Finite Group Databasebeta

← #34201060

Subgroups — #34201060

Order 20224.

400 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 59 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#23769065C211yesmaximal, characteristic
2#23762357C212yesmaximal
3#23761420C212yesmaximal
4#23761330C211yesmaximal, characteristic
5C79×#2030C211yesmaximal, characteristic
6#19098718—22no—
7#19098499—21no—
8#19098475C2212yes—
9#19098467—22no—
10#19098450—22no—
11C4×C22⋊Dic79—21no—
12#19098300—22no—
13#19098287C2212yes—
14#543×C79C2211yescharacteristic
15#381×C79—21no—
16#381×C79C2211yescharacteristic
17#342×C79—21no—
18#342×C79C2211yescharacteristic
19C22×Dic158D412yes—
20C2×#11929614—42no—
21C2×C4⋊Dic79—21no—
22C2×Dic79⋊C4—22no—
23C2×C22⋊Dic79—22no—
24C2×C4×Dic79—21no—
25C2×C4×Dic79—21no—
26#14579084—42no—
27C22⋊(C79⋊C8)—22no—
28C79⋊#95—22no—
29C79⋊#95—21no—
30C79⋊#95—22no—
31C79⋊#95D412yes—
32C23×C4×C79D411yescharacteristic
33C22⋊Q8×C79—22no—
34C22⋊Q8×C79—42no—
35C2×C22⋊C4×C79—21no—
36C2×C22⋊C4×C79—21no—
37C2×C22⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39#95×C79—22no—
40#95×C79—42no—
41C2×Dic158—42no—
42C2×Dic158—44no—
43C22×Dic79—22no—
44C22×Dic79—22no—
45#11929614—84no—
46C2×C79⋊C8—42no—
47C4⋊Dic79—81no—
48Dic79⋊C4—44no—
49C22⋊Dic79—44no—
50C4×Dic79—81no—
51C4×Dic79—81no—
52C24×C79C2×D411yescharacteristic
53C2×Q8×C79—22no—
54C22×C4×C79—21no—
55C22×C4×C79—21no—
56C22×C4×C79—21no—
57C22×C4×C79—22no—
58C22×C4×C79—41no—
59C22×C4×C79—41no—
60C22×C4×C79—41no—
61C22×C4×C79Q1612yescomplemented
62C22×C4×C79C2×D412yes—
63C4⋊C4×C79—22no—
64C4⋊C4×C79—42no—
65C22⋊C4×C79—41no—
66C22⋊C4×C79—41no—
67C22⋊C4×C79—41no—
68C22⋊C4×C79—41no—
69C22⋊C4×C79—41no—
70C22⋊C4×C79—41no—
71C23×C79—21no—
72C23×C79—21no—
73C23×C79—42no—
74C23×C79#12011yescharacteristic
75C23×C79C2×Q1611yescharacteristic
76C23×C79#13711yescharacteristic
77Q8×C79—82no—
78C2×C4×C79—22no—
79C2×C4×C79—22no—
80C2×C4×C79—41no—
81C2×C4×C79—41no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79—41no—
86C2×C4×C79—42no—
87C2×C4×C79—81no—
88C2×C4×C79—82no—
89C2×C4×C79—81no—
90C2×C4×C79—82no—
91C2×Dic79—24no—
92C2×Dic79—42no—
93C2×Dic79—42no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C2×Dic79—42no—
97Dic158—44no—
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—42no—
112C4×C79—81no—
113C4×C79—82no—
114C4×C79—161no—
115Dic79—44no—
116Dic79—82no—
117#12215—791nomaximal, Sylow
118C2×C79—21no—
119C2×C79—21no—
120C2×C79—41no—
121C2×C79—41no—
122C2×C79#160211yescharacteristic
123C2×C79#219712yes—
124#2607—791no—
125#2030D7911yescharacteristic
126#1602—792no—
127#1351—792no—
128#1317—791no—
129C79#1221511yescharacteristic, complemented, Sylow
130#543C2×D7911yescharacteristic
131#451—1582no—
132#399—1581no—
133#393—792no—
134#391—1582no—
135#387—1582no—
136#381—21no—
137#381C2×D7911yescharacteristic
138C4×C22⋊C4—1581no—
139#342—21no—
140#342C2×D7911yescharacteristic
141#328—1582no—
142#323—792no—
143C22×Q8—792no—
144C23×C4C22⋊D7911yescharacteristic
145C2×Q16—3162no—
146C22⋊Q8—22no—
147C22⋊Q8—42no—
148C2×C4⋊C4—1582no—
149C2×C4⋊C4—1581no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—1582no—
153C2×C22⋊C4C22⋊D7911yescharacteristic
154C2×C22⋊C4C22×D7911yescharacteristic
155C2×C42—1581no—
156C2×C42—1581no—
157#103—3162no—
158C22⋊C8—1582no—
159#95—22no—
160#95—42no—
161#95—792no—
162#95—1582no—
163#95—1581no—
164#95—1582no—
165C24C2×C22⋊D7911yescharacteristic
166C2×Q8—22no—
167C2×Q8—3162no—
168C2×Q8—3164no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—1582no—
177C22×C4—1582no—
178C22×C4#1192961412yescomplemented
179C22×C4D4×D7912yes—
180Q16—6324no—
181C2×C8—3162no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—3164no—
185C4⋊C4—6321no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C22⋊C4—41no—
191C22⋊C4—41no—
192C22⋊C4—3164no—
193C42—6321no—
194C42—6321no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#1457912911yescharacteristic
199C23C2×#1192961411yescharacteristic
200C23#1457919111yescharacteristic
201Q8—82no—
202Q8—3164no—
203Q8—6322no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—3162no—
218C2×C4—3162no—
219C2×C4—3162no—
220C2×C4—3162no—
221C2×C4—3162no—
222C2×C4—1584no—
223C8—6322no—
224C22—22no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22—81no—
231C22#1909871811yescharacteristic
232C22#1909874011yescharacteristic
233C22#1909874911yescharacteristic
234C4—42no—
235C4—42no—
236C4—81no—
237C4—82no—
238C4—161no—
239C4—3164no—
240C4—6322no—
241C2—21no—
242C2—21no—
243C2—41no—
244C2—41no—
245C2#2376235611yescharacteristic
246C2#2376538712yes—
subgroupG/H
centerC22#19098749
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2030C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2030D79
hypocenterC79#12215
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
61. 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
#237690652
#237623572
#237614202
#237613302
C79×#20302
#1221579