Finite Group Databasebeta

← #27141034

Subgroups — #27141034

Order 12416.

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#20261125C211yesmaximal, characteristic
2#20261101C212yesmaximal, complemented
3#20260477C212yesmaximal, complemented
4#20260475C211yesmaximal, characteristic, complemented
5#563×C97C211yesmaximal, characteristic, complemented
6C2×#13095498—21no—
7C2×#13095496—21no—
8#15690807C2211yescharacteristic
9#15690730—22no—
10#15690730—22no—
11#15690721C2211yescharacteristic
12#15690721C2212yes—
13C4⋊Q8×C97—21no—
14#126×C97—22no—
15#125×C97C2212yes—
16#124×C97C2211yescharacteristic
17#123×C97—21no—
18C22⋊Q8×C97—22no—
19C2×D388—21no—
20C2×D388—22no—
21#13095498—42no—
22#13095496—42no—
23C2×#11715895—21no—
24C2×#11715895—22no—
25C2×Q8×C97—21no—
26C2×Q8×C97D411yescharacteristic
27C2×D4×C97D411yescharacteristic
28C22×C4×C97—21no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97—22no—
31C4⋊C4×C97—22no—
32C4⋊C4×C97D412yes—
33C4⋊C4×C97D412yes—
34C22⋊C4×C97—22no—
35C22⋊C4×C97—21no—
36C22⋊C4×C97—22no—
37C42×C97—21no—
38C42×C97C2311yescharacteristic
39C23×C97—21no—
40C22×D97—41no—
41Q8×C97—41no—
42Q8×C97—81no—
43D4×C97—41no—
44C2×C4×C97—22no—
45C2×C4×C97—21no—
46C2×C4×C97—22no—
47C2×C4×C97—81no—
48C2×C4×C97C2×D412yes—
49C2×C4×C97C2×D411yescharacteristic
50D388—41no—
51D388—41no—
52D388—44no—
53#11715895—22no—
54#11715895—42no—
55C22×C97—41no—
56C22×C97—42no—
57C22×C97#12011yescharacteristic
58C2×D97—81no—
59C2×D97—82no—
60C4×C97—21no—
61C4×C97—22no—
62C4×C97—42no—
63C4×C97—41no—
64C4×C97—42no—
65C2×C97—81no—
66C2×C97#45311yescharacteristic
67C2×C97#45412yes—
68D97—161no—
69#1657—971nomaximal, Sylow
70C97#165711yescharacteristic, complemented, Sylow
71#563D9711yescharacteristic, complemented
72#496—971no—
73#490—972no—
74#330—972no—
75#329—971no—
76C2×SD16—1941no—
77C2×D8—1941no—
78C4⋊Q8—21no—
79#127—971no—
80#126—22no—
81#125C2×D9712yes—
82#124C2×D9711yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—1942no—
86#102—1942no—
87#97—972no—
88#97—971no—
89C2×Q8—21no—
90C2×Q8C22⋊D9711yescharacteristic
91C2×D4—1942no—
92C2×D4—1941no—
93C2×D4C22⋊D9711yescharacteristic
94C22×C4—21no—
95SD16—3882no—
96D8—3882no—
97C2×C8—1942no—
98C2×C8—1941no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D9712yes—
103C4⋊C4C22⋊D9712yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D9711yescharacteristic
109C23—21no—
110C23—3881no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—3881no—
115D4—3881no—
116D4—3884no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D9712yes—
122C2×C4C2×C22⋊D9711yescharacteristic
123C8—3882no—
124C8—1942no—
125C22—41no—
126C22—42no—
127C22—7761no—
128C22—7762no—
129C22#1569077911yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—15521no—
137C2#2026090311yescharacteristic
138C2#2026090812yes—
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#563×C97C2
socleC22×C97#120
prime radical#20261125C2
hypercenter#563D97
hypocenterC97#1657
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #20261101C2semidirect
3. #20260477C2semidirect
4. #20260475C2semidirect
5. #563×C97C2semidirect
70. C97#1657semidirect
71. #563D97semidirect

Maximal subgroups

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

subgroupindex
#202611252
#202611012
#202604772
#202604752
#563×C972
#165797