Finite Group Databasebeta

← #34201329

Subgroups — #34201329

Order 20224.

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

normal subgroups only

This group has 342 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#23764745C211yesmaximal, characteristic
2#23762441C212yesmaximal
3#23761427C212yesmaximal
4#23761346C211yesmaximal, characteristic
5C79×#2571C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098898—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#400×C79—21no—
14#400×C79C2212yes—
15#389×C79—22no—
16#384×C79C2211yescharacteristic
17#382×C79—22no—
18C4×C4⋊C4×C79—21no—
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—
29C2×C4⋊C4×C79—21no—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79—22no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79D412yes—
34C2×C4⋊C4×C79D411yescharacteristic
35C2×C4⋊C4×C79C2311yescharacteristic
36C2×C42×C79—22no—
37C2×C42×C79D411yescharacteristic
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—
54C22×C4×C79—21no—
55C22×C4×C79—22no—
56C22×C4×C79—22no—
57C22×C4×C79—21no—
58C22×C4×C79C2×D411yescharacteristic
59C22×C4×C79C2×D412yes—
60C4⋊C4×C79—41no—
61C4⋊C4×C79—42no—
62C4⋊C4×C79—44no—
63C4⋊C4×C79—42no—
64C4⋊C4×C79—42no—
65C4⋊C4×C79—41no—
66C4⋊C4×C79—81no—
67C42×C79—21no—
68C42×C79—44no—
69C42×C79Q1612yes—
70C23×C79#12011yescharacteristic
71C2×C4×C79—22no—
72C2×C4×C79—22no—
73C2×C4×C79—22no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—22no—
77C2×C4×C79—42no—
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—41no—
85C2×C4×C79—42no—
86C2×C4×C79C2×Q1611yescharacteristic
87C2×C4×C79#13711yescharacteristic
88C2×Dic79—41no—
89C2×Dic79—41no—
90Dic158—82no—
91C79⋊C8—42no—
92C79⋊C8—82no—
93C22×C79—21no—
94C22×C79—21no—
95C22×C79#45111yescharacteristic
96C22×C79#45611yescharacteristic
97C22×C79B(4,𝔽2)11yescharacteristic
98C4×C79—24no—
99C4×C79—21no—
100C4×C79—21no—
101C4×C79—42no—
102C4×C79—82no—
103C4×C79—81no—
104C4×C79—82no—
105Dic79—81no—
106Dic79—81no—
107#12257—791nomaximal, Sylow
108C2×C79—21no—
109C2×C79—21no—
110C2×C79#160511yescharacteristic
111C2×C79#219811yescharacteristic
112C2×C79#220411yescharacteristic
113#2571D7911yescharacteristic
114#2060—791no—
115#1617—792no—
116#1353—792no—
117#1325—791no—
118C79#1225711yescharacteristic, complemented, Sylow
119#544—791no—
120#484—1582no—
121#428—1581no—
122#400—21no—
123#400C2×D7912yes—
124#389—22no—
125#384C2×D7911yescharacteristic
126#382—22no—
127C4×C4⋊C4—21no—
128#343—791no—
129#339—1581no—
130#328—1582no—
131#324—792no—
132C2×M16—1581no—
133C4⋊Q8—3161no—
134#125—3161no—
135C22⋊Q8—1582no—
136#117—1581no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C22⋊D7911yescharacteristic
142C2×C4⋊C4C22⋊D7912yes—
143C2×C4⋊C4C22×D7911yescharacteristic
144C2×C42—22no—
145C2×C42C22⋊D7911yescharacteristic
146#107—3161no—
147#107—3162no—
148#106—3161no—
149#103—3162no—
150C22⋊C8—1582no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95C22⋊D7912yes—
156C2×Q8—1582no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4C2×C22⋊D7911yescharacteristic
162C22×C4C2×C22⋊D7912yes—
163M16—3161no—
164M16—1582no—
165C2×C8—3161no—
166C2×C8—3162no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—3162no—
175C4⋊C4—1581no—
176C4⋊C4—1581no—
177C4⋊C4—1582no—
178C22⋊C4—3161no—
179C42—21no—
180C42—44no—
181C42—3161no—
182C42#1192961412yes—
183C23#1457913011yescharacteristic
184Q8—6322no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—3161no—
201C2×C4—3161no—
202C2×C4C2×#1192961411yescharacteristic
203C2×C4#1457919211yescharacteristic
204C8—3162no—
205C8—6322no—
206C22—21no—
207C22—21no—
208C22#1909871911yescharacteristic
209C22#1909874611yescharacteristic
210C22#1909875011yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—82no—
218C4—6321no—
219C4—6321no—
220C2—21no—
221C2—21no—
222C2#2376237411yescharacteristic
223C2#2376539611yescharacteristic
224C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
FittingC79×#2571C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2571D79
hypocenterC79#12257
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
118. C79#12257semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237624412
#237614272
#237613462
C79×#25712
#1225779