Finite Group Databasebeta

← #34199464

Subgroups — #34199464

Order 20224.

274 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 61 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#23773779C211yesmaximal, characteristic
2#23765210C212yesmaximal
3#23761435C211yesmaximal, characteristic, complemented
4#23761434C212yesmaximal, complemented
5C79×#1356C211yesmaximal, characteristic, complemented
6#19099119C411yescharacteristic
7C4⋊Q8×D79C411yescharacteristic
8#19098933C411yescharacteristic
9#19098932C2211yescharacteristic
10#19098931C411yescharacteristic
11#19098912—22no—
12#19098570—22no—
13#19098515C2211yescharacteristic
14#19098383—22no—
15#19098344—22no—
16#19098341C2212yes—
17#491×C79C2211yescharacteristic
18#347×C79C2212yes—
19C2×Q8×D79—21no—
20C2×#11929607D412yes—
21#14579161C2×C411yescharacteristic
22#14579160—21no—
23#14579160—41no—
24#14579159—21no—
25#14579153C2×C411yescharacteristic
26#14579139—41no—
27#14579126—41no—
28#14579119—21no—
29C4⋊C4×D79—41no—
30C42×D79C2×C411yescharacteristic
31#14579101—22no—
32#14579086—22no—
33#14579085—22no—
34#14579084—22no—
35#14579072C2×C411yescharacteristic
36C8×Dic79D412yes—
37C4⋊Q8×C79C2×C411yescharacteristic
38#125×C79C2×C411yescharacteristic
39M32×C79—22no—
40#103×C79—22no—
41#97×C79C2311yescharacteristic
42C2×Dic158—21no—
43C2×Dic158—41no—
44C2×Dic158—41no—
45Q8×D79—44no—
46C2×C4×D79—21no—
47C2×C4×D79—41no—
48C2×C4×D79C22⋊C411yescharacteristic
49#11929607—42no—
50C2×C79⋊C8C22⋊C412yes—
51C4⋊Dic79—21no—
52C4⋊Dic79—41no—
53Dic79⋊C4—41no—
54Dic79⋊C4—42no—
55Dic79⋊C4—41no—
56(C2×D79)⋊C4—41no—
57C4×Dic79—21no—
58C4×Dic79—41no—
59C4×Dic79C22⋊C411yescharacteristic
60C79⋊C16—42no—
61C2×Q8×C79—21no—
62C2×C8×C79C2×D412yes—
63C4⋊C4×C79—21no—
64C4⋊C4×C79—41no—
65C4⋊C4×C79—41no—
66C42×C79C22×C411yescharacteristic
67C16×C79—42no—
68C22×D79#10011yescharacteristic
69Q8×C79—42no—
70C2×C4×C79—21no—
71C2×C4×C79—41no—
72C2×C4×C79—41no—
73C2×C4×C79C2×C22⋊C411yescharacteristic
74C2×Dic79—21no—
75C2×Dic79—41no—
76C2×Dic79—41no—
77C2×Dic79#10011yescharacteristic
78C4×D79—22no—
79C4×D79—42no—
80C4×D79—82no—
81Dic158—42no—
82Dic158—82no—
83Dic158—82no—
84C8×C79—22no—
85C79⋊C8—22no—
86C22×C79C2×#10011yescharacteristic
87C2×D79—21no—
88C2×D79—41no—
89C4×C79—41no—
90C4×C79—81no—
91C4×C79—81no—
92C4×C79#42112yes—
93Dic79—22no—
94Dic79—21no—
95Dic79—81no—
96Dic79—81no—
97#11916—791nomaximal, Sylow
98C2×C79—21no—
99C2×C79#153311yescharacteristic
100D79—41no—
101#3139—791no—
102#2165—792no—
103#1356—791no—
104#1356—792no—
105#1356D7911yescharacteristic, complemented
106C79#1191611yescharacteristic, complemented, Sylow
107#533—791no—
108C2×C4⋊Q8—791no—
109#491—791no—
110#491—791no—
111#491—791no—
112#491C2×D7911yescharacteristic
113#487—1582no—
114#419—1582no—
115C4×M16—791no—
116#364—1582no—
117#349—1582no—
118#347—792no—
119#347C2×D7912yes—
120C22×Q8—1581no—
121C2×M16—792no—
122C4⋊Q8—3161no—
123C4⋊Q8—791no—
124C4⋊Q8—1581no—
125C4⋊Q8—1581no—
126C4⋊Q8C4×D7911yescharacteristic
127#125—791no—
128#125C4×D7911yescharacteristic
129C22⋊Q8—3161no—
130C4×Q8—3161no—
131#117—1581no—
132C2×C4⋊C4—3161no—
133C2×C42—791no—
134M32—22no—
135M32—1582no—
136#103—22no—
137#103—1582no—
138#103—1582no—
139#103—1582no—
140#97—791no—
141#97C22×D7911yescharacteristic
142C4×C8—792no—
143C2×Q8—21no—
144C2×Q8—3161no—
145C2×Q8—3161no—
146C2×Q8—3164no—
147C2×Q8—1581no—
148C22×C4—3161no—
149C22×C4—791no—
150C22×C4—1581no—
151M16—3162no—
152C2×C8—792no—
153C2×C8D4×D7912yes—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—3161no—
158C4⋊C4—3161no—
159C4⋊C4—3162no—
160C4⋊C4—3161no—
161C4⋊C4—1581no—
162C22⋊C4—3161no—
163C42—3161no—
164C42—791no—
165C42—1581no—
166C42C2×C4×D7911yescharacteristic
167C16—42no—
168C16—3162no—
169C23—791no—
170Q8—42no—
171Q8—3162no—
172Q8—6322no—
173Q8—6322no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—3162no—
178C2×C4—3161no—
179C2×C4—3161no—
180C2×C4—791no—
181C2×C4—6322no—
182C2×C4—1581no—
183C2×C4—1582no—
184C2×C4C22⋊C4×D7911yescharacteristic
185C8—22no—
186C8—1582no—
187C22—3161no—
188C22—1581no—
189C22#100×D7911yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—6321no—
194C4—6321no—
195C4—1582no—
196C4—1581no—
197C4#1909857812yes—
198C2—21no—
199C2—3161no—
200C2#2376202211yescharacteristic
subgroupG/H
centerC2#23762022
commutatorC42×C79C22×C4
Frattini#97C22×D79
FittingC79×#1356C2
socleC2×C79#1533
prime radicalC22×D79#100
cosocle#97×C79C23
hypercenter#1356D79
hypocenterC79#11916
LCSG ▷ C42×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
3. #23761435C2semidirect
4. #23761434C2semidirect
5. C79×#1356C2semidirect
105. #1356D79semidirect
106. C79#11916semidirect

Maximal subgroups

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

subgroupindex
#237737792
#237652102
#237614352
#237614342
C79×#13562
#1191679