C79—41no—
87C23×C79—42no—
88C23×C79#12111yescharacteristic
89D4×C79—24no—
90D4×C79—42no—
91C2×C4×C79—41no—
92C2×C4×C79—41no—
93C2×C4×C79—41no—
94C2×C4×C79—42no—
95C2×C4×C79—82no—
96C2×C4×C79—81no—
97C2×C4×C79—81no—
98C2×C4×C79#12011yescharacteristic
99C2×C4×C79#12012yes—
100C2×C4×C79#12112yes—
101C2×C4×C79#12711yescharacteristic
102C2×C4×C79C2×SD1612yes—
103C2×C4×C79#13712yes—
104C2×Dic79—42no—
105C2×Dic79—42no—
106Dic158—44no—
107Dic158—44no—
108C8×C79—24no—
109C8×C79—81no—
110C79⋊C8—44no—
111C22×C79—21no—
112C22×C79—21no—
113C22×C79—44no—
114C22×C79—42no—
115C22×C79—82no—
116C22×C79—81no—
117C22×C79—81no—
118C22×C79#39211yescharacteristic
119C22×C79#46112yes—
120C4×C79—22no—
121C4×C79—22no—
122C4×C79—81no—
123C4×C79#44812yes—
124C4×C79#49211yescharacteristic
125C4×C79#49711yescharacteristic
126Dic79—82no—
127Dic79—82no—
128#10036—791nomaximal, Sylow
129C2×C79—22no—
130C2×C79—82no—
131C2×C79#201011yescharacteristic
132C2×C79#201311yescharacteristic
133C2×C79#201711yescharacteristic
134#3162—791no—
135#3101—792no—
136#1955D7911yescharacteristic
137#1481—792no—
138#1275—791no—
139C79#1003611yescharacteristic, complemented, Sylow
140#533—792no—
141C2×C4×D4C2×D7911yescharacteristic
142#501—1582no—
143#501—1582no—
144#483—1582no—
145#476—1582no—
146#461—1582no—
147#450—1582no—
148C4×SD16—1584no—
149#431—792no—
150#427—1582no—
151C2×C4⋊C8C2×D7911yescharacteristic
152C2×C22⋊C8—22no—
153C2×#97C2×D7911yescharacteristic
154Q8⋊C8—1584no—
155C22×D4D31611yescharacteristic
156C23×C4—21no—
157C2×SD16—1584no—
158C22×C8—21no—
159C22×C8C22⋊D7912yes—
160C4⋊Q8—792no—
161C4⋊Q8—792no—
162#125—1582no—
163C22⋊Q8—1582no—
164C22⋊Q8—1582no—
165C4×Q8—1582no—
166C4×Q8—1582no—
167C4×D4—22no—
168C4×D4—22no—
169#117—1582no—
170C2×C4⋊C4D31611yescharacteristic
171C2×C22⋊C4—21no—
172C2×C42C22×D7911yescharacteristic
173#107—1582no—
174#107—1582no—
175#106—1584no—
176#106—1584no—
177C4⋊C8—22no—
178C4⋊C8—1582no—
179#103—1584no—
180#102—1584no—
181C22⋊C8—44no—
182C22⋊C8—1582no—
183#97—22no—
184C4×C8—794no—
185C24—21no—
186C2×Q8—1582no—
187C2×Q8—1582no—
188C2×D4—22no—
189C2×D4—42no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4C2×C22⋊D7911yescharacteristic
196C22×C4C2×D31611yescharacteristic
197C22×C4#1192962411yescharacteristic
198SD16—3168no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—1584no—
204C2×C8#1192961214yescomplemented
205C4⋊C4—22no—
206C4⋊C4—3162no—
207C4⋊C4—3162no—
208C4⋊C4—1582no—
209C4⋊C4—1582no—
210C4⋊C4—1582no—
211C4⋊C4—1582no—
212C22⋊C4—42no—
213C22⋊C4—3162no—
214C42—3162no—
215C42D4×D7912yes—
216C42D4×D7912yes—
217C23—22no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23#1457913511yescharacteristic
223Q8—3164no—
224Q8—3164no—
225D4—24no—
226D4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—81no—
234C2×C4—3162no—
235C2×C4—3162no—
236C2×C4#1457912911yescharacteristic
237C2×C4#1457913112yes—
238C2×C4#1457913412yes—
239C2×C4#1457915711yescharacteristic
240C2×C4C2×#1192961212yes—
241C2×C4#1457919112yes—
242C8—24no—
243C8—81no—
244C8—3164no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1909847211yescharacteristic
253C22#1909876812yes—
254C4—22no—
255C4—22no—
256C4—81no—
257C4—6322no—
258C4—6322no—
259C4#1909870112yes—
260C4#1909893511yescharacteristic
261C4#1909895811yescharacteristic
262C2—22no—
263C2—82no—
264C2#2376446711yescharacteristic
265C2#2376448211yescharacteristic
266C2#2376450411yescharacteristic
subgroupG/H
centerC2×C4#14579157
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1955C2
socleC22×C79#392
prime radicalC22×D4×C79D4
hypercenter#1955D79
hypocenterC79#10036
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #23773924C2semidirect
3. #23761802C2semidirect
4. #23761231C2semidirect
78. C2×C8×C79SD16semidirect
139. C79#10036semidirect
204. C2×C8#11929612semidirect

Maximal subgroups

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

subgroupindex
#237739242
#237734082
#237618022
#237612312
C79×#19552
#1003679