Finite Group Databasebeta

← #34201066

Subgroups — #34201066

Order 20224.

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

normal subgroups only

This group has 388 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#23762424C212yesmaximal
3#23761424C212yesmaximal
4#23761330C211yesmaximal, characteristic
5C79×#2607C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098898—22no—
8#19098426—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13#399×C79C2212yes—
14#393×C79—21no—
15#391×C79—22no—
16#387×C79—22no—
17#381×C79C2211yescharacteristic
18C4×C22⋊C4×C79—21no—
19#14579140—22no—
20#14579138—42no—
21C2×C22⋊Dic79—21no—
22C2×C22⋊Dic79—21no—
23#14579094—42no—
24#14579084—42no—
25C22⋊(C79⋊C8)—22no—
26C79⋊#95—42no—
27C22×Q8×C79—21no—
28C23×C4×C79D411yescharacteristic
29C2×C4⋊C4×C79—22no—
30C2×C4⋊C4×C79D412yes—
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—
44C2×C79⋊C8—42no—
45C4⋊Dic79—22no—
46Dic79⋊C4—42no—
47C22⋊Dic79—41no—
48C22⋊Dic79—41no—
49C22⋊Dic79—41no—
50C22⋊Dic79—41no—
51C24×C79C2×D411yescharacteristic
52C2×Q8×C79—44no—
53C2×Q8×C79—81no—
54C22×C4×C79—21no—
55C22×C4×C79—22no—
56C22×C4×C79—21no—
57C22×C4×C79—21no—
58C22×C4×C79—22no—
59C22×C4×C79—41no—
60C22×C4×C79—41no—
61C22×C4×C79Q1612yes—
62C22×C4×C79C2×D412yes—
63C4⋊C4×C79—44no—
64C4⋊C4×C79—42no—
65C22⋊C4×C79—41no—
66C22⋊C4×C79—44no—
67C22⋊C4×C79—41no—
68C42×C79—81no—
69C42×C79—81no—
70C23×C79—21no—
71C23×C79—21no—
72C23×C79—42no—
73C23×C79#12011yescharacteristic
74C23×C79C2×Q1611yescharacteristic
75C23×C79#13711yescharacteristic
76Q8×C79—84no—
77C2×C4×C79—24no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—42no—
81C2×C4×C79—41no—
82C2×C4×C79—42no—
83C2×C4×C79—41no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79—42no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79—81no—
92C2×Dic79—42no—
93C2×Dic79—82no—
94C2×Dic79—81no—
95C2×Dic79—82no—
96Dic158—82no—
97C79⋊C8—82no—
98C22×C79—22no—
99C22×C79—21no—
100C22×C79—21no—
101C22×C79—41no—
102C22×C79—41no—
103C22×C79—42no—
104C22×C79—81no—
105C22×C79#45111yescharacteristic
106C22×C79#45511yescharacteristic
107C22×C79B(4,𝔽2)11yescharacteristic
108C4×C79—44no—
109C4×C79—42no—
110C4×C79—42no—
111C4×C79—81no—
112C4×C79—82no—
113Dic79—82no—
114Dic79—161no—
115#12216—791nomaximal, Sylow
116C2×C79—21no—
117C2×C79—21no—
118C2×C79—41no—
119C2×C79—41no—
120C2×C79#160211yescharacteristic
121C2×C79#219812yes—
122#2607D7911yescharacteristic
123#2030—791no—
124#1614—792no—
125#1352—792no—
126#1317—791no—
127C79#1221611yescharacteristic, complemented, Sylow
128#543—791no—
129#484—1582no—
130#399C2×D7912yes—
131#393—21no—
132#391—22no—
133#387—22no—
134#381—1581no—
135#381C2×D7911yescharacteristic
136C4×C22⋊C4—21no—
137#342—791no—
138#342—1581no—
139#328—1582no—
140#323—792no—
141C22×Q8—21no—
142C23×C4C22⋊D7911yescharacteristic
143C22⋊Q8—3162no—
144C22⋊Q8—1582no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊D7912yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—1581no—
149C2×C22⋊C4—1581no—
150C2×C22⋊C4C22⋊D7911yescharacteristic
151C2×C22⋊C4C22×D7911yescharacteristic
152C2×C42—21no—
153C2×C42—21no—
154#107—3162no—
155#103—3162no—
156C22⋊C8—1582no—
157#95—22no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—3162no—
162#95C22⋊D7912yes—
163C24C2×C22⋊D7911yescharacteristic
164C2×Q8—44no—
165C2×Q8—81no—
166C2×Q8—1582no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—3161no—
175C22×C4—1582no—
176C22×C4#1192961412yes—
177C22×C4C2×C22⋊D7912yes—
178C2×C8—3162no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—3162no—
182C4⋊C4—1582no—
183C22⋊C4—41no—
184C22⋊C4—44no—
185C22⋊C4—41no—
186C22⋊C4—3161no—
187C22⋊C4—3161no—
188C22⋊C4—3161no—
189C22⋊C4—3161no—
190C42—81no—
191C42—81no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23#1457913011yescharacteristic
196C23C2×#1192961411yescharacteristic
197C23#1457919211yescharacteristic
198Q8—84no—
199Q8—6322no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—81no—
215C2×C4—3162no—
216C2×C4—6322no—
217C2×C4—6321no—
218C2×C4—6322no—
219C8—6322no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22#1909871911yescharacteristic
228C22#1909874111yescharacteristic
229C22#1909875011yescharacteristic
230C4—44no—
231C4—42no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—6322no—
236C4—12641no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2#2376235711yescharacteristic
242C2#2376539612yes—
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#12216
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
127. C79#12216semidirect

Maximal subgroups

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

subgroupindex
#237645672
#237624242
#237614242
#237613302
C79×#26072
#1221679