Finite Group Databasebeta

← #34201650

Subgroups — #34201650

Order 20224.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#23765617C212yesmaximal
2#23764975C211yesmaximal, characteristic
3#23761388C212yesmaximal
4#23761351C211yesmaximal, characteristic
5C79×#3255C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098385—22no—
8#19098379—22no—
9#19098345C2212yes—
10#19098335—21no—
11#19098323—21no—
12#19098320C2211yescharacteristic
13#533×C79—21no—
14#475×C79—22no—
15#470×C79—21no—
16#470×C79C2212yes—
17#462×C79—22no—
18#423×C79C2211yescharacteristic
19C2×#11929608—21no—
20#14579120—21no—
21#14579101—22no—
22#14579097—22no—
23C2×Q16×C79—21no—
24C2×Q16×C79D412yes—
25C2×M16×C79D411yescharacteristic
26C2×M16×C79C2311yescharacteristic
27C4⋊Q8×C79—21no—
28C4⋊Q8×C79—21no—
29#125×C79—41no—
30C22⋊Q8×C79—22no—
31C22⋊Q8×C79—41no—
32C4×Q8×C79—22no—
33C4×Q8×C79—41no—
34#117×C79—41no—
35C2×C42×C79D411yescharacteristic
36#106×C79—21no—
37#106×C79D412yes—
38C4⋊C8×C79—22no—
39#103×C79—22no—
40#103×C79—22no—
41#11929608—41no—
42#11929608—41no—
43C2×C79⋊C8—41no—
44C4⋊Dic79—41no—
45C22⋊Dic79—41no—
46C4×Dic79—41no—
47C79⋊C16—42no—
48C2×Q8×C79—22no—
49C2×Q8×C79—41no—
50C22×C4×C79—21no—
51C22×C4×C79C2×D411yescharacteristic
52Q16×C79—42no—
53Q16×C79—81no—
54M16×C79—41no—
55M16×C79—41no—
56C2×C8×C79—21no—
57C2×C8×C79C2×D412yes—
58C4⋊C4×C79—22no—
59C4⋊C4×C79—41no—
60C4⋊C4×C79—41no—
61C4⋊C4×C79—42no—
62C4⋊C4×C79—41no—
63C4⋊C4×C79—42no—
64C22⋊C4×C79—42no—
65C42×C79—21no—
66C42×C79—42no—
67C42×C79SD1611yescharacteristic
68C42×C79SD1611yescharacteristic
69C23×C79AGL(1,ℤ/8)11yescharacteristic
70Q8×C79—42no—
71Q8×C79—42no—
72Q8×C79—82no—
73C2×C4×C79—21no—
74C2×C4×C79—21no—
75C2×C4×C79—21no—
76C2×C4×C79—21no—
77C2×C4×C79—42no—
78C2×C4×C79—42no—
79C2×C4×C79—42no—
80C2×C4×C79#12011yescharacteristic
81C2×C4×C79C2×SD1611yescharacteristic
82C2×Dic79—81no—
83C8×C79—22no—
84C8×C79—41no—
85C79⋊C8—81no—
86C22×C79—21no—
87C22×C79—41no—
88C22×C79#45011yescharacteristic
89C4×C79—22no—
90C4×C79—21no—
91C4×C79—41no—
92C4×C79—82no—
93C4×C79—82no—
94C4×C79#45411yescharacteristic
95C4×C79#45811yescharacteristic
96Dic79—161no—
97#12308—791nomaximal, Sylow
98C2×C79—21no—
99C2×C79—41no—
100C2×C79#161611yescharacteristic
101#3255D7911yescharacteristic
102#2239—792no—
103#2102—791no—
104#1340—792no—
105#1327—791no—
106C79#1230811yescharacteristic, complemented, Sylow
107#533—21no—
108#475—22no—
109#470—21no—
110#470C2×D7912yes—
111#462—22no—
112#430—791no—
113#423C2×D7911yescharacteristic
114#365—1582no—
115#362—1582no—
116#349—792no—
117#344—1581no—
118#339—1581no—
119H(3,ℤ/4)—791no—
120C2×Q16—21no—
121C2×Q16C22⋊D7912yes—
122C2×M16—1581no—
123C2×M16C22⋊D7911yescharacteristic
124C2×M16C22×D7911yescharacteristic
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127#125—41no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C4×Q8—22no—
131C4×Q8—41no—
132#117—41no—
133#117—1581no—
134C2×C42C22⋊D7911yescharacteristic
135M32—1582no—
136#108—1582no—
137#106—21no—
138#106C22⋊D7912yes—
139C4⋊C8—22no—
140#103—22no—
141#103—22no—
142C2×Q8—22no—
143C2×Q8—41no—
144C22×C4—21no—
145C22×C4C2×C22⋊D7911yescharacteristic
146Q16—42no—
147Q16—81no—
148M16—41no—
149M16—41no—
150M16—3161no—
151M16—3161no—
152C2×C8—21no—
153C2×C8—3161no—
154C2×C8C2×C22⋊D7912yes—
155C4⋊C4—22no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—3161no—
162C22⋊C4—42no—
163C22⋊C4—3161no—
164C42—21no—
165C42—42no—
166C42—3161no—
167C42#1192961211yescharacteristic
168C42#1192961211yescharacteristic
169C16—3162no—
170C23#1457918711yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—6321no—
182C2×C4#1457913011yescharacteristic
183C2×C4C2×#1192961211yescharacteristic
184C8—22no—
185C8—41no—
186C8—6321no—
187C22—21no—
188C22—41no—
189C22#1909871411yescharacteristic
190C4—22no—
191C4—21no—
192C4—41no—
193C4—82no—
194C4—82no—
195C4—12641no—
196C4#1909873611yescharacteristic
197C4#1909875311yescharacteristic
198C2—21no—
199C2—41no—
200C2#2376243711yescharacteristic
subgroupG/H
centerC2#23762437
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#3255C2
socleC2×C79#1616
prime radicalC23×C79AGL(1,ℤ/8)
hypercenter#3255D79
hypocenterC79#12308
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3255

Complements

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

NH ≅ G/Ntype
106. C79#12308semidirect

Maximal subgroups

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

subgroupindex
#237656172
#237649752
#237613882
#237613512
C79×#32552
#1230879