Finite Group Databasebeta

← #30638806

Subgroups — #30638806

Order 16256.

300 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 77 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×#16861165C211yesmaximal, characteristic
2#22549737C212yesmaximal
3#22549643C211yesmaximal, characteristic
4#22549642C212yesmaximal
5#342×C127C211yesmaximal, characteristic
6C23×Dic127C2211yescharacteristic
7#16861165—24no—
8C2×Dic127⋊C4C412yescomplemented
9C2×C22⋊Dic127C2212yes—
10C2×C22⋊Dic127C2211yescharacteristic
11C2×C4×Dic127C412yescomplemented
12C127⋊#95—24no—
13C2×C22⋊C4×C127C2212yes—
14C2×C22⋊C4×C127C2211yescharacteristic
15C22×Dic127—22no—
16C22×Dic127—21no—
17C22×Dic127—22no—
18C22×Dic127—21no—
19C22×Dic127C2×C411yescharacteristic
20C22×Dic127C2×C411yescharacteristic
21C22×Dic127C2×C412yes—
22C22×Dic127D414yes—
23Dic127⋊C4—24no—
24C22⋊Dic127—22no—
25C22⋊Dic127—22no—
26C22⋊Dic127—22no—
27C4×Dic127—24no—
28C24×C127C2311yescharacteristic
29C22×C4×C127—22no—
30C22×C4×C127C2×C412yes—
31C22⋊C4×C127—22no—
32C22⋊C4×C127—22no—
33C22⋊C4×C127—22no—
34C2×Dic127—21no—
35C2×Dic127—21no—
36C2×Dic127—21no—
37C2×Dic127—21no—
38C2×Dic127—24no—
39C2×Dic127—24no—
40C2×Dic127—22no—
41C2×Dic127—42no—
42C2×Dic127—44no—
43C2×Dic127—42no—
44C2×Dic127C22⋊C412yes—
45C2×Dic127C22⋊C412yes—
46C23×C127—42no—
47C23×C127C22×C411yescharacteristic
48C23×C127C2×D412yes—
49C23×C127#4111yescharacteristic
50C23×C127#4112yes—
51C23×C127#4111yescharacteristic
52C2×C4×C127—22no—
53C2×C4×C127—42no—
54C2×C4×C127—44no—
55C2×C4×C127—42no—
56C22×C127—22no—
57C22×C127—24no—
58C22×C127—41no—
59C22×C127—42no—
60C22×C127—41no—
61C22×C127C2×C22⋊C411yescharacteristic
62C22×C127#11711yescharacteristic
63C22×C127#11711yescharacteristic
64C22×C127#12312yes—
65C22×C127#12312yes—
66C4×C127—42no—
67C4×C127—82no—
68Dic127—22no—
69Dic127—22no—
70Dic127—42no—
71Dic127—82no—
72C2×C127—22no—
73C2×C127—22no—
74C2×C127—21no—
75C2×C127—21no—
76C2×C127#38111yescharacteristic
77C2×C127#41012yes—
78#1820—1271nomaximal, Sylow
79C127#182011yescharacteristic, complemented, Sylow
80C2×#117—1271no—
81#381—1272no—
82#342—1272no—
83#342—1271no—
84#342D12711yescharacteristic
85C23×C4—1271no—
86#117—2544no—
87C2×C4⋊C4—1272no—
88C2×C22⋊C4—1271no—
89C2×C22⋊C4—1272no—
90C2×C22⋊C4C2×D12712yes—
91C2×C22⋊C4C2×D12711yescharacteristic
92C2×C42—1272no—
93#95—2544no—
94C24C22×D12711yescharacteristic
95C22×C4—22no—
96C22×C4—1271no—
97C22×C4—1271no—
98C22×C4—1274no—
99C22×C4—1272no—
100C22×C4—2542no—
101C22×C4—2541no—
102C22×C4—2542no—
103C22×C4—2541no—
104C22×C4C4×D12712yes—
105C4⋊C4—2544no—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—2542no—
110C22⋊C4—2542no—
111C22⋊C4—2542no—
112C42—2544no—
113C23—42no—
114C23C2×C4×D12711yescharacteristic
115C23D4×D12712yes—
116C23#1452324711yescharacteristic
117C23#1452324812yes—
118C23#1452324911yescharacteristic
119C2×C4—22no—
120C2×C4—42no—
121C2×C4—44no—
122C2×C4—42no—
123C2×C4—1272no—
124C2×C4—1272no—
125C2×C4—5084no—
126C2×C4—5082no—
127C2×C4—5082no—
128C2×C4—2542no—
129C2×C4—2544no—
130C2×C4—2544no—
131C2×C4—2541no—
132C2×C4—2541no—
133C2×C4—2541no—
134C2×C4—2541no—
135C22—22no—
136C22—24no—
137C22—41no—
138C22—42no—
139C22—41no—
140C22C22⋊C4×D12711yescharacteristic
141C22#1686116611yescharacteristic
142C22#1686116711yescharacteristic
143C22#1686119112yes—
144C22#1686119412yes—
145C4—42no—
146C4—82no—
147C4—10162no—
148C4—5082no—
149C4—2542no—
150C4—2542no—
151C2—22no—
152C2—22no—
153C2—21no—
154C2—21no—
155C2#2254973811yescharacteristic
156C2#2254984812yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#342×C127C2
socleC22×C127C2×C22⋊C4
prime radical, cosocleC24×C127C23
hypercenter#342D127
hypocenterC127#1820
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
8. C2×Dic127⋊C4C4semidirect
11. C2×C4×Dic127C4semidirect
79. C127#1820semidirect

Maximal subgroups

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

subgroupindex
C2×#168611652
#225497372
#225496432
#225496422
#342×C1272
#1820127