Finite Group Databasebeta

← #34201348

Subgroups — #34201348

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#23764750C211yesmaximal, characteristic
2#23762373C212yesmaximal
3#23761430C212yesmaximal
4#23761344C211yesmaximal, characteristic
5C79×#2844C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098719—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13#400×C79—21no—
14#400×C79—22no—
15#395×C79—22no—
16#395×C79C2212yes—
17#382×C79—21no—
18#382×C79C2211yescharacteristic
19C2×#11929614—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579094—41no—
26#14579092—41no—
27#14579085—42no—
28C22⋊(C79⋊C8)—22no—
29C22×Q8×C79D412yes—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79—21no—
32C2×C4⋊C4×C79—21no—
33C2×C4⋊C4×C79—22no—
34C2×C4⋊C4×C79D411yescharacteristic
35C2×C4⋊C4×C79C2311yescharacteristic
36C2×C42×C79D411yescharacteristic
37#95×C79—22no—
38#95×C79—22no—
39#95×C79—21no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44#11929614—84no—
45#11929608—22no—
46#11929608—41no—
47C2×C79⋊C8—41no—
48C2×C79⋊C8—42no—
49C4⋊Dic79—22no—
50C4⋊Dic79—21no—
51C4⋊Dic79—21no—
52Dic79⋊C4—42no—
53C22⋊Dic79—41no—
54C4×Dic79—41no—
55C2×Q8×C79—42no—
56C2×Q8×C79—82no—
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—22no—
60C22×C4×C79—22no—
61C22×C4×C79C2×D411yescharacteristic
62C22×C4×C79C2×D412yes—
63C4⋊C4×C79—41no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—81no—
66C4⋊C4×C79—81no—
67C4⋊C4×C79—82no—
68C4⋊C4×C79—81no—
69C42×C79—21no—
70C42×C79Q1612yescomplemented
71C23×C79#12011yescharacteristic
72Q8×C79—44no—
73Q8×C79—82no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—22no—
77C2×C4×C79—22no—
78C2×C4×C79—41no—
79C2×C4×C79—41no—
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—42no—
87C2×C4×C79—41no—
88C2×C4×C79—42no—
89C2×C4×C79C2×Q1611yescharacteristic
90C2×C4×C79#13711yescharacteristic
91C2×Dic79—41no—
92C2×Dic79—41no—
93Dic158—82no—
94C79⋊C8—42no—
95C79⋊C8—82no—
96C22×C79—21no—
97C22×C79—21no—
98C22×C79#45111yescharacteristic
99C22×C79#45611yescharacteristic
100C22×C79B(4,𝔽2)11yescharacteristic
101C4×C79—21no—
102C4×C79—21no—
103C4×C79—42no—
104C4×C79—42no—
105C4×C79—82no—
106C4×C79—82no—
107C4×C79—81no—
108Dic79—81no—
109Dic79—81no—
110#12260—791nomaximal, Sylow
111C2×C79—21no—
112C2×C79—21no—
113C2×C79#160511yescharacteristic
114C2×C79#219911yescharacteristic
115C2×C79#220311yescharacteristic
116#2844D7911yescharacteristic
117#2061—791no—
118#1605—792no—
119#1354—792no—
120#1324—791no—
121C79#1226011yescharacteristic, complemented, Sylow
122#544—791no—
123#451—1582no—
124#428—1581no—
125#400—21no—
126#400—22no—
127#395—22no—
128#395C2×D7912yes—
129#382—21no—
130#382C2×D7911yescharacteristic
131#343—791no—
132#339—1581no—
133#328—1582no—
134#324—792no—
135C22×Q8C22⋊D7912yes—
136C2×Q16—3162no—
137C2×M16—1581no—
138C4⋊Q8—3161no—
139#125—3161no—
140C22⋊Q8—1582no—
141#117—1581no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊D7911yescharacteristic
147C2×C4⋊C4C22×D7911yescharacteristic
148C2×C42C22⋊D7911yescharacteristic
149#107—3161no—
150#106—3161no—
151#103—3162no—
152C22⋊C8—1582no—
153#95—22no—
154#95—22no—
155#95—21no—
156#95—22no—
157#95—22no—
158#95C22⋊D7912yes—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—1582no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D7911yescharacteristic
167C22×C4C2×C22⋊D7912yes—
168Q16—6324no—
169M16—3161no—
170M16—1582no—
171C2×C8—3161no—
172C2×C8—3162no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—3162no—
180C4⋊C4—1582no—
181C4⋊C4—1581no—
182C4⋊C4—1581no—
183C22⋊C4—3161no—
184C42—21no—
185C42—3161no—
186C42#1192961412yescomplemented
187C23#1457913011yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—6322no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—3161no—
207C2×C4—3161no—
208C2×C4C2×#1192961411yescharacteristic
209C2×C4#1457919211yescharacteristic
210C8—3162no—
211C8—6322no—
212C22—21no—
213C22—21no—
214C22#1909871911yescharacteristic
215C22#1909874611yescharacteristic
216C22#1909875011yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—6321no—
225C4—6321no—
226C2—21no—
227C2—21no—
228C2#2376237411yescharacteristic
229C2#2376540111yescharacteristic
230C2#2376542111yescharacteristic
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#12260
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
70. C42×C79Q16semidirect
121. C79#12260semidirect
186. C42#11929614semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237623732
#237614302
#237613442
C79×#28442
#1226079