Finite Group Databasebeta

← #30640170

Subgroups — #30640170

Order 16256.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#16861143C211yesmaximal, characteristic, complemented
2#22549877C212yesmaximal
3#22549644C212yesmaximal, complemented
4#22549641C211yesmaximal, characteristic, complemented
5#408×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2212yes—
8#16861143—22no—
9#16861133—22no—
10#16861130—22no—
11C2×#41×C127C2211yescharacteristic
12C2×M16×C127C2211yescharacteristic
13C22×C8×C127C2211yescharacteristic
14C22⋊C8×C127—22no—
15C22×Dic127—22no—
16#14523232—22no—
17#14523232—22no—
18C2×C127⋊C8—22no—
19C4⋊Dic127—22no—
20C4⋊Dic127—22no—
21#41×C127—22no—
22#41×C127—22no—
23C2×Q8×C127D411yescharacteristic
24C2×D4×C127—21no—
25C2×D4×C127D411yescharacteristic
26C22×C4×C127—21no—
27C22×C4×C127C2311yescharacteristic
28M16×C127—22no—
29C2×C8×C127—22no—
30C2×C8×C127—21no—
31C2×C8×C127D412yes—
32C2×Dic127—42no—
33C2×Dic127—82no—
34C127⋊C8—42no—
35C23×C127—21no—
36C23×C127#4111yescharacteristic
37Q8×C127—22no—
38D4×C127—22no—
39D4×C127—42no—
40C2×C4×C127—21no—
41C2×C4×C127—41no—
42C2×C4×C127—41no—
43C2×C4×C127C2×D411yescharacteristic
44C2×C4×C127C2×D411yescharacteristic
45C2×C4×C127#4112yes—
46C2×C4×C127#4112yes—
47C8×C127—22no—
48C8×C127—41no—
49C22×C127—21no—
50C22×C127—21no—
51C22×C127—41no—
52C22×C127—41no—
53C22×C127—41no—
54C22×C127#12111yescharacteristic
55C22×C127#12612yes—
56C4×C127—41no—
57C4×C127#12311yescharacteristic
58C4×C127#12312yes—
59C4×C127#12411yescharacteristic
60Dic127—82no—
61C2×C127—22no—
62C2×C127—81no—
63C2×C127#39711yescharacteristic
64C2×C127#47211yescharacteristic
65C2×C127#47311yescharacteristic
66#2078—1271nomaximal, Sylow
67C127#207811yescharacteristic, complemented, Sylow
68C2×#107—1271no—
69#417—1272no—
70#408D12711yescharacteristic
71#343—1272no—
72#341—1271no—
73C2×#41C2×D12711yescharacteristic
74C2×M16—1272no—
75C2×M16C2×D12711yescharacteristic
76C22×C8C2×D12711yescharacteristic
77C2×C4⋊C4—1272no—
78#107—2542no—
79#103—2542no—
80#102—2542no—
81C22⋊C8—22no—
82#41—22no—
83#41—22no—
84C2×Q8D50811yescharacteristic
85C2×D4—21no—
86C2×D4D50811yescharacteristic
87C22×C4—21no—
88C22×C4—2542no—
89C22×C4C22×D12711yescharacteristic
90M16—22no—
91M16—2542no—
92M16—2542no—
93C2×C8—22no—
94C2×C8—21no—
95C2×C8—2542no—
96C2×C8C22⋊D12712yes—
97C4⋊C4—2542no—
98C4⋊C4—2542no—
99C23—21no—
100C23#1452324811yescharacteristic
101Q8—22no—
102D4—22no—
103D4—42no—
104C2×C4—21no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—10162no—
108C2×C4—5082no—
109C2×C4C2×C22⋊D12711yescharacteristic
110C2×C4C2×D50811yescharacteristic
111C2×C4#1452324712yes—
112C2×C4#1452324712yes—
113C8—22no—
114C8—41no—
115C8—5082no—
116C22—21no—
117C22—21no—
118C22—41no—
119C22—41no—
120C22—41no—
121C22#1686118311yescharacteristic
122C22#1686120212yes—
123C4—41no—
124C4—10162no—
125C4#1686119011yescharacteristic
126C4#1686119312yes—
127C4#1686119611yescharacteristic
128C2—22no—
129C2—81no—
130C2#2254980111yescharacteristic
131C2#2255014611yescharacteristic
132C2#2255015111yescharacteristic
subgroupG/H
centerC22#16861183
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenter#408D127
hypocenterC127#2078
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16861143C2semidirect
3. #22549644C2semidirect
4. #22549641C2semidirect
67. C127#2078semidirect

Maximal subgroups

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

subgroupindex
C2×#168611432
#225498772
#225496442
#225496412
#408×C1272
#2078127