Finite Group Databasebeta

← #34201015

Subgroups — #34201015

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#23762390C212yesmaximal
3#23761400C212yesmaximal
4#23761330C211yesmaximal, characteristic
5C79×#2817C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098893—22no—
8#19098426—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13C2×C22⋊Q8×C79—21no—
14#400×C79—22no—
15#397×C79—21no—
16#397×C79C2212yes—
17#395×C79—22no—
18#381×C79C2211yescharacteristic
19#14579140—22no—
20#14579138—42no—
21C2×C22⋊Dic79—21no—
22C2×C22⋊Dic79—21no—
23#14579092—42no—
24#14579084—42no—
25C22⋊(C79⋊C8)—22no—
26C79⋊#95—42no—
27C22×Q8×C79—21no—
28C23×C4×C79D411yescharacteristic
29C22⋊Q8×C79—42no—
30C22⋊Q8×C79—42no—
31C2×C4⋊C4×C79—22no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—21no—
34C2×C4⋊C4×C79D412yes—
35C2×C22⋊C4×C79—22no—
36C2×C22⋊C4×C79D411yescharacteristic
37C2×C22⋊C4×C79C2311yescharacteristic
38#95×C79—22no—
39#95×C79—21no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44C22×Dic79—22no—
45C22×Dic79—41no—
46C2×C79⋊C8—42no—
47C4⋊Dic79—22no—
48Dic79⋊C4—42no—
49C22⋊Dic79—41no—
50C22⋊Dic79—41no—
51C22⋊Dic79—41no—
52C22⋊Dic79—41no—
53C24×C79C2×D411yescharacteristic
54C2×Q8×C79—42no—
55C2×Q8×C79—81no—
56C2×Q8×C79—81no—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—22no—
61C22×C4×C79—22no—
62C22×C4×C79—41no—
63C22×C4×C79—41no—
64C22×C4×C79SD1612yes—
65C22×C4×C79C2×D412yes—
66C4⋊C4×C79—42no—
67C4⋊C4×C79—44no—
68C4⋊C4×C79—42no—
69C4⋊C4×C79—81no—
70C22⋊C4×C79—41no—
71C22⋊C4×C79—42no—
72C22⋊C4×C79—42no—
73C22⋊C4×C79—41no—
74C23×C79—21no—
75C23×C79—21no—
76C23×C79—42no—
77C23×C79#12011yescharacteristic
78C23×C79C2×SD1611yescharacteristic
79C23×C79AGL(1,ℤ/8)11yescharacteristic
80Q8×C79—84no—
81C2×C4×C79—22no—
82C2×C4×C79—22no—
83C2×C4×C79—41no—
84C2×C4×C79—41no—
85C2×C4×C79—42no—
86C2×C4×C79—42no—
87C2×C4×C79—42no—
88C2×C4×C79—44no—
89C2×C4×C79—42no—
90C2×C4×C79—42no—
91C2×C4×C79—41no—
92C2×C4×C79—42no—
93C2×C4×C79—42no—
94C2×C4×C79—81no—
95C2×Dic79—42no—
96C2×Dic79—82no—
97C2×Dic79—81no—
98C2×Dic79—82no—
99Dic158—82no—
100C79⋊C8—82no—
101C22×C79—22no—
102C22×C79—21no—
103C22×C79—21no—
104C22×C79—41no—
105C22×C79—41no—
106C22×C79—42no—
107C22×C79—81no—
108C22×C79#45011yescharacteristic
109C22×C79#45511yescharacteristic
110C22×C79B(4,𝔽2)11yescharacteristic
111C4×C79—42no—
112C4×C79—42no—
113C4×C79—82no—
114C4×C79—81no—
115C4×C79—82no—
116Dic79—82no—
117Dic79—161no—
118#12207—791nomaximal, Sylow
119C2×C79—21no—
120C2×C79—21no—
121C2×C79—41no—
122C2×C79—41no—
123C2×C79#160111yescharacteristic
124C2×C79#220112yes—
125#2817D7911yescharacteristic
126#2030—791no—
127#1608—792no—
128#1344—792no—
129#1317—791no—
130C79#1220711yescharacteristic, complemented, Sylow
131#543—791no—
132C2×C22⋊Q8—21no—
133#483—1582no—
134#400—22no—
135#397—21no—
136#397C2×D7912yes—
137#395—22no—
138#381—1581no—
139#381C2×D7911yescharacteristic
140#342—791no—
141#342—1581no—
142#328—1582no—
143#323—792no—
144C22×Q8—21no—
145C23×C4C22⋊D7911yescharacteristic
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—3162no—
149C22⋊Q8—1582no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4C22⋊D7912yes—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—1581no—
156C2×C22⋊C4—1581no—
157C2×C22⋊C4C22⋊D7911yescharacteristic
158C2×C22⋊C4C22×D7911yescharacteristic
159#106—3162no—
160#103—3162no—
161C22⋊C8—1582no—
162#95—22no—
163#95—21no—
164#95—22no—
165#95—22no—
166#95—3162no—
167#95C22⋊D7912yes—
168C24C2×C22⋊D7911yescharacteristic
169C2×Q8—42no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—1582no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—3161no—
181C22×C4—1582no—
182C22×C4#1192961212yes—
183C22×C4C2×C22⋊D7912yes—
184C2×C8—3162no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—81no—
189C4⋊C4—3162no—
190C4⋊C4—1582no—
191C22⋊C4—41no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—41no—
195C22⋊C4—3161no—
196C22⋊C4—3161no—
197C22⋊C4—3161no—
198C22⋊C4—3161no—
199C23—21no—
200C23—21no—
201C23—42no—
202C23#1457913011yescharacteristic
203C23C2×#1192961211yescharacteristic
204C23#1457918711yescharacteristic
205Q8—84no—
206Q8—6322no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—81no—
221C2×C4—3162no—
222C2×C4—6322no—
223C2×C4—6321no—
224C2×C4—6322no—
225C8—6322no—
226C22—22no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—81no—
233C22#1909871411yescharacteristic
234C22#1909874111yescharacteristic
235C22#1909875011yescharacteristic
236C4—42no—
237C4—42no—
238C4—82no—
239C4—81no—
240C4—82no—
241C4—6322no—
242C4—12641no—
243C2—21no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2#2376235211yescharacteristic
248C2#2376541312yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2817C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2817D79
hypocenterC79#12207
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
130. C79#12207semidirect

Maximal subgroups

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

subgroupindex
#237645672
#237623902
#237614002
#237613302
C79×#28172
#1220779