Finite Group Databasebeta

← #34201061

Subgroups — #34201061

Order 20224.

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

normal subgroups only

This group has 400 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#23764567C211yesmaximal, characteristic
2#23762356C212yesmaximal
3#23761421C212yesmaximal
4#23761330C211yesmaximal, characteristic
5C79×#2607C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098719—22no—
8#19098426—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13#399×C79—21no—
14#393×C79C2212yes—
15#391×C79—22no—
16#387×C79—22no—
17#381×C79C2211yescharacteristic
18C4×C22⋊C4×C79—21no—
19C2×#11929614—42no—
20#14579140—22no—
21#14579138—42no—
22C2×C22⋊Dic79—21no—
23C2×C22⋊Dic79—21no—
24#14579085—42no—
25C22⋊(C79⋊C8)—22no—
26C79⋊#95—42no—
27C22×Q8×C79D412yes—
28C23×C4×C79D411yescharacteristic
29C2×C4⋊C4×C79—22no—
30C2×C4⋊C4×C79—21no—
31C2×C22⋊C4×C79—22no—
32C2×C22⋊C4×C79D411yescharacteristic
33C2×C22⋊C4×C79C2311yescharacteristic
34C2×C42×C79—21no—
35C2×C42×C79—21no—
36#95×C79—22no—
37#95×C79—21no—
38#95×C79—22no—
39#95×C79—22no—
40#95×C79D412yes—
41C2×Dic158—22no—
42C22×Dic79—22no—
43C22×Dic79—41no—
44#11929614—84no—
45C2×C79⋊C8—42no—
46C4⋊Dic79—22no—
47Dic79⋊C4—42no—
48C22⋊Dic79—41no—
49C22⋊Dic79—41no—
50C22⋊Dic79—41no—
51C22⋊Dic79—41no—
52C24×C79C2×D411yescharacteristic
53C2×Q8×C79—44no—
54C2×Q8×C79—42no—
55C22×C4×C79—21no—
56C22×C4×C79—22no—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—22no—
60C22×C4×C79—41no—
61C22×C4×C79—41no—
62C22×C4×C79Q1612yescomplemented
63C22×C4×C79C2×D412yes—
64C4⋊C4×C79—44no—
65C4⋊C4×C79—81no—
66C22⋊C4×C79—41no—
67C22⋊C4×C79—44no—
68C22⋊C4×C79—41no—
69C42×C79—81no—
70C42×C79—81no—
71C23×C79—21no—
72C23×C79—21no—
73C23×C79—42no—
74C23×C79#12011yescharacteristic
75C23×C79C2×Q1611yescharacteristic
76C23×C79#13711yescharacteristic
77Q8×C79—44no—
78Q8×C79—82no—
79C2×C4×C79—24no—
80C2×C4×C79—22no—
81C2×C4×C79—22no—
82C2×C4×C79—42no—
83C2×C4×C79—41no—
84C2×C4×C79—41no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79—42no—
89C2×C4×C79—42no—
90C2×C4×C79—42no—
91C2×C4×C79—42no—
92C2×C4×C79—41no—
93C2×C4×C79—81no—
94C2×Dic79—42no—
95C2×Dic79—82no—
96C2×Dic79—81no—
97C2×Dic79—82no—
98Dic158—82no—
99C79⋊C8—82no—
100C22×C79—22no—
101C22×C79—21no—
102C22×C79—21no—
103C22×C79—41no—
104C22×C79—41no—
105C22×C79—42no—
106C22×C79—81no—
107C22×C79#45111yescharacteristic
108C22×C79#45511yescharacteristic
109C22×C79B(4,𝔽2)11yescharacteristic
110C4×C79—42no—
111C4×C79—44no—
112C4×C79—42no—
113C4×C79—82no—
114C4×C79—81no—
115Dic79—82no—
116Dic79—161no—
117#12215—791nomaximal, Sylow
118C2×C79—21no—
119C2×C79—21no—
120C2×C79—41no—
121C2×C79—41no—
122C2×C79#160211yescharacteristic
123C2×C79#219712yes—
124#2607D7911yescharacteristic
125#2030—791no—
126#1602—792no—
127#1351—792no—
128#1317—791no—
129C79#1221511yescharacteristic, complemented, Sylow
130#543—791no—
131#451—1582no—
132#399—21no—
133#393C2×D7912yes—
134#391—22no—
135#387—22no—
136#381—1581no—
137#381C2×D7911yescharacteristic
138C4×C22⋊C4—21no—
139#342—791no—
140#342—1581no—
141#328—1582no—
142#323—792no—
143C22×Q8C22⋊D7912yes—
144C23×C4C22⋊D7911yescharacteristic
145C2×Q16—3162no—
146C22⋊Q8—3162no—
147C22⋊Q8—1582no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—1581no—
152C2×C22⋊C4—1581no—
153C2×C22⋊C4C22⋊D7911yescharacteristic
154C2×C22⋊C4C22×D7911yescharacteristic
155C2×C42—21no—
156C2×C42—21no—
157#103—3162no—
158C22⋊C8—1582no—
159#95—22no—
160#95—21no—
161#95—22no—
162#95—22no—
163#95—3162no—
164#95C22⋊D7912yes—
165C24C2×C22⋊D7911yescharacteristic
166C2×Q8—44no—
167C2×Q8—42no—
168C2×Q8—1582no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—3161no—
177C22×C4—1582no—
178C22×C4#1192961412yescomplemented
179C22×C4C2×C22⋊D7912yes—
180Q16—6324no—
181C2×C8—3162no—
182C4⋊C4—44no—
183C4⋊C4—81no—
184C4⋊C4—3162no—
185C4⋊C4—1582no—
186C22⋊C4—41no—
187C22⋊C4—44no—
188C22⋊C4—41no—
189C22⋊C4—3161no—
190C22⋊C4—3161no—
191C22⋊C4—3161no—
192C22⋊C4—3161no—
193C42—81no—
194C42—81no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#1457913011yescharacteristic
199C23C2×#1192961411yescharacteristic
200C23#1457919211yescharacteristic
201Q8—44no—
202Q8—82no—
203Q8—6322no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—3162no—
220C2×C4—6322no—
221C2×C4—6321no—
222C2×C4—6322no—
223C8—6322no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#1909871911yescharacteristic
232C22#1909874111yescharacteristic
233C22#1909875011yescharacteristic
234C4—42no—
235C4—44no—
236C4—42no—
237C4—82no—
238C4—81no—
239C4—6322no—
240C4—12641no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#2376235711yescharacteristic
246C2#2376539012yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2607C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2607D79
hypocenterC79#12215
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2607

Complements

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

NH ≅ G/Ntype
62. C22×C4×C79Q16semidirect
129. C79#12215semidirect
178. C22×C4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237645672
#237623562
#237614212
#237613302
C79×#26072
#1221579