Finite Group Databasebeta

← #34201353

Subgroups — #34201353

Order 20224.

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

normal subgroups only

This group has 330 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#23764750C211yesmaximal, characteristic
2#23762441C212yesmaximal
3#23761427C212yesmaximal
4#23761344C211yesmaximal, characteristic
5C79×#2844C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098898—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#400×C79—22no—
14#400×C79C2212yes—
15#395×C79—22no—
16#395×C79—21no—
17#382×C79—21no—
18#382×C79C2211yescharacteristic
19C2×#11929608—21no—
20#14579160—41no—
21#14579152—41no—
22#14579140—22no—
23#14579120—21no—
24#14579094—42no—
25#14579094—41no—
26#14579092—41no—
27#14579084—42no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79—21no—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—22no—
33C2×C4⋊C4×C79D412yes—
34C2×C4⋊C4×C79D411yescharacteristic
35C2×C4⋊C4×C79C2311yescharacteristic
36C2×C42×C79D411yescharacteristic
37#95×C79—22no—
38#95×C79—21no—
39#95×C79—22no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44#11929608—22no—
45#11929608—41no—
46C2×C79⋊C8—41no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—21no—
49C4⋊Dic79—21no—
50C4⋊Dic79—22no—
51Dic79⋊C4—42no—
52C22⋊Dic79—41no—
53C4×Dic79—41no—
54C2×Q8×C79—82no—
55C2×Q8×C79—81no—
56C22×C4×C79—21no—
57C22×C4×C79—22no—
58C22×C4×C79—22no—
59C22×C4×C79—21no—
60C22×C4×C79C2×D411yescharacteristic
61C22×C4×C79C2×D412yes—
62C4⋊C4×C79—41no—
63C4⋊C4×C79—42no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—81no—
66C4⋊C4×C79—81no—
67C4⋊C4×C79—82no—
68C42×C79—21no—
69C42×C79Q1612yes—
70C23×C79#12011yescharacteristic
71Q8×C79—84no—
72C2×C4×C79—21no—
73C2×C4×C79—21no—
74C2×C4×C79—22no—
75C2×C4×C79—22no—
76C2×C4×C79—41no—
77C2×C4×C79—41no—
78C2×C4×C79—42no—
79C2×C4×C79—42no—
80C2×C4×C79—42no—
81C2×C4×C79—42no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—42no—
85C2×C4×C79—42no—
86C2×C4×C79—41no—
87C2×C4×C79C2×Q1611yescharacteristic
88C2×C4×C79#13711yescharacteristic
89C2×Dic79—41no—
90C2×Dic79—41no—
91Dic158—82no—
92C79⋊C8—42no—
93C79⋊C8—82no—
94C22×C79—21no—
95C22×C79—21no—
96C22×C79#45111yescharacteristic
97C22×C79#45611yescharacteristic
98C22×C79B(4,𝔽2)11yescharacteristic
99C4×C79—21no—
100C4×C79—21no—
101C4×C79—42no—
102C4×C79—42no—
103C4×C79—82no—
104C4×C79—82no—
105C4×C79—81no—
106Dic79—81no—
107Dic79—81no—
108#12261—791nomaximal, Sylow
109C2×C79—21no—
110C2×C79—21no—
111C2×C79#160511yescharacteristic
112C2×C79#220111yescharacteristic
113C2×C79#220211yescharacteristic
114#2844D7911yescharacteristic
115#2061—791no—
116#1617—792no—
117#1353—792no—
118#1324—791no—
119C79#1226111yescharacteristic, complemented, Sylow
120#544—791no—
121#484—1582no—
122#428—1581no—
123#400—22no—
124#400C2×D7912yes—
125#395—22no—
126#395—21no—
127#382—21no—
128#382C2×D7911yescharacteristic
129#343—791no—
130#339—1581no—
131#328—1582no—
132#324—792no—
133C22×Q8—21no—
134C2×M16—1581no—
135C4⋊Q8—3161no—
136#125—3161no—
137C22⋊Q8—1582no—
138#117—1581no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C22⋊D7911yescharacteristic
143C2×C4⋊C4C22⋊D7912yes—
144C2×C4⋊C4C22×D7911yescharacteristic
145C2×C42C22⋊D7911yescharacteristic
146#107—3161no—
147#107—3162no—
148#106—3161no—
149#103—3162no—
150C22⋊C8—1582no—
151#95—22no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95C22⋊D7912yes—
157C2×Q8—82no—
158C2×Q8—81no—
159C2×Q8—1582no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C2×C22⋊D7911yescharacteristic
165C22×C4C2×C22⋊D7912yes—
166M16—3161no—
167M16—1582no—
168C2×C8—3161no—
169C2×C8—3162no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—82no—
176C4⋊C4—3162no—
177C4⋊C4—1581no—
178C4⋊C4—1581no—
179C4⋊C4—1582no—
180C22⋊C4—3161no—
181C42—21no—
182C42—3161no—
183C42#1192961412yes—
184C23#1457913011yescharacteristic
185Q8—84no—
186Q8—6322no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—3161no—
203C2×C4—3161no—
204C2×C4C2×#1192961411yescharacteristic
205C2×C4#1457919211yescharacteristic
206C8—3162no—
207C8—6322no—
208C22—21no—
209C22—21no—
210C22#1909871911yescharacteristic
211C22#1909874611yescharacteristic
212C22#1909875011yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4—81no—
220C4—6321no—
221C4—6321no—
222C2—21no—
223C2—21no—
224C2#2376237411yescharacteristic
225C2#2376541311yescharacteristic
226C2#2376541711yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2844C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2844D79
hypocenterC79#12261
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
119. C79#12261semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237624412
#237614272
#237613442
C79×#28442
#1226179