Finite Group Databasebeta

← #23765421

Subgroups — #23765421

Order 10112.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#19098745C212yesmaximal, complemented
2#19098735C211yesmaximal, characteristic
3#19098362C212yesmaximal, complemented
4#19098360C211yesmaximal, characteristic, complemented
5#563×C79C211yesmaximal, characteristic, complemented
6#14579209C2211yescharacteristic
7#14579192—22no—
8#14579187—21no—
9#14579089—21no—
10#14579089—22no—
11#14579080C2211yescharacteristic
12#14579080C2212yes—
13C4⋊Q8×C79C2212yes—
14#126×C79—22no—
15#125×C79—21no—
16#124×C79C2211yescharacteristic
17#123×C79—21no—
18C22⋊Q8×C79—22no—
19#11929625—41no—
20#11929624—21no—
21#11929624—22no—
22Q8×D79—41no—
23#11929614—42no—
24#11929612—41no—
25#11929611—42no—
26#11929609—41no—
27#11929608—21no—
28#11929608—22no—
29C2×Q8×C79D412yes—
30C2×Q8×C79C2311yescharacteristic
31C2×D4×C79D411yescharacteristic
32C22×C4×C79—21no—
33C4⋊C4×C79—21no—
34C4⋊C4×C79—22no—
35C4⋊C4×C79—21no—
36C4⋊C4×C79—22no—
37C4⋊C4×C79—22no—
38C22⋊C4×C79—22no—
39C22⋊C4×C79—21no—
40C22⋊C4×C79—22no—
41C42×C79D411yescharacteristic
42C42×C79D412yes—
43C23×C79—21no—
44Q8×C79—41no—
45Q8×C79—42no—
46D4×C79—41no—
47C2×C4×C79—21no—
48C2×C4×C79—22no—
49C2×C4×C79—22no—
50C2×C4×C79—81no—
51C2×C4×C79C2×D412yes—
52C2×C4×C79C2×D411yescharacteristic
53C22⋊D79—41no—
54D316—21no—
55D316—22no—
56C4×D79—41no—
57C4×D79—42no—
58Dic158—21no—
59Dic158—22no—
60C79⋊C8—41no—
61C79⋊C8—42no—
62C22×C79—42no—
63C22×C79—41no—
64C22×C79#12011yescharacteristic
65C2×D79—41no—
66C4×C79—21no—
67C4×C79—22no—
68C4×C79—41no—
69C4×C79—42no—
70C4×C79—42no—
71Dic79—41no—
72C2×C79—21no—
73C2×C79—81no—
74C2×C79B(4,𝔽2)11yescharacteristic
75D79—81no—
76#2203—791nomaximal, Sylow
77C79#220311yescharacteristic, complemented, Sylow
78#563D7911yescharacteristic, complemented
79#456—792no—
80#454—791no—
81#356—792no—
82#355—791no—
832−1+4—791no—
84#137—1582no—
85AGL(1,ℤ/8)—1581no—
86C4⋊Q8C2×D7912yes—
87#126—22no—
88#125—21no—
89#124C2×D7911yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—1581no—
93#104—1582no—
94#101—792no—
95#101—791no—
96#41—3161no—
97#41—1581no—
98#41—1582no—
99C2×Q8—3161no—
100C2×Q8C22⋊D7912yes—
101C2×Q8C22×D7911yescharacteristic
102C2×D4C22⋊D7911yescharacteristic
103C22×C4—21no—
104Q16—3162no—
105SD16—3161no—
106SD16—3162no—
107D8—3161no—
108M16—1581no—
109M16—1582no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D7911yescharacteristic
119C42C22⋊D7912yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—1581no—
124Q8—1582no—
125D4—41no—
126D4—3161no—
127D4—1581no—
128D4—1582no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—3161no—
134C2×C4—3162no—
135C2×C4C2×C22⋊D7912yes—
136C2×C4C2×C22⋊D7911yescharacteristic
137C8—3161no—
138C8—3162no—
139C22—42no—
140C22—41no—
141C22—3161no—
142C22#1457913011yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—3161no—
149C2—21no—
150C2—81no—
151C2—6321no—
152C2#1909875011yescharacteristic
subgroupG/H
centerC2#19098750
commutator, cosocleC2×Q8×C79C23
FrattiniC2×Q8C22×D79
Fitting#563×C79C2
socleC2×C79B(4,𝔽2)
prime radical#19098735C2
hypercenter#563D79
hypocenterC79#2203
LCSG ▷ C2×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #19098745C2semidirect
3. #19098362C2semidirect
4. #19098360C2semidirect
5. #563×C79C2semidirect
77. C79#2203semidirect
78. #563D79semidirect