Finite Group Databasebeta

← #34201633

Subgroups — #34201633

Order 20224.

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

normal subgroups only

This group has 262 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#23764980C211yesmaximal, characteristic
3#23761394C212yesmaximal
4#23761353C211yesmaximal, characteristic
5C79×#3261C211yesmaximal, characteristic
6#19098612C2211yescharacteristic
7#19098385—22no—
8#19098379—22no—
9#19098345C2212yes—
10#19098333—21no—
11#19098328—21no—
12C79⋊#338C2211yescharacteristic
13#541×C79—21no—
14#479×C79—22no—
15#470×C79C2212yes—
16#467×C79—21no—
17#462×C79—22no—
18#424×C79C2211yescharacteristic
19C2×#11929608—21no—
20C2×#11929608—21no—
21#14579101—22no—
22#14579097—22no—
23C2×Q16×C79—21no—
24C2×Q16×C79D412yes—
25C2×M16×C79C2311yescharacteristic
26C22×C8×C79D411yescharacteristic
27C4⋊Q8×C79—21no—
28#126×C79—42no—
29#125×C79—21no—
30#125×C79—42no—
31C22⋊Q8×C79—22no—
32C4×Q8×C79—22no—
33#117×C79D411yescharacteristic
34#107×C79—21no—
35#106×C79D412yes—
36C4⋊C8×C79—22no—
37#103×C79—22no—
38#103×C79—22no—
39#11929608—41no—
40#11929608—41no—
41#11929608—41no—
42#11929608—41no—
43C2×C79⋊C8—41no—
44C2×C79⋊C8—41no—
45C79⋊C16—42no—
46C2×Q8×C79—22no—
47C22×C4×C79C2×D411yescharacteristic
48Q16×C79—42no—
49Q16×C79—81no—
50M16×C79—41no—
51C2×C8×C79—21no—
52C2×C8×C79—41no—
53C2×C8×C79C2×D412yes—
54C4⋊C4×C79—21no—
55C4⋊C4×C79—22no—
56C4⋊C4×C79—42no—
57C4⋊C4×C79—42no—
58C4⋊C4×C79—42no—
59C4⋊C4×C79—42no—
60C22⋊C4×C79—42no—
61C22⋊C4×C79SD1612yes—
62C42×C79—21no—
63C42×C79—42no—
64C23×C79C2×SD1611yescharacteristic
65Q8×C79—42no—
66Q8×C79—42no—
67C2×C4×C79—21no—
68C2×C4×C79—21no—
69C2×C4×C79—22no—
70C2×C4×C79—42no—
71C2×C4×C79—42no—
72C2×C4×C79#12011yescharacteristic
73C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
74C8×C79—22no—
75C8×C79—41no—
76C79⋊C8—81no—
77C79⋊C8—81no—
78C22×C79—21no—
79C22×C79—41no—
80C22×C79#45011yescharacteristic
81C4×C79—21no—
82C4×C79—42no—
83C4×C79—82no—
84C4×C79—82no—
85C4×C79#45411yescharacteristic
86C4×C79#45811yescharacteristic
87#12305—791nomaximal, Sylow
88C2×C79—21no—
89C2×C79—41no—
90C2×C79#161611yescharacteristic
91#3261D7911yescharacteristic
92#2239—792no—
93#2103—791no—
94#1342—792no—
95#1328—791no—
96C79#1230511yescharacteristic, complemented, Sylow
97#541—21no—
98#479—22no—
99#470C2×D7912yes—
100#467—21no—
101#462—22no—
102#430—791no—
103#424C2×D7911yescharacteristic
104#365—1582no—
105#362—1582no—
106#349—792no—
107#344—1581no—
108#341—1581no—
109#338—791no—
110C2×Q16—21no—
111C2×Q16C22⋊D7912yes—
112C2×M16—1581no—
113C2×M16—1581no—
114C2×M16C22×D7911yescharacteristic
115C22×C8C22⋊D7911yescharacteristic
116C4⋊Q8—21no—
117#126—42no—
118#125—21no—
119#125—42no—
120C22⋊Q8—22no—
121C4×Q8—22no—
122#117C22⋊D7911yescharacteristic
123M32—1582no—
124#108—1582no—
125#107—21no—
126#106C22⋊D7912yes—
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130C2×Q8—22no—
131C22×C4C2×C22⋊D7911yescharacteristic
132Q16—42no—
133Q16—81no—
134M16—41no—
135M16—3161no—
136M16—3161no—
137M16—3161no—
138M16—3161no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—3161no—
142C2×C8—3161no—
143C2×C8C2×C22⋊D7912yes—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4#1192961212yes—
152C42—21no—
153C42—42no—
154C16—3162no—
155C23C2×#1192961211yescharacteristic
156Q8—42no—
157Q8—42no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#1457913011yescharacteristic
164C2×C4#1457918711yescharacteristic
165C8—22no—
166C8—41no—
167C8—6321no—
168C8—6321no—
169C22—21no—
170C22—41no—
171C22#1909871411yescharacteristic
172C4—21no—
173C4—42no—
174C4—82no—
175C4—82no—
176C4#1909873611yescharacteristic
177C4#1909875311yescharacteristic
178C2—21no—
179C2—41no—
180C2#2376243711yescharacteristic
subgroupG/H
centerC2#23762437
commutator, cosocleC2×M16×C79C23
FrattiniC2×M16C22×D79
FittingC79×#3261C2
socleC2×C79#1616
prime radicalC23×C79C2×SD16
hypercenter#3261D79
hypocenterC79#12305
LCSG ▷ C2×M16×C79 ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3261

Complements

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

NH ≅ G/Ntype
96. C79#12305semidirect

Maximal subgroups

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

subgroupindex
#237656172
#237649802
#237613942
#237613532
C79×#32612
#1230579