Finite Group Databasebeta

← #23762698

Subgroups — #23762698

Order 10112.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#19098953C211yesmaximal, characteristic
2#19098929C212yesmaximal, complemented
3#19098305C212yesmaximal, complemented
4#19098303C211yesmaximal, characteristic, complemented
5#563×C79C211yesmaximal, characteristic, complemented
6C2×#11929611—21no—
7C2×#11929609—21no—
8#14579158C2211yescharacteristic
9#14579081—22no—
10#14579081—22no—
11#14579072C2212yes—
12#14579072C2211yescharacteristic
13C4⋊Q8×C79—21no—
14#126×C79—22no—
15#125×C79C2212yes—
16#124×C79C2211yescharacteristic
17#123×C79—21no—
18C22⋊Q8×C79—22no—
19C2×D316—22no—
20C2×D316—21no—
21#11929611—42no—
22#11929609—42no—
23C2×C79⋊C8—21no—
24C2×C79⋊C8—22no—
25C2×Q8×C79—21no—
26C2×Q8×C79D411yescharacteristic
27C2×D4×C79D411yescharacteristic
28C22×C4×C79—21no—
29C4⋊C4×C79—22no—
30C4⋊C4×C79—22no—
31C4⋊C4×C79—22no—
32C4⋊C4×C79D412yes—
33C4⋊C4×C79D412yes—
34C22⋊C4×C79—22no—
35C22⋊C4×C79—21no—
36C22⋊C4×C79—22no—
37C42×C79—21no—
38C42×C79C2311yescharacteristic
39C23×C79—21no—
40C22×D79—41no—
41Q8×C79—41no—
42Q8×C79—81no—
43D4×C79—41no—
44C2×C4×C79—22no—
45C2×C4×C79—21no—
46C2×C4×C79—22no—
47C2×C4×C79—81no—
48C2×C4×C79C2×D412yes—
49C2×C4×C79C2×D411yescharacteristic
50D316—41no—
51D316—41no—
52D316—44no—
53C79⋊C8—22no—
54C79⋊C8—42no—
55C22×C79—41no—
56C22×C79—42no—
57C22×C79#12011yescharacteristic
58C2×D79—81no—
59C2×D79—82no—
60C4×C79—21no—
61C4×C79—22no—
62C4×C79—42no—
63C4×C79—41no—
64C4×C79—42no—
65C2×C79—81no—
66C2×C79#45311yescharacteristic
67C2×C79#45412yes—
68D79—161no—
69#1657—791nomaximal, Sylow
70C79#165711yescharacteristic, complemented, Sylow
71#563D7911yescharacteristic, complemented
72#496—791no—
73#490—792no—
74#330—792no—
75#329—791no—
76C2×SD16—1581no—
77C2×D8—1581no—
78C4⋊Q8—21no—
79#127—791no—
80#126—22no—
81#125C2×D7912yes—
82#124C2×D7911yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—1582no—
86#102—1582no—
87#97—792no—
88#97—791no—
89C2×Q8—21no—
90C2×Q8C22⋊D7911yescharacteristic
91C2×D4—1582no—
92C2×D4—1581no—
93C2×D4C22⋊D7911yescharacteristic
94C22×C4—21no—
95SD16—3162no—
96D8—3162no—
97C2×C8—1582no—
98C2×C8—1581no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D7912yes—
103C4⋊C4C22⋊D7912yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D7911yescharacteristic
109C23—21no—
110C23—3161no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—3161no—
115D4—3161no—
116D4—3164no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D7912yes—
122C2×C4C2×C22⋊D7911yescharacteristic
123C8—3162no—
124C8—1582no—
125C22—41no—
126C22—42no—
127C22—6321no—
128C22—6322no—
129C22#1457913011yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—12641no—
137C2#1909873111yescharacteristic
138C2#1909873612yes—
subgroupG/H
centerC22#14579130
commutator, cosocleC42×C79C23
FrattiniC42C22×D79
Fitting#563×C79C2
socleC22×C79#120
prime radical#19098953C2
hypercenter#563D79
hypocenterC79#1657
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #19098929C2semidirect
3. #19098305C2semidirect
4. #19098303C2semidirect
5. #563×C79C2semidirect
70. C79#1657semidirect
71. #563D79semidirect