Finite Group Databasebeta

← #30640175

Subgroups — #30640175

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×#16861141C211yesmaximal, characteristic, complemented
2#22549877C212yesmaximal
3#22549644C212yesmaximal, complemented
4#22549641C211yesmaximal, characteristic, complemented
5#408×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2212yes—
8#16861141—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—22no—
50C22×C127—41no—
51C22×C127—42no—
52C22×C127#12111yescharacteristic
53C22×C127#12612yes—
54C4×C127—41no—
55C4×C127#12311yescharacteristic
56C4×C127#12312yes—
57C4×C127#12411yescharacteristic
58Dic127—82no—
59C2×C127—22no—
60C2×C127—81no—
61C2×C127#39711yescharacteristic
62C2×C127#47112yes—
63#2079—1271nomaximal, Sylow
64C127#207911yescharacteristic, complemented, Sylow
65C2×#106—1271no—
66#417—1272no—
67#408D12711yescharacteristic
68#343—1272no—
69#341—1271no—
70C2×#41C2×D12711yescharacteristic
71C2×M16—1272no—
72C2×M16C2×D12711yescharacteristic
73C22×C8C2×D12711yescharacteristic
74C2×C4⋊C4—1272no—
75#106—2542no—
76#103—2542no—
77#102—2542no—
78C22⋊C8—22no—
79#41—22no—
80#41—22no—
81C2×Q8D50811yescharacteristic
82C2×D4—21no—
83C2×D4D50811yescharacteristic
84C22×C4—21no—
85C22×C4—2542no—
86C22×C4C22×D12711yescharacteristic
87M16—22no—
88M16—2542no—
89M16—2542no—
90C2×C8—22no—
91C2×C8—21no—
92C2×C8—2542no—
93C2×C8C22⋊D12712yes—
94C4⋊C4—2542no—
95C4⋊C4—2542no—
96C23—21no—
97C23#1452324811yescharacteristic
98Q8—22no—
99D4—22no—
100D4—42no—
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—10162no—
105C2×C4—5082no—
106C2×C4C2×C22⋊D12711yescharacteristic
107C2×C4C2×D50811yescharacteristic
108C2×C4#1452324712yes—
109C2×C4#1452324712yes—
110C8—22no—
111C8—41no—
112C8—5082no—
113C22—22no—
114C22—41no—
115C22—42no—
116C22#1686118311yescharacteristic
117C22#1686120212yes—
118C4—41no—
119C4—10162no—
120C4#1686119011yescharacteristic
121C4#1686119312yes—
122C4#1686119611yescharacteristic
123C2—22no—
124C2—81no—
125C2#2254980111yescharacteristic
126C2#2255013912yes—
subgroupG/H
centerC22#16861183
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127#121
prime radicalC2×#41×C127C22
hypercenter#408D127
hypocenterC127#2079
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×#16861141C2semidirect
3. #22549644C2semidirect
4. #22549641C2semidirect
64. C127#2079semidirect

Maximal subgroups

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

subgroupindex
C2×#168611412
#225498772
#225496442
#225496412
#408×C1272
#2079127