Finite Group Databasebeta

← #27143619

Subgroups — #27143619

Order 12416.

144 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261165C211yesmaximal, characteristic
2#20260852C211yesmaximal, characteristic, complemented
3#20260851C211yesmaximal, characteristic, complemented
4#20260564C211yesmaximal, characteristic, complemented
5#20260543C211yesmaximal, characteristic, complemented
6#20260510C211yesmaximal, characteristic, complemented
7#359×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9#15690818—21no—
10#15690817—21no—
11#15690756C411yescharacteristic, complemented
12#15690754C411yescharacteristic, complemented
13#15690753C411yescharacteristic, complemented
14#15690751C411yescharacteristic, complemented
15C2×#13095486C2211yescharacteristic
16#15690746C2211yescharacteristic
17#15690737—21no—
18#15690736—21no—
19C8×Dic97C2211yescharacteristic
20#135×C97C2211yescharacteristic
21C2×C16×C97C2211yescharacteristic
22#108×C97C2211yescharacteristic
23#13095511—21no—
24Dic388C2×C411yescharacteristic
25#13095500—21no—
26D776C2×C411yescharacteristic
27#13095495—21no—
28#13095495—41no—
29#13095494—41no—
30C2×#11715895—41no—
31C2×#11715895D411yescharacteristic
32C8×D97—41no—
33C4×Dic97D411yescharacteristic
34#13095486C2×C411yescharacteristic
35#13095486C2×C411yescharacteristic
36#41×C97—21no—
37Q16×C97C2×C411yescharacteristic
38SD16×C97—21no—
39D8×C97C2×C411yescharacteristic
40M16×C97—21no—
41C2×C8×C97C2311yescharacteristic
42C16×C97—21no—
43Q8×C97—21no—
44D4×C97—21no—
45D4×C97—41no—
46C2×C4×C97—41no—
47C2×C4×C97C2×D411yescharacteristic
48C22⋊D97—41no—
49C2×Dic97—21no—
50D388—21no—
51C4×D97—41no—
52Dic194—21no—
53C8×C97—41no—
54C8×C97C22×C411yescharacteristic
55C8×C97#4111yescharacteristic
56#11715895—41no—
57#11715895D811yescharacteristic
58#11715895D811yescharacteristic
59C22×C97—41no—
60C22×C97#13511yescharacteristic
61C2×D97—41no—
62C4×C97—41no—
63C4×C97C4×D411yescharacteristic
64C4×C97C2×D811yescharacteristic
65Dic97—41no—
66Dic97—41no—
67C2×C97—21no—
68C2×C97—81no—
69C2×C97C4×D811yescharacteristic
70D97—81no—
71#2179—971nomaximal, Sylow
72C97#217911yescharacteristic, complemented, Sylow
73#508—971no—
74#443—971no—
75#443—971no—
76#368—971no—
77#359—971no—
78#359D9711yescharacteristic, complemented
79#346—971no—
80#135—971no—
81#135C2×D9711yescharacteristic
82#131—1941no—
83#131—1941no—
84Q32—971no—
85SD32—971no—
86SD32—971no—
87D16—971no—
88C2×C16—971no—
89C2×C16C2×D9711yescharacteristic
90#108—971no—
91#108C2×D9711yescharacteristic
92#104—1941no—
93#104—1941no—
94C4×C8—971no—
95#41—21no—
96#41—1941no—
97Q16—971no—
98Q16C4×D9711yescharacteristic
99SD16—21no—
100SD16—1941no—
101D8—971no—
102D8C4×D9711yescharacteristic
103M16—21no—
104M16—3881no—
105M16—3881no—
106M16—1941no—
107C2×C8—971no—
108C2×C8—3881no—
109C2×C8—3881no—
110C2×C8C22×D9711yescharacteristic
111C42—971no—
112C16—21no—
113C16—971no—
114C16—971no—
115Q8—21no—
116Q8—1941no—
117D4—21no—
118D4—41no—
119D4—3881no—
120D4—1941no—
121C2×C4—41no—
122C2×C4—3881no—
123C2×C4—1941no—
124C2×C4D4×D9711yescharacteristic
125C8—41no—
126C8—971no—
127C8—971no—
128C8—3881no—
129C8C2×C4×D9711yescharacteristic
130C8#1309551011yescharacteristic
131C22—41no—
132C22—3881no—
133C22#1569083111yescharacteristic
134C4—41no—
135C4—3881no—
136C4—3881no—
137C4#1569077011yescharacteristic
138C4D8×D9711yescharacteristic
139C2—21no—
140C2—81no—
141C2—7761no—
142C2#2026081511yescharacteristic
subgroupG/H
centerC4D8×D97
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#359×C97C2
socleC2×C97C4×D8
prime radical#20261165C2
cosocleC2×C8×C97C23
hypercenter#359D97
hypocenterC97#2179
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #20260852C2semidirect
3. #20260851C2semidirect
4. #20260564C2semidirect
5. #20260543C2semidirect
6. #20260510C2semidirect
7. #359×C97C2semidirect
11. #15690756C4semidirect
12. #15690754C4semidirect
13. #15690753C4semidirect
14. #15690751C4semidirect
72. C97#2179semidirect
78. #359D97semidirect

Maximal subgroups

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

subgroupindex
#202611652
#202608522
#202608512
#202605642
#202605432
#202605102
#359×C972
#217997