Finite Group Databasebeta

← #34201324

Subgroups — #34201324

Order 20224.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764745C211yesmaximal, characteristic
2#23762373C212yesmaximal
3#23761430C212yesmaximal
4#23761346C211yesmaximal, characteristic
5C79×#2551C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098719—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×C4⋊Q8×C79—22no—
14#395×C79—21no—
15#395×C79C2212yes—
16#389×C79—22no—
17#384×C79C2211yescharacteristic
18C4×C4⋊C4×C79—21no—
19C2×#11929614—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579094—41no—
26#14579092—41no—
27#14579085—42no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79—21no—
30C22×Q8×C79D412yes—
31C4⋊Q8×C79—48no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—22no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C4⋊C4×C79C2311yescharacteristic
39C2×C42×C79—22no—
40C2×C42×C79D411yescharacteristic
41#95×C79—21no—
42#95×C79—22no—
43#95×C79D412yes—
44C2×Dic158—22no—
45#11929614—84no—
46#11929608—22no—
47#11929608—41no—
48C2×C79⋊C8—41no—
49C2×C79⋊C8—42no—
50C4⋊Dic79—22no—
51C4⋊Dic79—21no—
52C4⋊Dic79—21no—
53Dic79⋊C4—42no—
54C22⋊Dic79—41no—
55C4×Dic79—41no—
56C2×Q8×C79—44no—
57C2×Q8×C79—44no—
58C2×Q8×C79—42no—
59C2×Q8×C79—81no—
60C22×C4×C79—21no—
61C22×C4×C79—22no—
62C22×C4×C79—22no—
63C22×C4×C79—21no—
64C22×C4×C79C2×D411yescharacteristic
65C22×C4×C79C2×D412yes—
66C4⋊C4×C79—24no—
67C4⋊C4×C79—24no—
68C4⋊C4×C79—41no—
69C4⋊C4×C79—44no—
70C4⋊C4×C79—44no—
71C4⋊C4×C79—44no—
72C4⋊C4×C79—41no—
73C42×C79—21no—
74C42×C79—44no—
75C42×C79Q1612yescomplemented
76C23×C79#12011yescharacteristic
77Q8×C79—44no—
78Q8×C79—82no—
79Q8×C79—84no—
80C2×C4×C79—22no—
81C2×C4×C79—22no—
82C2×C4×C79—22no—
83C2×C4×C79—21no—
84C2×C4×C79—21no—
85C2×C4×C79—24no—
86C2×C4×C79—24no—
87C2×C4×C79—22no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—42no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79—42no—
94C2×C4×C79—41no—
95C2×C4×C79C2×Q1611yescharacteristic
96C2×C4×C79#13711yescharacteristic
97C2×Dic79—41no—
98C2×Dic79—41no—
99Dic158—82no—
100C79⋊C8—42no—
101C79⋊C8—82no—
102C22×C79—21no—
103C22×C79—21no—
104C22×C79#45111yescharacteristic
105C22×C79#45611yescharacteristic
106C22×C79B(4,𝔽2)11yescharacteristic
107C4×C79—24no—
108C4×C79—21no—
109C4×C79—21no—
110C4×C79—44no—
111C4×C79—44no—
112C4×C79—42no—
113C4×C79—81no—
114Dic79—81no—
115Dic79—81no—
116#12256—791nomaximal, Sylow
117C2×C79—21no—
118C2×C79—21no—
119C2×C79#160511yescharacteristic
120C2×C79#219611yescharacteristic
121C2×C79#220511yescharacteristic
122#2551D7911yescharacteristic
123#2060—791no—
124#1605—792no—
125#1354—792no—
126#1325—791no—
127C79#1225611yescharacteristic, complemented, Sylow
128#544—791no—
129C2×C4⋊Q8—22no—
130#451—1582no—
131#428—1581no—
132#395—21no—
133#395C2×D7912yes—
134#389—22no—
135#384C2×D7911yescharacteristic
136C4×C4⋊C4—21no—
137#343—791no—
138#339—1581no—
139#328—1582no—
140#324—792no—
141C22×Q8—21no—
142C22×Q8C22⋊D7912yes—
143C2×Q16—3162no—
144C2×M16—1581no—
145C4⋊Q8—48no—
146C4⋊Q8—3161no—
147#125—3161no—
148C22⋊Q8—1582no—
149#117—1581no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D7911yescharacteristic
156C2×C4⋊C4C22×D7911yescharacteristic
157C2×C42—22no—
158C2×C42C22⋊D7911yescharacteristic
159#107—3161no—
160#106—3161no—
161#103—3162no—
162C22⋊C8—1582no—
163#95—21no—
164#95—22no—
165#95C22⋊D7912yes—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—1582no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D7911yescharacteristic
176C22×C4C2×C22⋊D7912yes—
177Q16—6324no—
178M16—3161no—
179M16—1582no—
180C2×C8—3161no—
181C2×C8—3162no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—3162no—
190C4⋊C4—1582no—
191C4⋊C4—1581no—
192C4⋊C4—1581no—
193C22⋊C4—3161no—
194C42—21no—
195C42—44no—
196C42—3161no—
197C42#1192961412yescomplemented
198C23#1457913011yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—6322no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—3161no—
219C2×C4—3161no—
220C2×C4C2×#1192961411yescharacteristic
221C2×C4#1457919211yescharacteristic
222C8—3162no—
223C8—6322no—
224C22—21no—
225C22—21no—
226C22#1909871911yescharacteristic
227C22#1909874611yescharacteristic
228C22#1909875011yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—6321no—
237C4—6321no—
238C2—21no—
239C2—21no—
240C2#2376237411yescharacteristic
241C2#2376538411yescharacteristic
242C2#2376543011yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2551C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2551D79
hypocenterC79#12256
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
75. C42×C79Q16semidirect
127. C79#12256semidirect
197. C42#11929614semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237623732
#237614302
#237613462
C79×#25512
#1225679