Finite Group Databasebeta

← #27143630

Subgroups — #27143630

Order 12416.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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#20261176C211yesmaximal, characteristic
2#20260856C211yesmaximal, characteristic
3#20260842C211yesmaximal, characteristic, complemented
4#20260559C211yesmaximal, characteristic, complemented
5#20260546C211yesmaximal, characteristic
6#20260512C211yesmaximal, characteristic, complemented
7#362×C97C211yesmaximal, characteristic, complemented
8#15690832C2211yescharacteristic
9#15690818—21no—
10Q8×Dic97—21no—
11#15690756C412yescomplemented
12#15690753C412yescomplemented
13#15690750C2211yescharacteristic
14#15690742C2211yescharacteristic
15#15690736—21no—
16#15690734—21no—
17#15690720C2211yescharacteristic
18C2×Q16×C97C2211yescharacteristic
19M32×C97C2211yescharacteristic
20#108×C97C2211yescharacteristic
21#13095511—21no—
22Dic388C2×C411yescharacteristic
23#13095500—21no—
24D776C2×C411yescharacteristic
25#13095494—41no—
26C2×#11715895D411yescharacteristic
27C8×D97—41no—
28C4⋊Dic97—21no—
29C4⋊Dic97—41no—
30C4×Dic97—41no—
31C4×Dic97D411yescharacteristic
32#13095486C2×C412yes—
33C2×Q8×C97—21no—
34Q16×C97—21no—
35Q16×C97C2×C412yes—
36M16×C97—21no—
37C2×C8×C97C2311yescharacteristic
38C16×C97—21no—
39Q8×C97—22no—
40Q8×C97—41no—
41C2×C4×C97—41no—
42C2×C4×C97C2×D411yescharacteristic
43C22⋊D97—41no—
44C2×Dic97—21no—
45C2×Dic97—41no—
46D388—21no—
47C4×D97—41no—
48Dic194—21no—
49C8×C97—41no—
50C8×C97C22×C411yescharacteristic
51C8×C97#4111yescharacteristic
52#11715895—21no—
53C22×C97AGL(1,ℤ/8)11yescharacteristic
54C2×D97—41no—
55C4×C97—42no—
56C4×C97C4×D411yescharacteristic
57C4×C97AGL(1,ℤ/8)11yescharacteristic
58Dic97—41no—
59Dic97—41no—
60Dic97—81no—
61C2×C97—21no—
62C2×C97#44211yescharacteristic
63D97—81no—
64#2180—971nomaximal, Sylow
65C97#218011yescharacteristic, complemented, Sylow
66#510—971no—
67#444—971no—
68#441—971no—
69#365—971no—
70#362D9711yescharacteristic, complemented
71#360—971no—
72#347—971no—
73#135—971no—
74C2×Q16C2×D9711yescharacteristic
75#131—1941no—
76C4×Q8—1941no—
77Q32—972no—
78SD32—972no—
79M32—971no—
80M32C2×D9711yescharacteristic
81#108C2×D9711yescharacteristic
82#106—971no—
83#104—1941no—
84#103—1941no—
85#97—971no—
86#41—1941no—
87C2×Q8—21no—
88Q16—21no—
89Q16—971no—
90Q16C4×D9712yes—
91SD16—1941no—
92D8—971no—
93M16—21no—
94M16—3881no—
95C2×C8—971no—
96C2×C8—3881no—
97C2×C8C22×D9711yescharacteristic
98C4⋊C4—3881no—
99C4⋊C4—1941no—
100C42—971no—
101C42—3881no—
102C16—21no—
103C16—972no—
104Q8—22no—
105Q8—41no—
106Q8—1941no—
107D4—3881no—
108D4—1941no—
109C2×C4—41no—
110C2×C4—3881no—
111C2×C4—3881no—
112C2×C4—1941no—
113C2×C4D4×D9711yescharacteristic
114C8—41no—
115C8—1941no—
116C8C2×C4×D9711yescharacteristic
117C8#1309551011yescharacteristic
118C22—3881no—
119C22#1569083711yescharacteristic
120C4—42no—
121C4—7761no—
122C4—3881no—
123C4—3881no—
124C4#1569077011yescharacteristic
125C4#1569083411yescharacteristic
126C2—21no—
127C2—7761no—
128C2#2026084411yescharacteristic
subgroupG/H
centerC2#20260844
commutatorC8×C97C22×C4
FrattiniC2×C8C22×D97
Fitting#362×C97C2
socleC2×C97#442
prime radical#15690832C22
cosocleC2×C8×C97C23
hypercenter#362D97
hypocenterC97#2180
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #362

Complements

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

NH ≅ G/Ntype
3. #20260842C2semidirect
4. #20260559C2semidirect
6. #20260512C2semidirect
7. #362×C97C2semidirect
11. #15690756C4semidirect
12. #15690753C4semidirect
65. C97#2180semidirect
70. #362D97semidirect

Maximal subgroups

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

subgroupindex
#202611762
#202608562
#202608422
#202605592
#202605462
#202605122
#362×C972
#218097