Finite Group Databasebeta

← #27140879

Subgroups — #27140879

Order 12416.

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

normal subgroups only

This group has 218 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#20261451C211yesmaximal, characteristic
2#20261022C212yesmaximal, complemented
3#20260484C211yesmaximal, characteristic, complemented
4#20260464C212yesmaximal, complemented
5#494×C97C211yesmaximal, characteristic, complemented
6#15690799—22no—
7#15690785—22no—
8C4×D388C2212yes—
9#15690768—22no—
10#15690741—22no—
11C4⋊#11715895C2212yes—
12#15690730—22no—
13C2×Q16×C97—22no—
14C4⋊Q8×C97C2211yescharacteristic
15C4⋊Q8×C97C2211yescharacteristic
16C4×C8×C97C2211yescharacteristic
17C2×D388—22no—
18C2×D388D412yes—
19C2×C4×D97—22no—
20C2×#11715895—22no—
21C4⋊Dic97D412yes—
22#13095488—24no—
23#13095488—22no—
24C4×Dic97—22no—
25C2×Q8×C97—21no—
26C2×Q8×C97D412yes—
27Q16×C97—44no—
28C2×C8×C97—21no—
29C4⋊C4×C97—22no—
30C4⋊C4×C97—22no—
31C42×C97C2311yescharacteristic
32C22×D97—22no—
33Q8×C97—42no—
34Q8×C97—42no—
35C2×C4×C97—22no—
36C2×C4×C97—41no—
37C2×C4×C97C2×D412yes—
38C2×C4×C97C2×D411yescharacteristic
39C2×Dic97—22no—
40D388—22no—
41D388—22no—
42D388—82no—
43C4×D97—42no—
44C4×D97—42no—
45C8×C97—22no—
46#11715895—42no—
47C22×C97#12011yescharacteristic
48C2×D97—42no—
49C2×D97—42no—
50C2×D97—42no—
51C4×C97—21no—
52C4×C97—42no—
53C4×C97—81no—
54C4×C97#13512yes—
55C4×C97#13712yes—
56Dic97—42no—
57C2×C97#45212yes—
58C2×C97#45311yescharacteristic
59D97—82no—
60#1634—971nomaximal, Sylow
61C97#163411yescharacteristic, complemented, Sylow
62#555—971no—
63#494D9711yescharacteristic, complemented
64#476—972no—
65#333—971no—
66D4⋊C8—972no—
67C2×Q16—22no—
68C4⋊Q8C2×D9711yescharacteristic
69C4⋊Q8C2×D9711yescharacteristic
70#124—1942no—
71#121—1942no—
72C4×D4—972no—
73#117—1942no—
74#106—1942no—
75C4⋊C8—972no—
76#102—1942no—
77C4×C8C2×D9711yescharacteristic
78C2×Q8—21no—
79C2×Q8C22⋊D9712yes—
80C2×D4—972no—
81C2×D4—1942no—
82C22×C4—1942no—
83Q16—44no—
84C2×C8—21no—
85C2×C8—1942no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—972no—
89C22⋊C4—1944no—
90C22⋊C4—1942no—
91C42—1942no—
92C42C22×D9711yescharacteristic
93C23—1942no—
94Q8—42no—
95Q8—42no—
96D4—7762no—
97D4—1942no—
98D4—1942no—
99C2×C4—22no—
100C2×C4—41no—
101C2×C4—3882no—
102C2×C4—3882no—
103C2×C4—1942no—
104C2×C4D4×D9712yes—
105C2×C4C2×C22⋊D9711yescharacteristic
106C8—22no—
107C8—3882no—
108C22—3882no—
109C22—3882no—
110C22—3882no—
111C22#1569077811yescharacteristic
112C4—21no—
113C4—42no—
114C4—81no—
115C4—3882no—
116C4#1569083312yes—
117C4#1569084212yes—
118C2—7762no—
119C2#2026089612yes—
120C2#2026090211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#494×C97C2
socleC22×C97#120
prime radical#20261451C2
hypercenter#494D97
hypocenterC97#1634
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
2. #20261022C2semidirect
3. #20260484C2semidirect
4. #20260464C2semidirect
5. #494×C97C2semidirect
61. C97#1634semidirect
63. #494D97semidirect

Maximal subgroups

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

subgroupindex
#202614512
#202610222
#202604842
#202604642
#494×C972
#163497